Middle School Math: Ratios, Rates & Percents

Two true sentences about the same class

A class has 24 students in it: 10 boys and 14 girls. Two students are asked to describe the class in one sentence.

The first says: the ratio of boys to girls is 10 to 14. The second says: boys are 10 out of 24 of the class. Both sentences are correct. Both are about the same 24 people. But the numbers underneath them are different, and if you mix them up you will get wrong answers for the rest of your life in ways that are hard to notice.

This lesson is about the difference between those two sentences. It sounds like a small thing. It is not.

Key idea: A ratio compares one group to another group. A fraction compares one group to the whole. They can describe the same situation and still be different numbers.

Part to part, and part to whole

Line the two sentences up.

Sentence	What is being compared	The numbers
The ratio of boys to girls is 10 to 14	one part against another part	10 and 14
Boys are 10 out of 24 students	one part against the whole group	10 and 24

The 14 in the first row is the number of girls. The 24 in the second row is the number of everybody. Those are not the same number and they are not describing the same comparison. A ratio is usually a part-to-part comparison: this many of these for every that many of those. A fraction is usually a part-to-whole comparison: this many out of the total.

Here is a quick way to hear the difference. The word "to" signals a ratio. The words "out of" signal a fraction. Say them out loud: "10 to 14" and "10 out of 24." Different words, different comparison, different numbers.

Now, the honest complication. A ratio can also be written part to whole, and when it is, it matches a fraction exactly. "The ratio of boys to all students is 10 to 24" is a perfectly good ratio, and it is the same comparison as the fraction 10/24. So the real rule is not that ratios and fractions are different kinds of object. The real rule is: read what is being compared to what. Never assume. The words in the sentence tell you.

The point: Before you write any numbers down, name the two things being compared. Boys to girls, or boys to everyone? The answer changes the second number.

Three ways to write the same comparison

Take a fruit bowl with 6 apples and 4 oranges. The ratio of apples to oranges can be written three ways, and they are read the same out loud.

• With the word to: 6 to 4
• With a colon: 6 : 4
• As a fraction bar: 6/4

Read the colon as the word "to." So 6 : 4 is spoken "six to four," never "six colon four." The fraction-bar form is where students get tangled, because 6/4 looks like the fraction six fourths. In ratio work it is not a fraction of the bowl. There is no way for apples to be six fourths of anything. It is a comparison: six apples for every four oranges. The bar is doing the same job the colon does.

Because of that trap, a useful habit is to write ratios with a colon while you are learning, and save the fraction bar for when you are doing arithmetic with them. You will meet the fraction-bar form constantly, so you need to recognise it, but you do not have to choose it.

Simplifying without changing the mix

Here is a bright orange paint mixed from 12 spoons of yellow and 8 spoons of red. Ratio of yellow to red: 12 : 8.

Can we say that more simply? Yes, and the method is the same one you use on fractions. Divide both numbers by the same thing.

Step 1. Look for a number that divides into both 12 and 8. Both are even, so 2 works. Both also divide by 4, and 4 is the biggest one that does, so use 4 and finish in one move.

Step 2. Divide the first number: 12 ÷ 4 = 3.

Step 3. Divide the second number: 8 ÷ 4 = 2.

Step 4. Write the simplified ratio: 3 : 2.

So 12 : 8 and 3 : 2 are the same orange. Not a similar orange. The identical colour. If you mix 3 spoons of yellow with 2 spoons of red you get a smaller batch of exactly the same shade, because what makes the colour is the relationship between the two amounts, not the amounts themselves.

Check it by going the other way. Multiply 3 : 2 by 4 and you get 12 : 8. Multiply by 10 and you get 30 : 20, a bucketful of the same orange. Every one of those pairs describes one colour.

Worth holding on to: Dividing both parts of a ratio by the same number does not change the ratio. It only changes the size of the batch you are talking about.

Order matters, and it matters a lot

Go back to the paint. Yellow to red is 3 : 2. What is red to yellow?

It is 2 : 3, not 3 : 2. You have to flip the numbers when you flip the words. This is the single most common slip in ratio problems, and it produces answers that are not slightly wrong but badly wrong. Mix 3 spoons of red into 2 spoons of yellow and you get a dark brick colour, nothing like the bright orange.

So build a habit right now: write the labels down before the numbers. Not "3 : 2" floating on the page, but "yellow : red = 3 : 2." When the question later asks about red to yellow, your own notes will stop you from grabbing the numbers in the wrong order.

A ratio with three parts

Ratios are not limited to two things. Concrete for a garden path is often described as 1 : 2 : 3, meaning 1 part cement to 2 parts sand to 3 parts gravel. All the rules still hold. The order of the words fixes the order of the numbers, and you can scale the whole thing by multiplying every part by the same number.

Suppose you want six times as much. Multiply each part by 6: 1 × 6 = 6, 2 × 6 = 12, 3 × 6 = 18. So 6 : 12 : 18, which is 6 buckets of cement, 12 of sand, and 18 of gravel. The mix is unchanged.

Here is a question that catches people out. In a 1 : 2 : 3 mix, what fraction of the total is sand? You cannot read that off the ratio directly, because 2 is being compared to the other parts, not to the whole. Add the parts up first: 1 + 2 + 3 = 6 parts in total. Sand is 2 of those 6, so sand is 2/6 of the mix, which simplifies to 1/3.

That step, adding the parts to find the whole, is how you cross from ratio language into fraction language. It is worth practising until it is automatic, because a lot of test questions are built on exactly that bridge.

In short: To turn a part-to-part ratio into a part-to-whole fraction, add all the parts together to get the denominator.

Working one all the way through

A pet shelter has 15 dogs and 25 cats. Answer three questions about it, showing every step.

Question 1: what is the ratio of dogs to cats, in simplest form?

Step 1. Write the labels: dogs : cats.

Step 2. Put the numbers in that order: 15 : 25.

Step 3. Find the biggest number that divides both. 5 divides 15 and 25. Nothing bigger does.

Step 4. Divide: 15 ÷ 5 = 3, and 25 ÷ 5 = 5.

Step 5. Answer: 3 : 5. Three dogs for every five cats.

Question 2: what fraction of the animals are dogs?

Step 1. Find the whole: 15 + 25 = 40 animals.

Step 2. Write the part over the whole: 15/40.

Step 3. Simplify by dividing top and bottom by 5: 15 ÷ 5 = 3, 40 ÷ 5 = 8. Answer: 3/8 of the animals are dogs.

Question 3: what is the ratio of cats to all the animals?

Step 1. Labels: cats : all animals.

Step 2. Numbers: 25 : 40.

Step 3. Divide both by 5: 5 : 8.

Look at the three answers together: 3 : 5, then 3/8, then 5 : 8. Same shelter, same 40 animals, three different pairs of numbers, because three different comparisons were asked for. Nothing went wrong. That is just what happens when you change what you are comparing to what.

Common misconceptions

"A ratio is just a fraction with a colon instead of a bar." Not quite. The written forms overlap, but the comparison usually does not. In the class of 24, the ratio of boys to girls is 10 : 14 while the fraction of the class that is boys is 10/24. Writing 10 : 14 as 10/14 does not make it the fraction of the class. It is still boys compared to girls.

"Simplifying a ratio makes it smaller, so it must mean less." It means the same. 12 : 8 and 3 : 2 are one colour of paint. The simplified version names the smallest batch that has the right relationship; every larger batch is a multiple of it.

"6 : 4 and 4 : 6 are basically the same, it is just which way you say it." They are opposite comparisons. If juice to water is 6 : 4 the drink is strong, and if it is 4 : 6 the drink is weak. Same two numbers, different drink. Always write the labels.

"To find what fraction of a 2 : 3 mix is the first thing, I use 2/3." No. 2/3 compares the first part to the second part. For the fraction of the whole, add the parts first: 2 + 3 = 5, so the first thing is 2/5 of the mix.

Pulling it together

You now have the vocabulary that the next four lessons are built out of.

• A ratio compares two amounts. Name what is being compared to what before you write anything down.
• Part to part is the usual meaning of a ratio; part to whole is the usual meaning of a fraction; either can be written either way, so read the words.
• Three notations, one meaning: 6 to 4, 6 : 4, and 6/4.
• Divide both parts by the same number to simplify. The relationship does not change, only the batch size.
• Order is not decoration. Flip the words, flip the numbers.
• To go from a part-to-part ratio to a fraction of the whole, add all the parts to get the denominator.

Next lesson takes the other direction. Instead of shrinking a ratio down to its simplest form, you will grow it, building a ratio table that scales a recipe, a map, or a paint mix to any size you need.

Sources

1. Khan Academy. (n.d.). Intro to ratios. 6th grade math: Ratios, rates, and percentages. khanacademy.org
2. Marecek, L., and Mathis, A. H. (2020). Ratios and rate. Prealgebra 2e. OpenStax, Rice University. openstax.org
3. Wikipedia contributors. (n.d.). Ratio. Wikipedia. en.wikipedia.org
4. Illustrative Mathematics. (n.d.). Problem-based mathematics curriculum, grades 6 to 8. illustrativemathematics.org

The pancake recipe that has to feed eleven

A pancake recipe makes 8 pancakes using 1 cup of flour and 2 eggs. Eleven people are coming and you want three pancakes each, so you need 33 pancakes. The recipe card cannot help you. It only knows about 8.

You could guess. Four times the recipe gives 32 pancakes, close enough, so 4 cups of flour and 8 eggs. That happens to work here because the numbers are friendly. Next week the numbers will not be friendly, and guessing will produce a batter that is wrong in a way you can taste.

What you want instead is a piece of paper that grows the recipe in a way that cannot go wrong. That piece of paper is a ratio table, and it is the single most useful tool in this whole course.

Why this matters: Scaling a ratio is multiplication, never addition. A table makes that impossible to forget.

What makes two ratios equivalent

Two ratios are equivalent when one is the other one multiplied straight through by the same number. Take 2 : 3.

• Multiply both by 2: 4 : 6
• Multiply both by 3: 6 : 9
• Multiply both by 10: 20 : 30
• Divide both by 2: 1 : 1.5

All five of those describe the same relationship. If 2 : 3 is squash to water in a drink, every one of those mixes tastes identical. Only the glass size changed.

The word "straight through" is doing real work. Both numbers, same multiplier, every time. Multiply the 2 by 3 and the 3 by 2 and you get 6 : 6, which is a completely different drink.

Why adding breaks it

Here is the mistake that costs the most marks, and it is worth seeing it fail rather than just being told not to do it.

Squash to water is 2 : 3. You want more drink, so you add 2 to each part and get 4 : 5. Is that the same drink?

Test it. In 2 : 3, the water is 1.5 times the squash, because 3 ÷ 2 = 1.5. In 4 : 5, the water is 1.25 times the squash, because 5 ÷ 4 = 1.25. The second drink has proportionally less water in it. It is stronger. Adding the same amount to both parts made the smaller part grow by a bigger share of itself, which is exactly why adding is not allowed.

Push it further and the point gets obvious. Add 100 to both parts of 2 : 3 and you get 102 : 103, which is very nearly 1 : 1, half squash and half water. That is not a bigger version of the original drink. It is a different drink entirely.

Remember: Multiplying both parts keeps the relationship. Adding to both parts destroys it.

The ratio table, built one column at a time

A ratio table is just a two-row table where every column holds one equivalent version of the same ratio. Back to the pancakes: 1 cup of flour to 8 pancakes.

Cups of flour	1	2	3	4
Pancakes	8	16	24	32

Read any column and you get a true statement: 3 cups of flour makes 24 pancakes. Read across the top row and you see it climbing by 1. Read across the bottom and it climbs by 8. That is the pattern, and it will always be there in a ratio table, but the pattern is a consequence, not the rule. The rule is that every column is the first column times something.

Now the harder question. You need 33 pancakes, and 33 is not in the table. Two moves get you there.

Step 1. Find how many cups make one pancake. Take a column you trust, 1 cup to 8 pancakes, and divide both parts by 8: 1 ÷ 8 = 0.125 cups, and 8 ÷ 8 = 1 pancake. So one pancake takes 0.125 cups, which is one eighth of a cup.

Step 2. Multiply by 33: 0.125 × 33 = 4.125 cups.

So 33 pancakes needs about 4 and one eighth cups of flour. In a kitchen you would round to 4 cups and accept 32 pancakes. On a test you would write 4.125. Either way you now know the number instead of guessing it.

That middle step, going through the amount for one, has a name you will meet next lesson: the unit rate. It is the master key for ratio tables, because once you know the value for 1 you can get the value for anything by multiplying once.

Filling a gap in the middle of a table

A shop sells rope. The sign says 6 metres for $9. How much is 15 metres?

Build the table and put in what you know.

Metres	6	1	15
Dollars	9	?	?

Step 1. Get to the 1 column. To turn 6 into 1 you divide by 6, so divide the dollars by 6 as well: 9 ÷ 6 = 1.5. One metre costs $1.50.

Step 2. Get from the 1 column to the 15 column. To turn 1 into 15 you multiply by 15, so multiply the dollars by 15: 1.5 × 15 = 22.5. Fifteen metres costs $22.50.

Step 3. Sanity check. 15 metres is two and a half times 6 metres, and two and a half times $9 is $22.50. It matches, so the answer is safe.

There is a shortcut here worth noticing. You did not have to go through 1. You could have jumped straight from 6 to 15 by multiplying by 2.5, since 6 × 2.5 = 15, and then 9 × 2.5 = 22.5. Both routes are legal. Go through 1 when the multiplier between the two columns is awkward, and jump directly when it is clean.

The upshot: In a ratio table you may multiply or divide any column by any number, as long as you do it to both rows. That single permission solves every problem in this lesson.

A table where the answer is not obvious

A car travels 250 kilometres on 20 litres of fuel. The tank holds 45 litres. How far can it go on a full tank?

Litres	20	1	45
Kilometres	250	?	?

Step 1. Divide by 20 to reach the 1 column: 20 ÷ 20 = 1 litre, and 250 ÷ 20 = 12.5 kilometres. The car does 12.5 km on one litre.

Step 2. Multiply by 45: 12.5 × 45. Work it in parts if that helps. 12.5 × 40 = 500, and 12.5 × 5 = 62.5. Add them: 500 + 62.5 = 562.5.

Step 3. A full tank takes the car 562.5 kilometres.

Step 4. Check that the size makes sense. 45 litres is a bit more than twice 20 litres, and 562.5 is a bit more than twice 250. Good.

Notice what would have happened with the addition mistake. Twenty litres plus 25 more litres, so 250 kilometres plus 25 more kilometres, giving 275 km. That answer says the extra 25 litres carried the car only 25 kilometres, which would be a spectacularly thirsty car. Estimating first is how you catch that before it costs you.

Where this goes next

Setting two equivalent ratios equal to each other and solving for the missing one has a name: a proportion. There is a standard method for it, cross multiplication, and it is taught in Pre-Algebra and Math Foundations, the course that follows this one. You do not need it yet. Everything in this course can be done with a ratio table, and doing it that way first means that when cross multiplication arrives you will already understand what it is doing, which is exactly what the table does, written in one line instead of three columns.

Common misconceptions

"To make a ratio bigger, add the same amount to both sides." This is the big one. Adding changes the relationship, because the same addition is a bigger share of the smaller number. 2 : 3 with 2 added to each becomes 4 : 5, a different mix. Only multiplying and dividing preserve a ratio.

"The multiplier has to be a whole number." It does not. Going from 6 metres to 15 metres means multiplying by 2.5, and that is perfectly legal. Multiplying by a decimal or a fraction is still multiplying both rows by the same thing.

"A ratio table has to start at 1." It can start anywhere. The 1 column is useful, not required. If the jump between two known columns is clean, take it directly and skip the 1 entirely.

What to carry forward

• Equivalent ratios are the same relationship at different sizes. You make them by multiplying or dividing both terms by the same number.
• Adding the same number to both terms is not allowed and produces a genuinely different mix.
• A ratio table stores equivalent ratios in columns. Any column can be scaled to any other column.
• Going through the value for 1 will always work, even when the jump between two columns is ugly.
• Estimate before you compute so that a wrong-sized answer stands out.

The 1 column you kept passing through has a name and a lot of power. That is the next lesson.

Sources

1. Khan Academy. (n.d.). Equivalent ratios. 6th grade math: Ratios, rates, and percentages. khanacademy.org
2. PhET Interactive Simulations, University of Colorado Boulder. (n.d.). Proportion playground. phet.colorado.edu
3. Marecek, L., and Mathis, A. H. (2020). Ratios and rate. Prealgebra 2e. OpenStax, Rice University. openstax.org
4. Wikipedia contributors. (n.d.). Ratio. Wikipedia. en.wikipedia.org

Two boxes of the same cereal

A 20 ounce box of cereal costs $4.29. Beside it, a 14 ounce box of the identical cereal costs $3.19. Which should you buy?

You cannot tell by looking. The big box costs more but holds more. The small box costs less but you get less. The two numbers in each price tag are locked together, and until you unlock them there is nothing to compare.

The tool that unlocks them takes about ten seconds and works on every price, every speed, and every rate you will ever meet. It is called the unit rate.

Key idea: A unit rate answers the question "how much for exactly one?" Once two things are both expressed per one, they can be compared directly.

A rate is a ratio whose two parts have different units

Last lesson compared cups of squash to cups of water. Same unit on both sides, so it was a plain ratio. Now compare dollars to ounces, or kilometres to hours, or heartbeats to minutes. When the two quantities are measured in different units, the comparison is called a rate.

• $4.29 for 20 ounces is a rate: dollars per ounce.
• 180 kilometres in 3 hours is a rate: kilometres per hour.
• 72 heartbeats in 60 seconds is a rate: beats per second.

A unit rate is a rate whose second number has been driven down to 1. Kilometres per one hour. Dollars per one ounce. Beats per one minute. The word "per" is the giveaway, and it means "for each one."

You already meet unit rates constantly without calling them that. A speedometer reads a unit rate. A wage of $15 an hour is a unit rate. A phone plan advertised as 10 gigabytes a month is a unit rate. Someone did the dividing for you.

Doing the dividing yourself

Back to the cereal. To find dollars per ounce, divide the dollars by the ounces.

Big box.

Step 1. Write the rate as a division: 4.29 ÷ 20.

Step 2. Divide. 4.29 ÷ 20 = 0.2145.

Step 3. Round sensibly for money: about 21.5 cents per ounce.

Small box.

Step 1. Write the division: 3.19 ÷ 14.

Step 2. Divide. 3.19 ÷ 14 = 0.2278 and a bit.

Step 3. Round: about 22.8 cents per ounce.

Compare. 21.5 cents beats 22.8 cents, so the 20 ounce box is the better buy by about 1.3 cents an ounce. Over the whole 20 ounce box that saves you around 26 cents. Small, but it is the same 26 cents every week for a year.

Notice which number went on top. Dollars on top, ounces on the bottom, because the question was "how much money for one ounce." Had you divided the other way, 20 ÷ 4.29 = 4.66, you would get ounces per dollar, which is also a perfectly good unit rate. It just answers a different question, and this time bigger is better instead of smaller. Both are correct as long as you say which one you computed.

What matters here: Decide what "one" means before you divide. Per one ounce, or per one dollar? The label on the answer tells you which way the comparison runs.

Reading the shelf label instead

Many shops print the unit price in small type on the shelf label, beside the big price. That number is exactly the division you just did. In the United States, the model rules that states adopt for retail package labelling and for the scales and scanners in shops are published by the Office of Weights and Measures at the National Institute of Standards and Technology, and unit pricing is part of that world.

Two cautions about shelf labels, both worth knowing before you trust one.

1. Check the unit. One label may say "per 100 g" and the one beside it "per kg." Those differ by a factor of ten, and comparing them straight across will tell you the opposite of the truth.
2. Check that it is the same product. Unit price compares quantity, not quality. Two hundred grams of something you will not eat is not a bargain at any price per gram.

Comparing three at once

A shop sells the same juice in three sizes. Which is the best value?

Bottle	Price	Volume	Price per litre
Small	$1.20	0.5 L	1.20 ÷ 0.5 = $2.40
Medium	$2.10	1 L	2.10 ÷ 1 = $2.10
Large	$4.50	2 L	4.50 ÷ 2 = $2.25

Read the last column. The medium bottle is cheapest per litre at $2.10, then the large at $2.25, then the small at $2.40. The large bottle is not the best deal, even though shops train you to assume the biggest package always is. Sometimes it is. Here it is not, and the only way to know is to divide.

Work the saving. Buying two litres as one large bottle costs $4.50. Buying two litres as two medium bottles costs $4.20. You save 30 cents by carrying two bottles instead of one.

When the cheaper unit price is still the wrong buy

Now the case that stops unit price from being a magic answer. A 5 kilogram sack of rice is $8, which is $1.60 per kilogram. A 1 kilogram bag is $2.20 per kilogram. The sack wins on unit price by a wide margin.

But suppose you live alone, you eat rice twice a month, and it takes two years to get through 5 kilograms. Rice will keep, so maybe that is fine. Now make it strawberries instead. A large punnet at a lower price per gram is worthless if half of it goes soft before you eat it. The moment you throw food away, the real price per gram of what you actually ate jumps.

The mathematics is not wrong. The mathematics answered exactly the question it was asked, which was cost per unit purchased. It was never asked about cost per unit eaten, or about whether you have a cupboard big enough. Knowing what a calculation does not cover is part of using it well.

The point: Unit price tells you which is cheaper per unit. It does not tell you which is the better decision. Those are different questions and you have to ask the second one yourself.

Speed is a unit rate too

A train covers 240 kilometres in 3 hours. What is its speed?

Step 1. Speed means kilometres per one hour, so divide kilometres by hours: 240 ÷ 3.

Step 2. 240 ÷ 3 = 80.

Step 3. The speed is 80 kilometres per hour.

Turn it around. If the train keeps that speed, how far in 7 hours? Multiply the unit rate by the number of hours: 80 × 7 = 560 kilometres. And how long to cover 400 kilometres? Divide by the unit rate: 400 ÷ 80 = 5 hours.

That is the pattern for every rate problem you will ever see. Find the value for one, then multiply to go up or divide to come back down. Three questions, one unit rate, no new method needed for any of them.

Common misconceptions

"The bigger package is always cheaper per unit." Usually, but not always, as the juice table shows. Shops know shoppers assume it. Dividing takes ten seconds and occasionally saves real money.

"A lower unit price always means a better deal." Only if you use what you buy. Cost per unit purchased and cost per unit actually used are different numbers whenever anything gets wasted.

"It does not matter which number goes on top." It changes what the answer means. Dollars per ounce and ounces per dollar are both valid, but with dollars per ounce you want the smaller number and with ounces per dollar you want the bigger one. Label your answer and you will not get this backwards.

"Unit rate is a different topic from ratios." It is the same topic. A unit rate is the ratio table's 1 column, computed directly instead of walked to.

The short version

• A rate compares two quantities with different units. A unit rate has 1 as its second quantity.
• Find a unit rate by dividing. Decide first which quantity you want "one" of, then divide by that one.
• To compare prices, put them both on the same per-unit footing, then compare.
• Check the units on shelf labels before trusting them, and check that they really are the same product.
• Once you have a unit rate you can answer every related question by multiplying up or dividing down.
• A cheaper unit price is not automatically the better choice.

Dividing to get "per one" is also the trick that converts miles to kilometres and hours to seconds. That is the next lesson.

Sources

1. Khan Academy. (n.d.). Intro to rates. 6th grade math: Ratios, rates, and percentages. khanacademy.org
2. National Institute of Standards and Technology. (n.d.). Office of Weights and Measures. NIST Physical Measurement Laboratory. nist.gov
3. Wikipedia contributors. (n.d.). Unit price. Wikipedia. en.wikipedia.org
4. PhET Interactive Simulations, University of Colorado Boulder. (n.d.). Unit rates. phet.colorado.edu

The orbiter that flew too close to Mars

NASA launched the Mars Climate Orbiter on 11 December 1998. On 23 September 1999, after nine and a half months of flight, it fired its engine to slide into orbit around Mars and was never heard from again.

The investigation found the cause, and it was not a broken part. One piece of ground software reported the small pushes from the spacecraft's thrusters in pound-force seconds, an imperial unit. The navigation software receiving those numbers assumed they were newton-seconds, the metric unit. Nobody converted. The numbers looked reasonable, so nothing rang an alarm, and over months of tiny navigation corrections the error accumulated until the spacecraft arrived at Mars far lower than planned.

A conversion nobody performed destroyed a spacecraft. This lesson is about performing it, and about a way of writing the work down that makes forgetting almost impossible.

Key idea: A conversion factor is a rate that equals 1. Multiplying by it changes the units without changing the amount.

Why a conversion factor is secretly the number 1

There are 100 centimetres in 1 metre. That is a fact about the metric system, and you can write it as a rate two different ways:

• 100 cm / 1 m
• 1 m / 100 cm

Here is the strange and useful thing. Both of those fractions equal 1. Not approximately. Exactly. The top and the bottom are the same length, written in different units, in the same way that 4/4 equals 1. A hundred centimetres and one metre are the same distance, so dividing one by the other gives 1.

That is why conversion works. Multiplying any number by 1 leaves it unchanged, so multiplying by a conversion factor changes only the label, never the amount. Six metres and 600 centimetres are the same length. You have relabelled it, not resized it.

Remember: Every conversion factor can be written two ways. Choose the one that puts the unit you want to get rid of on the bottom.

Cancelling the unit you do not want

Convert 6 metres into centimetres, writing every step.

Step 1. Start with what you have, written as a fraction over 1: 6 m / 1.

Step 2. Choose the version of the conversion factor with metres on the bottom, so that metres will cancel: 100 cm / 1 m.

Step 3. Multiply: (6 m / 1) × (100 cm / 1 m).

Step 4. Cancel. There is an m on the top and an m on the bottom, so they cancel each other out, exactly the way a 5 on the top and a 5 on the bottom would. What is left is centimetres.

Step 5. Do the arithmetic: 6 × 100 = 600. Answer: 600 cm.

Now watch what happens if you pick the wrong version of the factor. (6 m / 1) × (1 m / 100 cm) gives 6 ÷ 100 = 0.06, and the units left over are m squared per cm, which is not a length at all. The units themselves tell you that you made a mistake. That is the whole reason for writing conversions this way: the labels check your work for you.

Four conversions, worked

1. 3 hours into seconds. This needs two factors, because you go hours to minutes to seconds.

Step 1. 3 h × (60 min / 1 h) = 180 min. The hours cancel.

Step 2. 180 min × (60 s / 1 min) = 10,800 s. The minutes cancel.

Step 3. Answer: 3 hours is 10,800 seconds.

2. 250 centimetres into metres. This time you want metres on top.

Step 1. 250 cm × (1 m / 100 cm). The centimetres cancel.

Step 2. 250 ÷ 100 = 2.5. Answer: 2.5 m.

3. 12 inches into centimetres. One inch is exactly 2.54 cm, a definition rather than a measurement.

Step 1. 12 in × (2.54 cm / 1 in). Inches cancel.

Step 2. 12 × 2.54 = 30.48. Answer: 30.48 cm.

4. 5 miles into kilometres. One mile is exactly 1.609344 km, so round the factor to 1.609 for everyday work.

Step 1. 5 mi × (1.609 km / 1 mi). Miles cancel.

Step 2. 5 × 1.609 = 8.045. Answer: about 8.05 km.

Estimate to check: a mile is a bit over one and a half kilometres, so five miles should be a bit over 8. It is.

The two-step conversion that trips everyone: km per hour into metres per second

A car is doing 90 kilometres per hour. How many metres does it cover each second?

Two units have to change here, one on the top and one on the bottom. Do them one at a time and nothing goes wrong.

Step 1. Write the starting rate as a fraction: 90 km / 1 h.

Step 2. Convert the top from kilometres to metres. Multiply by 1000 m / 1 km. The km cancel, leaving 90,000 m / 1 h.

Step 3. Convert the bottom from hours to seconds. One hour is 3600 seconds, so multiply by 1 h / 3600 s. The hours cancel, leaving 90,000 m / 3600 s.

Step 4. Divide: 90,000 ÷ 3600 = 25.

Step 5. Answer: 90 km per hour is 25 metres per second.

Look at what happened to the two conversions. Multiplying by 1000 made the number bigger; dividing by 3600 made it smaller; the second effect was stronger, so the final number came out smaller than 90. If you had guessed instead of computing, you would have had no idea which way it would go.

What matters here: When the unit on the bottom changes, the conversion factor goes in upside down compared with the top. Let the cancelling decide, not your instinct.

A rate that changes: currency

Suppose a bank offers 1 US dollar for 0.92 euros. That is a rate, and it converts exactly like centimetres per metre.

Convert $150 into euros. 150 × 0.92 = 138 euros.

Convert 60 euros back into dollars. Now you need the factor the other way up: 1 euro buys 1 ÷ 0.92 = 1.087 dollars, so 60 × 1.087 = about $65.22.

One thing is different here, and it matters. Metres per centimetre never changes. Exchange rates change every day, and the bank usually gives you slightly less than the published rate because it keeps a small cut. So the arithmetic is exactly the same, but the input is only good for today, and the answer is an estimate of what you will actually receive. Knowing that a number is a snapshot rather than a constant is part of using it honestly.

Common misconceptions

"Converting changes the amount." It changes the label only. 6 metres and 600 centimetres are the same distance. The number got bigger because the unit got smaller, and those two always move in opposite directions.

"Bigger unit, bigger number." The reverse. Measuring in a smaller unit means you need more of them, so centimetres give a bigger number than metres for the same length. If your converted number moved the wrong way, you used the factor upside down.

"I can multiply by 2.54 and not worry about which unit that is for." The Mars Climate Orbiter is what that looks like at full size. Write the units on every line and cancel them; the units will refuse to cancel when the factor is the wrong way round, which is a free error check.

"To convert km per hour to m per second, convert both parts the same way." No. The top unit and the bottom unit need factors pointing in opposite directions, because one is being multiplied into the numerator and the other into the denominator.

Looking back

• A conversion factor is a rate whose top and bottom are the same amount in different units, so it equals 1.
• Write it the way round that puts the unwanted unit on the bottom, then cancel.
• The leftover units tell you whether you set it up correctly. Wrong units mean a wrong setup, every time.
• Two-step conversions are just two multiplications, done one at a time.
• When both the top and bottom units change, the two factors go in opposite ways up.
• Some rates, like currency, are true only on the day you looked them up.

Next comes the rate people use more than any other, and it has its own symbol: the one that fixes the bottom of the fraction at exactly one hundred.

Sources

1. NASA. (n.d.). Mars Climate Orbiter. NASA Science. science.nasa.gov
2. National Institute of Standards and Technology. (n.d.). Unit conversion. Metric (SI) resources, Office of Weights and Measures. nist.gov
3. Wikipedia contributors. (n.d.). Conversion of units. Wikipedia. en.wikipedia.org
4. Wikipedia contributors. (n.d.). Mars Climate Orbiter. Wikipedia. en.wikipedia.org

Three tests, three totals, one question

Three students sit three different tests. Ana scores 17 out of 20. Ben scores 21 out of 25. Cleo scores 43 out of 50. Who did best?

You cannot line those up. Seventeen is the smallest score and 43 is the biggest, but 17 came out of the smallest total. The scores are not comparable because the totals are not the same.

Last lesson solved this kind of problem by dividing everything down to "per one." That works here too, giving 0.85, 0.84 and 0.86, but those numbers are awkward to say and easy to misread. So the world agreed on a different landing spot: not per one, but per one hundred. Rewrite all three scores out of 100 and they become 85, 84 and 86. Cleo won, by one mark out of a hundred. Now you can see it.

The core of it: Percent is a rate with the bottom number locked at 100, which is what makes any two quantities comparable at a glance.

What the word says

Percent comes from the Latin phrase meaning "by the hundred." The symbol % is a squashed way of writing the digits and the two zeros of 100. So "37 percent" means "37 for every 100," and it can be written all of these ways, which are the same number:

Form	Written	What it says
Percent	37%	37 out of every 100
Fraction	37/100	37 parts of a 100-part whole
Decimal	0.37	37 hundredths
Ratio	37 : 100	37 compared with 100

All four are the same quantity wearing different clothes. Which one you use depends on what you are about to do with it: percents for reporting, decimals for calculating, fractions for exact work.

Moving between the three forms

Percent to decimal: divide by 100. On paper that means sliding the decimal point two places to the left.

45% becomes 0.45. 7% becomes 0.07, because you must fill the empty place with a zero. 120% becomes 1.20. 0.5% becomes 0.005.

Decimal to percent: multiply by 100. Slide the point two places right.

0.62 becomes 62%. 0.9 becomes 90%, not 9%, because 0.9 is nine tenths, which is ninety hundredths. That one catches people constantly, so pause on it: 0.9 = 0.90 = 90%.

Fraction to percent: divide top by bottom, then multiply by 100.

Take 3/8. Step 1: 3 ÷ 8 = 0.375. Step 2: 0.375 × 100 = 37.5%. So three eighths is 37.5 percent.

Take 17/20 from the test problem. Step 1: 17 ÷ 20 = 0.85. Step 2: 0.85 × 100 = 85%. Ana scored 85 percent.

There is a shortcut when the denominator divides into 100 cleanly. For 17/20, ask what turns 20 into 100: multiply by 5. Do the same to the top: 17 × 5 = 85. So 17/20 = 85/100 = 85%, with no decimals at all. That trick works for denominators of 2, 4, 5, 10, 20, 25 and 50.

Percent to fraction: put it over 100 and simplify.

60% = 60/100. Divide both by 20: 3/5. And 12% = 12/100, divide both by 4: 3/25.

Pre-Algebra and Math Foundations drills these conversions much further, including repeating decimals. For this course you need the four moves above and the confidence to do them without a calculator.

In short: Percent to decimal, point moves left twice. Decimal to percent, point moves right twice. Fraction to percent, divide then multiply by 100.

The benchmarks worth memorising

Fluent people do not compute most percents. They recognise them. These are worth knowing cold.

Percent	Fraction	Decimal	Fast way to find it
1%	1/100	0.01	Move the point two places left
10%	1/10	0.1	Move the point one place left
20%	1/5	0.2	Find 10%, then double it
25%	1/4	0.25	Halve, then halve again
50%	1/2	0.5	Halve it
75%	3/4	0.75	Halve, halve, then add the two together with the first half
100%	1	1.0	The whole thing, unchanged

The 10 percent row is the workhorse. Once you can take 10 percent of anything by moving a decimal point, you can build most other percents out of it. Fifteen percent of $60? Ten percent is $6, half of that is $3 for the 5 percent, and $6 + $3 = $9. No calculator, four seconds.

Percents above 100 and below 1

Nothing stops a percent from going past 100. If a bakery sold 40 loaves on Monday and 100 loaves on Tuesday, Tuesday's sales are 250 percent of Monday's, because 100 ÷ 40 = 2.5, and 2.5 × 100 = 250. A percent over 100 simply means "more than the whole thing you are comparing to."

Percents can also be tiny. A savings account paying 0.5 percent interest pays half of one percent, which as a decimal is 0.005. Read that carefully: 0.5% is not 0.5, and it is not 5%. It is one two-hundredth. Writing the decimal out is the safest way to be sure.

One place a percent genuinely cannot go past 100 is when it names a share of a fixed whole. You cannot have 130 percent of the students in a room, because the room's students are the whole. But you can absolutely have 130 percent of last year's students, because last year's total is a comparison point, not a container.

Back to the three tests

Finish the opening problem properly, with all steps shown.

Ana, 17 out of 20. 20 × 5 = 100, so 17 × 5 = 85. She scored 85%.

Ben, 21 out of 25. 25 × 4 = 100, so 21 × 4 = 84. He scored 84%.

Cleo, 43 out of 50. 50 × 2 = 100, so 43 × 2 = 86. She scored 86%.

Ranking: Cleo 86, Ana 85, Ben 84. The gaps are one mark in a hundred each, so it was close, and the raw scores of 17, 21 and 43 gave you no way to see that. Converting to a shared denominator of 100 is what made three unlike things comparable, and that is the entire job percent was invented to do.

Common misconceptions

"0.9 is 9 percent." It is 90 percent. Nine tenths equals ninety hundredths. Write in the missing zero, 0.90, and the two places are obvious.

"0.5% is the same as 0.5." They differ by a factor of 200. As a decimal, 0.5% is 0.005. Convert small percents to decimals in writing rather than in your head.

"A percent can never be more than 100." Only when it names a share of a fixed whole. As a comparison between two quantities, 250 percent is ordinary and just means two and a half times as much.

"Percent is a completely different topic from ratios." It is a ratio whose second term is fixed at 100. Everything you learned about scaling ratios still applies, which is why 17/20 can be scaled to 85/100 by multiplying both parts by 5.

Putting it together

• Percent means per hundred. It is a rate with the denominator locked at 100.
• Fixing the denominator is what makes unlike quantities comparable, which is why scores, surveys and interest rates all use it.
• Percent to decimal, divide by 100. Decimal to percent, multiply by 100. Fraction to percent, divide then multiply by 100.
• Scale the fraction instead when the denominator divides into 100 neatly.
• Know 1, 10, 25 and 50 percent by heart and build the rest from them.
• Percents may exceed 100 or fall below 1, and both are ordinary.

Knowing what 15 percent means is one thing. Getting 15 percent of an actual number is the next lesson.

Sources

1. Khan Academy. (n.d.). Intro to percents. 6th grade math: Ratios, rates, and percentages. khanacademy.org
2. Marecek, L., and Mathis, A. H. (2020). Understand percent. Prealgebra 2e. OpenStax, Rice University. openstax.org
3. Wikipedia contributors. (n.d.). Percentage. Wikipedia. en.wikipedia.org

The card machine that wants an answer in four seconds

A meal comes to $46. The card machine flashes three buttons: 18%, 20%, 22%. The waiter is standing there. You have about four seconds to decide whether 20% is $9 or $12 and whether that is more than you meant to leave.

There is a method for this that runs entirely in your head, and there is a method for the same problem that gets an exact answer on paper. Both are in this lesson, and knowing which to reach for is most of the skill.

Bottom line: The word "of" in "20 percent of 46" means multiply. Everything else is a choice of how to do that multiplication.

Method 1: turn the percent into a decimal and multiply

This is the method that always works, including for awkward percents, and it is the one to use when you need an exact answer.

Find 20% of 46.

Step 1. Convert the percent to a decimal by dividing by 100: 20% becomes 0.20.

Step 2. Replace the word "of" with a multiplication sign: 0.20 × 46.

Step 3. Multiply: 0.2 × 46 = 9.2.

Step 4. Answer: 20% of $46 is $9.20.

Find 18% of 46.

Step 1. 18% becomes 0.18.

Step 2. 0.18 × 46.

Step 3. Multiply in parts if that is easier: 0.1 × 46 = 4.6, and 0.08 × 46 = 3.68. Add: 4.6 + 3.68 = 8.28.

Step 4. Answer: $8.28.

Step 5. Check the size. 18% is a bit less than 20%, and $8.28 is a bit less than $9.20. Sensible.

Method 2: build it from 10% and 1%

This is the mental method, and it is the reason some people can do percents at a till while everyone else reaches for a phone.

Two building blocks:

• 10% of a number: move the decimal point one place left. 10% of 46 is 4.6.
• 1% of a number: move the decimal point two places left. 1% of 46 is 0.46.

Now assemble anything you like.

20%: two lots of 10%. 4.6 + 4.6 = $9.20.

5%: half of 10%. Half of 4.6 is $2.30.

15%: 10% plus 5%. 4.6 + 2.3 = $6.90.

18%: 20% minus 2%. Two percent is two lots of 0.46, which is 0.92. So 9.20 minus 0.92 = $8.28. Same answer as method 1, done in your head.

35%: three lots of 10% is 13.8, plus a 5% of 2.3, giving $16.10.

The pattern is always the same: get to 10% and 1%, then add, subtract, double or halve your way to the percent you were asked for. Notice that 18% was easier to reach by going down from 20% than by going up from 10%. Choosing the shorter route is part of the method.

The upshot: Ten percent and one percent are decimal-point moves. Every other percent is a few additions away from them.

Method 3: use the fraction when the fraction is friendly

Some percents are famous fractions in disguise, and using the fraction is faster than either other method.

Percent	Fraction	What you actually do	Example on 60
50%	1/2	Halve it	30
25%	1/4	Halve, then halve again	15
75%	3/4	Quarter it, then take three of those	45
20%	1/5	Divide by 5	12
33 and a third %	1/3	Divide by 3	20

Asked for 25% of 84, do not reach for 0.25 × 84. Halve 84 to get 42, halve again to get 21. Done, and it is exact.

Estimating first, so that nonsense stands out

Before computing any percent, decide roughly where the answer should land. Three anchors do the job.

• A percent under 100 gives an answer smaller than the original number.
• Exactly 100% gives the original number back.
• A percent over 100 gives an answer bigger than the original.

So 40% of 250 must be less than 250, and since 40% is a bit less than half, the answer should be a bit under 125. Compute: 0.4 × 250 = 100. Under 125, so it passes.

Suppose you had slipped and written 4 × 250 = 1000. The estimate catches it instantly, because 40 percent of something can never be four times bigger than it. Almost every percent disaster is a misplaced decimal point, and estimating is what catches misplaced decimal points.

Worked: a sales tax and a discount together

A jacket costs $80. It is 30% off, and then 8% sales tax is added to the reduced price. What do you pay?

Step 1. Find the discount. 10% of 80 is 8, so 30% is 3 × 8 = $24.

Step 2. Subtract it from the original: 80 minus 24 = $56. That is the sale price.

Step 3. Find the tax on the sale price, not on the original. 1% of 56 is 0.56, so 8% is 8 × 0.56 = $4.48.

Step 4. Add: 56 + 4.48 = $60.48.

Step 5. Check the size. You should pay well under $80 because a 30% discount is much bigger than an 8% tax. $60.48 is well under. Good.

The trap in step 3 is applying the tax to $80 instead of $56. Tax is charged on what you actually pay, so the base changed after step 2. Whenever a problem has two percents in it, stop and ask what each one is a percent of. That question is the whole of the next lesson and most of the one after it.

Percents that are not whole numbers

Real percents are often untidy. A tax rate of 8.25%, a survey result of 62.4%, an interest rate of 4.75%. None of those break anything, but they do rule out some of the mental shortcuts, so method 1 becomes the sensible choice.

Find 8.25% of 400.

Step 1. Convert: 8.25% becomes 0.0825. Count the places carefully. Moving two places left from 8.25 gives 0.0825, with a zero filling the gap.

Step 2. Multiply: 0.0825 × 400. Break it up if it helps. 0.08 × 400 = 32, and 0.0025 × 400 = 1. Add: 33.

Step 3. Answer: $33.

Step 4. Estimate check. Eight percent of 400 is 32, and 8.25% should be a whisker more. It is.

You can still use benchmarks on a fractional percent if you are willing to work in quarters. For 8.25%, take 8% (which is 8 lots of 1%, so 8 × 4 = 32) and then add a quarter of 1%, which is a quarter of 4, so 1. Total 33 again. Two routes, one answer, which is the reassurance you want when the numbers get ugly.

Common misconceptions

"Percent of always makes something bigger." Only when the percent is above 100. Fifteen percent of 200 is 30, far smaller than 200. The percent tells you what share you are taking, and most shares are less than the whole.

"To find 20% of 46, divide 46 by 20." That gives 2.3, which is 5% of 46, not 20%. Dividing by the percent number is a different operation. Multiply by the decimal instead, or build it from 10%.

"You can add two percents of different amounts." Only if they are percents of the same base. Thirty percent off $80 and 8% tax on $56 are percents of different numbers, so their percent values cannot simply be netted to 22%.

"The decimal method and the mental method can disagree." They cannot. 18% of 46 is $8.28 by either route. If two methods give different answers, one of them has an arithmetic slip in it, and redoing the shorter one usually finds it.

What you now know

• "Percent of" means multiply. Convert to a decimal and multiply for an exact answer.
• Ten percent is one decimal-point move; one percent is two. Build everything else from those.
• Use the fraction form for 50, 25, 75, 20 and a third, because halving and dividing beat decimal multiplication.
• Estimate first. Under 100 percent means a smaller answer; over 100 percent means a bigger one.
• When a problem contains two percents, check what each is a percent of before combining them.

Everything so far has given you the whole and asked for the part. The next module turns that around and gives you the part.

Sources

1. Marecek, L., and Mathis, A. H. (2020). Solve general applications of percent. Prealgebra 2e. OpenStax, Rice University. openstax.org
2. Khan Academy. (n.d.). Percent word problems. 7th grade math. khanacademy.org
3. Wikipedia contributors. (n.d.). Sales tax. Wikipedia. en.wikipedia.org

A receipt with the price torn off

You saved $27 on a coat. The tag said 30% off. What did the coat originally cost?

Every problem in the last lesson gave you the whole and asked for the part. This one gives you the part and asks for the whole. It is the same relationship, read from the other end, and it is where most percent errors in real life happen, because the obvious move, taking 30% of 27, is wrong and produces an answer that looks almost reasonable.

Try it and see what goes wrong. Thirty percent of $27 is $8.10. So the coat cost $8.10? It cannot. The coat has to cost more than the discount you got on it. The answer is not just wrong, it is on the wrong side of the number you started with, which is the clue that the operation was backwards.

So what?: When you are given a part, the whole is always bigger. If your answer came out smaller, you multiplied where you should have divided.

The 1 percent method, which is hard to get wrong

This method has two steps and it works on every problem of this shape.

Step A. Divide by the percent to find 1 percent. Step B. Multiply by 100 to reach the whole.

Work the coat.

Step 1. $27 is 30% of the price. Divide by 30 to find what 1% is: 27 ÷ 30 = 0.9. So 1% of the original price is $0.90.

Step 2. The whole price is 100%, so multiply by 100: 0.9 × 100 = 90.

Step 3. The coat originally cost $90.

Step 4. Check it forwards. Is 30% of 90 equal to 27? Ten percent of 90 is 9, so 30% is 27. Yes. The answer survives the check.

The reason this method resists mistakes is that step 1 always makes the number smaller and step 2 always makes it bigger, and you can feel both steps happening. Nothing is hidden.

The division method, which is faster once you trust it

The same job in one step. Since part = percent (as a decimal) × whole, you can undo the multiplication by dividing.

whole = part ÷ decimal form of the percent

Coat again: 27 ÷ 0.30 = 90. One step, same answer.

Dividing by a decimal smaller than 1 makes the number bigger, which is exactly what you want here and is worth pausing on if it feels odd. Asking "how many 0.3s fit into 27" is asking how many thirty-percents make up the whole, and the answer is more than 27 because each 0.3 is a small slice.

Another: 24 is 40% of what number?

Step 1. Write it: whole = 24 ÷ 0.40.

Step 2. Divide: 24 ÷ 0.4 = 60.

Step 3. Check forwards: 40% of 60 is 0.4 × 60 = 24. Correct.

By the 1 percent method: 24 ÷ 40 = 0.6 for one percent, then 0.6 × 100 = 60. Same. Use whichever you like, but check with the other one when the answer matters.

The trap that costs the most: what the percent is describing

Here is a nastier version. A jumper is on sale for $42 after 30% off. What was the original price?

The instinct is to divide 42 by 0.30, giving $140. That is wrong, and the reason is worth reading twice.

The $42 is not 30% of the original price. Thirty percent is what was taken away. What you are holding, the $42, is what is left, and what is left is 100% minus 30% = 70% of the original.

Redo it properly.

Step 1. Identify the percent that $42 represents: 100 − 30 = 70%.

Step 2. Divide: 42 ÷ 0.70 = 60.

Step 3. The original price was $60.

Step 4. Check forwards. Thirty percent of 60 is 18, and 60 minus 18 is 42. It matches the sale price exactly.

Compare the two answers: $140 against $60. Not a small difference. The error came entirely from misreading which percent the given number stood for, not from any arithmetic.

What matters here: Before dividing, ask what percent the number you have actually represents. A discount problem hands you the remaining percent, not the discount percent.

The same trap with tax on top

A bill including 20% tax comes to $84. What was the bill before tax?

Now the given number is bigger than the original, so the percent it represents is over 100.

Step 1. The total is the original plus 20% of the original, which is 100% + 20% = 120% of the original.

Step 2. Divide: 84 ÷ 1.20 = 70.

Step 3. The bill before tax was $70.

Step 4. Check: 20% of 70 is 14, and 70 + 14 = 84. Correct.

The common wrong move is to take 20% of $84, getting $16.80, and subtract it to get $67.20. That is not the same answer, and it is not right, because the 20% was charged on $70, never on $84. This single confusion is behind a great many wrong invoices.

A ratio table does it too

If the algebra feels slippery, the table from Module 1 handles every one of these problems without any new ideas.

For the jumper: $42 is 70%, and you want 100%.

Percent	70	1	100
Dollars	42	0.6	60

Divide 70 by 70 to get the 1 column, so divide 42 by 70 as well: 42 ÷ 70 = 0.6. Then multiply the 1 column by 100: 0.6 × 100 = 60. The table is the 1 percent method with the bookkeeping written down, which is why the two never disagree.

Tracing a wrong answer to the exact line that broke

A student is asked: 35 students are absent, and that is 14% of the school. How many students are in the school? Their work looks like this.

1. 14% of 35
2. 0.14 × 35 = 4.9
3. Answer: 4.9 students

Line 3 is obviously wrong, and not by a little. You cannot have 4.9 students, and a school cannot be smaller than the number of children absent from it. But knowing the answer is wrong is not the same as knowing where it broke, so find the line.

Line 2 is fine. If the question really had been "what is 14% of 35," then 4.9 is correct arithmetic. Line 1 is where it went wrong. The problem never said 14% of 35. It said 35 is 14% of something. The 35 is the part, and the unknown school is the whole.

Fix line 1 and the rest follows.

Step 1. 35 is 14%, so 1% is 35 ÷ 14 = 2.5.

Step 2. The whole is 100%, so 2.5 × 100 = 250.

Step 3. There are 250 students in the school.

Step 4. Check forwards: 14% of 250 is 0.14 × 250 = 35. Correct.

The lesson from the trace is that the mistake was in reading, not in calculating. A useful habit: underline the word "of" in the question. Whatever follows "of" is the whole. In "14% of the school," the whole is the school, which is what you were asked to find, so it cannot also be something you multiply by.

Common misconceptions

"To find the original, take the same percent of the number I have." This is the single most common percent error there is. Taking 30% of $42 gives $12.60, which is 30% of the wrong number. The percent was always a percent of the original, which is the thing you do not yet know.

"A sale price of $42 after 30% off means $42 is 30% of the original." It is 70% of the original. The discount percent names what left; the price names what stayed.

"Adding 20% tax then removing 20% gets you back to the start." It does not, because the 20% you add is a percent of the smaller pre-tax figure and the 20% you remove is a percent of the larger total. The next lesson is entirely about why.

"Dividing should make the answer smaller." Dividing by a number less than 1 makes it bigger. 27 ÷ 0.3 = 90. If that still feels wrong, use the 1 percent method, where both steps are visible.

Where this leaves us

• Given a part and its percent, the whole is found by dividing, not multiplying.
• The 1 percent method: divide by the percent, then multiply by 100.
• The division method: whole = part divided by the decimal form of the percent.
• Always ask what percent the given number stands for. After a 30% discount you hold 70%; after 20% tax you hold 120%.
• Check every answer by running it forwards. It takes ten seconds and catches nearly everything.

Next: what happens when a price goes up and then comes back down by the same percent, and why you do not end up where you started.

Sources

1. Marecek, L., and Mathis, A. H. (2020). Solve sales tax, commission, and discount applications. Prealgebra 2e. OpenStax, Rice University. openstax.org
2. Khan Academy. (n.d.). Ratios, rates, and percentages. 6th grade math. khanacademy.org
3. Wikipedia contributors. (n.d.). Discounting. Wikipedia. en.wikipedia.org

The pay cut that was supposed to be reversed

A workshop cuts everyone's weekly hours by 20 percent in March because orders have dried up. In September the orders come back, and the manager announces that hours are going up by 20 percent, back to where they were.

They are not back to where they were. A worker on $500 a week in February is on $480 a week in October, and that is not a mistake by the payroll clerk. It is arithmetic, and the manager almost certainly does not know it.

This lesson works out why, and gives you the tool that makes it obvious in one line instead of two paragraphs.

Key idea: A percent change is always a percent of something. Two changes in a row are percents of two different somethings, which is why they do not cancel.

Measuring a change as a percent

Start with the basic calculation. A jacket was $80 and is now $92. By what percent did it rise?

Step 1. Find the change: 92 − 80 = 12.

Step 2. Compare that change with the original amount, as a fraction: 12/80.

Step 3. Turn it into a percent: 12 ÷ 80 = 0.15, and 0.15 × 100 = 15%.

Step 4. Answer: a 15% increase.

The rule, in words: percent change = change divided by the original amount, times 100.

The word "original" is the whole lesson in one word. Not the new amount. Not the average of the two. The amount you started from. If you had divided 12 by 92 instead you would get about 13%, which is the wrong answer to this question, though it is the right answer to a different one.

Now a decrease. A bike was $250 and is now $200.

Step 1. Change: 250 − 200 = 50.

Step 2. Over the original: 50/250.

Step 3. 50 ÷ 250 = 0.2, so 20%.

Step 4. Answer: a 20% decrease.

The multiplier, which turns two steps into one

Finding a percent then adding it on is two operations. There is a way to do both at once, and it is the single most useful trick in this module.

To increase by 15%, you want 100% of the original plus 15% more, which is 115% of it. As a decimal that is 1.15. So:

80 × 1.15 = 92. One step, no adding.

To decrease by 20%, you want 100% minus 20%, which is 80% of the original, or 0.80.

250 × 0.80 = 200. One step, no subtracting.

Change	Percent you end up with	Multiplier
Up 5%	105%	1.05
Up 20%	120%	1.20
Up 100%	200%	2.00
Down 10%	90%	0.90
Down 25%	75%	0.75
Down 50%	50%	0.50

Read the table until the pattern is automatic. Increases give a multiplier above 1; decreases give one below 1. And a multiplier is reversible: to undo a × 1.15 you divide by 1.15, which is exactly the reverse problem from last lesson written in a tidier way.

In short: Up p percent means multiply by 1 + p/100. Down p percent means multiply by 1 − p/100.

Now the pay cut, in one line

A wage of $500. Down 20%, then up 20%.

Step 1. Down 20% is a multiplier of 0.80: 500 × 0.80 = 400.

Step 2. Up 20% is a multiplier of 1.20: 400 × 1.20 = 480.

Step 3. Final wage: $480, which is $20 short of the original $500.

Here is the same thing in one line, and it is where the reason lives:

500 × 0.80 × 1.20 = 500 × 0.96 = 480.

The two multipliers combined into a single multiplier of 0.96, which is a 4% decrease. Not zero. Multiply 0.8 by 1.2 and you get 0.96, not 1. That is the entire explanation, and you can check it on a calculator in two seconds.

Why does it come out short? Because the 20% that came off was 20% of $500, which is $100, while the 20% that went back on was 20% of $400, which is only $80. Same percent, smaller base, smaller amount. The percent looks symmetric. The dollars are not.

Does the order matter?

Suppose the manager had done it the other way round: up 20% first, then down 20%.

500 × 1.20 = 600, then 600 × 0.80 = 480.

The same $480. That is not a coincidence: multiplication does not care about order, so 1.2 × 0.8 and 0.8 × 1.2 are both 0.96. Whichever way round you do it, you land 4% below where you started.

And the size of the loss follows a pattern. Up and down by 10% gives 1.1 × 0.9 = 0.99, a 1% loss. Up and down by 50% gives 1.5 × 0.5 = 0.75, a 25% loss. The bigger the percent, the worse the damage, and it grows fast. A 100% rise followed by a 100% fall leaves you with nothing at all, since 2 × 0 = 0.

Why this matters: Percent changes never cancel by having equal percents. They cancel only when the multipliers multiply to exactly 1.

What would actually restore the wage

To get from $400 back to $500 you need a rise of 100, and 100 out of the current 400 is 100 ÷ 400 = 0.25, which is 25%. So a 20% cut needs a 25% rise to undo it. Check: 400 × 1.25 = 500. Correct.

This asymmetry is worth carrying around. A 50% loss needs a 100% gain to recover. A 10% cut needs an 11.1% rise. The recovery percent is always larger than the loss percent, because it is measured against a smaller base.

Percent, and percentage points

One more distinction, because news reports get it wrong constantly. Suppose a class's pass rate goes from 40% to 50%.

• It rose by 10 percentage points, because 50 minus 40 is 10.
• It rose by 25 percent, because the change of 10 compared with the original 40 is 10 ÷ 40 = 0.25.

Both statements are true and they are not the same number. "Percentage points" measures the gap between two percents. "Percent" measures the change relative to where it started. A headline that says "unemployment up 25 percent" and one that says "unemployment up 10 percentage points" describe very different situations, and swapping them is a good way to mislead people without printing anything false.

Common misconceptions

"Up 20 percent then down 20 percent returns you to the start." It leaves you 4% down, every time, because the second percent is taken from a different base. The multipliers 1.2 and 0.8 multiply to 0.96, not 1.

"Percent change is the change divided by the new amount." It is divided by the original. Dividing by the new amount answers a different question and gives a different number.

"A 50 percent loss needs a 50 percent gain to recover." It needs 100 percent. Halving $200 gives $100, and getting from $100 back to $200 means doubling, which is a 100 percent rise.

"A rise from 40 percent to 50 percent is a 10 percent rise." It is a rise of 10 percentage points, which is a 25 percent rise. Both descriptions are legitimate; using the wrong word for the one you mean is how statistics get twisted.

The takeaway

• Percent change equals the change divided by the original amount, times 100.
• Apply a change in one step with a multiplier: 1 plus the decimal for a rise, 1 minus it for a fall.
• Combine two changes by multiplying their multipliers. Only a product of exactly 1 returns you to the start.
• An equal percent up and down always loses, and the loss grows quickly with the size of the percent.
• Undoing a p percent fall needs a rise bigger than p, because the base shrank.
• Percentage points and percent are different measurements of the same change.

That is percent finished. The next module leaves it behind and goes somewhere new: below zero.

Sources

1. Marecek, L., and Mathis, A. H. (2020). Solve sales tax, commission, and discount applications. Prealgebra 2e. OpenStax, Rice University. openstax.org
2. US Bureau of Labor Statistics. (n.d.). Consumer Price Index: Questions and answers. bls.gov
3. Wikipedia contributors. (n.d.). Percentage. Wikipedia. en.wikipedia.org

Standing 282 feet below the sea

There is a salt flat in Death Valley National Park called Badwater Basin. A sign on the cliff above it marks sea level, and the sign is high over your head, because the ground you are standing on is 282 feet below sea level. It is the lowest land in North America.

How do you write that as a number? Not 282. That is the height of a building. What you need is a number that says "282, in the downward direction," and mathematics writes it with a minus sign in front: −282 feet.

The minus sign here is not an instruction to subtract anything. It is part of the number, the way the dot in 3.5 is part of the number. It means "on the other side of zero."

Key idea: A negative number is a position on the far side of zero, and the minus sign is part of its name.

Extending the number line

You have used a number line with 0, 1, 2, 3 running to the right. Now continue it to the left of zero, evenly spaced, and label those positions −1, −2, −3, and so on.

−5	−4	−3	−2	−1	0	1	2	3	4	5
|	|	|	|	|	|	|	|	|	|	|

Three facts fall straight out of that picture, and they carry the whole lesson.

1. Zero is not the smallest number. It is the middle. Everything to the left of it is smaller.
2. Further right always means greater. This one rule settles every comparison, including the confusing ones.
3. Each number has an opposite the same distance from zero on the other side. The opposite of 4 is −4. The opposite of −7 is 7. The opposite of 0 is 0.

Numbers of this kind, the whole numbers together with their negatives, have a name: the integers. So −3, 0 and 12 are integers, while 2.5 and three quarters are not, although those also have negatives and also sit on the line.

The comparison that feels wrong

Which is greater, −4 or −9?

Almost everyone's first instinct is −9, because 9 is bigger than 4. The instinct is wrong, and the number line explains why in one look. Walking left from zero, you pass −4 first and keep going to reach −9. So −9 sits further left, and further left means smaller. Therefore −4 is greater than −9.

Put a thermometer next to it and the strangeness disappears. Would you rather be outside at −4 degrees or −9 degrees? At −4. It is warmer. Warmer means the temperature is greater. So −4 is greater than −9, and it never felt confusing when it was about coats.

Order these from smallest to largest: 3, −7, 0, −2, 5, −10.

Step 1. Split them. Negatives: −7, −2, −10. Zero: 0. Positives: 3, 5.

Step 2. Order the negatives. The one with the biggest digit is furthest left and therefore smallest: −10, then −7, then −2.

Step 3. Assemble: −10, −7, −2, 0, 3, 5.

The point: Among negative numbers, the bigger the digit, the smaller the number. Among positive numbers, the bigger the digit, the bigger the number. The number line reconciles both without you having to remember two rules.

Reading direction instead of size

A positive number and a negative number often describe the same measurement in opposite directions, and which direction counts as positive is a decision somebody made.

Situation	Positive means	Negative means
Elevation	Above sea level	Below sea level, like −282 feet at Badwater Basin
Temperature in Celsius	Above the freezing point of water	Below it
Bank account	Money you have	Money you owe
A lift in a building	Floors above the ground	Basement levels
Time before and after an event	Minutes after launch	Minutes before launch, as in a countdown

Notice that zero is a chosen reference point in every row, not an absence of anything. Sea level is a real height; zero Celsius is a real temperature; the ground floor is a real floor. The negatives are not nothing. They are somewhere.

Moving along the line

You do not need the full rules for integer arithmetic yet, and Pre-Algebra and Math Foundations covers them properly in its integer weeks. What you do need is the picture, because the picture makes the rules obvious later.

Adding a positive means walking right. Subtracting a positive means walking left.

Example: the temperature is 5 degrees and drops by 8. Where does it land?

Step 1. Start at 5.

Step 2. Walk 8 steps left.

Step 3. Count them: 4, 3, 2, 1, 0, then keep going, −1, −2, −3.

Step 4. You land on −3, so the temperature is −3 degrees.

Notice you passed straight through zero without stopping. That is the whole trick. Zero is a place on the road, not a wall.

Another: a diver at −15 metres rises 6 metres.

Step 1. Start at −15. Step 2. Rising means walking right, 6 steps. Step 3. −14, −13, −12, −11, −10, −9. The diver is at −9 metres, still below the surface but nearer to it.

Distance between two points on the line

How much colder is −9 degrees than 4 degrees?

Count the gap in two pieces, because zero is a convenient stopping point.

Step 1. From 4 down to 0 is 4 degrees.

Step 2. From 0 down to −9 is 9 degrees.

Step 3. Add them: 4 + 9 = 13 degrees.

So there is a 13 degree gap. Splitting at zero works every time and avoids the sign errors that come from trying to do it in one move. The next lesson gives the tool that makes this official.

Common misconceptions

"−9 is bigger than −4 because 9 is bigger than 4." The digit tells you how far from zero, not how great. Further left is always smaller, so −9 is the smaller number. The thermometer test settles it: −9 degrees is colder.

"Zero is the smallest number there is." Zero is the middle of the line. Every negative number is smaller than zero.

"The minus sign means subtract." It has two jobs. In 7 − 3 it is an operation. In −3 it is part of the number's name, telling you which side of zero it lives on.

"You cannot have less than nothing, so negatives are not real." They describe direction from a chosen zero, and that zero is a real reference point. Badwater Basin is a real place at −282 feet, and an overdrawn account is a real debt.

What to remember

• The number line continues left of zero, and those positions are the negative numbers.
• Further right means greater, always, and that one rule handles every comparison.
• Among negatives, a bigger digit means a smaller number.
• Every number has an opposite, the same distance from zero on the other side.
• Negative numbers describe direction from a chosen zero: below sea level, below freezing, money owed, floors underground.
• To find the gap between a positive and a negative, count to zero and then onward, and add the two pieces.

That idea of "distance from zero, direction ignored" turns out to be useful enough to get its own name and its own symbol. That is next.

Sources

1. National Park Service. (n.d.). Death Valley National Park. US Department of the Interior. nps.gov
2. Marecek, L., and Mathis, A. H. (2020). Introduction to integers. Prealgebra 2e. OpenStax, Rice University. openstax.org
3. Khan Academy. (n.d.). Negative numbers. 6th grade math. khanacademy.org
4. Wikipedia contributors. (n.d.). Badwater Basin. Wikipedia. en.wikipedia.org

The bag of flour that has to be close enough

A bag on a supermarket shelf is labelled 1000 grams. No packing machine on Earth puts exactly 1000 grams in every bag, so the rules the packer works to allow a small margin. Suppose the margin is 5 grams either way.

A bag at 1003 grams is fine. A bag at 996 grams is fine. A bag at 1009 grams is not, and neither is one at 989. Notice what the inspector actually cares about: not whether the bag is over or under, but how far off it is. Being 9 grams heavy and being 9 grams light are equally wrong.

That question, "how far off, ignoring which way," comes up constantly, and mathematics has a name and a symbol for it.

Key idea: Absolute value is distance from zero. It answers how far, never which way.

The definition, and the two vertical bars

The absolute value of a number is its distance from zero on the number line. It is written with a vertical bar on each side.

• The absolute value of 7 is written as the bars around 7, and it equals 7, because 7 sits seven steps from zero.
• The absolute value of −7 is also 7, because −7 also sits seven steps from zero. It is on the other side, but the walk is the same length.
• The absolute value of 0 is 0, because zero is standing on zero and has not gone anywhere.

So the operation strips off any minus sign and hands back the size. In practice: if the number is positive or zero, leave it alone. If it is negative, drop the sign.

Now the sentence that matters most in this lesson. An absolute value can never be negative. Distance is not a direction. You can be nine grams under target, but you cannot walk a negative number of steps, and there is no such thing as being minus nine grams away from anything.

Working with it

Evaluate each one, showing the reasoning.

The absolute value of −12. Distance from zero: 12. Answer 12.

The absolute value of 4.5. Already positive, so it stays. Answer 4.5.

The absolute value of −3, then take the opposite of the result. Two steps here, and order matters. Step 1: the absolute value of −3 is 3. Step 2: the opposite of 3 is −3. Answer −3. So a minus sign written outside the bars survives, while one written inside them does not. That distinction is worth reading twice, because it is the only genuinely fiddly thing about the whole topic.

The absolute value of −8, added to the absolute value of 5. Step 1: the pieces are 8 and 5. Step 2: 8 + 5 = 13. Answer 13.

Size against position, side by side

Line up a few numbers with their absolute values and the difference between the two questions becomes visible.

Number	Absolute value	Position on the line
−15	15	Furthest left, so the smallest of these
−6	6	Left of zero
0	0	The middle
3	3	Right of zero
11	11	Furthest right, so the largest of these

Read the first column top to bottom and it climbs: −15, −6, 0, 3, 11. Read the second column and it does something else entirely: 15, 6, 0, 3, 11. It falls, then rises. The number with the largest absolute value here, −15, is the smallest number in the list.

Two different questions. "Which is greater?" is about position, so use the number line. "Which is bigger in size?" is about absolute value, so ignore the signs. Confusing the two is the single reliable way to get these problems wrong.

What matters here: A large absolute value tells you a number is far from zero. It does not tell you whether the number is large.

Distance between two numbers

Last lesson found the gap between two numbers by counting to zero and adding the pieces. Absolute value gives the official version: the distance between two numbers is the absolute value of their difference.

How far apart are −3 and 5?

Step 1. Subtract in either order: 5 − (−3) = 8, or −3 − 5 = −8.

Step 2. Take the absolute value: both give 8.

Step 3. The distance is 8.

Look at what the absolute value did there. Subtracting in the two different orders gave answers with opposite signs, but distance cannot depend on which number you name first, and the absolute value is what removes that dependence. It is why the definition is written with the bars rather than told as a rule about which number to put first.

Back to the flour

Now the opening problem can be written in one line. If a bag holds w grams, the amount it is off target is the absolute value of w − 1000, and the packer's rule is that this must be no more than 5.

Test some bags.

Bag weight	w − 1000	Absolute value	Within 5 grams?
1003 g	3	3	Yes
996 g	−4	4	Yes
1009 g	9	9	No, too heavy
989 g	−11	11	No, too light
1000 g	0	0	Yes, exactly on target

The third column is the only one the inspector needs. That is the point of the tool: it converts two separate worries, too heavy and too light, into one number that can be compared against one limit.

The same shape turns up everywhere once you look for it. How far is this reading from the true value? How far is today's temperature from the seasonal average? How far is this machine part from the size on the drawing? Each is an absolute value of a difference.

Common misconceptions

"The absolute value of a negative number is negative." It is never negative. Distance from zero cannot be less than zero, and −7 is seven steps away just as surely as 7 is.

"Absolute value means change the sign." It means make it positive. Applying it to 7 leaves 7, not −7. Numbers that are already positive are untouched.

"The number with the bigger absolute value is the bigger number." Only among positives. Among negatives it is the reverse: −15 has a bigger absolute value than −6 and is the smaller number.

"A minus sign outside the bars disappears too." It does not. The bars only govern what is inside them. The absolute value of −3 is 3, but the opposite of that absolute value is −3.

Summing up

• Absolute value is distance from zero, so it answers how far and never which way.
• It is written with a vertical bar on each side of the number.
• It is never negative. Positive numbers and zero are unchanged; negative numbers lose their sign.
• The distance between two numbers is the absolute value of their difference, which is why the order of subtraction does not matter.
• Comparing size and comparing position are different questions, and only one of them ignores signs.
• Real uses are everywhere a tolerance or an error is measured: how far off, in either direction.

One number line handled position. The next lesson crosses two of them and gets a whole plane.

Sources

1. Khan Academy. (n.d.). Absolute value. 6th grade math: Negative numbers. khanacademy.org
2. Wikipedia contributors. (n.d.). Absolute value. Wikipedia. en.wikipedia.org
3. National Institute of Standards and Technology. (n.d.). Office of Weights and Measures. NIST Physical Measurement Laboratory. nist.gov

The appendix that changed geometry

In 1637 Rene Descartes published a book about how to think clearly, the Discourse on Method. Bolted onto the back of it were three essays meant to show the method in action, and the last of them, La Geometrie, contained an idea so useful that the plane it describes is now named after him.

The idea: if you lay two number lines across each other, every point in the flat space between them can be named by exactly two numbers. Shapes become numbers. Numbers become shapes. Every graph you have ever seen, every map grid, every pixel on the screen you are reading this on, is that idea still running.

Key idea: One number line names a position on a line. Two crossed number lines name a position on a whole plane.

The parts, named

Draw a horizontal number line and a vertical number line so that they cross at each other's zero.

• The horizontal line is the x axis. Right is positive, left is negative.
• The vertical line is the y axis. Up is positive, down is negative.
• The point where they cross is the origin, written (0, 0).
• A point is named by an ordered pair, written (x, y), with the x value first.

The word "ordered" is a warning. The pair (3, 5) and the pair (5, 3) are two different points, not two ways of writing one. Getting them the wrong way round is the most common error on this whole topic, and it survives into algebra and beyond, so it is worth fixing now.

The mnemonic that works: along the corridor, then up the stairs. You walk along a corridor before you climb the stairs, and you read the x coordinate before the y. Always across first, then up or down.

Plotting a point, one step at a time

Plot the point (4, 3).

Step 1. Start at the origin.

Step 2. The first number is 4, and it is positive, so move 4 units to the right along the x axis.

Step 3. The second number is 3, and it is positive, so from there move 3 units up.

Step 4. Mark the point. That is (4, 3).

Plot the point (−2, 5).

Step 1. Start at the origin. Step 2. The x value is −2, so move 2 units left. Step 3. The y value is 5, so move 5 units up. Step 4. Mark it.

Plot the point (0, −4).

Step 1. The x value is 0, so do not move sideways at all; you stay on the y axis. Step 2. The y value is −4, so move 4 units down. Step 3. Mark it. Any point with an x of 0 sits on the y axis, and any point with a y of 0 sits on the x axis.

The four quadrants

The two axes cut the plane into four regions, called quadrants, numbered anticlockwise starting from the top right.

Read the diagram as a rule about signs rather than a picture to memorise. If both coordinates are positive, the point is in Quadrant I. If only the y is positive, Quadrant II. If both are negative, Quadrant III. If only the x is positive, Quadrant IV.

Quadrant	Sign of x	Sign of y	Example point
I	positive	positive	(4, 3)
II	negative	positive	(−2, 5)
III	negative	negative	(−3, −2)
IV	positive	negative	(6, −1)

A point sitting exactly on an axis is in no quadrant at all. The axes are the borders, and (0, −4) is on the border rather than inside one of the four regions.

In short: The two signs tell you the quadrant, and the quadrant tells you the two signs. It works in both directions.

Reflections, and what they do to the numbers

Take the point (5, 3) and flip it across the x axis, as though the axis were a mirror lying on the floor.

Step 1. The point was 5 to the right. A mirror on the horizontal axis does not change how far right you are, so x stays 5.

Step 2. The point was 3 above. Its mirror image is 3 below, so y becomes −3.

Step 3. The reflected point is (5, −3).

Now flip (5, 3) across the y axis instead, a mirror standing upright.

Step 1. Height is unchanged, so y stays 3. Step 2. Five to the right becomes five to the left, so x becomes −5. Step 3. The reflected point is (−5, 3).

The pattern is worth stating plainly: reflecting across the x axis flips the sign of y; reflecting across the y axis flips the sign of x. The coordinate that changes is the one measured perpendicular to the mirror, which is exactly what you would expect from holding a piece of paper up to a real mirror.

How far apart are two points?

When two points share a coordinate, the distance is easy, and it is the absolute value idea from the last lesson doing the work.

Distance from (2, 7) to (2, −3). Both have x = 2, so they sit on the same vertical line. Only the heights differ. The gap is the absolute value of 7 − (−3) = 10. They are 10 units apart.

Distance from (−6, 4) to (5, 4). Both have y = 4, so they lie on the same horizontal line. The gap is the absolute value of 5 − (−6) = 11. Eleven units apart.

When neither coordinate matches, the two points sit on a slant and the distance needs the Pythagorean theorem, which arrives in later courses. For now, stick to points that share a row or a column, and you can do it all in your head.

Common misconceptions

"(3, 5) and (5, 3) are the same point." They are different points in different places. The pair is ordered, and x always comes first. Along the corridor, then up the stairs.

"Quadrant II is the bottom right." The numbering runs anticlockwise from the top right, so II is top left, III bottom left, IV bottom right. Learning it as a sign rule rather than a picture makes it stick better.

"A point on the x axis is in Quadrant I or IV." Points on an axis are in no quadrant. The axes are the boundaries between the four regions, not part of them.

"Reflecting a point changes both coordinates." Only one changes: the one measured perpendicular to the mirror line. Across the x axis, y flips and x stays put.

Where this leaves us

• Two crossed number lines make a plane, with the x axis horizontal and the y axis vertical, meeting at the origin (0, 0).
• A point is an ordered pair (x, y). The order is not negotiable: across first, then up or down.
• The four quadrants are numbered anticlockwise from the top right, and the signs of the two coordinates identify which one a point is in.
• Points on an axis belong to no quadrant.
• Reflecting across an axis flips the sign of one coordinate, the one measured perpendicular to that axis.
• Points sharing a coordinate are separated by the absolute value of the difference in the other coordinate.

Coordinates put numbers into places. The next module puts letters into numbers.

Sources

1. Marecek, L., and Mathis, A. H. (2020). Use the rectangular coordinate system. Prealgebra 2e. OpenStax, Rice University. openstax.org
2. Khan Academy. (n.d.). Coordinate plane. 6th grade math: Negative numbers. khanacademy.org
3. Wikipedia contributors. (n.d.). Cartesian coordinate system. Wikipedia. en.wikipedia.org
4. Wikipedia contributors. (n.d.). Rene Descartes. Wikipedia. en.wikipedia.org

The car park sign with only two numbers on it

A car park charges $3 to enter, then $2 for every hour you stay. That is the whole sign. No table of prices, no list of durations.

The sign does not need a table, because those two numbers answer every possible question. Stay one hour and you pay 3 + 2 = $5. Stay four hours and you pay 3 + 2 × 4 = $11. Stay nine hours and you pay 3 + 2 × 9 = $21.

Notice what stayed the same in all three: the shape of the calculation. Three, plus two times however many hours. Write that shape down once, with a letter standing in for the number you do not know yet, and you have the whole price list in five characters: 3 + 2h.

Key idea: A letter in a calculation holds a place for a number that has not been chosen yet, which lets one line answer a question for every possible input.

The vocabulary, which you need because the questions use it

Take the expression 3 + 2h.

• A variable is a letter standing for a number that can change. Here it is h, for hours.
• A coefficient is the number multiplying a variable. Here it is 2, the hourly rate.
• A constant is a number on its own, not attached to any variable. Here it is 3, the entry fee, which you pay no matter how long you stay.
• A term is a chunk separated by plus or minus signs. This expression has two terms: 3 and 2h.
• An expression is the whole thing. It has no equals sign, so it is not asking you to find anything. It is a recipe waiting for an input.

One notation rule: 2h means 2 times h. The multiplication sign is dropped, because the letter x is used so often as a variable that writing x for multiplication too would be a disaster. So 5n is 5 times n, and 7ab is 7 times a times b. A number written next to a letter always means multiply.

Careful with the other direction, though. The two-digit number 23 does not mean 2 times 3. Digits written together make a number; a number written next to a letter means multiplication. Only the second case drops a hidden multiplication sign.

Turning words into expressions

Most of the difficulty in early algebra is translation, not calculation. Build a habit: read the sentence, decide what the unknown is, name it with a letter, then write the operations in the order the sentence describes them.

In words	As an expression	Why
Five more than a number	n + 5	More than means add
Five less than a number	n − 5	Less than means subtract, and the order flips
Three times a number	3n	Times means multiply, sign dropped
A number divided by four	n/4	Divided by means a fraction bar
Twice a number, then add seven	2n + 7	Two operations, in the order given
Seven added to a number, then doubled	2(n + 7)	The brackets force the addition first

Look at the last two rows together, because they are the trap. "Twice a number then add seven" and "add seven then double" contain the same words and are different expressions with different answers. Try n = 4: the first gives 2 × 4 + 7 = 15, the second gives 2 × (4 + 7) = 22. Brackets are how you record which happened first, and they are not optional decoration.

The row for "five less than a number" is the other classic. It becomes n − 5, not 5 − n, even though the 5 is spoken first. Read it as "start with the number, take five away."

What matters here: Translate the operations in the order they actually happen, not the order the words appear.

Substitution: putting a number back in

To evaluate an expression is to replace the variable with a value and work out the result.

Evaluate 3 + 2h when h = 6.

Step 1. Write the expression: 3 + 2h.

Step 2. Replace h with 6, putting brackets round it to keep the multiplication visible: 3 + 2(6).

Step 3. Multiplication runs before addition: 2 × 6 = 12.

Step 4. Now add: 3 + 12 = 15.

Step 5. Answer: $15 for six hours.

Step 3 is where results go wrong. Working left to right instead would give 3 + 2 = 5, then 5 × 6 = 30, which is double the real price. The order of operations, covered in full in Pre-Algebra and Math Foundations, is what settles it: brackets first, then multiplication and division, then addition and subtraction.

Evaluate 5n − 4 when n = 3.

Step 1. Substitute: 5(3) − 4.

Step 2. Multiply first: 15 − 4.

Step 3. Subtract: 11.

Evaluate 2(a + 7) when a = 5.

Step 1. Substitute: 2(5 + 7).

Step 2. Brackets first: 5 + 7 = 12.

Step 3. Then multiply: 2 × 12 = 24.

Evaluate 4x + 3y when x = 2 and y = 5. Two variables now, but nothing changes.

Step 1. Substitute both: 4(2) + 3(5).

Step 2. Both multiplications: 8 + 15.

Step 3. Add: 23.

Substituting a negative number

This is where brackets stop being optional. Evaluate 6 + 4t when t = −2.

Step 1. Substitute with brackets: 6 + 4(−2).

Step 2. Multiply: 4 × −2 = −8. A positive times a negative gives a negative.

Step 3. Add: 6 + (−8), which means starting at 6 and walking 8 to the left, landing on −2.

Step 4. Answer: −2.

Written without brackets, step 1 would read 6 + 4 − 2, which is 8, and it is wrong. The brackets are what keep the minus sign attached to the 2 instead of loose in the middle of the sum. Get in the habit now and the habit will save you repeatedly later.

An expression is not an equation

One distinction to carry into the next lesson.

• 3 + 2h is an expression. It has no equals sign. You cannot solve it, because it is not asking anything. You can only evaluate it, once you are told what h is.
• 3 + 2h = 15 is an equation. It makes a claim: this expression has the value 15. Now there is something to find, namely which h makes the claim true.

Expression, evaluate. Equation, solve. The two words are not interchangeable, and mixing them up makes questions incomprehensible.

Common misconceptions

"2h means twenty-something, or 2 and h stuck together." It means 2 multiplied by h. A number beside a letter is always a multiplication with the sign left out.

"Five less than a number is 5 − n." It is n − 5. The phrase describes taking 5 away from the number, so the number comes first. The wording puts them the other way round on purpose, and that is why it catches people.

"You can evaluate left to right." Only if the operations happen to allow it. In 3 + 2(6) the multiplication runs first, giving 15, while left to right gives 30.

"An expression can be solved." It can only be evaluated for a chosen value. Solving needs an equals sign, because solving means finding what makes a statement true, and an expression makes no statement.

Recap

• A variable is a letter holding a place for a number that can change.
• A coefficient multiplies a variable; a constant stands alone; terms are the chunks separated by plus and minus signs.
• A number written beside a letter means multiply.
• Translate word problems by the order operations happen, and use brackets to record what came first.
• Evaluate by substituting the value in brackets, then following the order of operations.
• Expressions are evaluated; equations are solved. Only equations have an equals sign.

Next: an expression, an equals sign, a number, and the question of which value makes it all true.

Sources

1. Marecek, L., and Mathis, A. H. (2020). Evaluate, simplify, and translate expressions. Prealgebra 2e. OpenStax, Rice University. openstax.org
2. Khan Academy. (n.d.). Parts of algebraic expressions. 6th grade math. khanacademy.org
3. Wikipedia contributors. (n.d.). Variable (mathematics). Wikipedia. en.wikipedia.org

A sealed box on a kitchen scale

A sealed box sits on a kitchen scale, which reads 940 grams. You know the empty box weighs 115 grams, because you weighed it before you packed it. How much is inside?

Almost everybody can do this without any algebra: 940 − 115 = 825 grams. Fine. But watch what your brain actually did. It knew that the contents plus the box equals 940, and it undid the "plus the box" by subtracting. That is the entire method of this lesson, and writing it down properly is what lets you use it later on problems you cannot do in your head.

In symbols, with c for the contents: c + 115 = 940.

The core of it: Solving an equation means undoing whatever was done to the variable, using the opposite operation, on both sides at once.

What "solve" actually means

An equation is a claim that two things are equal. The equals sign is the claim. To solve it is to find the value of the variable that makes the claim true.

Test that idea before doing any solving. Is x = 4 a solution of x + 3 = 7?

Step 1. Substitute: 4 + 3 = 7.

Step 2. Work out the left side: 7.

Step 3. Compare with the right side: 7. They match, so the claim is true, so 4 is a solution.

Try x = 5 in the same equation: 5 + 3 = 8, and 8 is not 7, so 5 is not a solution. Every equation of this kind has exactly one number that works, and the job is to find it without testing every number in the world.

The balance, and why both sides

Picture the equation as an old balance scale with two pans. The equals sign means the pans are level. Whatever is on the left weighs the same as whatever is on the right.

Now: if you take 115 grams off one pan and nothing off the other, the balance tips and the claim stops being true. If you take 115 grams off both pans, it stays level. The two sides changed, but they still match, so it is still a true equation, and it is now a simpler one.

That is the golden rule of equation solving, and there are no exceptions to it: whatever you do to one side, do to the other.

Solve the box.

Step 1. Write the equation: c + 115 = 940.

Step 2. The variable has had 115 added to it. The opposite of adding 115 is subtracting 115.

Step 3. Subtract 115 from both sides: c + 115 − 115 = 940 − 115.

Step 4. On the left, +115 − 115 cancels to nothing, leaving c. On the right, 940 − 115 = 825.

Step 5. So c = 825 grams.

Step 6. Check by substituting back into the original: 825 + 115 = 940. It matches the scale. Done.

The four one-step equations

There are only four things that can have been done to a variable, so there are only four kinds of one-step equation, and each is undone by its opposite.

What was done to the variable	Example equation	Inverse operation	Solution
Something was added	x + 7 = 12	Subtract 7 from both sides	x = 5
Something was subtracted	x − 4 = 9	Add 4 to both sides	x = 13
It was multiplied	6x = 42	Divide both sides by 6	x = 7
It was divided	x/5 = 8	Multiply both sides by 5	x = 40

Work the third row slowly, because it is the one people rush.

Solve 6x = 42.

Step 1. Read what is happening: x has been multiplied by 6.

Step 2. The inverse of multiplying by 6 is dividing by 6.

Step 3. Divide both sides by 6: 6x/6 = 42/6.

Step 4. On the left the 6s cancel, leaving x. On the right, 42 ÷ 6 = 7.

Step 5. x = 7.

Step 6. Check: 6 × 7 = 42. Correct.

And the fourth row, which feels backwards the first time.

Solve x/5 = 8.

Step 1. x has been divided by 5.

Step 2. The inverse is multiplying by 5.

Step 3. Multiply both sides by 5: (x/5) × 5 = 8 × 5.

Step 4. On the left the division and the multiplication cancel, leaving x. On the right, 40.

Step 5. x = 40. Check: 40 ÷ 5 = 8. Correct.

Remember: Look at what has been done to the variable, then do the opposite to both sides. Add undoes subtract; multiply undoes divide.

Equations with a negative in them

Solve x + 9 = 4.

Step 1. Nine has been added, so subtract 9 from both sides: x = 4 − 9.

Step 2. Start at 4 and walk 9 steps left, through zero: x = −5.

Step 3. Check: −5 + 9 = 4. Correct.

A negative answer is not a sign that something went wrong. It just means the number you were looking for lives on the left of zero, which the number line has already made an ordinary place for.

Solve x − 3 = −7.

Step 1. Three has been subtracted, so add 3 to both sides: x = −7 + 3.

Step 2. Start at −7 and walk 3 to the right: x = −4.

Step 3. Check: −4 − 3 = −7. Correct.

From a sentence to an equation

Most real problems arrive as sentences. Three steps get you from one to the other: name the unknown, write the equation, solve it, then answer the question that was actually asked.

A class collected 168 tins of food, which is 4 times what they collected last year. How many did they collect last year?

Step 1. Let t be last year's total.

Step 2. Four times last year equals this year: 4t = 168.

Step 3. Divide both sides by 4: t = 168 ÷ 4 = 42.

Step 4. Check: 4 × 42 = 168. Correct.

Step 5. Answer the actual question in words: they collected 42 tins last year.

After spending $23, Kai has $61 left. How much did he start with?

Step 1. Let s be the starting amount.

Step 2. Starting amount minus what he spent equals what is left: s − 23 = 61.

Step 3. Add 23 to both sides: s = 61 + 23 = 84.

Step 4. Check: 84 − 23 = 61. Correct. He started with $84.

Equations that need two moves, such as 3x + 4 = 19, work the same way with one extra step, and they are taught in Pre-Algebra and Math Foundations. Everything you need for them is in this lesson already.

Common misconceptions

"You only have to change the side with the variable on it." Then the two sides stop being equal and every later line is false. The balance has two pans, and both must be treated identically.

"To solve 6x = 42, subtract 6." Subtracting undoes adding, not multiplying. In 6x nothing was added to x; it was multiplied, so the inverse is division. Subtracting 6 would give 6x − 6, which is not x.

"A negative solution means I made a mistake." It means the answer lies left of zero. Checking by substitution is what tells you whether it is right, and −5 checks out perfectly in x + 9 = 4.

"Checking the answer is optional." It is the cheapest quality control in mathematics. One substitution, ten seconds, and it catches nearly every slip, including sign errors and using the wrong inverse.

Pulling it together

• An equation claims two things are equal; solving it means finding the value that makes the claim true.
• Whatever you do to one side, do to the other. That keeps the balance level.
• Identify what was done to the variable, then apply the inverse operation.
• Add and subtract are inverses; multiply and divide are inverses.
• Negative solutions are ordinary. So are checking steps, and they cost almost nothing.
• For word problems, name the unknown first, write the equation second, and answer in a sentence last.

That is the algebra in this course. The last module returns to shapes and to data.

Sources

1. Marecek, L., and Mathis, A. H. (2020). Solve equations using the subtraction and addition properties of equality. Prealgebra 2e. OpenStax, Rice University. openstax.org
2. Khan Academy. (n.d.). One-step multiplication and division equations. 6th grade math. khanacademy.org
3. Wikipedia contributors. (n.d.). Equation. Wikipedia. en.wikipedia.org

The gable end that needs one and a bit tins of paint

The end wall of a house is a rectangle 6 metres wide and 4 metres tall, with a triangular gable on top that rises another 2 metres to the roof peak. The tin of paint says one litre covers 12 square metres.

How many litres? The wall is not a rectangle and it is not a triangle. It is both, stacked. That is the situation this lesson is really about: almost nothing in the world is one of the shapes in the formula list, and the skill is cutting real shapes into ones that are.

Key idea: Area is the number of unit squares that fit inside a shape. Every area formula in this lesson is a shortcut for counting those squares.

Starting from the rectangle

Area of a rectangle is length times width, and the reason is worth saying once. A rectangle 5 cm by 3 cm can be divided into a grid with 3 rows of 5 squares. Counting them is 5 + 5 + 5, which is 5 × 3 = 15. The formula is repeated addition wearing a coat.

The unit matters as much as the number. Fifteen what? Fifteen squares, each 1 cm on a side, so 15 square centimetres, written 15 cm². That little raised 2 is not decoration. It records that two lengths were multiplied together, and an area answer without it is incomplete.

The parallelogram, and one clean cut

A parallelogram is a four-sided shape whose opposite sides are parallel: a rectangle that has been pushed over. Its area is base × perpendicular height, and here is why.

Cut a right-angled triangle off one end of the parallelogram, slide it round to the other end, and it fits exactly. What you are left with is a rectangle with the same base and the same height. No area was added or removed by moving a piece around, so the parallelogram's area equals the rectangle's: base times height.

Find the area of a parallelogram with base 9 cm and height 4 cm.

Step 1. Write the formula: area = base × height.

Step 2. Substitute: 9 × 4.

Step 3. Multiply: 36.

Step 4. Answer with units: 36 cm².

Now the trap, and it is the main one in this whole lesson. The height is the perpendicular distance between the two parallel sides, measured straight across, not the length of the slanted side. Suppose that same parallelogram has slanted sides of 5 cm. Using 5 instead of 4 gives 45 cm², which is wrong, and it is wrong because the slanted side leans, so it is longer than the actual gap between the two lines. Squares stack straight up, not at an angle.

What matters here: Height always means the perpendicular distance. If a diagram shows a dashed line inside the shape with a right angle marked, that dashed line is the height.

The triangle, which is half of something

Take any triangle. Make an identical copy, turn the copy upside down, and slot the two together along a matching side. They form a parallelogram, exactly, with the same base and the same height as the original triangle.

Two triangles make one parallelogram, so one triangle is half of one:

Area of a triangle = (base × height) ÷ 2

Find the area of a triangle with base 10 cm and height 6 cm.

Step 1. Multiply base by height: 10 × 6 = 60.

Step 2. Halve it: 60 ÷ 2 = 30.

Step 3. Answer: 30 cm².

Forgetting step 2 is the most common error in the topic, and the answer it gives, 60, is the rectangle the triangle sits inside. If you can picture the triangle filling half of that rectangle, you will not forget again.

One more thing about triangles: any of the three sides can be called the base, as long as the height is measured perpendicular to that same side. Choose whichever pairing the diagram actually gives you numbers for.

Composite shapes, method one: cut it up

Back to the gable wall: a 6 m by 4 m rectangle with a triangle on top, base 6 m and height 2 m.

Step 1. Split it into two shapes you know.

Step 2. Rectangle: 6 × 4 = 24 m².

Step 3. Triangle: (6 × 2) ÷ 2 = 12 ÷ 2 = 6 m².

Step 4. Add: 24 + 6 = 30 m².

Step 5. Paint: one litre covers 12 m², so 30 ÷ 12 = 2.5 litres. Buy three tins, because paint comes in tins and walls need a second coat anyway.

Notice that the last step was a unit rate, from Module 1. These topics are not separate compartments.

Composite shapes, method two: build it big and subtract

An L-shaped room. Imagine the outline: the whole space would be 8 m by 6 m if the corner were filled in, but a 3 m by 2 m rectangle has been bitten out of one corner.

Step 1. Find the area of the big rectangle as if it were whole: 8 × 6 = 48 m².

Step 2. Find the area of the missing bite: 3 × 2 = 6 m².

Step 3. Subtract: 48 − 6 = 42 m².

You could have split the L into two rectangles instead and added them, and you would get 42 either way. Both methods are correct, and choosing between them is about which one gives you fewer unknown lengths to work out. Subtracting is usually easier when a neat rectangular piece is missing; splitting is usually easier when the shape is a run of blocks.

The upshot: There is no formula for a composite shape. There is only cutting it into shapes that have formulas, and keeping track of which pieces to add and which to take away.

A worked composite with a triangle removed

A rectangular metal plate 12 cm by 7 cm has a triangular notch cut out of it. The notch has base 4 cm and height 3 cm. What area of metal is left?

Step 1. Whole plate: 12 × 7 = 84 cm².

Step 2. Notch: (4 × 3) ÷ 2 = 6 cm².

Step 3. Remaining metal: 84 − 6 = 78 cm².

Step 4. Sanity check: the notch is small, so the answer should be a little under 84. It is.

Common misconceptions

"The height of a parallelogram is the slanted side." It is the perpendicular distance between the parallel sides. The slanted side is longer because it leans, and using it always overstates the area.

"Forgetting to halve is a small slip." It doubles the answer. A triangle with base 10 and height 6 has area 30, not 60, and 60 is the rectangle it fits inside.

"Area and perimeter are much the same." Perimeter is the distance round the outside, measured in cm; area is the surface inside, measured in cm squared. Two shapes can share a perimeter and have very different areas.

"Composite shapes need their own formula." They need decomposition. Split into known shapes and add, or complete the shape and subtract the missing piece.

The short version

• Area counts unit squares, and the answer is always in square units.
• Rectangle: length times width. Parallelogram: base times perpendicular height, because a cut-and-slide turns it into a rectangle.
• Triangle: base times height, then halved, because two identical triangles make a parallelogram.
• Height means the perpendicular distance every single time, never the slanted side.
• For a composite shape, split and add, or complete and subtract. Both work; pick the one with fewer unknowns.

Flat shapes have area. Solid ones have surface area and volume, and the way in to both is to unfold the solid flat.

Sources

1. Khan Academy. (n.d.). Areas of triangles. 6th grade math: Geometry. khanacademy.org
2. Marecek, L., and Mathis, A. H. (2020). Solve geometry applications: circles and irregular figures. Prealgebra 2e. OpenStax, Rice University. openstax.org
3. Wikipedia contributors. (n.d.). Parallelogram. Wikipedia. en.wikipedia.org
4. PhET Interactive Simulations, University of Colorado Boulder. (n.d.). Area builder. phet.colorado.edu

The cereal box, flat on the table

Take an empty cereal box, pull the glued flaps apart carefully, and press the whole thing flat. What you now have is a single connected sheet of cardboard with six rectangles in it, and it is the exact shape the factory cut from a roll.

That flat sheet is called a net, and it is genuinely useful, not just a way of showing off. Two of the hardest-sounding measurements in middle school geometry become easy the moment the solid is flat, because a net turns a three-dimensional problem into the two-dimensional problem you solved last lesson.

Key idea: Surface area is the area of the net. Once a solid is unfolded, there is nothing new to learn.

What a net is

A net is a flat pattern that folds up into a solid, with no overlaps and no gaps. Every face of the solid appears exactly once.

A cube has 6 identical square faces, so its net is 6 squares joined edge to edge. There are eleven genuinely different arrangements of six squares that fold into a cube, and plenty of arrangements that do not, which is why the paper-and-scissors test in this lesson's activity is worth actually doing rather than reading about.

A rectangular prism, which is the mathematical name for a box, has 6 rectangular faces in three matching pairs: front and back, left and right, top and bottom. Opposite faces are always identical, and that fact is what makes surface area quick.

Surface area, counted off the net

Find the surface area of a box 5 cm long, 4 cm wide and 3 cm tall.

Step 1. Identify the three pairs of faces and the dimensions of each.

Pair of faces	Dimensions	Area of one	Area of both
Front and back	5 by 3	15 cm2	30 cm2
Left and right	4 by 3	12 cm2	24 cm2
Top and bottom	5 by 4	20 cm2	40 cm2

Step 2. Add the last column: 30 + 24 + 40 = 94.

Step 3. Answer: 94 cm².

Check that you found six faces, not three. Counting each pair once is the commonest slip and gives 47, exactly half the right answer. The table above is designed so the doubling cannot be skipped.

A cube with edge 6 cm. All six faces are 6 by 6, so each is 36 cm², and 6 × 36 = 216 cm². That one is quick because there is only one face size to compute.

Volume, counted in unit cubes

Surface area is about the skin. Volume is about the inside: how much space the solid takes up, or how much would fit in it.

Picture filling the 5 by 4 by 3 box with centimetre cubes. Lay one flat layer on the bottom: 5 along by 4 across is 20 cubes. Now stack layers until the box is full: it is 3 cm tall, so 3 layers. Total cubes: 20 × 3 = 60.

So the volume is 60 cubic centimetres, written 60 cm³. The raised 3 records that three lengths were multiplied. The formula is just that stacking, written short:

Volume of a rectangular prism = length × width × height

Step 1. Substitute: 5 × 4 × 3.

Step 2. Multiply the first two: 20.

Step 3. Multiply by the third: 60.

Step 4. Answer: 60 cm³.

Because multiplication does not care about order, it does not matter which dimension you call length and which you call height. You can multiply them in any order and get 60 every time.

In short: Two lengths multiplied give square units and measure a surface. Three lengths multiplied give cubic units and measure a space.

Prisms that are not boxes

There is a version of the volume rule that covers far more shapes: volume = area of the cross-section × length. A prism is any solid whose cross-section stays the same all the way along, like a loaf of bread with identical slices, and the rule says to work out the area of one slice and multiply by how many centimetres of loaf there are.

A triangular prism: the triangular end has base 6 cm and height 4 cm, and the prism is 10 cm long.

Step 1. Area of the triangular end: (6 × 4) ÷ 2 = 12 cm².

Step 2. Multiply by the length: 12 × 10 = 120.

Step 3. Answer: 120 cm³.

Check it against the box rule. A rectangular prism is a prism whose cross-section is a rectangle, so area of cross-section times length gives (5 × 4) × 3 = 60, the same answer as before. One rule, two shapes.

Two boxes, same volume, different amounts of cardboard

Here is something worth knowing, because it is why packaging looks the way it does.

Box	Dimensions	Volume	Surface area
A	8 by 2 by 2 cm	32 cm3	2(16) + 2(16) + 2(4) = 72 cm2
B	4 by 4 by 2 cm	32 cm3	2(16) + 2(8) + 2(8) = 64 cm2

Both boxes hold exactly the same amount, 32 cubic centimetres. Box A needs 72 square centimetres of cardboard and box B needs only 64. The long thin box wastes 8 square centimetres of material for no extra capacity.

The general pattern: for a fixed volume, the closer a box gets to a cube, the less surface area it needs. That is why bulk goods are shipped in near-cubic cartons and why a long thin package feels wasteful when you open it. Volume and surface area are genuinely independent measurements, and knowing one does not tell you the other.

Which one does the question want?

Read the situation, not the numbers, and ask what is being measured.

• Wrapping paper for a present: surface area, because paper covers the outside.
• Sand to fill a sandpit: volume, because sand fills the inside.
• Paint for the outside of a shed: surface area.
• Water a fish tank holds: volume.
• Cardboard the factory cuts for a box: surface area.
• Cereal the box holds: volume.

The units give it away too. If the answer is in square units, it is a surface. If it is in cubic units, or in litres and millilitres, it is a space.

Common misconceptions

"A box has three faces, so add three areas." A box has six faces in three matching pairs. Forgetting to double gives exactly half the true surface area.

"Any six squares in a row fold into a cube." They do not. Only certain arrangements work, and six in a straight line is one of the ones that fails. Cutting one out and folding it is the fastest way to see this.

"Bigger volume means bigger surface area." Not necessarily. Two boxes with equal volume can need very different amounts of cardboard, and the nearer a box is to a cube, the less it needs.

"Cubic centimetres and square centimetres are roughly the same." They measure different things entirely. One counts flat squares covering a surface; the other counts solid cubes filling a space. Writing the raised 2 or 3 is how you record which.

What to carry forward

• A net is a solid unfolded flat, with every face appearing once.
• Surface area is the total area of the net, which for a box is three pairs of matching rectangles.
• Volume of a rectangular prism is length times width times height, which is layers of unit cubes counted quickly.
• For any prism, volume is the area of the cross-section times the length.
• Square units mean a surface; cubic units mean a space. Equal volumes can have very different surface areas.
• Decide which quantity the question wants by asking whether something covers the outside or fills the inside.

Shapes done. The last two lessons are about reading what other people have drawn, and about not being fooled by it.

Sources

1. Khan Academy. (n.d.). Nets of 3D figures. 6th grade math: Geometry. khanacademy.org
2. Marecek, L., and Mathis, A. H. (2020). Solve geometry applications: volume and surface area. Prealgebra 2e. OpenStax, Rice University. openstax.org
3. Wikipedia contributors. (n.d.). Net (polyhedron). Wikipedia. en.wikipedia.org
4. Wikipedia contributors. (n.d.). Cuboid. Wikipedia. en.wikipedia.org

Twenty-four students and one very long night

A teacher asks 24 students how many minutes they spent on homework last night. Sorted from smallest to largest, the answers are:

10, 15, 15, 20, 20, 25, 25, 25, 25, 25, 30, 30, 30, 30, 35, 35, 40, 45, 45, 50, 60, 70, 90, 120

You could read that list all evening and learn very little. Twenty-four numbers is already too many to hold in your head at once. So people draw them, and this lesson draws this one list three different ways to show what each drawing is good at, because the choice of display changes what you notice.

Key idea: A data display is a decision about what to show and what to hide. Three displays of the same numbers can leave three different impressions, all of them honest.

The dot plot: every value, exactly where it is

A dot plot puts one dot above the number line for each data value. Same value, dots stacked.

Minutes	Dots	How many
10	o	1
15	o o	2
20	o o	2
25	o o o o o	5
30	o o o o	4
35	o o	2
40	o	1
45	o o	2
50	o	1
60	o	1
70	o	1
90	o	1
120	o	1

Three things jump out of that picture that the list did not offer.

1. A cluster. Most of the class sits between 20 and 35 minutes, with the tallest stack at 25.
2. A tail. The values thin out to the right and keep going: 60, 70, 90.
3. An outlier. One student reported 120 minutes, sitting far away from everyone else with a big gap before it.

Dot plots keep every single value, so nothing is lost. That is their strength and also their limit: with 24 values it is readable, and with 2400 it would be a smear.

The histogram: values grouped into bins

A histogram gives up the individual values and counts how many fall into each interval, or bin. Using bins 20 minutes wide:

Bin (minutes)	Count	Bar
0 to 19	3	|||
20 to 39	13	|||||||||||||
40 to 59	4	||||
60 to 79	2	||
80 to 99	1	|
100 to 119	0
120 to 139	1	|

The shape is now unmistakable: one tall bar with a long thin tail trailing off to the right. Statisticians call that skewed to the right, and it means the far-away values sit on the high side.

Two features of histograms are worth knowing, because they are how you tell one from a bar chart.

• The bars touch. There are no gaps between them, because the bins are stretches of a number line with nothing in between. A bar chart shows separate categories, like favourite colours, so its bars are drawn apart.
• The order is fixed. Bins run in numerical order and you cannot rearrange them. A bar chart's categories can be shuffled without lying about anything.

Bin width is a real choice, and it changes the picture. Wide bins smooth the data and can hide a gap; narrow bins show detail and can make random wobbles look like patterns. There is no single correct width, which is worth remembering when someone shows you a histogram that proves their point unusually neatly.

Worth holding on to: A histogram trades individual values for a clear view of shape. Once the data is binned, you cannot get the original numbers back.

The box plot: five numbers and nothing else

A box plot throws away almost everything and keeps five numbers, called the five-number summary. Build it for the homework data.

Step 1. Minimum. The smallest value: 10.

Step 2. Median. The middle value. With 24 values there is no single middle, so average the 12th and 13th: both are 30, so the median is 30.

Step 3. Lower quartile, Q1. The median of the bottom half, which is the first 12 values. Average the 6th and 7th of those: both are 25, so Q1 = 25.

Step 4. Upper quartile, Q3. The median of the top half, the last 12 values. Average the 18th and 19th values overall: both are 45, so Q3 = 45.

Step 5. Maximum. The largest value: 120.

The five-number summary is therefore 10, 25, 30, 45, 120. The box is drawn from Q1 to Q3 with a line at the median, and whiskers run out to the minimum and maximum.

Part of the plot	Value	What it tells you
Left whisker end	10	The shortest homework time
Left edge of box, Q1	25	A quarter of the class did less than this
Line inside the box	30	Half the class did less, half did more
Right edge of box, Q3	45	Three quarters of the class did less than this
Right whisker end	120	The longest homework time

Each of the four sections holds a quarter of the students, six people each. That is the part everyone gets wrong at first: the sections are equal in count, not in width. The box runs from 25 to 45, twenty minutes wide and holding twelve students, while the right whisker runs from 45 to 120, seventy-five minutes wide and holding only six. A long whisker means the values in it are spread out, not that there are more of them.

The width of the box itself, Q3 minus Q1, is 45 − 25 = 20 minutes. That is the interquartile range, and it describes the middle half of the class while ignoring the extremes entirely, which is exactly why people use it when one strange value would otherwise dominate.

The outlier, the mean and the median

Add all 24 values and you get 915 minutes. Divide by 24 and the mean is 38.1 minutes. But the median is 30 minutes.

Why the gap? Because the mean has to share out the 120-minute student across everybody, while the median only cares about who is standing in the middle of the queue. Remove that one student and the mean of the remaining 23 drops to 795 ÷ 23, which is about 34.6, while the median barely shifts.

So "the average student did 38 minutes" is a defensible sentence, and it is also slightly misleading, because 16 of the 24 students did less than 38 minutes. The median describes the typical student better whenever the data has a long tail, which is why incomes and house prices are almost always reported as medians.

Which display for which job

Display	Shows well	Hides	Best when
Dot plot	Every value, gaps, repeats, outliers	Nothing, but it gets crowded	Small data sets, up to about 50 values
Histogram	Overall shape and where the bulk sits	Individual values, exact figures inside a bin	Large data sets where shape matters
Box plot	Median, spread, and comparisons between groups	Shape, clusters, gaps, how many values there are	Comparing two or more groups side by side

The last row is the box plot's real purpose. Three box plots stacked above one number line compare three classes instantly. Three histograms take much longer to read against each other, and three dot plots even longer.

Common misconceptions

"A longer section of a box plot has more data in it." All four sections hold a quarter of the values. A long whisker means those values are more spread out, not more numerous.

"A histogram and a bar chart are the same thing." Histogram bars touch and show numerical intervals in fixed order; bar chart bars are separated and show categories that could be reordered.

"The mean is the best average." It is the best when the data is roughly symmetric. With a long tail, one extreme value drags it away from the typical case, which is why medians are used for incomes.

"You can read individual values off a histogram." You cannot. Once the data is binned you know only how many fell in each range. A dot plot keeps the values; a histogram trades them for a clearer shape.

What you now know

• A dot plot shows every value and makes clusters, gaps and outliers visible.
• A histogram groups values into touching bins and shows the shape of the distribution; the bin width is a choice that changes the picture.
• A box plot draws the five-number summary: minimum, Q1, median, Q3 and maximum.
• Each of the four box plot sections holds a quarter of the data, so section length means spread, not count.
• The interquartile range, Q3 minus Q1, describes the middle half and ignores extremes.
• A long tail pulls the mean away from the median, and the median usually describes the typical case better.

All three of those displays were drawn honestly. The last lesson looks at what happens when they are not.

Sources

1. Illowsky, B., and Dean, S. (2023). Box plots. Introductory Statistics 2e. OpenStax, Rice University. openstax.org
2. Khan Academy. (n.d.). Box and whisker plots. 6th grade math: Data and statistics. khanacademy.org
3. Wikipedia contributors. (n.d.). Five-number summary. Wikipedia. en.wikipedia.org
4. US Census Bureau. (n.d.). Statistics in Schools. census.gov

Four extra units, drawn to look like a revolution

A company sells 100 units in one month and 104 the next. That is a rise of 4 percent, which is real but unexciting.

Now draw it as a bar chart with the vertical axis running from 99 to 105 instead of from 0. The first bar is one unit tall above the baseline. The second is five units tall. On the page, the second bar is five times the height of the first, and the eye reads it as an enormous jump.

Nothing on that chart is false. The bars are at the right heights for the axis drawn. The numbers are correct. And the impression it creates is completely wrong.

Key idea: A chart can be entirely accurate and still be designed to mislead. What deceives you is usually the axis, not the data.

Trick one: the axis that does not start at zero

On a bar chart, the length of the bar is the message. Bars work because the eye compares lengths, and that comparison is only meaningful if every bar is measured from zero. Cut the bottom off and the lengths no longer stand for the values; they stand for "the value minus 99," which is a quantity nobody asked about.

Work out how badly it distorts. With the axis at zero, the bars are in the ratio 100 : 104, so the second is 4 percent taller, a difference you can barely see. With the axis at 99, the visible parts are 1 and 5, a ratio of 1 : 5, so the second bar looks 400 percent taller. The chart exaggerated the difference by a factor of about a hundred.

An honest exception. Line graphs of temperature, of blood pressure, of anything where zero is not a meaningful floor, routinely start somewhere other than zero, and that is fine. A graph of daily temperatures in Celsius starting at 0 would waste most of the page and hide the pattern. The rule is not "always start at zero." The rule is: on a bar chart, where length carries the meaning, start at zero, and on any chart, look at the axis before you look at the picture.

Trick two: intervals that are not equal

A horizontal axis is supposed to be a ruler. Here is one that is not.

Label on the axis	1990	2000	2010	2015	2018	2020	2021
Spacing on the page	equal	equal	equal	equal	equal	equal	equal

The first gap is ten years and the last is one year, but they are drawn the same width. Any line running across that axis is being stretched on the left and squashed on the right, so recent changes look far steeper than older ones of the same size. The labels are all true. The ruler is broken.

The same trick appears on a vertical axis with gridlines at 0, 10, 20, 50, 100. Check that the numbers on any axis go up in equal steps before you read the shape of anything drawn against it.

Trick three: making a picture bigger in two directions at once

A newspaper reports that house prices have doubled, and illustrates it with two drawings of a house: the second twice as tall as the first.

Here is what your eye actually does. It does not measure the height of the drawing. It takes in the whole shape, and the second drawing is twice as tall and twice as wide, so it covers four times the area. Doubling in two directions quadruples the area, exactly as in the area lesson: a 2 by 2 square has area 4, and a 4 by 4 square has area 16.

So the picture says four when the data said two. The fix is to compare with a bar, which grows in one direction only, or with a set of small identical icons where you count them.

The point: Length doubles by doubling one dimension. Area doubles by multiplying each dimension by about 1.41, not by 2. Any chart that scales a picture in two directions overstates the change.

Trick four: choosing where the story starts

Take any wobbly quantity over twenty years. Start the chart at the lowest point and it rises. Start it at the highest point and it falls. Same data, opposite headlines, no false numbers anywhere.

This is the hardest trick to catch, because nothing about the chart itself looks wrong. The only defence is to ask why this particular window was chosen and to look for the longer series. A chart that begins at an oddly specific date, especially one that is not a round number of years back, is worth a second look.

A related version drops the base of a percent. "Crime up 100 percent in the town centre" is alarming until you learn that means two incidents became four. A percent with no base attached is a number you cannot evaluate, and leaving the base out is a decision somebody made.

The checklist

Six questions, roughly twenty seconds, on any chart that matters.

1. Where does the vertical axis start? If it is a bar chart and the answer is not zero, the differences are being magnified.
2. Are the intervals on both axes equal? Unequal steps bend the shape of the line.
3. Is anything drawn as a picture that grows in two directions? If so, the area is overstating the change.
4. What time range is shown, and why that one? Ask what is just outside the edges of the chart.
5. Percent of what? A percent change with no base number can hide a tiny raw number.
6. Where did the data come from, and who benefits from this reading? Not every misleading chart is dishonest, but the ones that are usually favour the person who drew them.

Reading charts sceptically is not the same as distrusting all of them. Most charts are honest, and a good one shows you something you could not have seen in the numbers. The checklist exists so that the honest ones can be trusted quickly and the rest can be caught.

Common misconceptions

"A misleading chart must contain false numbers." Almost never. The numbers are usually correct and the distortion comes from the axis, the scale, the picture size or the chosen window.

"Every chart must start its vertical axis at zero." Bar charts must, because bar length carries the meaning. Line graphs of temperature or blood pressure reasonably do not, because zero is not a meaningful floor for them.

"A big percent always means a big change." Not without the base. A rise from 2 to 4 is a 100 percent increase and also an increase of two.

"If it looks professional it is probably fine." Clean design and a misleading axis go together very comfortably. The check is the numbers on the axis, not the quality of the artwork.

The takeaway

• A chart can be factually correct and still create a false impression, and that is the usual case.
• A truncated vertical axis on a bar chart exaggerates differences, sometimes by enormous factors.
• Unequal intervals on an axis distort the shape of anything drawn against it.
• Scaling a picture in two directions multiplies its area and overstates the change.
• The chosen time window, and a percent with no base, are quiet ways to steer a conclusion.
• Read the axes before the picture. It takes twenty seconds and it is the whole defence.

That completes the course. Ratios and rates gave you a way to compare unlike things; percent fixed the comparison at a hundred; negatives and coordinates gave numbers a place to live; expressions and equations let you write a rule once; and area, volume and data displays turned all of it on the physical world. The final exam draws on every lesson.

Sources

1. Wikipedia contributors. (n.d.). Misleading graph. Wikipedia. en.wikipedia.org
2. Illowsky, B., and Dean, S. (2023). Histograms, frequency polygons, and time series graphs. Introductory Statistics 2e. OpenStax, Rice University. openstax.org
3. US Census Bureau. (n.d.). Census Academy: data literacy training. census.gov
4. Huff, D. (1954). How to Lie with Statistics. W. W. Norton and Company, New York. The original popular account of these tricks, still in print.