Pre-Algebra & Math Foundations

Every number is built from digits (0 through 9), and where a digit sits gives it its place value. In the number 4,275 the digit 4 sits in the thousands place, so it is worth 4,000. Reading a number from left to right, each place is ten times bigger than the place to its right: ones, tens, hundreds, thousands, and so on. This is why our system is called base ten.

The four operations

Addition combines amounts, subtraction takes away, multiplication is fast repeated addition, and division splits into equal groups. When a column adds up to 10 or more, you regroup (carry) into the next place. When a top digit is too small to subtract from, you borrow from the next place.

Here is a fully worked addition with regrouping:

	4	2	7	5
+	1	8	4	9
=	6	1	2	4

Add the ones: 5 + 9 = 14, write 4 and carry 1. Tens: 7 + 4 + 1 = 12, write 2 and carry 1. Hundreds: 2 + 8 + 1 = 11, write 1 and carry 1. Thousands: 4 + 1 + 1 = 6. The answer is 6,124. Always estimate first (4,000 + 2,000 is about 6,000) so you can tell whether your answer is reasonable.

When an expression has more than one operation, mathematicians agree on the order to do them so everyone gets the same answer. The rule is PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right). A helpful phrase is "Please Excuse My Dear Aunt Sally."

Two important tips

Multiplication and division are equal partners, so you do whichever comes first as you read left to right. The same is true for addition and subtraction. Parentheses always go first, and an exponent like 3^2 means 3 times itself, which is 9.

Here is a fully worked example. Simplify 4 + 3 x (8 - 6)^2.

Step 1, parentheses: 8 - 6 = 2, giving 4 + 3 x 2^2.
Step 2, exponent: 2^2 = 4, giving 4 + 3 x 4.
Step 3, multiply: 3 x 4 = 12, giving 4 + 12.
Step 4, add: 4 + 12 = 16.

Notice that if you had added 4 + 3 first by mistake you would have gotten a completely different answer. Following PEMDAS keeps everyone honest and consistent.

A factor of a number divides it evenly with no remainder. A multiple is what you get by multiplying that number by 1, 2, 3, and so on. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, while the first multiples of 12 are 12, 24, 36, and 48.

Primes and how they help

A prime number has exactly two factors, 1 and itself (like 2, 3, 5, 7, 11). A composite number has more than two factors. Breaking a number into its prime factorization makes it easy to find the greatest common factor (GCF) and the least common multiple (LCM).

Let us find the GCF and LCM of 12 and 18.

Prime factorization: 12 = 2 x 2 x 3 and 18 = 2 x 3 x 3.
The GCF uses the primes they share: one 2 and one 3, so GCF = 2 x 3 = 6.
The LCM uses the highest count of each prime: two 2s and two 3s, so LCM = 2 x 2 x 3 x 3 = 36.

You can check: 6 divides both 12 and 18, and 36 is the smallest number that both 12 and 18 divide into. GCF and LCM become very useful when you start simplifying and adding fractions.

A fraction shows part of a whole. The bottom number, the denominator, tells how many equal pieces the whole is cut into. The top number, the numerator, tells how many of those pieces you have. In 3/4 the whole is cut into 4 equal parts and you have 3 of them.

Equivalent fractions and simplifying

Equivalent fractions name the same amount, like 1/2 and 2/4. You get them by multiplying or dividing the top and bottom by the same number. To simplify (reduce to lowest terms), divide both parts by their GCF.

Simplify 12/18. The GCF of 12 and 18 is 6, so 12 / 6 = 2 and 18 / 6 = 3, giving 2/3.

An improper fraction has a numerator that is larger than its denominator, like 11/4. You can rewrite it as a mixed number. Divide 11 by 4: it goes in 2 times with 3 left over, so 11/4 = 2 3/4. To go back, multiply the whole number by the denominator and add the numerator: 2 x 4 + 3 = 11, giving 11/4. Picturing a pizza cut into equal slices makes all of this feel natural.

To add or subtract fractions, the pieces must be the same size, which means the fractions need the same denominator. When denominators already match, you simply add or subtract the numerators and keep the denominator. For example, 2/7 + 3/7 = 5/7.

Unlike denominators

When denominators differ, first find a common denominator, usually the LCM of the two denominators. Rewrite each fraction as an equivalent fraction with that denominator, then combine.

Add 1/4 + 1/6.

The LCM of 4 and 6 is 12. Rewrite each: 1/4 = 3/12 (multiply top and bottom by 3) and 1/6 = 2/12 (multiply top and bottom by 2). Now add: 3/12 + 2/12 = 5/12. Since 5 and 12 share no common factor, 5/12 is already in lowest terms.

One more, with subtraction: 5/6 - 1/3. The LCM is 6, so rewrite 1/3 = 2/6. Then 5/6 - 2/6 = 3/6, which simplifies to 1/2. Always check whether your answer can be simplified or written as a mixed number at the end.

Multiplying fractions is refreshingly direct: multiply the numerators together and the denominators together. No common denominator is needed. For example, 2/3 x 4/5 = (2 x 4)/(3 x 5) = 8/15.

Dividing with the reciprocal

The reciprocal of a fraction is that fraction flipped upside down, so the reciprocal of 3/4 is 4/3. To divide by a fraction, multiply by its reciprocal. This is often remembered as "keep, change, flip": keep the first fraction, change division to multiplication, and flip the second fraction.

Divide 3/4 / 2/5.

Keep 3/4, change to multiply, flip 2/5 to 5/2: 3/4 x 5/2 = (3 x 5)/(4 x 2) = 15/8. As a mixed number that is 1 7/8.

You can also cross-cancel before multiplying to keep numbers small. In 4/9 x 3/8, the 4 and 8 share a factor of 4 (becoming 1 and 2) and the 3 and 9 share a factor of 3 (becoming 1 and 3), leaving 1/3 x 1/2 = 1/6. Canceling early saves you from simplifying big numbers later.

A decimal extends place value to the right of the ones place using a decimal point. The places are tenths, hundredths, thousandths, and so on. In 3.masks the digit 7 is in the tenths place, worth 7/10. Decimals are just another way to write fractions whose denominators are 10, 100, 1000, and so on.

Operating with decimals

To add or subtract, line up the decimal points so like places sit in the same column, then compute as usual. For example, 3.4 + 2.75 becomes 3.40 + 2.75 = 6.15.

To multiply, ignore the points, multiply the whole numbers, then place the decimal so the product has as many decimal places as the two factors combined. Multiply 1.2 x 0.3: compute 12 x 3 = 36. The factors have one and one decimal place, two total, so the answer is 0.36.

To divide by a decimal, shift the decimal point in both numbers until the divisor is a whole number. For 4.5 / 0.5, shift each one place to get 45 / 5 = 9. Estimating first keeps your decimal point in the right spot.

Fractions, decimals, and percents are three languages for the same idea: a part of a whole. The word percent means "per hundred," so 50% means 50 out of 100. Being able to switch between the three forms is one of the most useful skills in all of math.

The conversion moves

Fraction to decimal: divide the numerator by the denominator. For 3/4, compute 3 / 4 = 0.75.
Decimal to percent: multiply by 100 (move the point two places right). So 0.75 = 75%.
Percent to decimal: divide by 100 (move the point two places left). So 75% = 0.75.
Percent to fraction: write it over 100 and simplify. So 75% = 75/100 = 3/4.

Fraction	Decimal	Percent
1/2	0.5	50%
1/4	0.25	25%
3/5	0.6	60%

Memorizing a few common conversions, like the ones above, makes mental math much faster when you shop, cook, or read statistics.

A ratio compares two quantities. If a recipe uses 2 cups of flour for every 3 cups of milk, the ratio of flour to milk is 2 to 3. You can write it three ways: 2 to 3, 2:3, or 2/3. A rate is a special ratio comparing different units, like 60 miles per hour.

Proportions

A proportion is an equation stating that two ratios are equal, like 2/3 = 4/6. A quick test is cross-multiplication: multiply diagonally, and if the two products match, it is a true proportion. Here 2 x 6 = 12 and 3 x 4 = 12, so it checks out.

Cross-multiplication also solves for a missing value. Suppose 3/4 = x/20.

Cross-multiply: 3 x 20 = 4 x x, so 60 = 4x. Divide both sides by 4: x = 15. You can verify that 15/20 simplifies back to 3/4.

Proportions are everywhere: scaling a recipe, reading a map, converting units, or figuring out a better deal at the store. Setting up the ratio carefully, with matching units on top and bottom, is the key to getting the right answer.

Percentages appear every time you shop, tip, or read the news. The core idea is finding a percent of a number. To do it, change the percent to a decimal and multiply. For example, 20% of 50 is 0.20 x 50 = 10.

Everyday percent problems

A discount is a percent taken off a price. Find the discount, then subtract. A tip or tax is a percent added on. A percent of change compares how much something grew or shrank relative to the original amount.

Worked example, a sale: a jacket costs $80 and is 25% off. The discount is 0.25 x 80 = 20, so the sale price is 80 - 20 = 60 dollars.

Worked example, a tip: a meal costs $40 and you leave an 18% tip. The tip is 0.18 x 40 = 7.20, so the total is 40 + 7.20 = 47.20 dollars.

Worked example, percent of change: a plant grows from 20 cm to 25 cm. The change is 25 - 20 = 5, and 5 / 20 = 0.25, which is a 25% increase. Notice you always divide the change by the original amount.

So far our numbers have been zero or positive. Integers include the negative whole numbers too, like -1, -2, and -3. On a number line, zero sits in the middle, positives go to the right, and negatives go to the left. Negative numbers describe things like temperatures below zero, money owed, or steps below ground level.

Comparing and measuring

On the number line, a number is greater than any number to its left. So -2 > -5, even though 5 is bigger than 2, because -2 sits to the right of -5. The opposite of a number is the same distance from zero on the other side, so the opposite of 4 is -4.

The absolute value of a number is its distance from zero, always zero or positive, written with bars. For example, the absolute value of -7 is 7, written |-7| = 7, because -7 is seven units from zero. Likewise |3| = 3.

Comparison example: order -3, 2, and -6 from least to greatest. On the number line -6 is farthest left, then -3, then 2, so the order is -6, -3, 2. Picturing the number line makes negative comparisons simple.

Now that we know negative numbers, let us do arithmetic with them. For adding, if the signs are the same, add the values and keep the sign: -4 + -3 = -7. If the signs are different, subtract the smaller value from the larger and take the sign of the larger: -8 + 5 = -3.

Subtracting and multiplying

To subtract, add the opposite. So 3 - 7 becomes 3 + (-7) = -4, and -2 - 5 becomes -2 + (-5) = -7. This "add the opposite" trick turns every subtraction into an addition you already know how to do.

For multiplication and division, the sign rule is short: same signs give a positive answer, different signs give a negative answer.

Problem	Signs	Answer
(-4) x (-3)	same	12
(-6) x (2)	different	-12
(-20) / (5)	different	-4

Real example: the temperature is -3 degrees and drops 4 more degrees. That is -3 - 4 = -3 + (-4) = -7 degrees. Sign rules feel tricky at first, but with a little practice they become second nature.

Algebra begins when we let a letter stand for a number we do not yet know. That letter is a variable. An algebraic expression combines variables, numbers, and operations, like 3x + 5. The number multiplied by a variable is its coefficient (here 3), and a number by itself is a constant (here 5).

From words to symbols

Word phrases translate into expressions. "5 more than a number" is n + 5. "Twice a number" is 2n. "3 less than a number" is n - 3. Writing 3x means 3 times x, since in algebra a number next to a variable means multiplication.

To evaluate an expression, substitute a value for the variable and follow the order of operations. Evaluate 3x + 5 when x = 4.

Replace x with 4: 3(4) + 5. Multiply first: 3 x 4 = 12. Then add: 12 + 5 = 17. So the expression equals 17 when x is 4.

You can also combine like terms, which are terms with the same variable. For example 2x + 3x = 5x. Variables let one short expression describe a rule that works for many different numbers.

An equation is a statement that two expressions are equal, like x + 3 = 10. To solve it means to find the value of the variable that makes both sides balance. The golden rule is that whatever you do to one side you must do to the other, keeping the equation balanced like a scale.

Inverse operations

We undo operations using their inverse: addition and subtraction undo each other, and multiplication and division undo each other. To solve x + 3 = 10, subtract 3 from both sides: x = 7. To solve 4x = 20, divide both sides by 4: x = 5.

A two-step equation needs two moves. Solve 2x + 5 = 13.

Step 1, undo the addition: subtract 5 from both sides to get 2x = 8.
Step 2, undo the multiplication: divide both sides by 2 to get x = 4.

Always check by substituting back: 2(4) + 5 = 8 + 5 = 13, which matches the right side, so x = 4 is correct. Undo addition and subtraction first, then multiplication and division, and your equations will come out right every time.

Geometry measures shapes. The perimeter is the distance all the way around a flat shape, found by adding the side lengths. The area is the amount of surface inside a flat shape, measured in square units. Volume is the amount of space inside a solid, measured in cubic units.

Key formulas

For a rectangle with length L and width W: perimeter is P = 2L + 2W and area is A = L x W. For a triangle with base b and height h: area is A = (1/2) x b x h. For a rectangular box: volume is V = L x W x H.

Worked example, a rectangle 8 cm long and 3 cm wide.
Perimeter: 2(8) + 2(3) = 16 + 6 = 22 cm.
Area: 8 x 3 = 24 square cm.

Worked example, a triangle with base 6 cm and height 4 cm.
Area: (1/2) x 6 x 4 = (1/2) x 24 = 12 square cm.

Worked example, a box 5 cm by 2 cm by 3 cm.
Volume: 5 x 2 x 3 = 30 cubic cm.

Remember the units: perimeter uses plain units, area uses square units, and volume uses cubic units. Labeling units correctly is part of getting the right answer.

Our final week is about making sense of data, the numbers we collect about the world, and celebrating everything you have learned. Four common measures describe a data set. The mean (average) is the sum divided by how many numbers there are. The median is the middle value when the numbers are in order. The mode is the value that appears most often. The range is the largest value minus the smallest.

Worked example

Find the mean, median, mode, and range of 4, 8, 6, 4, 8, 8.
Mean: the sum is 4 + 8 + 6 + 4 + 8 + 8 = 38, and 38 / 6 is about 6.33.
Median: in order the set is 4, 4, 6, 8, 8, 8; the two middle values are 6 and 8, and their average is (6 + 8) / 2 = 7.
Mode: 8 appears three times, more than any other, so the mode is 8.
Range: 8 - 4 = 4.

Bar graphs compare categories with bars, and line graphs show how something changes over time. Reading them means checking the axes and labels carefully. Look back at the sidebar links and the OpenStax chapters to review any topic. From place value to equations to data, you now have the foundation to succeed in algebra. Great work this term.