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College math review · Study sheet

Math refresher for adults going back to school

Six rule cards with worked examples: exponent rules, order of operations, fractions, decimals and percents, ratios and proportions, and basic algebra. Then 20 practice questions with step-by-step solutions and a summary that tells you which topics to review. Free, no sign-up needed.

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1 · Rule cards

The rules, with worked examples

Six cards, in the order a college math review usually takes them. Read the rule in symbols, then in plain words, then follow the worked examples. Jump to the card you need:

Rule card 1

Exponent rules

An exponent counts how many times the base is multiplied by itself: 23 = 2 · 2 · 2 = 8, not 2 · 3. These seven rules cover almost every exponent problem in a college algebra course. The product and quotient rules need the same base; the quotient, zero-exponent and negative-exponent rules need a base other than zero.

The rules

  1. Product rule

    am · an = am+n

    Same base, multiplying: add the exponents.

    Example: x3 · x4 = x7

  2. Power rule

    (am)n = amn

    A power raised to a power: multiply the exponents.

    Example: (x2)5 = x10

  3. Product to a power

    (ab)m = am bm

    Every factor inside the parentheses gets the exponent.

    Example: (3x)2 = 9x2

  4. Quotient rule

    am ÷ an = am−n

    Same base, dividing: subtract the bottom exponent from the top one.

    Example: x8 ÷ x3 = x5

  5. Quotient to a power

    (a/b)m = am / bm

    Top and bottom each get the exponent.

    Example: (x/2)3 = x3/8

  6. Zero exponent

    a0 = 1

    Any nonzero base to the zero power is 1.

    Example: 70 = 1 and (5y)0 = 1

  7. Negative exponent

    a−n = 1/an

    A negative exponent means a reciprocal, not a negative number.

    Example: 2−3 = 1/8 and x−4 = 1/x4

Worked examples

  1. Simplify x2 · x5 · x

    1. x on its own is x1.
    2. Same base, so add the exponents: 2 + 5 + 1 = 8.

    Answer: x8

  2. Simplify (2y3)2

    1. Every factor inside gets the exponent: 22 · (y3)2.
    2. 22 = 4, and (y3)2 = y6 by the power rule.

    Answer: 4y6

  3. Simplify (a2 b)3 ÷ a4

    1. Top first: (a2 b)3 = a6 b3.
    2. Divide the a terms: a6 ÷ a4 = a2. The b3 stays.

    Answer: a2 b3

  4. Write 5−2 without a negative exponent

    1. A negative exponent is a reciprocal: 1/52.
    2. 52 = 25.

    Answer: 1/25

Watch out: −22 is −4, because the exponent applies only to the 2. (−2)2 is 4, because the parentheses put the sign inside.

Rule card 2

Order of operations

When an expression mixes operations, this order decides what happens first. The memory aid is PEMDAS: Please Excuse My Dear Aunt Sally. The two pairs at the end are ties, worked left to right.

The rules

  1. 1. Parentheses

    Grouping symbols first

    Work inside brackets first, innermost first. A fraction bar groups its top and its bottom too.

    Example: (4 + 3) · 2 = 7 · 2 = 14

  2. 2. Exponents

    Powers next

    Simplify anything with an exponent (and roots).

    Example: 23 · 3 = 8 · 3 = 24

  3. 3. Multiplication and division

    Left to right

    They rank equally, so take them in the order they appear.

    Example: 12 ÷ 3 · 2 = 4 · 2 = 8

  4. 4. Addition and subtraction

    Left to right

    They rank equally too.

    Example: 20 − 8 + 1 = 12 + 1 = 13

Worked examples

  1. 4 + 3 · 2

    1. Multiply first: 3 · 2 = 6.
    2. Then add: 4 + 6.

    Answer: 10

  2. 23 · 3 − 4

    1. Exponent: 23 = 8.
    2. Multiply: 8 · 3 = 24.
    3. Subtract: 24 − 4.

    Answer: 20

  3. 12 ÷ 3 · 2

    1. Division and multiplication rank equally, so go left to right: 12 ÷ 3 = 4.
    2. Then 4 · 2.

    Answer: 8, not 2

  4. (6 + 4) / (5 − 3)

    1. The fraction bar groups: top 6 + 4 = 10, bottom 5 − 3 = 2.
    2. 10 ÷ 2.

    Answer: 5

Watch out: A common slip is doing all the multiplication before any division. They are a tie: whichever comes first from the left goes first.

Rule card 3

Fractions

A fraction is a division: 3/4 means 3 ÷ 4. A fraction is simplified (in lowest terms) when the top and bottom share no factor but 1. Every rule below ends the same way: simplify the answer.

The rules

  1. Simplify

    Divide top and bottom by the same number

    Use the greatest common factor to do it in one step.

    Example: 18/24 = 3/4 (both ÷ 6)

  2. Multiply

    a/b · c/d = ac/bd

    Straight across: tops together, bottoms together, then simplify. Cancelling a common factor first saves work.

    Example: 2/3 · 9/10 = 18/30 = 3/5

  3. Divide

    a/b ÷ c/d = a/b · d/c

    Keep the first fraction, flip the second (its reciprocal), then multiply.

    Example: 3/4 ÷ 2/5 = 3/4 · 5/2 = 15/8

  4. Add or subtract, same denominator

    a/d + b/d = (a + b)/d

    Add or subtract the tops; keep the bottom.

    Example: 3/8 + 1/8 = 4/8 = 1/2

  5. Add or subtract, different denominators

    Rewrite over the LCD first

    The least common denominator is the smallest number both bottoms divide into. Rewrite each fraction, then add or subtract the tops.

    Example: 1/4 + 1/6 = 3/12 + 2/12 = 5/12

  6. Mixed number ↔ improper fraction

    a b/c = (a · c + b)/c

    Whole number times the denominator, plus the numerator, over the denominator. Divide to go back.

    Example: 2 3/4 = 11/4, and 7/3 = 2 1/3

Worked examples

  1. Simplify 40/64

    1. The greatest common factor of 40 and 64 is 8.
    2. 40 ÷ 8 = 5 and 64 ÷ 8 = 8.

    Answer: 5/8

  2. 5/6 − 1/4

    1. The LCD of 6 and 4 is 12.
    2. Rewrite: 5/6 = 10/12 and 1/4 = 3/12.
    3. Subtract the tops: 10 − 3 = 7.

    Answer: 7/12

  3. 3/4 ÷ 2/5

    1. Flip the second fraction: 2/5 becomes 5/2.
    2. Multiply: 3/4 · 5/2 = 15/8.
    3. As a mixed number, 15 ÷ 8 is 1 with 7 left over.

    Answer: 15/8, or 1 7/8

  4. 1 1/2 · 2/3

    1. Convert the mixed number: 1 1/2 = 3/2.
    2. Multiply: 3/2 · 2/3 = 6/6.

    Answer: 1

Watch out: Never add the denominators: 1/2 + 1/3 is 5/6, not 2/5.

Rule card 4

Decimals and percents

Fractions, decimals and percents are three ways to write one number. Percent means per hundred: 45% = 45/100 = 0.45. The chart under this card lists the values worth knowing by heart.

The rules

  1. Fraction → decimal

    Divide the top by the bottom

    The fraction bar is a division sign. If the division never ends, the decimal repeats.

    Example: 3/8 = 3 ÷ 8 = 0.375, and 2/3 = 0.666…

  2. Decimal → fraction

    Read the place value, then simplify

    Two decimal places means hundredths, three means thousandths.

    Example: 0.45 = 45/100 = 9/20

  3. Decimal ↔ percent

    Move the point two places

    Decimal to percent: two places right. Percent to decimal: two places left.

    Example: 0.375 = 37.5%, and 6% = 0.06

  4. Fraction → percent

    Go through the decimal

    Divide, then move the point.

    Example: 3/4 = 0.75 = 75%

  5. Percent of a number

    percent (as a decimal) · number

    Of means multiply.

    Example: 15% of 60 = 0.15 · 60 = 9

  6. Percent change

    (new − original) ÷ original, × 100

    The change is always compared with the original amount, not the new one.

    Example: 40 → 50: 10 ÷ 40 = 0.25 = 25% increase

Worked examples

  1. Write 7/8 as a decimal and as a percent

    1. 7 ÷ 8 = 0.875.
    2. Move the point two places right: 87.5%.

    Answer: 0.875 = 87.5%

  2. Write 0.08 as a fraction

    1. 0.08 is 8 hundredths: 8/100.
    2. Simplify by 4.

    Answer: 2/25

  3. A $48 item is 25% off. What is the sale price?

    1. 25% of 48 = 0.25 · 48 = 12.
    2. Take it off: 48 − 12.

    Answer: $36

  4. A class grew from 20 students to 25. What was the percent increase?

    1. Increase: 25 − 20 = 5.
    2. Compare with the original: 5 ÷ 20 = 0.25.

    Answer: 25%

Watch out: Percent change divides by the original amount. A drop from 50 to 40 is a 20% decrease (10 ÷ 50), even though 40 to 50 was a 25% increase.

Fraction to decimal chart

Each row is the fraction divided out. Repeating decimals are shown with three dots and the percent rounded to one place.

FractionDecimalPercent
1/1000.011%
1/200.055%
1/160.06256.25%
1/120.0833…8.3%
1/100.110%
1/80.12512.5%
1/60.1666…16.7%
1/50.220%
1/40.2525%
3/100.330%
1/30.333…33.3%
3/80.37537.5%
FractionDecimalPercent
2/50.440%
1/20.550%
3/50.660%
5/80.62562.5%
2/30.666…66.7%
7/100.770%
3/40.7575%
4/50.880%
5/60.8333…83.3%
7/80.87587.5%
9/100.990%
1/11100%

Rule card 5

Ratios and proportions

A ratio compares two quantities with the same unit: 12 to 18, 12:18 or 12/18. A rate compares different units, such as 150 miles in 3 hours. A proportion is a statement that two ratios are equal, and it is how most word problems with "per" get solved.

The rules

  1. Simplify a ratio

    Like a fraction

    Divide both parts by their common factor.

    Example: 12:18 = 2:3

  2. Unit rate

    Divide so the second number is 1

    Miles per hour, dollars per ounce, cups per serving.

    Example: 150 miles in 3 hours = 50 miles per hour

  3. Proportion

    a/b = c/d

    Two equal ratios. Put the same unit on top in both.

    Example: cups/servings = cups/servings

  4. Cross products

    a/b = c/d means a · d = b · c

    Cross-multiply, then divide to find the unknown.

    Example: 3/8 = x/24 → 72 = 8x → x = 9

Worked examples

  1. Solve 3/8 = x/24

    1. Cross-multiply: 3 · 24 = 8 · x, so 72 = 8x.
    2. Divide both sides by 8.

    Answer: x = 9

  2. A recipe uses 2 cups of flour for 12 muffins. How much flour for 30 muffins?

    1. Cups over muffins on both sides: 2/12 = x/30.
    2. Cross-multiply: 2 · 30 = 12 · x, so 60 = 12x.
    3. Divide by 12.

    Answer: 5 cups

  3. Unit price: 24 ounces for $3.60

    1. Dollars over ounces: 3.60 ÷ 24.

    Answer: $0.15 per ounce

Watch out: Keep the same unit on top in both ratios. Flipping one side (muffins/cups against cups/muffins) gives a wrong answer that still looks tidy.

Rule card 6

Basic algebra

Solving for x means undoing what was done to x, one step at a time, and doing the same thing to both sides. A line written as y = mx + b (slope-intercept form) shows its slope m and the point (0, b) where it crosses the y-axis.

The rules

  1. Undo in reverse order

    Add or subtract first, then multiply or divide

    Move the plain numbers away from x first, then clear the number in front of x.

    Example: 3x + 5 = 20 → 3x = 15 → x = 5

  2. Variables on both sides

    Collect x terms on one side, numbers on the other

    Subtract the smaller x term from both sides, then the constants.

    Example: 5x − 7 = 2x + 8 → 3x = 15 → x = 5

  3. Parentheses

    Distribute first

    Multiply the number outside by each term inside, then solve as usual.

    Example: 2(x − 3) = 10 → 2x − 6 = 10 → x = 8

  4. Check

    Substitute the answer back

    Both sides of the original equation should come out equal.

    Example: 3(5) + 5 = 20 ✓

  5. Slope

    m = (y2 − y1) ÷ (x2 − x1)

    Rise over run between any two points on the line.

    Example: (1, 2) and (4, 8): (8 − 2) ÷ (4 − 1) = 2

  6. Slope-intercept form

    y = mx + b

    m is the slope, b is the y-intercept. To read an equation in another form, solve it for y.

    Example: 3x + y = 5 → y = −3x + 5: slope −3, intercept (0, 5)

Worked examples

  1. Solve 5x − 3 = 27

    1. Add 3 to both sides: 5x = 30.
    2. Divide both sides by 5.
    3. Check: 5(6) − 3 = 27.

    Answer: x = 6

  2. Solve 4(x − 1) = x + 11

    1. Distribute: 4x − 4 = x + 11.
    2. Subtract x from both sides: 3x − 4 = 11.
    3. Add 4: 3x = 15, then divide by 3.

    Answer: x = 5

  3. Find the slope and y-intercept of y = −3x + 4

    1. Compare with y = mx + b: m = −3 and b = 4.

    Answer: slope −3, y-intercept (0, 4)

  4. Write the equation of the line with slope 1/2 that crosses the y-axis at (0, −4)

    1. m = 1/2 and b = −4.
    2. Put them into y = mx + b.

    Answer: y = 1/2 x − 4

Watch out: Whatever you do to one side, do to the other. Subtracting 7 from the left only is the most common wrong step.

2 · Practice questions

20 practice questions

Twenty questions across the six cards, in card order. Type each answer (no calculator), check it, and read the steps. At the end you get a summary by topic that says which rule cards to review. Your result is free: no sign-up needed.

Question 1 of 20 · Exponent rules

Simplify x3 · x5

Type a power like x8

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Sources

Where the rules come from

Every rule on this page follows these two open textbooks from OpenStax, in our own words. The problems, the worked examples and the plain-words explanations are our own. Check anything important against your own course materials.

Heading back to school and not sure which program fits? Our free skills finder turns what you have already done into a skills list.

Questions

Common questions

Seven rules cover almost everything. Product rule: same base multiplied, add the exponents (x³ · x⁴ = x⁷). Power rule: a power of a power, multiply the exponents ((x²)⁵ = x¹⁰). Product to a power: every factor inside gets the exponent ((3x)² = 9x²). Quotient rule: same base divided, subtract the exponents (x⁸ ÷ x³ = x⁵). Quotient to a power: top and bottom each get the exponent. Zero exponent: any nonzero base to the zero power is 1. Negative exponent: a reciprocal, so 2⁻³ = 1/8. The first rule card on this page shows each one with a worked example.

Keep the first fraction, flip the second one (write its reciprocal), and multiply straight across. So 3/4 ÷ 2/5 becomes 3/4 · 5/2 = 15/8, which is 1 7/8 as a mixed number. Simplify the result if the top and bottom share a factor.

Divide the top and the bottom by the same number until they share no factor but 1. Using the greatest common factor does it in one step: 24/36 has a greatest common factor of 12, so it simplifies to 2/3. A fraction is in lowest terms when nothing but 1 divides both numbers.

Find the least common denominator (the smallest number both bottoms divide into), rewrite each fraction over it, add the tops and keep the bottom, then simplify. For 1/4 + 1/6 the least common denominator is 12: 3/12 + 2/12 = 5/12. Subtraction works the same way with a minus sign. Never add the denominators themselves.

Divide the top by the bottom, because the fraction bar is a division sign: 5/8 = 5 ÷ 8 = 0.625. If the division never ends, the decimal repeats (2/3 = 0.666…). To turn that decimal into a percent, move the decimal point two places to the right: 0.625 = 62.5%. The chart under the decimals card lists the common values.

Percent change = (new amount − original amount) ÷ original amount, then multiply by 100. Going from $80 to $60 is a decrease of 20, and 20 ÷ 80 = 0.25, so a 25% decrease. Always divide by the original amount, not the new one, which is why 40 to 50 is a 25% increase but 50 to 40 is only a 20% decrease.

No. The rule cards are a reference and the practice questions check an answer you type, so you work each step yourself, the way a first-week college math course expects. Fractions can be typed like 3/4, mixed numbers like 1 1/2, ratios like 3:4, and powers like x^8 or x⁸.

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