doing math

courses › Fractions, Decimals & Percents

Fraction arithmetic

level 12 course

Add, subtract, multiply, and divide fractions.

In your head, jot if needed · no calculator why?

Learn first (about 3 minutes)

Opens at level 12.

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Builds on: Fraction basics (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

Fractions follow a few fixed rules, and most mistakes come from using one operation's rule for another.

To add or subtract, the pieces must be the same size, so first rewrite both fractions over a common denominator, a bottom number that both denominators divide into. To multiply, multiply the tops and multiply the bottoms. To divide, flip the second fraction and multiply. A fraction of a number is a multiplication: 3/4 of 20 is 3/4 × 20.

Give every answer in lowest terms. An answer bigger than 1, like 5/4, stays a fraction.

Techniques

Find one part first

Finding a fraction of a whole number.

  1. Divide the number by the bottom of the fraction. That is one part.
  2. Multiply by the top to get that many parts.
worked example

Example: What is 5/6 of 42?

  1. One sixth: 42 ÷ 6 = 7.
  2. Five sixths: 5 × 7 = 35.

Answer: 35

Common denominator first

Adding or subtracting fractions with different bottoms.

  1. Find a number both bottoms divide into. The smallest one keeps the numbers small.
  2. Rewrite each fraction over it, multiplying its top and bottom by the same number.
  3. Add or subtract the tops, and keep the bottom.
  4. Simplify to lowest terms.
worked example

Example: 5/6 − 1/4. Answer as a fraction in lowest terms.

  1. 6 and 4 both divide into 12.
  2. 5/6 = 10/12, and 1/4 = 3/12.
  3. 10/12 − 3/12 = 7/12.

Answer: 7/12

Multiply across; flip to divide

Multiplying or dividing two fractions.

  1. To divide, flip the second fraction upside down and multiply instead.
  2. Cancel first: divide any top and any bottom by a factor they share.
  3. Multiply the tops, then the bottoms.
  4. Check that the answer is in lowest terms.
worked example

Example: 3/4 ÷ 9/10. Answer as a fraction in lowest terms.

  1. Flip and multiply: 3/4 × 10/9.
  2. Cancel 3 with 9, and 10 with 4: 1/2 × 5/3.
  3. Tops 1 × 5, bottoms 2 × 3: 5/6.

Answer: 5/6

Tips by skill

  • TipFraction of a number: Divide the whole number by the bottom to get one part, then multiply by the top.
  • TipAdd & subtract: Rewrite both fractions over a common denominator, add or subtract the tops, then simplify.
  • TipMultiply: Cancel any top with any bottom that shares a factor, then multiply tops and bottoms. Finish in lowest terms.
  • TipDivide: Flip the second fraction and multiply. Cancel before you multiply to keep the numbers small.

Watch out for

  • Adding the tops and adding the bottoms. 2/3 + 1/4 is 11/12, not 3/7, so rewrite both as twelfths first.
  • Multiplying without flipping when you divide. 5/6 ÷ 2/3 means 5/6 × 3/2, not 5/6 × 2/3.
  • Leaving an answer unsimplified. 6/36 is the right amount, but it must be written as 1/6.
  • Finding one part and stopping. 3/8 of 64 is three eighths: 3 × 8, not 8 alone.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

your rounds

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