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Foundations · Fractions

Adding, subtracting, multiplying and dividing fractions

A common denominator for adding, straight across for multiplying, flip and multiply for dividing — with the pictures that show why.

First, the rules

Add with the same cut, multiply straight across, divide by flipping

Three different rules, one reason each: you can only add pieces of the same size, "a fraction of a fraction" is a grid, and dividing asks how many times one fits into the other.

★ Add / subtract
Example 1
ab ± cd = ad ± bcbd
Use the LCM as the common denominator to keep numbers small: 16 + 34 = 212 + 912 = 1112.
Mixed numbers
Example 1
Either convert to improper fractions, or add wholes and fractions separately: 234 + 112 = 3 + 54 = 414.
★ Multiply
Example 2
ab × cd = acbd — cancel any top with any bottom before multiplying: 49 × 38 = 13 × 12 = 16.
★ Divide
Example 3
ab ÷ cd = ab × dc — "keep, change, flip". Dividing by ½ doubles: 3 ÷ ½ = 6.
Whole numbers
Example 2
A whole number is a fraction over 1: 5 × 23 = 51 × 23 = 103.

See it

Pictures for each operation

Adding recuts both bars on the common denominator; multiplying shades a grid; dividing counts how many copies of the second fraction fit into the first.

Step by step

Worked examples

1

Adding and subtracting

common denominator

Work out (a) 56 − 38 and (b) 223 + 134.

Ask yourself: What is the smallest number both denominators divide?

Show the solution
  1. (a) lcm(6, 8) = 24: 2024 − 924 = 1124.
  2. (b) Wholes: 2 + 1 = 3. Fractions: lcm(3, 4) = 12, 812 + 912 = 1712 = 1512. Total 4512.

Answer 1124 · 4512

2

Multiplying with cancelling

multiply

Work out 910 × 512 and then 112 × 223.

Ask yourself: Cancel first — any numerator against any denominator.

Show the solution
  1. 9 and 12 share 3; 5 and 10 share 5: 32 × 14 = 38.
  2. Mixed numbers must become improper first: 32 × 83 = 246 = 4 (or cancel: 3 with 3, 2 with 8 → 1 × 4).

Answer 38 · 4

3

Dividing — how many fit?

divide

A ribbon is 334 m long. How many pieces of 38 m can be cut from it?

Ask yourself: The question "how many of these fit in that" is a division.

Show the solution
  1. 334 ÷ 38 = 154 × 83.
  2. Cancel 15 with 3 and 8 with 4: 51 × 21 = 10.
  3. Check: 10 × 38 = 308 = 334 ✓.

Answer 10 pieces

Practice: fraction operations

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Add and subtract with a common denominator, multiply straight across, divide by flipping — and mixed numbers. Your best score is kept in this browser only.

About this lesson

Operations with fractions: adding and subtracting with a common denominator (the LCM), multiplying numerators and denominators, dividing by multiplying by the reciprocal, cancelling before you multiply, and working with mixed numbers. An area model you can move, worked examples and practice questions with explanations. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.