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Step 7: Fraction and decimal arithmetic

Order of operations and simple expressions

Step 7 · Fraction and decimal arithmetic topics

Work out expressions in the agreed order, use parentheses and brackets to change that order, and write simple expressions from words.

The order of operations

What is 3 + 4 × 2? Working left to right gives 14; multiplying first gives 11. So there is an agreed order: multiply and divide before you add and subtract.

Example: Multiplying first

What is 3 + 4 × 2?

  1. Multiply first: 4 × 2 = 8.
  2. Then add: 3 + 8 = 11.

Answer11

Multiplication and division have the same rank, so work them from left to right. The same goes for addition and subtraction: 20 − 5 + 3 = 18, not 12.

Example: Left to right

What is 24 ÷ 4 × 2?

  1. Division and multiplication rank the same, so go left to right.
  2. 24 ÷ 4 = 6.
  3. 6 × 2 = 12.

Answer12

Using parentheses

Parentheses ( ) say: do this first. (3 + 4) × 2 = 7 × 2 = 14. Brackets [ ] and braces { } do the same job when one group sits inside another.

Example: Groups inside groups

What is 2 × [5 + (6 − 2)]?

  1. Innermost first: 6 − 2 = 4.
  2. Then the brackets: 5 + 4 = 9.
  3. 2 × 9 = 18.

Answer18

OrderWhat to do
1Parentheses, brackets and braces, from the inside out
2Multiply and divide, left to right
3Add and subtract, left to right

Writing and reading expressions

An expression is a calculation written with numbers and signs, without an equals sign. Words turn into expressions: "add 8 and 7, then multiply by 2" is 2 × (8 + 7).

Example: Words to an expression

Write "3 times the sum of 18 and 932" as an expression.

  1. The sum of 18 and 932 is 18 + 932.
  2. 3 times it is 3 × (18 + 932).

Answer3 × (18 + 932)

You can read an expression without working it out: 3 × (18 + 932) is three times as big as 18 + 932.

Try it yourself

Work each one on paper first, then open the answer to check.

  1. What is 5 + 3 × 6?

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    23.

  2. What is (5 + 3) × 6?

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    48.

  3. What is 18 − 12 ÷ 3?

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    14.

  4. What is 4 × [10 − (2 + 3)]?

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    20.

  5. Write "subtract 4 from 10, then multiply by 5" as an expression.

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    (10 − 4) × 5, which is 30.

What next?