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Year 5 Maths Number Sentences & Order of Operations

Complete number sentences (find the unknown) - Using inverse operations

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Theory

Inverse operations undo each other: subtraction undoes addition, and division undoes multiplication. To find the gap in \(\;\rule{1.2cm}{0.4pt}\; + 56 = 140\), undo the \(+\,56\) by subtracting: \(140 - 56 = 84\). This Year 5 maths page shows which operation to use in each kind of sentence, including the tricky ones where the gap is in the middle.

An inverse operation undoes another. Adding \(56\) and then subtracting \(56\) brings you back to where you started.

Addition and subtraction are inverses. Multiplication and division are inverses.

When the gap is the first number, undo the operation: \(\;\rule{1.2cm}{0.4pt}\; - 38 = 47\) becomes \(47 + 38\).

When the gap comes after a minus or divide sign, it is different: in \(92 - \;\rule{1.2cm}{0.4pt}\; = 57\), the gap is \(92 - 57\); in \(168 \div \;\rule{1.2cm}{0.4pt}\; = 14\), the gap is \(168 \div 14\).

Undoing plus 56 A number machine: an unknown number goes in, 56 is added and 140 comes out. The red arrow underneath goes backwards, subtracting 56, so the unknown is 84. ? 140 + 56 − 56
Forwards \(+\,56\); backwards \(-\,56\): \(140 - 56 = 84\).
Undoing times 6 A number machine: an unknown number is multiplied by 6 to give 114. The red arrow underneath goes backwards, dividing by 6, so the unknown is 19. ? 114 × 6 ÷ 6
Forwards \(\times\,6\); backwards \(\div\,6\): \(114 \div 6 = 19\).

Each operation has an inverse:

\[ + \;\longleftrightarrow\; - \]
addition and subtraction are inverses
\[ \times \;\longleftrightarrow\; \div \]
multiplication and division are inverses
\[ \;\rule{1.2cm}{0.4pt}\; + 56 = 140 \;\Rightarrow\; 140 - 56 = 84 \]
84+56=140
Gap first: use the inverse. \(\;\rule{1.2cm}{0.4pt}\; \times 6 = 114\): divide, \(114 \div 6 = 19\).
Gap after minus or divide: keep the operation. \(92 - \;\rule{1.2cm}{0.4pt}\; = 57\): \(92 - 57 = 35\). \(168 \div \;\rule{1.2cm}{0.4pt}\; = 14\): \(168 \div 14 = 12\).

How to use inverse operations

  1. Look at what is done to the missing number.
  2. Do the inverse to the number on the other side of the equals sign.
  3. Substitute your answer to check.
Example 1 — Undo addition
Find the missing number: \(\;\rule{1.2cm}{0.4pt}\; + 56 = 140\).
Solution
\(140 - 56\)\(=\)\(84\)
84+56=140

\(84\). Check: \(84 + 56 = 140\).

Example 2 — Undo multiplication
Find the missing number: \(\;\rule{1.2cm}{0.4pt}\; \times 6 = 114\).
Solution
\(114 \div 6\)\(=\)\(19\)
19×6=114

\(19\). Check: \(19 \times 6 = 114\).

Example 3 — Undo subtraction
Find the missing number: \(\;\rule{1.2cm}{0.4pt}\; - 38 = 47\).
Solution
\(47 + 38\)\(=\)\(85\)
85−38=47

\(85\).

Example 4 — Gap after a divide sign
Find the missing number: \(168 \div \;\rule{1.2cm}{0.4pt}\; = 14\).
Solution
\(168 \div 14\)\(=\)\(12\)
168÷12=14

\(12\). Check: \(168 \div 12 = 14\).

Example 5 — Several sentences
Find each missing number.
(i) \(\;\rule{1.2cm}{0.4pt}\; \div 7 = 16\)
(ii) \(92 - \;\rule{1.2cm}{0.4pt}\; = 57\)
(iii) \(\;\rule{1.2cm}{0.4pt}\; - 1275 = 3480\)
Solution
\(16 \times 7\)\(=\)\(112\)
\(92 - 57\)\(=\)\(35\)
\(3480 + 1275\)\(=\)\(4755\)
112÷7=16,92−35=57,4755−1275=3480

(i) \(112\)
(ii) \(35\)
(iii) \(4755\)

Example 6 — Word problem
A rainwater tank had some water. After \(285\) L was used, \(1415\) L was left.
(i) How much water was in the tank at the start?
(ii) That starting amount would fill \(5\) equal drums. How much does each drum hold?
Solution

(i) \(\;\rule{1.2cm}{0.4pt}\; - 285 = 1415\)
(ii) \(\;\rule{1.2cm}{0.4pt}\; \times 5 = 1700\)

\(1415 + 285\)\(=\)\(1700\)
\(1700 \div 5\)\(=\)\(340\)
1700−285=1415,340×5=1700

(i) \(1700\) L
(ii) \(340\) L

Example 7 — Find the mistake
For \(\;\rule{1.2cm}{0.4pt}\; \times 8 = 104\), Ana multiplied and wrote \(832\). Explain the mistake and fix it.
Solution
\(104 \div 8\)\(=\)\(13\)
13×8=104

She used the same operation instead of the inverse. Divide: the missing number is \(13\), and \(13 \times 8 = 104\).

Common pitfalls

Using the same operation. \(\;\rule{1.2cm}{0.4pt}\; \times 8 = 104\) needs division, not multiplication.
Undoing when the gap is after a minus. \(92 - \;\rule{1.2cm}{0.4pt}\; = 57\) is \(92 - 57\), not \(92 + 57\).
Skipping the check. Substituting takes a few seconds and shows straight away if the answer is wrong.
Errors with large numbers. Set out large subtractions and additions in columns.

Frequently asked questions

What is an inverse operation?

An operation that undoes another. Subtraction undoes addition and division undoes multiplication.

How do you find the missing number in ___ + 56 = 140?

Subtract: \(140 - 56 = 84\).

How do you find the missing number in ___ × 6 = 114?

Divide: \(114 \div 6 = 19\).

What if the gap comes after a minus sign?

Subtract the answer from the first number: in \(92 - \;\rule{1.2cm}{0.4pt}\; = 57\), the gap is \(92 - 57 = 35\).

How do you check?

Put the number back in the gap and check that both sides have the same value.

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