Complete number sentences (find the unknown) - Multi-step
Theory
In \(4 \times \;\rule{1.2cm}{0.4pt}\; + 7 = 35\), two things happen to the missing number: it is multiplied by \(4\), then \(7\) is added. To find it, undo the steps in reverse order: subtract \(7\) first, then divide by \(4\). This Year 5 maths page shows how to do this, how brackets change the order, and how to use it for problems like taxi fares.
Think of the missing number going through a number machine: \(\times 4\), then \(+\,7\), giving \(35\). To go backwards, do the inverse of each step, starting with the last one.
Brackets show what is done first. In \((\;\rule{1.2cm}{0.4pt}\; - 5) \times 3 = 24\), the subtraction happens first and the multiplication last, so undo the \(\times 3\) first.
If part of a side has no gap, work it out first: in \(5 \times \;\rule{1.2cm}{0.4pt}\; + 2 \times 6 = 47\), \(2 \times 6 = 12\), so \(5 \times \;\rule{1.2cm}{0.4pt}\; = 35\).
The missing number can also be on one side with a calculation on the other: in \(3 \times \;\rule{1.2cm}{0.4pt}\; - 4 = 2 \times 10\), the right side is \(20\).
Undo the steps in reverse order:
How to solve a two-step number sentence
- Work out any part that has no gap.
- List the steps done to the missing number, in order.
- Undo them in reverse order using inverse operations.
- Substitute to check.
| \(35 - 7\) | \(=\) | \(28\) |
| \(28 \div 4\) | \(=\) | \(7\) |
\(7\). Check: \(4 \times 7 + 7 = 35\).
| \(24 \div 3\) | \(=\) | \(8\) |
| \(8 + 5\) | \(=\) | \(13\) |
\(13\).
| \(11 - 3\) | \(=\) | \(8\) |
| \(8 \times 4\) | \(=\) | \(32\) |
\(32\).
\(3 \times \;\rule{1.2cm}{0.4pt}\;\) is taken from \(50\) to leave \(26\).
| \(50 - 26\) | \(=\) | \(24\) |
| \(24 \div 3\) | \(=\) | \(8\) |
\(8\).
| \(2 \times 6\) | \(=\) | \(12\) |
| \(47 - 12\) | \(=\) | \(35\) |
| \(35 \div 5\) | \(=\) | \(7\) |
\(7\).
| \(2 \times 10\) | \(=\) | \(20\) |
| \(20 + 4\) | \(=\) | \(24\) |
| \(24 \div 3\) | \(=\) | \(8\) |
\(8\).
\(4 + 2 \times \;\rule{1.2cm}{0.4pt}\; = 30\).
| \(30 - 4\) | \(=\) | \(26\) |
| \(26 \div 2\) | \(=\) | \(13\) |
\(13\) hours.
He undid the steps in the wrong order. The \(+\,7\) was done last, so it must be undone first: \((35 - 7) \div 4 = 7\).
Common pitfalls
Frequently asked questions
How do you solve a two-step number sentence?
Undo the operations in reverse order. For \(4 \times \;\rule{1.2cm}{0.4pt}\; + 7 = 35\), subtract \(7\), then divide by \(4\).
Why undo the last step first?
The last thing done to the number is the outermost step. Undoing it first works backwards to the start.
What do brackets change?
They show what is done first, so the operation outside the brackets is undone first.
What if there are numbers on both sides?
Work out the side without a gap first, then solve: \(2 \times 10 = 20\).
How do you set up a taxi or hire problem?
Write fixed charge plus rate times the unknown equals the total: \(4 + 2 \times \;\rule{1.2cm}{0.4pt}\; = 30\).