Missing numbers in multiplication & division sentences
Theory
Missing numbers in multiplication and division sentences is a Year 4 maths number and algebra skill: finding the number that fits in a box, like \(7 \times \square = 28\) or \(36 \div \square = 4\). This page explains the inverse relationship between multiplication and division, how to use fact families and arrays, and what the equals sign really means. Solving for unknown values in multiplication and division sentences is formally Year 5 in both curricula (VC2M5A02 / AC9M5A02); this page uses Year 4 multiplication facts. Curriculum: VC2M4A02 / AC9M4A02, AC9M4N06.
A number sentence uses numbers, an operation and an equals sign, like \(2 \times 7 = 14\). Sometimes one number is hidden in a box, and your job is to find it: \(2 \times \square = 14\).
The equals sign means "the same as". Both sides of the sentence must have the same value, so a missing-number sentence asks which number makes the two sides balance.
Multiplication and division are inverse operations — they undo each other. This is the key to finding the box. The factors are the numbers you multiply, and the product is the answer. In \(8 \times 4 = 32\), the factors are 8 and 4 and the product is 32.
Missing-number sentences do not need formulas, but these facts help you fill in the box. One multiplication fact gives you a whole fact family.
How to find a missing number
- Read the sentence. Find where the box sits and remember the equals sign means both sides are the same.
- Choose the inverse. If a factor is missing, divide the product by the factor you know. If the product is missing, multiply the two factors.
- Check. Put your number back into the box and work it out to make sure both sides balance.
| \(\square\) | \(=\) | \(48 \div 8\) |
| \(\square\) | \(=\) | \(6\) |
| \(\text{check: } 8 \times 6\) | \(=\) | \(48\) |
A factor is missing, so divide. The box is \(6\).
| \(\square\) | \(=\) | \(35 \div 5\) |
| \(\square\) | \(=\) | \(7\) |
| \(\text{check: } 35 \div 7\) | \(=\) | \(5\) |
Split \(35\) into groups of \(5\). There are \(7\) groups.
| \(4 \times \square\) | \(=\) | \(28\) |
| \(\square\) | \(=\) | \(28 \div 4 = 7\) |
There are \(7\) rolls in each row.
| \(3 \times 8\) | \(=\) | \(24\) |
| \(\square \times 2\) | \(=\) | \(24\) |
| \(\square\) | \(=\) | \(24 \div 2 = 12\) |
Work out one side first, then balance the other. The box is \(12\).
Common pitfalls
Frequently asked questions
How do you find a missing number in a multiplication sentence?
Divide the product by the factor you already know. For \(9 \times \square = 72\), work out \(72 \div 9 = 8\). Then check by multiplying \(9 \times 8 = 72\).
What is a fact family?
A fact family is a group of number sentences made from the same three numbers. From \(8\), \(4\) and \(32\) you get \(8 \times 4 = 32\), \(4 \times 8 = 32\), \(32 \div 8 = 4\) and \(32 \div 4 = 8\).
What does the equals sign really mean?
It means "the same as". Both sides must have the same value, so \(3 \times \square = 21\) asks which number makes both sides balance. It does not mean "write the answer here".
How do you find a missing number in a division sentence?
It depends where the box is. For \(\square \div 5 = 6\) you multiply, \(5 \times 6 = 30\). For \(32 \div \square = 8\) you divide, \(32 \div 8 = 4\).
Are multiplication and division opposites?
Yes. They are inverse operations, which means they undo each other. If \(8 \times 4 = 32\) then \(32 \div 8 = 4\).
How can I check a missing number answer?
Put your number back into the sentence and work it out. If you found \(\square = 8\) in \(9 \times \square = 72\), check by multiplying \(9 \times 8 = 72\).