Unknown Factor Problems Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Unknown Factor Problems.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An unknown factor problem asks you to find a missing number in a multiplication equation, such as ?×6=48? \times 6 = 48 or 8×?=568 \times ? = 56.

If you know the total and one group size, division tells you how many groups — the missing factor is the answer to that division.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An unknown-factor problem hides one factor in a multiplication equation, and you recover it by dividing the product by the known factor.

Common stuck point: The procedure for unknown factor problems is the easy part; the trap is multiplying the product by the known factor. Asking "Is one factor hidden in a multiplication equation with the product known?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is one factor hidden in a multiplication equation with the product known?

Worked Examples

Example 1

easy
Step-by-step: solve ?×4=32? \times 4 = 32.

Answer

88

First step

1
Rewrite as division: ?=32÷4? = 32 \div 4.

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Example 2

medium
Maria has 84 stickers and gives the same number to each of her 7 friends with none left over. How many per friend? Write and solve.

Example 3

medium
Solve ?×4=100? \times 4 = 100 and check by multiplying.

Example 4

hard
A rectangle has area 60 and length 12. Find the width.

Example 5

hard
Find the missing factor in ?×25=5×100? \times 25 = 5 \times 100.

Example 6

challenge
A teacher wants to split 72 markers among ?? groups so each group gets exactly 6 markers. Then she splits 72 differently into groups of ?? markers with 9 groups. Find both missing factors.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Solve: ?×6=48? \times 6 = 48.

Example 2

easy
Solve: 8×?=568 \times ? = 56.

Example 3

easy
?×9=0? \times 9 = 0. Find ??.

Example 4

easy
1×?=121 \times ? = 12. Find ??.

Example 5

easy
Check: is ?=9?=9 correct for ?×4=36? \times 4 = 36?

Example 6

easy
Solve: ?×7=7? \times 7 = 7.

Example 7

easy
Write the division that solves ?×5=35? \times 5 = 35.

Example 8

easy
Solve: 3×?=213 \times ? = 21.

Example 9

medium
A box holds rows of 6 and has 54 items. How many rows? Write and solve.

Example 10

medium
Solve: ?×12=144? \times 12 = 144.

Example 11

medium
If 4×?=4×94 \times ? = 4 \times 9, what is ?? without multiplying?

Example 12

medium
Solve for the box: ?×8=8×5? \times 8 = 8 \times 5.

Example 13

medium
A number times 7 is 84. Find the number.

Example 14

medium
Solve: ?×6=90? \times 6 = 90.

Example 15

medium
?×4=100? \times 4 = 100. Is the factor a whole number? Find it.

Example 16

challenge
Find both missing factors: ?×?=36? \times ? = 36 with the two factors equal.

Example 17

challenge
?×5=5×(3×4)? \times 5 = 5 \times (3 \times 4). Find ??.

Example 18

challenge
For what whole numbers does ?×8=8×?? \times 8 = 8 \times ? hold true?

Example 19

medium
A baker packs 8 muffins per box and fills 72 muffins. How many boxes?

Example 20

medium
Solve: ?×11=121? \times 11 = 121.

Example 21

easy
Solve: ?×5=40? \times 5 = 40.

Example 22

easy
Solve: 9×?=279 \times ? = 27.

Example 23

easy
Solve: ?×10=80? \times 10 = 80.

Example 24

easy
Solve: 6×?=426 \times ? = 42.

Example 25

easy
Solve: ?×8=64? \times 8 = 64.

Example 26

easy
Solve: 2×?=182 \times ? = 18.

Example 27

easy
Solve: ?×3=24? \times 3 = 24.

Example 28

medium
A teacher arranges 63 chairs into rows of 7. How many rows? Write the unknown-factor equation and solve.

Example 29

medium
Solve: ?×9=108? \times 9 = 108.

Example 30

medium
A garden has 96 plants in equal rows of 8. How many rows?

Example 31

medium
Solve: ?×6=132? \times 6 = 132.

Example 32

medium
What number times 13 equals 169?

Example 33

medium
Solve: ?×7=7×11? \times 7 = 7 \times 11.

Example 34

medium
Solve: ?×5=200? \times 5 = 200.

Example 35

medium
A box of 12 pens costs \$36. How many boxes can you buy with \$108?

Example 36

medium
Solve: ?×15=90? \times 15 = 90.

Example 37

hard
Solve: ?×8=4×16? \times 8 = 4 \times 16.

Example 38

hard
Find ??: (?×6)+4=28(? \times 6) + 4 = 28.

Example 39

hard
Find ??: ?×?=49? \times ? = 49 where the two factors are equal whole numbers.

Example 40

hard
Solve: ?×9=9×(2×5)? \times 9 = 9 \times (2 \times 5).

Example 41

hard
A bakery bakes 144 cookies in batches of ??. There are 12 batches. Find ?? (the batch size).

Example 42

hard
What is the largest whole-number value of ?? so that ?×7<60? \times 7 < 60?

Example 43

challenge
Find two whole-number factors of 24 whose sum is 11.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

multiplicationdivision