Division asks: what times?
A division is a multiplication with a missing number.
Division hides a multiplication
We know 12 ÷ 3 = 4, because 3 × 4 = 12. A division asks: what number multiplied by the divisor gives the dividend?
For (−12) ÷ 3, we want a number that times 3 gives −12. What kind of number is it?
What this lesson covers
The idea
A division of integers a ÷ b is found by asking what number multiplied by b gives a, that is, by solving b × ? = a.
Division hides a multiplication
We know 12 ÷ 3 = 4, because 3 × 4 = 12. A division asks: what number multiplied by the divisor gives the dividend?
For (−12) ÷ 3, we want a number that times 3 gives −12. What kind of number is it?
- Positive
- Negative
- There is no such number
Find the missing number
Set the ? with ◀ ▶. A positive ? puts in that many sets of the divisor. A negative ? takes them out. Make the bar end on the flag.
b × ? = a
(−100) ÷ 25 asks: 25 × ? = −100. We know 25 × (−4) = −100, so (−100) ÷ 25 = −4.
Always put the divisor in front of the question mark. Check by multiplying back.
A division of integers a ÷ b is found by asking what number multiplied by b gives a, that is, by solving b × ? = a.
- Division | Missing factor | Answer
- (−100) ÷ 25 | 25 × ? = −100 | −4
- (−100) ÷ (−4) | (−4) × ? = −100 | 25
- 50 ÷ (−25) | (−25) × ? = 50 | −2
Notes
A division of integers a ÷ b is found by asking what number multiplied by b gives a, that is, by solving b × ? = a.
Check yourself
Find (−72) ÷ 9. Solve 9 × ? = −72.
Answer: -8
9 × (−8) = −72, so (−72) ÷ 9 = −8.
Find (−45) ÷ (−5). Solve (−5) × ? = −45.
Answer: 9
(−5) × 9 = −45, so (−45) ÷ (−5) = 9.
Ravi says 56 ÷ (−8) = 7, because 8 × 7 = 56. What is wrong?
Which multiplication does (−36) ÷ 4 ask us to solve?
- The divisor is −8: (−8) × 7 = −56, so the answer must be −7 — correct. Yes! Put the divisor first: (−8) × ? = 56 gives ? = −7.
- Nothing is wrong: the answer is 7. Check by multiplying back: (−8) × 7 = −56, not 56.
- The answer is −56. We want the number that times −8 gives 56. (−8) × (−56) is far too big.
- ? × (−36) = 4. The dividend is −36. It is the answer of the multiplication, not a factor.
- (−36) × 4 = ?. That is a multiplication, not a division. It does not have a missing factor.
- 4 × ? = −36 — correct. Yes! The divisor 4 multiplies the missing number to give the dividend −36.