Lesson 10 of 10 · Grades 3 to 4
Check with inverse operations
A multiplication and a division are opposites. That makes them perfect checking partners. When you finish a division, multiply to see whether you get the number you started with. When you finish a multiplication, divide to check.
Opposite operations
Multiplication and division pull in opposite directions. Multiplication builds a total from equal groups; division breaks a total back into those groups. Because they undo each other, they are called inverse operations.
That relationship is a gift for checking work. Finish a division, then multiply to see whether you get back to the start. Finish a multiplication, then divide to do the same. A correct answer survives the trip and comes home.
Check a division with multiplication
To check a division, multiply the quotient by the divisor. The product must equal the dividend. For 72 ÷ 8 = 9, multiply 8 × 9. The product is 72, which is the number you divided, so the quotient is correct. If the product does not match, one of the three numbers in your division is wrong.
This check also helps when a division fact does not come to mind. Instead of guessing, ask which number times 8 makes 72. Working from the multiplication side is often easier.
Check a multiplication with division
To check a multiplication, divide the product by one of the factors. The quotient must equal the other factor. For 6 × 9 = 54, divide 54 by 6. The quotient is 9, which is the other factor, so the multiplication is correct.
If you divide by the wrong factor, you still get a useful answer. Dividing 54 by 9 gives 6, which is the other factor from the multiplication. Either division works; pick the one that is easier for you.
Watch out for this
A check only works if you do it honestly. It is easy to see the number you want and nod along without multiplying. Cover the original answer first, work out the check from scratch, then compare. A check that copies the answer is no check at all.
Checks for word problems
A word problem deserves the same treatment. Suppose you work out that 96 pencils packed 8 to a box fill 12 boxes. The check is 8 × 12 = 96, which matches the number of pencils you started with. The check confirms that your division matched the story, not just the arithmetic.
When a check fails, do not just change the answer. Go back to the story or the fact family and find where the numbers came apart. The error is usually one small step, and finding it is how the arithmetic becomes solid.
Worked example
Work out 96 ÷ 8, then use a multiplication to check the answer.
Step 1: divide. 96 ÷ 8 = 12.
Step 2: check with the inverse operation. Multiply the quotient by the divisor: 8 × 12.
Step 3: work out the product: 8 × 12 = 96.
Step 4: the product matches the number you started with, so the quotient 12 is correct.
Answer: 12. Check: 8 × 12 = 96.
Remember
- Multiplication and division are inverse operations.
- Check a division by multiplying the quotient by the divisor.
- Check a multiplication by dividing the product by a factor.
- Work the check from scratch, without looking at the answer.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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Check 7 × 9 = 63 with a division.
Show answer
63 ÷ 7 = 9, so the product is correct.
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Check 48 ÷ 6 = 8 with a multiplication.
Show answer
6 × 8 = 48, so the quotient is correct.
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Sara works out 6 × 7 = 42. Which division checks her answer?
Show answer
42 ÷ 6 = 7, or 42 ÷ 7 = 6.
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Tomas works out 72 ÷ 9 = 8. Which multiplication checks his answer?
Show answer
9 × 8 = 72.
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Use the inverse to check 5 × 8 = 40.
Show answer
40 ÷ 5 = 8, so the product is correct.
Puzzle · The right check
Puzzle time
Lena works out 72 ÷ 8 = 9. Which multiplication checks her answer?
Show a hint
To check a division, multiply the quotient by the divisor. The product must be the dividend.
Show the answer
Answer: 8 × 9 = 72. The divisor is 8 and the quotient is 9, so the check is 8 × 9 = 72. The product matches the dividend, so Lena is right. The other three statements are false: 8 × 8 = 64, 9 × 9 = 81, and 72 × 8 is far more than 9.