Related Calculations and Inverse Operations

Division undoes multiplication.

Undo the subtraction with the opposite operation.

Sharing is dividing, so undo it with the opposite operation.

Compare 260 with 26: what has it been multiplied by?

Multiplication and division are inverses, so the same three numbers make a division fact.

Which operation undoes dividing by 9?

1. Multiplying by 9 undoes dividing by 9.
2. So Ravi should work out 68\times9 and see if he gets 612.
3. 68\times9=612, so his answer is right. The calculation is 68\times9.

96 shared into some equal groups gives 8 in each group. How many groups?

1. If 96\div\square=8, then \square\times8=96.
2. So the missing number is 96\div8=12.
3. Check: 96\div12=8. The missing number is 12.

Compare 4.7 with 47. What has 47 been divided by?

1. 4.7 is 47\div10, and 23 is unchanged.
2. So the answer is 1081\div10=108.1.
3. 4.7\times23=108.1.

Both numbers have been multiplied by 10. What does that do to the answer?

1. 180=18\times10 and 350=35\times10.
2. So the answer is 630\times10\times10=630\times100.
3. 180\times350=63\,000.

Turn the division fact round into a multiplication fact first.

1. From 552\div24=23, the inverse gives 23\times24=552.
2. 2.3 is 23\div10, so the answer is 552\div10=55.2.
3. 2.3\times24=55.2.

If one number is divided by 10, what must happen to the other number to keep the same answer?

1. 3.6=36\div10 and 140=14\times10.
2. Dividing by 10 and multiplying by 10 cancel out, so 3.6\times140=504.
3. The others are 50\,400, 5.04 and 50.4. The answer is 3.6\times140.

Compare 1.6 with 16. To keep the answer at 368, how must the other number change?

1. From the fact, 16\times23=368.
2. 1.6 is 16\div10, so the other number must be 10 times as big to keep the answer the same.
3. 23\times10=230. Check: 1.6\times230=368. The missing number is 230.

41 lots of 37 is one more lot of 37 than 40 lots of 37.

1. 37\times41 is 37\times40 plus one more 37.
2. 1480+37=1517.
3. 37\times41=1517.

49 crates is one crate fewer than 50 crates.

1. 50 crates hold 1200 bottles.
2. 49 crates is one crate fewer, so take away 24 bottles.
3. 1200-24=1176. There are 1176 bottles.

Start from the answer and undo each step, last step first.

1. Undo the last step first: the inverse of subtracting 9 is adding 9. 6+9=15.
2. Then undo dividing by 4 by multiplying by 4. 15\times4=60.
3. Check: 60\div4=15 and 15-9=6. Kemi thought of 60.

Multiplying by 4 undoes dividing by 4. Use it to check, then find the correct answer.

Part (a)

1. 254\times4=1016.
2. 1016 is not 936, so Lena's answer is wrong.

Part (b)

1. The correct answer is 936\div4=234. Check: 234\times4=936.
2. Lena's answer is 254-234=20 too big.

Count how many times each number has been multiplied or divided by 10.

Part (a)

1. 6.3=63\div10 and 0.48=48\div100.
2. So divide 3024 by 10 and then by 100, which is dividing by 1000.
3. 3024\div1000=3.024.

Part (b)

1. 0.63=63\div100 and 480=48\times10.
2. Divide by 100 and multiply by 10: overall divide by 10.
3. 3024\div10=302.4.

Part (c)

1. 6300=63\times100 and 4.8=48\div10.
2. Multiply by 100 and divide by 10: overall multiply by 10.
3. 3024\times10=30\,240.

First find how many pieces Jess cuts: you need 172.8\div2.7. Multiply both numbers by 10 to get a division you can find from the fact.

1. Number of pieces =172.8\div2.7.
2. Multiplying both numbers by 10 gives the same answer: 172.8\div2.7=1728\div27.
3. From the fact 27\times64=1728, so 1728\div27=64. Check: 2.7\times64=172.8.
4. Jess cuts 64 pieces, so she needs 75-64=11 more pieces.

Work backwards: undo the sharing first, then undo the spending.

1. Before the sharing, Dev had 3\times15.20=45.60, so £45.60.
2. Before buying the present, he had 45.60+18.40=64.
3. Check: 64-18.40=45.60 and 45.60\div3=15.20. Dev had £64 at the start.

£4.50 is 45 divided by 10.

Part (a)

1. £4.50 is 45\div10 pounds, so the cost is 1620\div10.
2. 36\times4.50=162. The total cost is £162.

Part (b)

1. Four £50 notes make 4\times50=200, so £200.
2. Change: 200-162=38. The change is £38.