 # Find Inverse Operations

In this worksheet, students will find inverse operations and use these to work backwards from outcomes to starting points. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Number, Number Operations and Integers

Curriculum subtopic:   Structure and Calculation, Inverse Operations

Difficulty level:   ### QUESTION 1 of 10

Inverse operations are really important in maths.

For example, when you have to solve equations (which is a huge topic), you can't do it without these inverse operations.

Inverse operations require us to understand the opposite operation of what is shown.

The opposite of + is -

The opposite of - is +

The opposite of x is ÷

The opposite of ÷ is x

e.g. What is the opposite of +7?

-7

e.g. What is the opposite of dividing by 2?

Multiplying by 2

Let's put an inverse operation into context now...

I add 7 and then multiply by 2.

If my answer is 18, what was my original number?

This one is a bit more challenging.

We have two operations: + 7 and x 2.

The inverse operations for these would be - 7 and ÷ 2.

But which operation do we use first?

If we are doing the opposite, we are moving backwards through the sum.

In the question, we added 7 first then multiplied by 2.

When we are doing the inverse, we swap this order as well.

So to solve this we divide by 2 first, then subtract 7:

18 ÷ 2 = 9

9 - 7 = 2

The original starting number was 2.

Now it's your turn to have a go!

In this activity, you will find inverse operations and use these to work backwards from outcomes to starting points.

What is the inverse of + 4?

What is the inverse operation of ÷ 6?

Match the operations with their inverses.

## Column B

+ 2
- 2
- 6
x 3
x 5
+ 6
÷ 3
÷ 5

I divide a number by 7 then subtract 4, what are the inverse operations?

## Column B

+ 2
- 2
- 6
x 3
x 5
+ 6
÷ 3
÷ 5

I add 7 to a number and get 15.

I multiply a number by 6 to get 18.

What was my original number?

Match these double operations with their inverses.

## Column B

x 2 + 3
x 2 ÷ 5
+ 3 ÷ 5
x 5 - 3
x 5 ÷ 2
-3 ÷ 2
-7 ÷ 3
x 3 + 7

I subtract 7 from a number and then half it.

Which of the options below is the correct inverse operation?

x 2 + 7

+ 7 x 2

÷ 2 + 7

I double a number and add 6 to get 18.

What was my original number?

I half a number and add 4 to get 9.

What was my original number?

• Question 1

What is the inverse of + 4?

-4
- 4
EDDIE SAYS
The opposite of + is -. So the opposite of +4 is -4. Simple!
• Question 2

What is the inverse operation of ÷ 6?

EDDIE SAYS
The opposite operation of divide is multiply. So the opposite of divide by 6 is multiply by 6.
• Question 3

Match the operations with their inverses.

## Column B

+ 2
- 2
- 6
+ 6
x 5
÷ 5
÷ 3
x 3
EDDIE SAYS
You need to remember the opposites from our Introduction. Review these now if you need to. + and - are opposites. ÷ and x are opposites. Once you know the inverse operation, is the number important in this question?
• Question 4

I divide a number by 7 then subtract 4, what are the inverse operations?

EDDIE SAYS
The inverse operation of subtract 4 is add 4. The inverse operation for divide by 7 is multiply by 7. But did you swap the order? The subtract 4 should have come first.
• Question 5

I add 7 to a number and get 15.

8
EDDIE SAYS
The opposite of adding 7 is subtracting 7. If we take 7 away from 15, we reach the starting number. So the starting number is 8.
• Question 6

I multiply a number by 6 to get 18.

What was my original number?

3
EDDIE SAYS
The opposite of multiplying by 6 is dividing by 6. If we divide 18 by 6, we reach the starting number. So the starting number was 3. You can always check your answer by putting it into the original question: 3 x 6 = 18 This statement is true so we know that we have the correct answer!
• Question 7

Match these double operations with their inverses.

## Column B

x 2 + 3
-3 ÷ 2
+ 3 ÷ 5
x 5 - 3
x 5 ÷ 2
x 2 ÷ 5
-7 ÷ 3
x 3 + 7
EDDIE SAYS
Its all about remembering those inverse operations: + and - are opposites; ÷ and x are opposites. As well as swapping the signs for their opposites, did you remember to reverse the order? Remember that we will be working backwards through each sum when we are calculating inverses.
• Question 8

I subtract 7 from a number and then half it.

Which of the options below is the correct inverse operation?

x 2 + 7
EDDIE SAYS
- 7 ÷ 2 becomes x 2 + 7 as an inverse. The key rule to remember is that when finding the inverse, you must remember to reverse the order of operations as well. Great work if you remembered that!
• Question 9

I double a number and add 6 to get 18.

What was my original number?

6
EDDIE SAYS
Remember, to find the inverses of the operations and then reverse the order. x 2 + 6 = 18 So (18 - 6) ÷ 2 = our starting number The starting number was 6.
• Question 10

I half a number and add 4 to get 9.

What was my original number?

10
EDDIE SAYS
Remember to find the inverses of the operations and then reverse the order. ÷ 2 + 4 = 9 So (9-4) x 2 = original number Original number = 10 Well done, that’s another activity completed!
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