9.5.2 Doing and Undoing

The next activity gives you some practice with doing and undoing. You will find these techniques helpful for rearranging formulas later. Part (c) involves two stages.

Activity Symbol Activity: Doing and Undoing

Try to work out what number I was thinking of in the following problems. You may find it helpful to use some doing-undoing diagrams.

(a) Think of a number, subtract 3, and the answer is 2.

Answer Symbol


So, the number I thought of was 5.

Remember, you can check your answer by using it in the calculation: five minus three equals two.

(b) Think of a number, multiply by 5, and the answer is 35.

Answer Symbol


So, the number I thought of was 7. (Check: seven multiplication five equals 35 ).

(c) Think of a number, add 4, double it, and the answer is 14.

Answer Symbol


So, working from the right, 14 division two equals seven and seven minus four equals three.

If you like, you can write the output from each box on the arrow as shown above.

So, the number I thought of was 3. (Check: three plus four equals seven, then seven multiplication two equals 14 ).

When you have to deal with more than one operation, just take each step in turn. Write down the doing diagram and draw the undoing one underneath, working backward to undo each operation.

To summarize, the following operations undo each other:

  • Addition and subtraction.

  • Multiplication and division.

  • Squaring and taking the positive square root.

Activity Symbol Activity: A Multistep Problem

Think of a number. Add 5. Multiply by 3. Subtract 4.

If the answer is 17, can you work out what number I was thinking of?

Here there are several steps, so to find the number, it will be necessary to undo each of these steps in turn, starting with the last step. Draw the doing diagram and then the undoing one beneath it.

Answer Symbol


The diagram below shows the steps taken in this case and then how to undo each one.

So, if the answer is 17, undoing the steps by working from right to left gives:

Add 4:
17 plus four equals 21
Divide by 3:
21 division three equals seven
Subtract 5:
seven minus five equals two

So, the number I thought of was 2. (Check: two plus five equals seven, then seven multiplication three equals 21, and 21 minus four equals 17 ).

You might like to go back to the BMI activity (Section 9.3.2) now and see if you can work out the weight at which the man would be classed as normal weight.

We will now look at how we can use this same technique to change formulas.

9.5.1 Another Number Trick

9.5.3 Changing Formulas