Video: GCSE Mathematics Foundation Tier Pack 2 • Paper 2 • Question 20

GCSE Mathematics Foundation Tier Pack 2 • Paper 2 • Question 20


Video Transcript

This number machine can be used to calculate the cost, in pounds, of renting a car. Input, number of days, multiplied by 30.7 add 27 output, cost. Max hires a car. It costs him 272 pounds and 60 pence. Part a) Calculate how many days Max hires the car for.

Let’s look at this number machine more closely. It tells us that to work out the cost of hiring a car, we first multiply the number of days by 30.7 and then add 27 to this value. In this case though, we’ve been told the output of the number machine, the cost of Max hiring the car. And we need to work backwards through the number machine to work out how many days he hired the car for.

We need to start on the right of the number machine at the output and work our way back through, performing the opposite or inverse operations in each step. The inverse or opposite of adding 27 is subtracting 27. So we begin by subtracting 27 from 272 pounds 60. This gives 245 pounds 60. So this is the value after the first stage of working backwards through our number machine.

The inverse or opposite of multiplying by 30.7 is dividing by 30.7. So the final step is to divide our value of 245 pounds 60 by 30.7. This gives eight. Now, this is a sensible value because remember this input is the number of days. So we need it to be an integer or a whole number. If we got a value of, for example, 8.2, this would suggest that something has gone wrong when we worked backwards through the number machine.

Now, we should do a quick check of our answer. And to do this, we can use eight as the input for our number machine and check that it does give an output of 272 pounds 60. First, we multiply eight by 30.7 and it gives 245.6. Then, we preform the second stage of the original number machine, which is adding 27, and it gives 272.6.

When we write this as an answer in pounds, so that’s with two decimal places, it gives 272 pounds 60, which is the correct cost for Max hiring the car. So this tells us that our answer of eight days is correct.

Caitlin hires a car for 𝑑 days. It costs her 𝑃 pounds. Part b) Write down a formula for 𝑃 in terms of 𝑑.

This time, we’ve been given both the input and the output of the number machine. But we’ve been given them algebraically. They’re letters. We need to write down the formula that connects them. So we start off on the left of our number machine with our input of 𝑑.

And the first thing we need to do is multiply it by 30.7. Now, this gives 30.7 multiplied by 𝑑. But we don’t need the multiplication sign in algebra. So we can just write this as 30.7𝑑. The next stage in our number machine is to add 27. So adding 27 to 30.7𝑑 gives the expression 30.7𝑑 plus 27.

Now, this is equal to our output value of 𝑃 pounds. So we can set this expression equal to 𝑃. And it gives us our formula for 𝑃 in terms of 𝑑. 𝑃 is equal to 30.7𝑑 plus 27.

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