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.