3.3 K) Ratio when Adding or Subtracting
The ratio of boys to girls in a running club is 6 : 5.
8 more boys join the running club and the ratio of boys to girls becomes 4 : 3.
How many girls are in the running club?
We are given two different ratios in this question. The first ratio that we are given is the ratio of the number of boys to the number of girls, which is 6 : 5. I am going label boys as B and girls as G.
We can make G 15 in the first ratio by multiplying the components in the first ratio by 3. And, we can make G 15 in the second ratio by multiplying the components in the second ratio by 5. The working is shown below:
We are now able to work out the number of girls that are in the running club. We work out the number of girls in the running club by multiplying the number of parts for the girls in the combined ratio (15) by what 1 part in the combined ratio represents (4).
The next example is a very complex one and we solve it by using simultaneous equations. This question is taken from a past paper.
p and q are two numbers such that p > q
When you subtract 5 from p and subtract 5 from q the answers are in the ratio 5 : 1
When you add 20 to p and add 20 to q the answers are in the ratio 5 : 2
Find the ratio p : q
Give your answer in its simplest form.
Before we go through the method used to answer this question, it is worth going through why we cannot answer this question using the method that we used in the previous example. We are unable to use the combining ratios method because there are no consistent values between the two ratios. This is because for both of the ratios, we are subtracting or adding to both p and q. Whereas in the previous example, the number of girls in the running club remained constant between the two ratios (before the 8 boys joined and after the 8 boys joined).
I am now going to display the information that we are given in the question in a better way.
The first ratio that we are given is that when 5 is subtracted from p and q, the ratio is 5 : 1.
Let’s do this for the first ratio, which is shown below.
I will now go through why this equation makes sense. “p – 5” in this ratio is represented by 5 parts, and “q – 5” is represented by 1 part. 5 parts is 5 times bigger than 1 part, which means that “p – 5” is 5 times bigger than “q – 5” . Therefore, if we were to divide the actual value for “p – 5” by the actual value for “q – 5”, we would get 5 (which is 5/1), and this is exactly what the equation above shows.
I am now going to tidy this equation up so that we do not have any fractions. We can ignore the denominator on the right side of the equation because a denominator of 1 has no effect. We can get rid of the denominator on the left side of the equation by multiplying both sides of the equation by the denominator. The denominator on the left side of the equation is “q – 5”. Therefore, we multiply both sides of the equation by “q – 5”. The working and the expanding of the bracket is shown below.
We now do the same with the other ratio, which is:
In order to make 5q the subject of the first equation, I need to move the -25 from the right side of the equation to the left. We can do this by doing the opposite and the opposite of taking 25 is adding 25. Therefore, we add 25 to both sides of the equation.
We now need to find the value of q and we do this by subbing p as 80 into either the first or second equation. I am going to sub p as 80 into the first equation.
We now have the actual values for both p and q; p is 80 and q is 20. We can now say that the ratio for p to q is:
This was a very long and very tricky question, so well done for getting through to the end of it. It may be the case that you will need/ want to go through the working for this question a few times to fully understand how to answer it.