17 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. - Edexcel - GCSE Maths - Question 18 - 2018 - Paper 2

For the second condition, we consider:

$p + 20$ $q + 20$

Setting the ratio gives us:

$\frac{p + 20}{q + 20} = \frac{5}{2}$

This leads to:

$2(p + 20) = 5(q + 20)$

Expanding this equation results in:

$2p + 40 = 5q + 100$ $2p = 5q + 60$

Simplifying gives us:

$p = \frac{5q + 60}{2}$

(2)

Now, we can substitute equation (1) into equation (2):

$\frac{5q + 60}{2} = 5q - 20$

Multiply through by 2 to eliminate the fraction:

$5q + 60 = 10q - 40$

Rearranging gives:

$60 + 40 = 10q - 5q$ $100 = 5q$ $q = 20$

Substituting back to find p using equation (1):

$p = 5(20) - 20$ $p = 100 - 20 = 80${p = 80}$$

Thus, we have found that:

$p = 80 \text{ and } q = 20$

Finally, we can find the ratio:

$p : q = 80 : 20$

To simplify, divide both terms by 20:

$\frac{80}{20} : \frac{20}{20} = 4 : 1$

Thus, the required ratio is:

Answer: $p : q = 4 : 1$