Relation between ratio and percentage

Question

I would like to know easiest method to solve following:

Que:

If $p$ is $128$% of $r$, $q$ is $96$% of $r$ and $r$ is $250$% of $s$, find the ratio of $p$:$q$:$s$.

My Approach:

Step 1:

$p =\frac{128r}{100}$

$q = \frac{96r}{100}$

$r = \frac{250s}{100}$

So, $s = \frac{100r}{250}$

I do not know what to do further. I know the answer but don't know how to achieve Ans: $16$:$12$:$5$

Thank You!

Answer 1

You have:

• $\frac{p}{r}=\frac{128}{100}$
• $\frac{q}{r}=\frac{ 96}{100}$
• $\frac{s}{r}=\frac{100}{250}$

Adjust them to the same denominator:

• $\frac{p}{r}=\frac{128}{100}\times\frac55=\frac{640}{500}$
• $\frac{q}{r}=\frac{ 96}{100}\times\frac55=\frac{480}{500}$
• $\frac{s}{r}=\frac{100}{250}\times\frac22=\frac{200}{500}$

And you get $p:q:s=640:480:200$.

Divide each factor by their greatest common divisor ($40$).

And you get $p:q:s=640/40:480/40:200/40=16:12:5$.

Answer 2

s is 40% of r, divide everything in units of 8% of r.

Answer 3

Substitute third equation ($r = 2.5s$) into first and second equations. That will give you ratio of p:s and q:s.

Now either substitute first equation into second or vice versa. That will give you p:q.

Last step is to make sure that the ratio's are given in integers (so for example $1:2$ instead of $\frac{1}{2} :1$). Note, you are doing it only for readability of the text.

Answer 4

Notice, $p, q, r$ all depend on $s$ as follows $$r=\frac{250}{100}\times s=\frac{250s}{100}$$ $$q=\frac{96}{100}\times r=\frac{96}{100}\times \frac{250s}{100}$$

$$p=\frac{128}{100}\times r=\frac{128}{100}\times \frac{250s}{100}$$ hence, $$\color{red}{p:q:s}=\left(\frac{128}{100}\times \frac{250s}{100}\right):\left(\frac{96}{100}\times \frac{250s}{100}\right):(s)$$ $$=128:96:40$$ $$=\color{red}{16:12:5}$$