Question

The compound ratio of 3:4 and the inverse ratio of 4:5 is 45: x. Find x.

Answer 1

Hint:
Here in the above question, we need to find a compound ratio. But first of all, we should know what the compound ratio is. A compound ratio is defined as the ratios obtained when we compound two or more ratios through multiplication. For two ratios $m:n$ and $p:q$, then the compound ratio is $mp:nq$. In this question, we will find the compound ratio in the same way. We also need to find the inverse of the ratio, then we know that reciprocal of the ratio is the inverse of a ratio. Let’s see how we can find it.

Complete step by step solution:
Inverse ratio of 4:5 is 5:4 (Reciprocal of 4:5)
Now, we know that Compound ratio is;
$a:b\,and\,c:d = ac:bd$
Where $a:b$ = 3:4 and $c:d$ = 5:4
$3:4\,and\,5:4 = 45:x$
$ \Rightarrow 3 \times 5:4 \times 4 = 45:x$ (here we multiply a X c and b X d)
$ \Rightarrow 15:16 = 45:x$
Now, we will solve this equation for x
$\dfrac{{15}}{{16}} = \dfrac{{45}}{x}$
$ \Rightarrow 15 \times x = 16 \times 45$
$x = \dfrac{{16 \times 45}}{{15}} = $48

Hence, the value of x is 48 and the ratio is 45:48.

Note:
Here we see a few more examples of compound ratio: What is the compound ratio of $10:30$ & $60:80$.
Compound Ratio = $10:30::60:80$= $\dfrac{{10}}{{30}} \times \dfrac{{60}}{{80}} = \dfrac{1}{4}$= 1:4.
Students should keep in mind that while solving the questions of compound ratio you should take antecedent as product of antecedents of the ratios and consequent as product of consequences of the ratios, then the ratio thus formed is called compound ratio.
Inverse Ratio: If two ratios, the antecedent and the consequent of one are respectively the consequent and antecedent of the other, they are said to be ‘inverse ratio’ or ‘reciprocal’ to one another. Let’s see an example- inverse of 3:4 will be 4:3.