Question

If the relation between a, b and c are given as $2a=3b=4c$, then find the value of $a:b:c$?
(a) $2:3:4$,
(b) $3:4:6$,
(c) $4:3:2$,
(d) $6:4:3$.

Answer 1

Hint: We start solving the problem by converting b in terms of a by using the relation $2a=3b$. Then we convert c in terms of a by using the relation $2a=4c$. Once we convert all the variables in terms of a, we start finding the ratio $a:b:c$. We make the required calculations and arrangements to get the required answer.

Complete step by step answer:
According to the problem, the relation between a, b and c is defined as $2a=3b=4c$. We need to find the value of $a:b:c$. Let us first convert b and c in terms of a before solving for the ratio.
We have $2a=3b$.
$\Rightarrow b=\dfrac{2a}{3}$ ---(1).
We have $2a=4c$.
$\Rightarrow c=\dfrac{2a}{4}$.
$\Rightarrow c=\dfrac{a}{2}$---(2).
So, we need to find the value of $a:b:c$ using the result obtained from (1) and (2).
We have $a:b:c=a:\dfrac{2a}{3}:\dfrac{a}{2}$.
$\Rightarrow a:b:c=\dfrac{6a}{6}:\dfrac{2a\times 2}{3\times 2}:\dfrac{a\times 3}{2\times 3}$.
$\Rightarrow a:b:c=\dfrac{6a}{6}:\dfrac{4a}{6}:\dfrac{3a}{6}$.
$\Rightarrow a:b:c=6:4:3$.
We have found the value of $a:b:c$ as $6:4:3$.
∴ The value of $a:b:c$ as $6:4:3$.

So, the correct answer is “Option D”.

Note: We need not convert all the variables in terms of a. We can also convert all the variables in terms of either b or c to solve for the desired ratio. We can also solve the problem alternatively as follows:
Let us assume $2a=3b=4c=r$. Let us convert all in terms of r.
We have $2a=r$.
$\Rightarrow a=\dfrac{r}{2}$ ---(3).
We have $3b=r$.
$\Rightarrow b=\dfrac{r}{3}$ ---(4).
We have $4c=r$.
$\Rightarrow c=\dfrac{r}{4}$ ---(5).
Now, we find the value of $a:b:c$ using the result obtained from (3), (4) and (5).
We have $a:b:c=\dfrac{r}{2}:\dfrac{r}{3}:\dfrac{r}{4}$.
$\Rightarrow a:b:c=\dfrac{r\times 6}{2\times 6}:\dfrac{r\times 4}{3\times 4}:\dfrac{r\times 3}{4\times 3}$.
$\Rightarrow a:b:c=\dfrac{6r}{12}:\dfrac{4r}{12}:\dfrac{3r}{12}$.
$\Rightarrow a:b:c=6:4:3$.