Question

If $A:B=\dfrac{1}{2}:\dfrac{1}{3},B:C=\dfrac{1}{2}:\dfrac{1}{3}$ , then A:B:C is equal to
(a) 2:3:3
(b)1:2:6
(c) 3:2:6
(d) 9:6:4

Answer 1

Hint: We have to write the ratios in fractional form and simplify using the property $\dfrac{\dfrac{a}{b}}{\dfrac{c}{d}}=\dfrac{a}{b}\times \dfrac{d}{c}$ . Then, we will obtain \[A:B=3:2\] and \[B:C=3:2\] . Then, we can find the component of A in A:B:C, by multiplying the component of A in A:B and the component of B in B:C. We can find the component of B in A:B:C by multiplying the component of B in A:B and the component of B in B:C. We will get the component of C in A:B:C by multiplying the component of B in A:B and the component of C in B:C.

Complete step by step answer:
We are given that $A:B=\dfrac{1}{2}:\dfrac{1}{3}$ . We write this ratio as follows.
\[\Rightarrow \dfrac{A}{B}=\dfrac{\dfrac{1}{2}}{\dfrac{1}{3}}\]
We know that $\dfrac{\dfrac{a}{b}}{\dfrac{c}{d}}=\dfrac{a}{b}\times \dfrac{d}{c}$ . Therefore, we can write the above ratio as
\[\begin{align}
  & \Rightarrow \dfrac{A}{B}=\dfrac{1}{2}\times \dfrac{3}{1} \\
 & \Rightarrow \dfrac{A}{B}=\dfrac{3}{2} \\
 & \Rightarrow A:B=3:2 \\
\end{align}\]

Now, we have to consider $B:C=\dfrac{1}{2}:\dfrac{1}{3}$ . We write this ratio as follows.
\[\begin{align}
  & \Rightarrow \dfrac{B}{C}=\dfrac{\dfrac{1}{2}}{\dfrac{1}{3}} \\
 & \Rightarrow \dfrac{B}{C}=\dfrac{3}{2} \\
 & \Rightarrow B:C=3:2 \\
\end{align}\]
Now, we have to find A:B:C. In this ratio, we will get A by multiplying the component of A in A:B and the component of B in B:C.
$\Rightarrow 3\times 3=9$
We can find the component of B in A:B:C by multiplying the component of B in A:B and the component of B in B:C.
$\Rightarrow 2\times 3=6$
We can find the component of C in A:B:C by multiplying the component of B in A:B and the component of C in B:C.
$\Rightarrow 2\times 2=4$
Therefore, we can write A:B:C as 9:6:4.

So, the correct answer is “Option d”.

Note: Students must know how to write the ratios in the fractional form and to simplify it. We can also solve this question in an alternate way.
We obtained \[A:B=3:2\] and \[B:C=3:2\] .
Let us multiply and divide the ratio A:B by 3.
\[\begin{align}
  & \Rightarrow \dfrac{A}{B}=\dfrac{3}{2}\times \dfrac{3}{3} \\
 & \Rightarrow \dfrac{A}{B}=\dfrac{9}{6} \\
 & \Rightarrow A:B=9:6 \\
\end{align}\]
Now, we have to multiply the ratio B:C by 2 so that the component of B in this ratio and A:B will be the same.
\[\begin{align}
  & \Rightarrow \dfrac{B}{C}=\dfrac{3}{2}\times \dfrac{2}{2} \\
 & \Rightarrow B:C=6:4 \\
\end{align}\]
 In this ratio, we can see that the mean terms are the same, that is the B components of the ratios A:B and B:C are the same. Therefore, we can write the result of A:B:C as
$A:B:C=9:6:4$