Question

If A:B = 1:2, B:C = 3:4 and C:D = 5:6, then D:A is equal to
[a] 6:5
[b] 5:16
[c] 5:6
[d] 16:5

Answer 1

Hint: Express B in terms of A using $A:B=1:2$. Then express C in terms of B and hence in terms of A using $B:C=3:4$. Finally, express D in terms of C using C:D = 5:6 and hence in terms of A. Hence find the ratio D:A.
Alternatively, use the fact that if a:b = c:d, then $\dfrac{a}{b}=\dfrac{c}{d}$. Hence express all the ratios in fractional form and multiply the equations and hence find the ratio A:D and hence the ratio D:A.

Complete step-by-step solution -
We have $A:B=1:2$
We know that product of extremes = product of means
Here extremes are A,2 and means are B, 1
Hence, we have $2A=B\Rightarrow B=2A\text{ ............................. }\left( i \right)$
We have $B:C=3:4$
We know that product of extremes = product of means
Here extremes are B and 4 and means are 3 and C
Hence, we have
$4B=3C$
Dividing both sides by 3, we get
$C=\dfrac{4B}{3}$
Substituting the value of B from equation (i), we get
$C=\dfrac{4}{3}\left( 2A \right)=\dfrac{8A}{3}\text{..................................... }\left( ii \right)$
We have $C:D=5:6$
We know that product of extremes= product of means
Here extremes are C , 6 and means are D,5
Hence, we have $6C=5D$
Dividing both sides by 5, we get
$D=\dfrac{6}{5}C$
Substituting the value of C from equation (ii), we get
$D=\dfrac{6}{5}\left( \dfrac{8}{3}A \right)=\dfrac{16}{5}A$
Dividing both sides by A, we get
$\dfrac{D}{A}=\dfrac{16}{5}$
Hence, we have D:A = 16:5.
Hence option [d] is correct.

Note: Alternative Solution: Best Method:
We have
 $\begin{align}
  & A:B=1:2 \\
 & \Rightarrow \dfrac{A}{B}=\dfrac{1}{2}\text{.......................... }\left( a \right) \\
\end{align}$
Similarly, we have
$\dfrac{B}{C}=\dfrac{3}{4}\text{.................................... }\left( b \right)$
And $\dfrac{C}{D}=\dfrac{5}{6}\text{............................... }\left( c \right)$
Multiplying equation(a), equation (b) and equation (c), we get
$\begin{align}
  & \dfrac{A}{B}\times \dfrac{B}{C}\times \dfrac{C}{D}=\dfrac{1}{2}\times \dfrac{3}{4}\times \dfrac{5}{6} \\
 & \Rightarrow \dfrac{A}{D}=\dfrac{5}{16} \\
\end{align}$
Hence, we have
$\dfrac{D}{A}=\dfrac{16}{5}$
Hence, we have D:A = 16:5, which is the same as obtained above.
Hence option [d] is correct.