Q. Ram and Shyam are running on different circular tracks. In the time that Ram completes 4 rounds of his track, Shyam completes 3 rounds of his track. If Ram and Shyam interchange their tracks, the time taken by both of them to complete 1 round is same. What is the ratio of speeds of Ram and Shyam?

A. \(\sqrt{\frac{3}{4}}\)

B. \(\sqrt{\frac{7}{9}}\)

C. \(\frac{2}{\sqrt{3}}\)

D. \(\frac{\sqrt{7}}{4}\)

E. Cannot be determined

C. \(\frac{2}{\sqrt{3}}\)

On first reading, the way forward is not crystal clear but what makes it doable is that time being same, track length is proportional on speeds; and there is a second relation also … so two variables, two data seems doable.

Let the speeds of Ram and Shyam be x and y.

Using the first equation and equating the time,

\(\frac{4\;×\;\text{length of track 1}}{x}\) = \(\frac{3\;×\;\text{length of track 2}}{y}\)

\(\Rightarrow \frac{\text{length of track 1}}{\text{length of track 2}}=\frac{3x}{4y}\)

Next, the second data …

\(\frac{1\;×\;\text{length of track 1}}{y}\) = \(\frac{1\;×\;\text{length of track 2}}{x}\)

\(\Rightarrow \frac{\text{length of track 1}}{\text{length of track 2}}=\frac{y}{x}\)

From the two relations,

\(\frac{y}{x}\) = \(\frac{3×x}{4 \times y}\)

\(\frac{x}{y}\) = \(\frac{2}{\sqrt{3}}\)