The problem asks for the time Mahima takes to complete one round of track X. We are given the speeds of Ankur and Mahima, the total length of their tracks, and the relationship between the number of rounds they complete in the same time.

Analyze Given Information

• Ankur's speed, $v_A = 1 \frac{2}{3} \text{ m/s} = \frac{5}{3} \text{ m/s}$.
• Mahima's speed, $v_M = 2 \frac{1}{12} \text{ m/s} = \frac{25}{12} \text{ m/s}$.
• Total length of Track X and Track Y, $L_X + L_Y = 0.325 \text{ km} = 325 \text{ m}$.
• Ankur completes 9 rounds of Track X in the same time Mahima completes 5 rounds of Track Y.

Calculate Track Lengths

Let $T$ be the time taken. The distance covered is Speed $\times$ Time.

• Distance covered by Ankur in time $T$: $D_A = 9 \times L_X$. Time taken by Ankur: $T = \frac{9 \times L_X}{v_A}$.
• Distance covered by Mahima in time $T$: $D_M = 5 \times L_Y$. Time taken by Mahima: $T = \frac{5 \times L_Y}{v_M}$.

Since the times are equal:

$ \frac{9 \times L_X}{v_A} = \frac{5 \times L_Y}{v_M} $

Substitute the speeds:

$ \frac{9 \times L_X}{\frac{5}{3}} = \frac{5 \times L_Y}{\frac{25}{12}} $ $ \frac{27 \times L_X}{5} = \frac{60 \times L_Y}{25} = \frac{12 \times L_Y}{5} $

This simplifies to:

$ 27 \times L_X = 12 \times L_Y $

Divide by 3:

$ 9 \times L_X = 4 \times L_Y $

We also know $L_X + L_Y = 325$. Express $L_Y$ in terms of $L_X$ from the ratio:

$ L_Y = \frac{9}{4} L_X $

Substitute this into the total length equation:

$ L_X + \frac{9}{4} L_X = 325 $ $ \frac{4 L_X + 9 L_X}{4} = 325 $ $ \frac{13 L_X}{4} = 325 $ $ L_X = \frac{325 \times 4}{13} = 25 \times 4 = 100 \text{ m} $

Now find $L_Y$:

$ L_Y = 325 - L_X = 325 - 100 = 225 \text{ m} $

Calculate Mahima's Time for Track X

We need to find the time taken by Mahima to complete one round of track X. This is given by:

$ T_{M, X} = \frac{\text{Length of Track X}}{\text{Mahima's Speed}} = \frac{L_X}{v_M} $

Substitute the values:

$ T_{M, X} = \frac{100 \text{ m}}{\frac{25}{12} \text{ m/s}} $ $ T_{M, X} = \frac{100 \times 12}{25} \text{ seconds} $ $ T_{M, X} = 4 \times 12 = 48 \text{ seconds} $