From the uploaded image, the problem states:

Three athletes, Samrat, Sarath, and Sridhar, run a race. Sarath finishes 24 meters ahead of Sridhar and 36 meters ahead of Samrat, while Sridhar finishes 16 meters ahead of Samrat. If each athlete runs the entire distance at their respective constant speeds, what is the length of the race?

Solution:

Let the length of the race be $L$.

• Assume their speeds:
  • Speed of Samrat = $S_{\text{Samrat}}$,
  • Speed of Sarath = $S_{\text{Sarath}}$,
  • Speed of Sridhar = $S_{\text{Sridhar}}$.

Each athlete completes the race in their own time. The relationship between speed, distance, and time is $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$.

Key relationships given:

1. Sarath finishes 24 meters ahead of Sridhar:

   • When Sarath finishes the race (i.e., runs $L$ meters), Sridhar has run $L - 24$ meters.

   Hence, their times are equal: $\frac{L}{S_{\text{Sarath}}} = \frac{L - 24}{S_{\text{Sridhar}}}.$

2. Sarath finishes 36 meters ahead of Samrat:

   • When Sarath finishes the race, Samrat has run $L - 36$ meters.

   Hence: $\frac{L}{S_{\text{Sarath}}} = \frac{L - 36}{S_{\text{Samrat}}}.$

3. Sridhar finishes 16 meters ahead of Samrat:

   • When Sridhar finishes the race, Samrat has run $L - 16$ meters.

   Hence: $\frac{L}{S_{\text{Sridhar}}} = \frac{L - 16}{S_{\text{Samrat}}}.$

Eliminating speeds:

From these equations, solve step-by-step:

(i) From the first equation: $\frac{L}{S_{\text{Sarath}}} = \frac{L - 24}{S_{\text{Sridhar}}} \implies \frac{S_{\text{Sridhar}}}{S_{\text{Sarath}}} = \frac{L - 24}{L}.$

(ii) From the second equation: $\frac{L}{S_{\text{Sarath}}} = \frac{L - 36}{S_{\text{Samrat}}} \implies \frac{S_{\text{Samrat}}}{S_{\text{Sarath}}} = \frac{L - 36}{L}.$

(iii) From the third equation: $\frac{L}{S_{\text{Sridhar}}} = \frac{L - 16}{S_{\text{Samrat}}} \implies \frac{S_{\text{Samrat}}}{S_{\text{Sridhar}}} = \frac{L - 16}{L}.$

Combining ratios:

Using the above relationships, solve for $L$:

1. Combine: $\frac{S_{\text{Sridhar}}}{S_{\text{Sarath}}} \cdot \frac{S_{\text{Sarath}}}{S_{\text{Samrat}}} \cdot \frac{S_{\text{Samrat}}}{S_{\text{Sridhar}}} = 1.$

2. Substitute the ratios: $\frac{L - 24}{L} \cdot \frac{L - 36}{L} \cdot \frac{L}{L - 16} = 1.$

3. Simplify: $\frac{(L - 24)(L - 36)}{L(L - 16)} = 1.$

4. Expand and solve: $(L - 24)(L - 36) = L(L - 16).$ Expand both sides: $L^2 - 60L + 864 = L^2 - 16L.$ Simplify: $-60L + 864 = -16L.$ $864 = 44L.$ $L = \frac{864}{44} = 19.636 \, \text{meters}.$

Thus, the length of the race is approximately 19.64 meters.

Would you like me to provide further explanations or verify the calculations? Let me know!