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1. In a 400 m race, A gives B a start of 5 seconds and beats him by 15 m. In another race of 400 m, A beats B by $7 \frac{1}{7}$ seconds. Their respective speed are -

A. 6 m/sec, 7 m/sec

B. 5 m/sec, 7 m/sec

C. 8 m/sec, 7 m/sec

D. 9 m/sec, 7 m/sec

Solution:

Suppose A covers 400 m in t seconds
Then, B covers 385 m in (t + 5) seconds

\begin{aligned} ∴ B covers 400 m in \\ = {\frac{(t + 5)}{385} \times 400} sec \\ = \frac{80 (t + 5)}{77} sec \\ Also, B covers 400 m in \\ = (t + 7 \frac{1}{7}) sec \\ = \frac{(7 t + 50)}{7} sec \\ ∴ \frac{80 (t + 5)}{77} = \frac{7 t + 50}{7} \\ \Rightarrow 80 (t + 5) = 11 (7 t + 50) \\ \Rightarrow (80 t - 77 t) = (550 - 400) \\ \Rightarrow 3 t = 150 \\ \Rightarrow t = 50 \\ ∴ A's speed \\ = \frac{400}{50} m / \sec \\ = 8 m / \sec \\ ∴ B's speed \\ = \frac{385}{55} m / \sec \\ = 7 m / \sec \end{aligned}

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