In a 1-kilometer race, if A gives B a 40 m head start, A wins by 19 s.
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In a 1-kilometer race, if A gives B a 40 m head start, A wins by 19 s. But if A gives B a 30 s start, B wins by 40 m. Find the time taken by B to run 5,000 m.
Show answer & explanation
Let B take 150 s for 1000 m, so for 5000 m he takes 5 × 150 = 750 s. This follows from solving the two head-start conditions using speeds or total race times.
Step-by-step Derivation:
Step 1: Let A's time for 1000 m be T_A and B's time for 1000 m be T_B.
Step 2: If A gives B a 40 m head start, B runs only 960 m. A wins by 19 s, so:
T_A + 19 = 0.96 T_B
=> T_A = 0.96 T_B - 19.
Step 3: If A gives B a 30 s start, B wins by 40 m. When B finishes 1000 m, A has run for T_B - 30 seconds and covered 960 m. Since 960 m takes 0.96 T_A for A:
0.96 T_A = T_B - 30
=> T_A = (T_B - 30) / 0.96.
Step 4: Equate the two expressions for T_A:
0.96 T_B - 19 = (T_B - 30) / 0.96.
Multiply by 0.96:
0.9216 T_B - 18.24 = T_B - 30
=> 11.76 = 0.0784 T_B
=> T_B = 150 s.
Step 5: B runs 1000 m in 150 s, so time for 5000 m = 5 × 150 = 750 s.