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A beats B by 13 sec in a 250 m race, B beats C by 18 sec in a 400 m race, C beats D by 21 sec in 600 m race and D beats E by 36 sec in a 750 m race. What must be the speed of A in order to beat E by 200 m in a 1 km race? 

A beats B by 13 sec in a 250 m race, B beats C by 18 sec in a 400 m race, C beats D by 21 sec in 600 m race and D beats E by 36 sec in a 750 m race. What must be the speed of A in order to beat E by 200 m in a 1 km race?  Correct Answer 5 km/h

Let VA, VB, VC, VD and VE be the speeds(m/s) of A, B, C, D and E respectively.

Since A beats E by 200 m in 1000 m race, so the ratio of their speeds,

⇒  VA/ VE = 1000/800 = 5/4      ---- (1)

Now, according to question,

A beats B: 250/VB – 250/VA = 13

⇒ 1/VB – 1/VA = 13/250      ---- (2)

B beats C: 400/VC – 400/VB = 18

⇒ 1/VC – 1/VB = 18/400 = 9/200      ---- (3)

C beats D: 600/VD – 600/VC = 21

⇒ 1/VD – 1/VC = 21/600 = 7/200      ---- (4)

D beats E: 750/VE – 750/VD = 36

⇒ 1/VE – 1/VD = 36/750 = 12/250     ---- (5)

Adding eq.(2), eq.(3), eq.(4) and eq.(5) together

⇒ 1/VE – 1/VA = 13/250 + 9/200 + 7/200 + 12/250 = 25/250 + 16/200

⇒ (VA /VE – 1)/VA = 1/10 + 2/25 = 9/50

⇒ VA = (5/4 – 1) × 50/9 = ¼ × 50/9 = 25/18 m/s = 25/18 × 18/5 km/h = 5 km/h

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