Race Time!: Solution

It was a 2 km race. Let Rohan's speed be "r" km/hr and Amit's speed be "a" km/hr. Since Rohan's speed is "r", he runs 2 km in 2/r hours. Likewise, Amit runs 2 km in 2/a hours.

Amit took two minutes longer to complete the race. In terms of hours, he took 2/60 or 1/30 hours longer to complete the race. Therefore, 2/a - 2/r = 1/30.

So, (2r - 2a)/ar = 1/30.

So, 2r - 2a = ar/30; this implies, r - a = ar/60.

Thus, 60(r - a) = ar.

If Rohan had run slower by 2 km/hr, his speed would have been (r - 2) km/hr, and he would have run 2 km in 2/(r - 2) hours. Likewise, if Amit had increased his speed by 2 km/hr his speed would have been (a + 2) km/hr and he would have run 2 km in 2/(a + 2) hours. In this case, Amit would have won the race by two minutes. Amit wins the race means he takes less time, so 2/(a + 2) is less than 2/(r - 2). So, in this case, 2/(r - 2) - 2/(a + 2) = 1/30.

Therefore, [2(a + 2) - 2(r - 2)] / [(r - 2)(a + 2)] = 1/30.

So, 2(a + 2) - 2(r - 2) = [(r - 2)(a + 2)]/30.

Therefore, 60(a + 2) - 60(r - 2) = (r - 2)(a + 2).

So, 60a + 120 - 60r + 120 = ra + 2r - 2a - 4.

So, 62a - 62r + 244 = ra.

But from the first case we have seen that 60(r - a) = ar.

Let's substitute. We get, 62a - 62r + 244 = 60(r - a).

So, 62a - 62r + 244 = 60r - 60a.

So, 122a - 122r + 244 = 0.

Thus, 122 (a - r) = -244. This means, (a - r) = -2. We can rewrite this as -(r - a) = -2, or (r - a) = 2.

But, 60(r - a) = ar; so, 60 × 2 = ar, or ar = 120.

Now I know that (x - y)2 = x2 - 2xy + y2; thus, (r - a)2 = r2 - 2ra + a2. I can rewrite this as: (r - a)2 = r2 + 2ra + a2 -4ra.

So, (r - a)2 = (r+a)2 - 4ra.

But (r - a) = 2 and ar = 120, so, 22 = (r+a)2 - 4(120).

So, 4 = (r+a)2 - 480; therefore, (r+a)2 = 484.

Solving this, we get (r + a) = +22 or (r + a) = -22. But we are dealing with speeds, which cannot be negative. Therefore, r + a = 22.

Thus, we have (r - a) = 2 and (r + a) = 22; solving these two equations we get r = 12 and a = 10. Thus, Rohan's speed is 12 km/hr and Amit's speed is 10 km/hr. Did I do it right? I seem to have taken ages and ages to solve this! Is there any simple method to solve it?

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