While A covers 1000 m, B covers ( 1000 - 40) m = 960 m and  C covers (1000 - 64) m = 936 m.

When B covers 960 m , C covers 936 m.

When B covers 1000 m,     C  covers `(936/960 xx 100) `m = 975 m

`:.`    B can give C a start of (1000 - 975) =  25 m.

A : B = 80 : 75

A : C = 80 : 65.

`B/C = ( B/A xx A/C)    = (75/80 xx 80/65)`

        = `15/13  = 60/52`  =  60 : 52

`:.`     In a game of 60 , B can give  C 8 points.

Distance covered by B in 9 sec. = `(100/45 xx 9)`m = 20 m.

`:.`     A beats B by 20 metres.

Time taken by A to  cover  100 m = `((60 xx 60)/(8000) xx 100)` sec= 45 sec.

`:.`    B covers ( 100 -  4) m = 96 m in (45 + 15) sec  = 60 sec.

`:.`   B's speed  = `((96 xx 60 xx 60)/(60 xx 1000))` m/hr = 5.76 km/hr.

Clearly , A beats  B by 10 sec.

Distance covered by B in 10 sec. =  `(1000/200  xx 10) m ` = 50 m.

`:.`     A beats B by  50 metres .

Ratio of the rates of A and  B = `7/4 : 1` =  7 : 4

So , in a race of 7 m , A gains 3 m over  B.

`:.`     3 m are gained by A  in a race of 7 m

`:.`    84 m are gained by A in a race of `(7/3 xx 84)`m = `96 m

`:.`     Winning post must be 196 m away from the starting point .

Clearly , B covers  28 min.7 seconds.

`:.`    B's time over the course = `(7/28 xx 1000)` sec. = 250 seconds.

`:.`     A's time over the course = (250 - 7) sec. = 243 sec. =  4 min.3 sec.

Answer : Option D

Explanation :

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Solution 1
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"A can give 15 points to B in 60".
=> In a game of 60, A starts with 0 points where B starts with 15 points
"A can give 20 points to C in 60".
=> In a game of 60, A starts with 0 points where C starts with 20 points
ie, in a game of 60-15-45, B can give C 20-15=5 points
=> ie, in a game of 90, B can give C 5×2=10 points
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Solution 2
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"A can give 15 points to B in 60".
=> In a game of 60, A will have to get 60 points while B will have to get only 60-15=45 points
"A can give 20 points to C in 60".
=> In a game of 60, A will have to get 60 points while C will have to get only 60-20=40 points
i.e., when B scores 45 points, C scores 40 points..
$MF#%\text{when B scores 90 points, C scores}\dfrac{40 \times 90}{45}\text{= 80 points}$MF#%
i.e.,in a game of 90, B can give C 90-80=10 points

A : B = 60 : 45.

A : C = 60 : 40.

B = B x A = 45 x 60 = 45 = 90 = 90 : 80. C A C 60 40 40 80

B can give C 10 points in a game of 90.

Answer : Option A

Explanation :

Let x be the length of the course
A beats B by 15 metres and C by 29 metres
=> When A runs x metre, B runs (x-15) metre and C runs (x-29) metre
When B and C run over the course together, B wins by 15 metres
=> when B runs x metre, C run (x-15) metre
$MF#%\begin{align}&\text{=> when B runs 1 metre, C run }\dfrac{x-15}{x}\text{ metre}\\\\ &\text{=> when B runs (x-15) metre, C run }\dfrac{x-15}{x} \times (x-15) = \dfrac{(x-15)^2}{x}\text{ metre}\\\\ &\text{i.e., }(x-29) = \dfrac{(x-15)^2}{x}\\\\ &\Rightarrow x^2 - 29x = x^2 - 30x + 225 \\\\ &\Rightarrow - 29x = - 30x + 225 \\\\ &\Rightarrow x = 225\end{align}$MF#%