At a game of billiards, A can give B 6 points in 50 and he can give C 13 in 65. In a game of 55, number of points B can give C is:

The speed of A and B are in the ratio 11 : 8.
Let, speeds be 11s and 8s (in m/sec).
Let, race be of P meter.
Then, time taken by A to run P meter is same as that of B to run (P - 120) meter.
? P / 11 = (P - 120) / 8
? P / 11 = (P - 120 / 8
? 8P = 11 x (P - 120 )
? 8P = 11P - 120 x 11
? 11P - 8P = 120 x 11
? 3P = 11 × 120
? P = 440.

15 meter is covered by B in 5 sec
? 300 meter is covered by B in 5 × 300/15 = 100 seconds
? A takes 100 - 5 = 95 seconds.

Raju runs on Monday and Friday = 1250 × 2 = 2500 m
On Tuesday, Wednesday, Thursday and Saturday,
Raju runs 1500 × 4 = 6000 m
In 1 week Raju runs 6000 + 2500 = 8500 m
In 3 weeks Raju runs 3 × 8500 = 25500 m = 25.5 K

L.C.M. of 252, 308 and 198 = 2772 seconds
= 46 × 60 + 12 = 46 min 12 seconds.
.

race remains to be run = 5 - ⁵?₄ = ¹⁵?₄ .

If A runs 600 m, then B runs 570 m.
If A runs 400 m, B then runs (570/600) x 400 m = 380 m
When B runs 500 m, then C runs 475 m,
When B runs 380 m, then C runs (475/500) x 380 = 361 m
? A beats C by (400 - 361) = 39 m

A is 1³/₄ faster than B.
Then, ratio of the rates of A and B = 7 : 4
3 m are gained in a race of 7 m .
30 m are gained in a race of (7/3) x 30 = 70 m

To reach the winning post, A has to cover (250 - 140)m = 110 m
From the the ratio of 1 : 2 ,
When A covers 1m, then B covers 2 m.
When A covers 110 m, then B covers
( 2 x 110)m = 220 m
? A wins by (250 - 220) = 30 m

According to the formula,
Time taken by the winner/Distance covered by the loser = Difference of winning time/ Difference of winning distance
? Time taken by P//(1000 - 72) = 12/72
? Time taken by P = (12/72) x 928
= 464/3 = 154²/₃ s

Clearly, X beats Y by 8 s.
Distance covered by Y in 600 s = 1000 m
(? 10 min = 600 s)
Distance covered by Y 80 s. = (1000/600) x 80 = 400/3 m = 133¹/₃ m