In a 100 m race, A beats B by 10 m and C by 13 m. In a race of 180 m, B will beat C by:

To solve this problem, we need to understand the relative speeds of A, B, and C.

1. In a 100m race, A beats B by 10m. This means when A runs 100m, B runs 90m. So, the speed ratio of A to B is 100:90 or 10:9.

2. Similarly, in a 100m race, A beats C by 13m. This means when A runs 100m, C runs 87m. So, the speed ratio of A to C is 100:87.

3. Now, we need to find the speed ratio of B to C. We know that A:B = 10:9 and A:C = 100:87. We can equate these to find B:C.

   10/9 = 100/87 B/C = (1087)/(9100) = 29/30

4. This means when B runs 30m, C runs 29m.

5. In a 180m race, how far will C run when B runs the whole distance? We can set up a proportion to find out:

   30/29 = 180/x x = (29*180)/30 = 174m

So, in a 180m race, B will beat C by 180m - 174m = 6m.