In a 10 km race, A, B and C each run at uniform speed, get the first, second and third position respectively. If A beats B by 1 km and B beats C by 1 km, then, by how much distance A beat C?

1.9 km

The correct answer is Option (2) → 1.9 km

Let the speeds of A, B, C be $v_A, v_B, v_C$ respectively.

Race distance = 10 km

A beats B by 1 km → when A runs 10 km, B runs 9 km:

$\frac{v_B}{v_A} = \frac{9}{10}$

B beats C by 1 km → when B runs 10 km, C runs 9 km:

$\frac{v_C}{v_B} = \frac{9}{10} \Rightarrow v_C = \frac{9}{10} v_B = \frac{9}{10} \cdot \frac{9}{10} v_A = \frac{81}{100} v_A$

Distance C runs when A runs 10 km:

$d_C = v_C \cdot \frac{10}{v_A} = \frac{81}{100} v_A \cdot \frac{10}{v_A} = 8.1 \text{ km}$

A beats C by: $10 - 8.1 = 1.9$ km