in a race of 1000 m, a can beat b by 100 m, in a race of 800m, b can beat c by 100m. by how many meters will a beat c in a race of 600 m?

A will beat C by 127.5 m.

Step-by-step reasoning

Let the speeds of A, B, C be a,b,ca,b,ca,b,c m/s.

1. From A vs B (1000 m race)
   • When A runs 1000 m, B runs 900 m.
   • They take the same time, so

1000a=900b⇒ab=1000900=109.\frac{1000}{a}=\frac{900}{b}\Rightarrow \frac{a}{b}=\frac{1000}{900}=\frac{10}{9}.a1000​=b900​⇒ba​=9001000​=910​.

2. From B vs C (800 m race)
   • When B runs 800 m, C runs 700 m.
   • Same time again, so

800b=700c⇒bc=800700=87.\frac{800}{b}=\frac{700}{c}\Rightarrow \frac{b}{c}=\frac{800}{700}=\frac{8}{7}.b800​=c700​⇒cb​=700800​=78​.

3. Find A : C speed ratio

ac=ab×bc=109×87=8063.\frac{a}{c}=\frac{a}{b}\times \frac{b}{c}=\frac{10}{9}\times \frac{8}{7}=\frac{80}{63}.ca​=ba​×cb​=910​×78​=6380​.

4. Use the 600 m race between A and C
   • Time taken by A to run 600 m is t=600at=\frac{600}{a}t=a600​.
   • In the same time, C runs distance

Distance by C=c⋅t=c⋅600a=600⋅ca=600⋅6380=600×6380=7.5×63=472.5 m.\text{Distance by C}=c\cdot t=c\cdot \frac{600}{a} =600\cdot \frac{c}{a} =600\cdot \frac{63}{80} =\frac{600\times 63}{80} =7.5\times 63 =472.5\text{ m}.Distance by C=c⋅t=c⋅a600​=600⋅ac​=600⋅8063​=80600×63​=7.5×63=472.5 m.

 * So A beats C by

600−472.5=127.5 m.600-472.5=127.5\text{ m}.600−472.5=127.5 m.

Answer: A will beat C by 127.5 meters in a 600 m race.