Let's analyze the problem step-by-step.
Problem Restatement
- In a 1000 m race, A beats B by 100 m.
- In an 800 m race, B beats C by 100 m.
- We need to find: In a race of 1000 m, by how many meters will A beat C?
Step 1: Understand the given information
A beats B by 100 m in 1000 m race
- When A runs 1000 m, B runs only 900 m (because A beats B by 100 m).
- So, the ratio of speeds of A to B is:
Speed of ASpeed of B=1000900=109\frac{\text{Speed of A}}{\text{Speed of B}}=\frac{1000}{900}=\frac{10}{9}Speed of BSpeed of A=9001000=910
Step 2: B beats C by 100 m in 800 m race
- When B runs 800 m, C runs only 700 m (because B beats C by 100 m).
- So, the ratio of speeds of B to C is:
Speed of BSpeed of C=800700=87\frac{\text{Speed of B}}{\text{Speed of C}}=\frac{800}{700}=\frac{8}{7}Speed of CSpeed of B=700800=78
Step 3: Find the ratio of speeds of A to C
Speed of ASpeed of C=Speed of ASpeed of B×Speed of BSpeed of C=109×87=8063\frac{\text{Speed of A}}{\text{Speed of C}}=\frac{\text{Speed of A}}{\text{Speed of B}}\times \frac{\text{Speed of B}}{\text{Speed of C}}=\frac{10}{9}\times \frac{8}{7}=\frac{80}{63}Speed of CSpeed of A=Speed of BSpeed of A×Speed of CSpeed of B=910×78=6380
Step 4: Calculate how far C runs when A runs 1000 m
- Let the distance C runs when A runs 1000 m be xxx.
- Since speed ratio is 8063\frac{80}{63}6380, time taken by A to run 1000 m is:
t=DistanceSpeed=1000Speed of At=\frac{\text{Distance}}{\text{Speed}}=\frac{1000}{\text{Speed of A}}t=SpeedDistance=Speed of A1000
- Distance covered by C in the same time is:
x=Speed of C×t=Speed of C×1000Speed of A=1000×Speed of CSpeed of A=1000×6380=787.5 mx=\text{Speed of C}\times t=\text{Speed of C}\times \frac{1000}{\text{Speed of A}}=1000\times \frac{\text{Speed of C}}{\text{Speed of A}}=1000\times \frac{63}{80}=787.5\text{ m}x=Speed of C×t=Speed of C×Speed of A1000=1000×Speed of ASpeed of C=1000×8063=787.5 m
Step 5: Find by how many meters A beats C
Distance by which A beats C=1000−787.5=212.5 meters\text{Distance by which A beats C}=1000-787.5=212.5\text{ meters}Distance by which A beats C=1000−787.5=212.5 meters
Final Answer:
A will beat C by 212.5 meters in a 1000 m race.
