In a race of 200 m B can give a start of 10 m to A and C can give a start of 20 m to B. Find the start that C can give to A, in the same race.

Options

A. 30 m
B. 25m
C. 29m
D. 27m

Answer

To solve the problem step by step, we need to analyze the information provided about the race between A, B, and C.

### Step 1: Understand the given information
- In a race of 200 meters, B gives a start of 10 meters to A. This means when B finishes 200 meters, A has only run 190 meters.
- C can give a start of 20 meters to B. This means when C finishes 200 meters, B has only run 180 meters.

### Step 2: Determine the speeds of A, B, and C
Let's denote the speeds of A, B, and C as \( v_A, v_B, \) and \( v_C \) respectively.

From the information:
- When B runs 200 meters, A runs 190 meters.
\[ \frac{v_A}{v_B} = \frac{190}{200} = \frac{19}{20} \]

- When C runs 200 meters, B runs 180 meters.
\[ \frac{v_B}{v_C} = \frac{180}{200} = \frac{18}{20} = \frac{9}{10} \]

### Step 3: Relate the speeds of A and C
We can express \( v_A \) in terms of \( v_C \) using the ratios we found:
1. From \( \frac{v_B}{v_C} = \frac{9}{10} \), we can express \( v_B \):
\[ v_B = \frac{9}{10} v_C \]

2. Substituting \( v_B \) into \( \frac{v_A}{v_B} = \frac{19}{20} \):
\[ \frac{v_A}{\frac{9}{10} v_C} = \frac{19}{20} \]
\[ v_A = \frac{19}{20} \cdot \frac{9}{10} v_C = \frac{171}{200} v_C \]

### Step 4: Find the distance A runs when C runs 200 meters
Now, we need to find out how far A runs when C runs 200 meters. Using the ratios of their speeds:
\[ \frac{A's \ distance}{C's \ distance} = \frac{v_A}{v_C} \]
Let \( d_A \) be the distance A runs when C runs 200 meters:
\[ \frac{d_A}{200} = \frac{171}{200} \]
\[ d_A = 200 \cdot \frac{171}{200} = 171 \text{ meters} \]

### Step 5: Calculate the start that C can give to A
Since C runs 200 meters and A runs 171 meters, the start that C can give to A is:
\[ \text{Start} = 200 - 171 = 29 \text{ meters} \]

### Conclusion
C can give a start of **29 meters** to A in the same race.

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