In a race of 600 m, A can beat B by 60 m and in a race of 500 m, B can beat C by 50 m. By how many metres will A beat C in a race of 400 m ?

Options

A. 76 m
B. `(76) 1/3` m
C. 77 m
D. `(77) 1/3` m

Answer

To solve the problem step by step, we will analyze the information given and calculate the distances A, B, and C can cover in the races.

### Step 1: Determine the speed ratio of A and B
In a race of 600 m, A beats B by 60 m. This means when A finishes 600 m, B has only run 540 m (600 - 60).

- **Speed Ratio of A to B**:
\[ \text{Speed of A} : \text{Speed of B} = 600 : 540 = 10 : 9 \]

### Step 2: Determine the speed ratio of B and C
In a race of 500 m, B beats C by 50 m. This means when B finishes 500 m, C has only run 450 m (500 - 50).

- **Speed Ratio of B to C**:
\[ \text{Speed of B} : \text{Speed of C} = 500 : 450 = 10 : 9 \]

### Step 3: Combine the ratios to find the speed ratio of A, B, and C
From the above steps, we have:
- A : B = 10 : 9
- B : C = 10 : 9

To combine these ratios, we express B in terms of a common factor:
- Let B = 9x (from A : B)
- Then A = 10x and C = 9y (from B : C)

Since B is the same in both ratios, we can equate:
\[ 9y = 9x \implies y = x \]

Thus, we can express the speeds as:
- A = 10x
- B = 9x
- C = 9x

### Step 4: Calculate the distance A beats C in a 400 m race
Now we need to find out how far A beats C when A runs 400 m.

Using the ratio:
- A runs 400 m, we need to find out how far C runs in the same time.
- The ratio of A to C is 10 : 9.

Let’s find out how far C runs when A runs 400 m:
\[ \text{Distance covered by C} = \frac{9}{10} \times 400 = 360 \text{ m} \]

### Step 5: Calculate the distance by which A beats C
To find out by how many meters A beats C:
\[ \text{Distance A beats C} = 400 - 360 = 40 \text{ m} \]

### Final Answer
A beats C by **40 meters** in a race of 400 meters.

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