In a flat race, A beats B by 15 metres and C by 29 metres. When B and C run over the course together, B wins by 15 metres. Find the length of the course—

Options

A. 220 m
B. 325 m
C. 225 m
D. 250 m

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Define the Length of the Course
Let the length of the course be \( D \) meters.

### Step 2: Determine Distances Run by B and C
- A beats B by 15 meters, which means when A runs \( D \) meters, B runs \( D - 15 \) meters.
- A beats C by 29 meters, which means when A runs \( D \) meters, C runs \( D - 29 \) meters.

### Step 3: Analyze the Race Between B and C
When B and C run together, B wins by 15 meters. This means when B finishes the race (runs \( D \) meters), C has run \( D - 15 \) meters.

### Step 4: Set Up the Relationship Between B and C's Distances
From the earlier analysis:
- When B runs \( D \) meters, C runs a distance proportional to the distances A, B, and C have run.
- The ratio of distances run by C when B runs \( D \) meters can be expressed as:
\[ \text{Distance run by C} = \frac{D - 29}{D - 15} \times D \]

### Step 5: Set Up the Equation
Since we know that when B runs \( D \) meters, C runs \( D - 15 \) meters, we can equate the two expressions:
\[ D - 15 = \frac{D - 29}{D - 15} \times D \]

### Step 6: Cross Multiply to Solve for D
Cross-multiplying gives:
\[ (D - 15)(D - 15) = D(D - 29) \]
Expanding both sides:
\[ D^2 - 30D + 225 = D^2 - 29D \]

### Step 7: Simplify the Equation
Now, we can simplify the equation by canceling \( D^2 \) from both sides:
\[ -30D + 225 = -29D \]
Rearranging gives:
\[ -30D + 29D + 225 = 0 \]
\[ -D + 225 = 0 \]
\[ D = 225 \]

### Conclusion
The length of the course is \( D = 225 \) meters.

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