A, B and C participated in a race. A covers the same distance in 49 steps, as B covers in 50 steps and C in 51 steps. A takes 10 steps in the same time as B takes 9 steps and C takes 8 steps. Who is the winner of the rece?

Options

A. a)A
B. b)B
C. c)C
D. d)can't be determined

Answer

To solve the problem, we need to determine the speed of A, B, and C based on the information given about their steps and the time taken. Let's break down the solution step by step.

### Step 1: Establish the relationship between steps and distance
We know that:
- A covers the same distance in 49 steps as B does in 50 steps and C does in 51 steps.

Let's denote the distance covered by A in 49 steps as \(d\). Therefore, we can express the distances covered by B and C in terms of \(d\):
- Distance covered by B in 50 steps = \(d\)
- Distance covered by C in 51 steps = \(d\)

From this, we can find the step sizes:
- Step size of A = \( \frac{d}{49} \)
- Step size of B = \( \frac{d}{50} \)
- Step size of C = \( \frac{d}{51} \)

### Step 2: Determine the time taken for steps
Next, we know that:
- A takes 10 steps in the same time that B takes 9 steps and C takes 8 steps.

Let’s denote the time taken for A to take 10 steps as \(t\). Therefore, we can express the time taken by B and C in terms of \(t\):
- Time taken by B for 9 steps = \(t\)
- Time taken by C for 8 steps = \(t\)

### Step 3: Calculate the speed of A, B, and C
Speed is defined as distance covered per unit time. We can calculate the speed of each participant as follows:

1. **Speed of A**:
\[ \text{Speed of A} = \frac{\text{Distance covered in 10 steps}}{\text{Time taken for 10 steps}} = \frac{10 \times \frac{d}{49}}{t} = \frac{10d}{49t} \]

2. **Speed of B**:
\[ \text{Speed of B} = \frac{\text{Distance covered in 9 steps}}{\text{Time taken for 9 steps}} = \frac{9 \times \frac{d}{50}}{t} = \frac{9d}{50t} \]

3. **Speed of C**:
\[ \text{Speed of C} = \frac{\text{Distance covered in 8 steps}}{\text{Time taken for 8 steps}} = \frac{8 \times \frac{d}{51}}{t} = \frac{8d}{51t} \]

### Step 4: Compare the speeds
Now we have the speeds:
- Speed of A = \( \frac{10d}{49t} \)
- Speed of B = \( \frac{9d}{50t} \)
- Speed of C = \( \frac{8d}{51t} \)

To compare these speeds, we can ignore \(d\) and \(t\) since they are common factors. We can compare the ratios:
- A's speed ratio: \( \frac{10}{49} \)
- B's speed ratio: \( \frac{9}{50} \)
- C's speed ratio: \( \frac{8}{51} \)

### Step 5: Find a common denominator and compare
To compare these fractions, we can find a common denominator or cross-multiply:
1. Compare A and B:
\[ \frac{10}{49} \text{ vs } \frac{9}{50} \implies 10 \times 50 = 500 \text{ and } 9 \times 49 = 441 \implies 500 > 441 \implies A > B \]

2. Compare A and C:
\[ \frac{10}{49} \text{ vs } \frac{8}{51} \implies 10 \times 51 = 510 \text{ and } 8 \times 49 = 392 \implies 510 > 392 \implies A > C \]

3. Compare B and C:
\[ \frac{9}{50} \text{ vs } \frac{8}{51} \implies 9 \times 51 = 459 \text{ and } 8 \times 50 = 400 \implies 459 > 400 \implies B > C \]

### Conclusion
From the comparisons, we can conclude that:
- A is the fastest, followed by B, and then C.

Thus, **A is the winner of the race**.