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 race?

Options

A. A
B. B
C. C
D. can't be determined

Answer

To determine the winner of the race among A, B, and C, we need to analyze the information given step by step.

### Step 1: Understand the distance covered by each participant
- A covers the distance in 49 steps.
- B covers the same distance in 50 steps.
- C covers the same distance in 51 steps.

**Hint:** Compare the number of steps taken by each participant to cover the same distance.

### Step 2: Determine the time taken for each step
We know that:
- A takes 10 steps in the same time B takes 9 steps.
- B takes 9 steps in the same time C takes 8 steps.

To find the time for one step for each participant, we can set up the following relationships:
- Let the time taken for A to take 10 steps be \( t_A \).
- Then, the time taken for B to take 9 steps is also \( t_A \), so the time for B to take 1 step is \( \frac{t_A}{9} \).
- Similarly, let the time taken for B to take 9 steps be \( t_B \), which means the time for C to take 8 steps is also \( t_B \), so the time for C to take 1 step is \( \frac{t_B}{8} \).

**Hint:** Establish a relationship between the time taken for each participant to complete their steps.

### Step 3: Calculate the time taken by each participant to finish the race
Now we can calculate the total time taken by each participant to finish the race based on the number of steps they take and the time per step.

1. **For A:**
- Steps taken = 49
- Time per step = \( \frac{t_A}{10} \)
- Total time = \( 49 \times \frac{t_A}{10} = \frac{49t_A}{10} \)

2. **For B:**
- Steps taken = 50
- Time per step = \( \frac{t_A}{9} \)
- Total time = \( 50 \times \frac{t_A}{9} = \frac{50t_A}{9} \)

3. **For C:**
- Steps taken = 51
- Time per step = \( \frac{t_B}{8} \) (where \( t_B = \frac{t_A}{9} \))
- Total time = \( 51 \times \frac{t_B}{8} = 51 \times \frac{t_A}{72} = \frac{51t_A}{72} \)

**Hint:** Use the relationships established to calculate the total time taken by each participant.

### Step 4: Compare the total times
Now we need to compare the total times calculated for A, B, and C.

1. **Total time for A:** \( \frac{49t_A}{10} \)
2. **Total time for B:** \( \frac{50t_A}{9} \)
3. **Total time for C:** \( \frac{51t_A}{72} \)

To compare these fractions, we can convert them to a common denominator or simply evaluate which is smaller.

**Hint:** Simplifying or converting to a common denominator will help in comparing the times directly.

### Step 5: Determine the winner
After comparing the total times, we find that:
- A has the least time taken, followed by B, and then C.

Thus, A is the winner of the race.

**Final Answer:** A is the winner of the race.