Two cyclists start simultaneously from two points A and B, their destinations being B and A, respectively. After crossing each other, they precisely take 2 hours 33 minutes 36 seconds and 1 hour 26 minutes 24 seconds respectively to reach their destinations. What is the ratio of the speed of the first to that of the second cyclist?

Options

A. ` 2:3`
B. `3:2`
C. `4:3`
D. `3:4`

Answer

To solve the problem of finding the ratio of the speed of the first cyclist to that of the second cyclist, we can follow these steps:

### Step 1: Convert the time taken by each cyclist to reach their destinations into hours.

1. **For the first cyclist (from A to B)**:
- Time taken = 2 hours 33 minutes 36 seconds
- Convert this time to hours:
- 2 hours = 2 hours
- 33 minutes = 33/60 hours = 0.55 hours
- 36 seconds = 36/3600 hours = 0.01 hours
- Total time = 2 + 0.55 + 0.01 = 2.56 hours

2. **For the second cyclist (from B to A)**:
- Time taken = 1 hour 26 minutes 24 seconds
- Convert this time to hours:
- 1 hour = 1 hour
- 26 minutes = 26/60 hours = 0.4333 hours
- 24 seconds = 24/3600 hours = 0.00667 hours
- Total time = 1 + 0.4333 + 0.00667 = 1.44 hours

### Step 2: Use the relationship between speed and time.

The speed of each cyclist is inversely proportional to the time they take to reach their destination. Therefore, we can express the ratio of their speeds as follows:

\[ \frac{S_A}{S_B} = \frac{T_B}{T_A} \]

Where:
- \( S_A \) = speed of the first cyclist
- \( S_B \) = speed of the second cyclist
- \( T_A \) = time taken by the first cyclist = 2.56 hours
- \( T_B \) = time taken by the second cyclist = 1.44 hours

### Step 3: Calculate the ratio of the speeds.

Substituting the values we found:

\[ \frac{S_A}{S_B} = \frac{1.44}{2.56} \]

### Step 4: Simplify the ratio.

To simplify \( \frac{1.44}{2.56} \):

1. Divide both the numerator and the denominator by 0.16 (to make calculations easier):
- \( 1.44 \div 0.16 = 9 \)
- \( 2.56 \div 0.16 = 16 \)

Thus, we have:

\[ \frac{S_A}{S_B} = \frac{9}{16} \]

### Step 5: Express the ratio in simplest form.

The ratio of the speed of the first cyclist to that of the second cyclist is:

\[ \frac{9}{16} \]

### Final Answer:

The ratio of the speed of the first cyclist to that of the second cyclist is \( 9:16 \).

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