A steamer and a motor boat cover the stretch between a seaport and a dock in 7 hours 20 minutes and 10 hours 40 minutes, respectively. What is the ratio of their speeds?

Options

A. `15:11`
B. `3:2`
C. `16:11`
D. `8:5`

Answer

To find the ratio of the speeds of a steamer and a motorboat that cover the same distance in different times, we can follow these steps:

### Step 1: Convert the time taken by each vessel into minutes.
- **Steamer time:** 7 hours 20 minutes
- Convert hours to minutes: \(7 \times 60 = 420\) minutes
- Add the additional 20 minutes: \(420 + 20 = 440\) minutes

- **Motorboat time:** 10 hours 40 minutes
- Convert hours to minutes: \(10 \times 60 = 600\) minutes
- Add the additional 40 minutes: \(600 + 40 = 640\) minutes

### Step 2: Determine the relationship between speed and time.
Speed is inversely proportional to time. This means that if one vessel takes less time, it moves faster, and vice versa.

### Step 3: Set up the ratio of speeds.
Let the speed of the steamer be \(S_s\) and the speed of the motorboat be \(S_m\). The ratio of their speeds can be expressed as:
\[ \frac{S_s}{S_m} = \frac{T_m}{T_s} \]
where \(T_s\) is the time taken by the steamer and \(T_m\) is the time taken by the motorboat.

### Step 4: Substitute the values of time into the ratio.
Substituting the times we calculated:
\[ \frac{S_s}{S_m} = \frac{640}{440} \]

### Step 5: Simplify the ratio.
To simplify \(\frac{640}{440}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 80:
\[ \frac{640 \div 80}{440 \div 80} = \frac{8}{5.5} = \frac{16}{11} \]

### Final Ratio:
Thus, the ratio of the speeds of the steamer to the motorboat is:
\[ \frac{S_s}{S_m} = 16 : 11 \]