A boat can travel 6.4 km downstream in 16 minutes and 9.9 km upstream in 33 minutes. What is the total time taken by the boat to travel 48 km upstream and the same distance downstream together?

Options

A. 4 hr 40 min
B. 4 hr
C. 5 hr
D. 5 hr 20 min

Answer

To solve the problem, we need to calculate the speeds of the boat downstream and upstream, and then use those speeds to find the total time taken for the boat to travel 48 km upstream and 48 km downstream.

### Step 1: Calculate Downstream Speed
The boat travels 6.4 km downstream in 16 minutes.
- Convert 16 minutes to hours:
\[ 16 \text{ minutes} = \frac{16}{60} \text{ hours} = \frac{4}{15} \text{ hours} \]
- Calculate the downstream speed:
\[ \text{Downstream speed} = \frac{\text{Distance}}{\text{Time}} = \frac{6.4 \text{ km}}{\frac{4}{15} \text{ hours}} = 6.4 \times \frac{15}{4} = 24 \text{ km/h} \]

### Step 2: Calculate Upstream Speed
The boat travels 9.9 km upstream in 33 minutes.
- Convert 33 minutes to hours:
\[ 33 \text{ minutes} = \frac{33}{60} \text{ hours} = \frac{11}{20} \text{ hours} \]
- Calculate the upstream speed:
\[ \text{Upstream speed} = \frac{\text{Distance}}{\text{Time}} = \frac{9.9 \text{ km}}{\frac{11}{20} \text{ hours}} = 9.9 \times \frac{20}{11} = 18 \text{ km/h} \]

### Step 3: Calculate Time for 48 km Upstream
Using the upstream speed calculated:
- Time taken to travel 48 km upstream:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{48 \text{ km}}{18 \text{ km/h}} = \frac{48}{18} \text{ hours} = \frac{8}{3} \text{ hours} \approx 2.67 \text{ hours} \]

### Step 4: Calculate Time for 48 km Downstream
Using the downstream speed calculated:
- Time taken to travel 48 km downstream:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{48 \text{ km}}{24 \text{ km/h}} = 2 \text{ hours} \]

### Step 5: Calculate Total Time Taken
Now, add the time taken for both upstream and downstream:
\[ \text{Total Time} = \text{Time Upstream} + \text{Time Downstream} = \frac{8}{3} \text{ hours} + 2 \text{ hours} = \frac{8}{3} + \frac{6}{3} = \frac{14}{3} \text{ hours} \approx 4.67 \text{ hours} \]

### Final Answer
The total time taken by the boat to travel 48 km upstream and the same distance downstream together is approximately \( \frac{14}{3} \) hours or 4 hours and 40 minutes.

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