A man can row 7.5 km/h in still water. If the river is running at 1.5 km/h, it takes him 50 min to row to a place and back, how far off is the place?

Options

A. 3 km
B. 4 km
C. 1 km
D. 2 km

Answer

To solve the problem step by step, we will break down the information given and use the relevant formulas.

### Step 1: Understand the speeds
- The speed of the man in still water (x) = 7.5 km/h
- The speed of the river (y) = 1.5 km/h

### Step 2: Calculate downstream and upstream speeds
- **Downstream speed** = Speed of man + Speed of river
\[ \text{Downstream speed} = 7.5 + 1.5 = 9 \text{ km/h} \]

- **Upstream speed** = Speed of man - Speed of river
\[ \text{Upstream speed} = 7.5 - 1.5 = 6 \text{ km/h} \]

### Step 3: Set up the time relationship
Let the distance to the place be \( d \) km. The time taken to row downstream to the place and back upstream can be expressed as:
- Time downstream = \( \frac{d}{9} \) hours
- Time upstream = \( \frac{d}{6} \) hours

### Step 4: Total time for the round trip
The total time for the round trip is given as 50 minutes, which we need to convert into hours:
\[ 50 \text{ minutes} = \frac{50}{60} \text{ hours} = \frac{5}{6} \text{ hours} \]

Now, we can set up the equation:
\[ \frac{d}{9} + \frac{d}{6} = \frac{5}{6} \]

### Step 5: Solve for \( d \)
To solve the equation, we need a common denominator for the fractions on the left side. The least common multiple of 9 and 6 is 18. Thus, we can rewrite the equation:
\[ \frac{2d}{18} + \frac{3d}{18} = \frac{5}{6} \]
Combining the fractions:
\[ \frac{5d}{18} = \frac{5}{6} \]

Now, cross-multiply to solve for \( d \):
\[ 5d \cdot 6 = 5 \cdot 18 \]
\[ 30d = 90 \]
\[ d = \frac{90}{30} = 3 \text{ km} \]

### Final Answer
The distance to the place is **3 km**.

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