A person rows a distance of `3 ( 3)/(4)` km upstream in `1(1)/(2)` hours and a distance of 13 km downstream in 2 hours. How much time ( in hours) will the person take to row a distance of 90 km in still water ?

Options

A. 15
B. 24
C. 18
D. 20

Answer

To solve the problem step by step, we will first define the variables and then use the given information to set up equations based on the upstream and downstream rowing.

### Step 1: Define Variables
Let:
- \( x \) = speed of the person in still water (in km/h)
- \( y \) = speed of the river flow (in km/h)

### Step 2: Convert Mixed Numbers to Improper Fractions
The distances and times given in the problem need to be converted for easier calculations:
- Upstream distance: \( 3 \frac{3}{4} \) km = \( \frac{15}{4} \) km
- Upstream time: \( 1 \frac{1}{2} \) hours = \( \frac{3}{2} \) hours
- Downstream distance: 13 km
- Downstream time: 2 hours

### Step 3: Set Up Equations
Using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \):

**For Upstream:**
The effective speed upstream is \( x - y \).
\[ \frac{15}{4} = (x - y) \times \frac{3}{2} \]
Rearranging gives:
\[ x - y = \frac{15/4}{3/2} = \frac{15}{4} \times \frac{2}{3} = \frac{30}{12} = \frac{5}{2} \]
So, we have our first equation:
\[ x - y = 2.5 \quad \text{(Equation 1)} \]

**For Downstream:**
The effective speed downstream is \( x + y \).
\[ 13 = (x + y) \times 2 \]
Rearranging gives:
\[ x + y = \frac{13}{2} = 6.5 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have a system of equations:
1. \( x - y = 2.5 \)
2. \( x + y = 6.5 \)

Adding both equations:
\[ (x - y) + (x + y) = 2.5 + 6.5 \]
This simplifies to:
\[ 2x = 9 \implies x = 4.5 \]

Now substitute \( x \) back into Equation 1:
\[ 4.5 - y = 2.5 \implies y = 4.5 - 2.5 = 2 \]

### Step 5: Calculate Time to Row 90 km in Still Water
Now that we have \( x = 4.5 \) km/h (the speed in still water), we can find the time taken to row 90 km:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{90}{4.5} = 20 \text{ hours} \]

### Final Answer
The person will take **20 hours** to row a distance of 90 km in still water.

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