The speed of a boat downstream is 2.5 times its speed upstream. If the total time taken by the boat for going 15 km downstream and the same distance upstream is `4 (2)/(3)` hours, then what is the speed (in km/h) of the boat downstream?

Options

A. A)`11 (1)/(2)`
B. B)`11 (1)/(4)`
C. C)`10 (5)/(8)`
D. D)`10`

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Define Variables
Let the speed of the boat upstream be \( x \) km/h. Then, the speed of the boat downstream will be \( 2.5x \) km/h, as given in the problem.

**Hint:** Define variables for unknowns to simplify calculations.

### Step 2: Determine Time Taken for Each Journey
The distance for both upstream and downstream is 15 km. The time taken to travel downstream is given by:
\[ \text{Time downstream} = \frac{\text{Distance}}{\text{Speed}} = \frac{15}{2.5x} \]
The time taken to travel upstream is given by:
\[ \text{Time upstream} = \frac{\text{Distance}}{\text{Speed}} = \frac{15}{x} \]

**Hint:** Use the formula for time, which is distance divided by speed.

### Step 3: Set Up the Total Time Equation
According to the problem, the total time taken for both journeys is \( 4 \frac{2}{3} \) hours, which can be converted to an improper fraction:
\[ 4 \frac{2}{3} = \frac{14}{3} \text{ hours} \]
Thus, we can set up the equation:
\[ \frac{15}{2.5x} + \frac{15}{x} = \frac{14}{3} \]

**Hint:** Combine the times into a single equation to solve for the unknown speed.

### Step 4: Simplify the Equation
To simplify the equation, find a common denominator. The common denominator for \( 2.5x \) and \( x \) is \( 2.5x \):
\[ \frac{15x}{2.5x^2} + \frac{15 \cdot 2.5}{2.5x} = \frac{14}{3} \]
This simplifies to:
\[ \frac{15 + 37.5}{2.5x} = \frac{14}{3} \]
\[ \frac{52.5}{2.5x} = \frac{14}{3} \]

**Hint:** Ensure all fractions are simplified and combined correctly.

### Step 5: Cross Multiply to Solve for \( x \)
Cross-multiplying gives:
\[ 52.5 \cdot 3 = 14 \cdot 2.5x \]
\[ 157.5 = 35x \]
Now, solve for \( x \):
\[ x = \frac{157.5}{35} = 4.5 \text{ km/h} \]

**Hint:** Use cross-multiplication to eliminate fractions and isolate the variable.

### Step 6: Calculate Downstream Speed
Now that we have \( x \), we can find the downstream speed:
\[ \text{Downstream speed} = 2.5x = 2.5 \cdot 4.5 = 11.25 \text{ km/h} \]

**Hint:** Substitute back into the original relationship to find the desired speed.

### Conclusion
The speed of the boat downstream is \( 11.25 \) km/h.

### Final Answer
**The speed of the boat downstream is \( 11 \frac{1}{4} \) km/h or \( 11.25 \) km/h.**