A man goes to his office at `2/3` of the speed at which he returns from the office. If his average speed is during the whole journey is 24 km/hr, at what speed the man goes to office?

Options

A. 20 km/hr
B. 21 km/hr
C. 22 km/hr
D. 23 km/hr

Answer

To solve the problem step by step, we will denote the speed at which the man returns from the office as \( x \) km/hr. The speed at which he goes to the office will then be \( \frac{2}{3}x \) km/hr.

### Step 1: Define the speeds
Let:
- Speed going to the office = \( s_1 = \frac{2}{3}x \)
- Speed returning from the office = \( s_2 = x \)

### Step 2: Use the average speed formula
The average speed for the entire journey can be calculated using the formula:
\[ \text{Average Speed} = \frac{2 \cdot s_1 \cdot s_2}{s_1 + s_2} \]
Given that the average speed is 24 km/hr, we can set up the equation:
\[ \frac{2 \cdot \left(\frac{2}{3}x\right) \cdot x}{\left(\frac{2}{3}x + x\right)} = 24 \]

### Step 3: Simplify the equation
First, simplify the denominator:
\[ s_1 + s_2 = \frac{2}{3}x + x = \frac{2}{3}x + \frac{3}{3}x = \frac{5}{3}x \]
Now substitute this back into the average speed formula:
\[ \frac{2 \cdot \left(\frac{2}{3}x\right) \cdot x}{\frac{5}{3}x} = 24 \]

### Step 4: Solve for \( x \)
Now, simplify the left-hand side:
\[ \frac{\frac{4}{3}x^2}{\frac{5}{3}x} = \frac{4x^2}{5x} = \frac{4x}{5} \]
Setting this equal to 24:
\[ \frac{4x}{5} = 24 \]
Now, multiply both sides by 5:
\[ 4x = 120 \]
Now, divide by 4:
\[ x = 30 \text{ km/hr} \]

### Step 5: Find the speed going to the office
Now that we have \( x \), we can find the speed at which the man goes to the office:
\[ s_1 = \frac{2}{3}x = \frac{2}{3} \cdot 30 = 20 \text{ km/hr} \]

### Final Answer
The speed at which the man goes to the office is **20 km/hr**.

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