A car travels a distance of x km at a speed of `5 5/9` m/sec and returns at 5 m/sec to the starting point. If the total time taken by the car is `7 3/5` hours then the value of x (in km) is:

Options

A. 78
B. 72
C. 80
D. 66

Answer

To solve the problem step by step, we will first convert all units to be consistent and then use the formula for time, which is distance divided by speed.

### Step 1: Convert the total time from hours to seconds
The total time taken by the car is given as \(7 \frac{3}{5}\) hours. We can convert this mixed fraction to an improper fraction:

\[ 7 \frac{3}{5} = \frac{7 \times 5 + 3}{5} = \frac{35 + 3}{5} = \frac{38}{5} \text{ hours} \]

Now, convert hours to seconds (1 hour = 3600 seconds):

\[ \text{Total time in seconds} = \frac{38}{5} \times 3600 = \frac{38 \times 3600}{5} = 27360 \text{ seconds} \]

### Step 2: Calculate the time taken for the onward journey
The car travels a distance of \(x\) km at a speed of \(5 \frac{5}{9}\) m/sec. First, convert \(5 \frac{5}{9}\) to an improper fraction:

\[ 5 \frac{5}{9} = \frac{5 \times 9 + 5}{9} = \frac{45 + 5}{9} = \frac{50}{9} \text{ m/sec} \]

Now, we need to convert \(x\) km to meters (1 km = 1000 m):

\[ \text{Distance in meters} = 1000x \text{ m} \]

The time taken for the onward journey \(T_1\) is:

\[ T_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{1000x}{\frac{50}{9}} = 1000x \times \frac{9}{50} = \frac{9000x}{50} = 180x \text{ seconds} \]

### Step 3: Calculate the time taken for the return journey
The return speed is \(5\) m/sec. The time taken for the return journey \(T_2\) is:

\[ T_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{1000x}{5} = 200x \text{ seconds} \]

### Step 4: Set up the equation for total time
The total time taken for the journey is the sum of \(T_1\) and \(T_2\):

\[ T_1 + T_2 = 27360 \]

Substituting the expressions for \(T_1\) and \(T_2\):

\[ 180x + 200x = 27360 \]

Combine like terms:

\[ 380x = 27360 \]

### Step 5: Solve for \(x\)
Now, divide both sides by \(380\):

\[ x = \frac{27360}{380} \]

Calculating the right side:

\[ x = 72 \text{ km} \]

### Final Answer
The value of \(x\) is \(72\) km.

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