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 + 3) is :

Options

A. 78
B. 80
C. 75
D. 72

Answer

To solve the problem step by step, we will first convert the speeds from meters per second to kilometers per hour, then calculate the time taken for each leg of the journey, and finally solve for \( x \).

### Step 1: Convert Speeds to Kilometers per Hour
The speed of the car while going to point B is given as \( 5 \frac{5}{9} \) m/sec. To convert this to km/hr, we use the conversion factor \( \frac{18}{5} \).

\[ \text{Speed} = 5 \frac{5}{9} \text{ m/sec} = \left(5 + \frac{5}{9}\right) \text{ m/sec} = \frac{45}{9} + \frac{5}{9} = \frac{50}{9} \text{ m/sec} \]

Now converting to km/hr:

\[ \text{Speed in km/hr} = \frac{50}{9} \times \frac{18}{5} = \frac{50 \times 18}{9 \times 5} = \frac{180}{9} = 20 \text{ km/hr} \]

The speed while returning from point B to A is \( 5 \) m/sec.

\[ \text{Speed in km/hr} = 5 \times \frac{18}{5} = 18 \text{ km/hr} \]

### Step 2: Calculate Time Taken for Each Leg of the Journey
Let \( x \) be the distance in kilometers.

- Time taken to travel to point B:

\[ T_1 = \frac{x}{20} \text{ hours} \]

- Time taken to return to point A:

\[ T_2 = \frac{x}{18} \text{ hours} \]

### Step 3: Total Time Taken
The total time taken for the journey is given as \( 7 \frac{3}{5} \) hours, which can be converted to an improper fraction:

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

Setting up the equation for total time:

\[ T_1 + T_2 = \frac{x}{20} + \frac{x}{18} = \frac{38}{5} \]

### Step 4: Finding a Common Denominator
The least common multiple of \( 20 \) and \( 18 \) is \( 180 \).

\[ \frac{x}{20} = \frac{9x}{180}, \quad \frac{x}{18} = \frac{10x}{180} \]

Thus, we can rewrite the equation:

\[ \frac{9x + 10x}{180} = \frac{38}{5} \]

This simplifies to:

\[ \frac{19x}{180} = \frac{38}{5} \]

### Step 5: Cross-Multiplying to Solve for \( x \)
Cross-multiplying gives:

\[ 19x \cdot 5 = 38 \cdot 180 \]

Calculating \( 38 \cdot 180 \):

\[ 38 \cdot 180 = 6840 \]

So we have:

\[ 95x = 6840 \]

Dividing both sides by \( 95 \):

\[ x = \frac{6840}{95} = 72 \text{ km} \]

### Step 6: Finding \( x + 3 \)
Finally, we need to find \( x + 3 \):

\[ x + 3 = 72 + 3 = 75 \]

### Final Answer
Thus, the value of \( x + 3 \) is \( \boxed{75} \).

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