A car travel between city A to B with the speed of 60 km/hr but in returning car lost its speed by x km/hr and take`1(1)/(3)` hr more time. If total distance between two cities is 240 km, then find the average speed of car in whole journey?

Options

A. A)`(370)/(7)` km/hr
B. B)`(380)/(7)` km/hr
C. C)`(390)/(7)` km/hr
D. D)`(360)/(7)` km/hr

Answer

To solve the problem step by step, we will break down the information given and apply the relevant formulas.

### Step 1: Understand the Problem
The car travels from City A to City B at a speed of 60 km/hr. On the return journey, it loses speed by \( x \) km/hr and takes \( 1 \frac{1}{3} \) hours more than the time taken to travel from A to B.

### Step 2: Calculate the Distance
The total distance between the two cities is given as 240 km.

### Step 3: Calculate Time for Journey A to B
The time taken to travel from A to B can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
For the journey from A to B:
\[ \text{Time}_{A \to B} = \frac{240 \text{ km}}{60 \text{ km/hr}} = 4 \text{ hours} \]

### Step 4: Calculate Speed for Journey B to A
The speed for the return journey from B to A is:
\[ \text{Speed}_{B \to A} = 60 - x \text{ km/hr} \]

### Step 5: Calculate Time for Journey B to A
The time taken for the return journey is:
\[ \text{Time}_{B \to A} = \frac{240 \text{ km}}{60 - x \text{ km/hr}} \]

### Step 6: Set Up the Equation for Time Difference
According to the problem, the return journey takes \( 1 \frac{1}{3} \) hours more than the journey from A to B. Convert \( 1 \frac{1}{3} \) hours to an improper fraction:
\[ 1 \frac{1}{3} = \frac{4}{3} \text{ hours} \]
Thus, we can set up the equation:
\[ \frac{240}{60 - x} = 4 + \frac{4}{3} \]
Convert \( 4 \) to a fraction:
\[ 4 = \frac{12}{3} \]
So,
\[ \frac{240}{60 - x} = \frac{12}{3} + \frac{4}{3} = \frac{16}{3} \]

### Step 7: Cross Multiply to Solve for \( x \)
Cross multiply to eliminate the fraction:
\[ 240 \cdot 3 = 16 \cdot (60 - x) \]
This simplifies to:
\[ 720 = 960 - 16x \]
Rearranging gives:
\[ 16x = 960 - 720 \]
\[ 16x = 240 \]
\[ x = \frac{240}{16} = 15 \]

### Step 8: Calculate the Speed for Journey B to A
Now that we have \( x \):
\[ \text{Speed}_{B \to A} = 60 - 15 = 45 \text{ km/hr} \]

### Step 9: Calculate the Average Speed for the Whole Journey
The average speed for the entire journey can be calculated using the formula:
\[ \text{Average Speed} = \frac{2 \cdot S_1 \cdot S_2}{S_1 + S_2} \]
Substituting \( S_1 = 60 \) km/hr and \( S_2 = 45 \) km/hr:
\[ \text{Average Speed} = \frac{2 \cdot 60 \cdot 45}{60 + 45} = \frac{5400}{105} \]
This simplifies to:
\[ \text{Average Speed} = \frac{360}{7} \text{ km/hr} \]

### Final Answer
The average speed of the car for the whole journey is \( \frac{360}{7} \) km/hr.

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