Vinay and Varsha run a race with their speeds in the ratio of 5:3 they prefer to run on a circular track of circumfrence 1.5 km.what is the distance covered by Vinay when he passes Varsha for the seventh time?

Options

A. 25.25 km
B. 26.25 km
C. 132 m
D. none of these

Answer

To solve the problem step by step, we will analyze the information given and calculate the distance covered by Vinay when he passes Varsha for the seventh time.

### Step 1: Understand the speed ratio
Vinay and Varsha's speeds are in the ratio of 5:3.
Let Vinay's speed be \( 5x \) km/h and Varsha's speed be \( 3x \) km/h.

**Hint:** The ratio of speeds helps us express their speeds in terms of a common variable.

### Step 2: Calculate the relative speed
The relative speed between Vinay and Varsha is the difference in their speeds since they are running in the same direction.
\[ \text{Relative Speed} = 5x - 3x = 2x \text{ km/h} \]

**Hint:** When two objects move in the same direction, their relative speed is the difference of their speeds.

### Step 3: Determine the time taken to meet
They meet when Vinay has completed one extra lap compared to Varsha. The circumference of the track is 1.5 km.

To find the time taken for Vinay to pass Varsha for the first time, we need to calculate how long it takes for Vinay to gain one full lap (1.5 km) over Varsha:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{1.5 \text{ km}}{2x \text{ km/h}} = \frac{0.75}{x} \text{ hours} \]

**Hint:** The time taken to meet can be calculated by dividing the distance gained (one lap) by their relative speed.

### Step 4: Calculate the distance covered by Vinay in that time
In the time it takes for Vinay to pass Varsha for the first time, the distance covered by Vinay is:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 5x \times \frac{0.75}{x} = 3.75 \text{ km} \]

**Hint:** To find the distance covered, multiply the speed by the time.

### Step 5: Calculate the distance covered by Vinay when he passes Varsha for the seventh time
Since Vinay passes Varsha every time he gains a full lap (1.5 km), by the seventh time he passes Varsha, he will have gained 7 laps:
\[ \text{Total Distance} = 7 \times 1.5 \text{ km} = 10.5 \text{ km} \]

**Hint:** To find the total distance when passing multiple times, multiply the number of passes by the circumference of the track.

### Step 6: Calculate the distance covered by Vinay
Now, we can find the total distance covered by Vinay when he passes Varsha for the seventh time:
\[ \text{Distance covered by Vinay} = \text{Distance gained} + \text{Distance covered in laps} = 10.5 \text{ km} \]

**Final Answer:** The distance covered by Vinay when he passes Varsha for the seventh time is **10.5 km**.