There are two concentric circular tracks of radii 50 m and 53 m. Karan takes as much time to complete 6 rounds of the inner track in a clockwise direction as Arjun takes to complete 8 rounds of the outer track in a counterclockwise direction. Approximately by what percent is Karan's speed lower than Arjun's

To find approximately by what percent Karan's speed is lower than Arjun's, let's break down the problem step by step:

### Step 1: Determine the lengths of the tracks.
- **Inner Track Radius (r1):** 50 m
- **Outer Track Radius (r2):** 53 m

The circumference \( C \) of a circle is given by the formula:
\[
C = 2\pi r
\]
- **Inner Track Circumference \( C_1 \):**
\[
C_1 = 2\pi \times 50 = 100\pi \text{ meters}
\]
- **Outer Track Circumference \( C_2 \):**
\[
C_2 = 2\pi \times 53 = 106\pi \text{ meters}
\]

### Step 2: Calculate the distances covered by Karan and Arjun.
- **Karan completes 6 rounds of the inner track:**
\[
\text{Distance covered by Karan} = 6 \times C_1 = 6 \times 100\pi = 600\pi \text{ meters}
\]
- **Arjun completes 8 rounds of the outer track:**
\[
\text{Distance covered by Arjun} = 8 \times C_2 = 8 \times 106\pi = 848\pi \text{ meters}
\]

### Step 3: Use the fact that they take the same time to complete their respective rounds.
Let the time taken by both Karan and Arjun be \( t \).

- **Speed of Karan \( v_K \):**
\[
v_K = \frac{\text{Distance covered by Karan}}{t} = \frac{600\pi}{t}
\]

- **Speed of Arjun \( v_A \):**
\[
v_A = \frac{\text{Distance covered by Arjun}}{t} = \frac{848\pi}{t}
\]

### Step 4: Find the percentage by which Karan's speed is lower than Arjun's.
The percentage by which Karan's speed is lower than Arjun's can be calculated as:
\[
\text{Percentage difference} = \left(\frac{v_A - v_K}{v_A}\right) \times 100\%
\]

Substituting the values:
\[
\text{Percentage difference} = \left(\frac{\frac{848\pi}{t} - \frac{600\pi}{t}}{\frac{848\pi}{t}}\right) \times 100\%
\]

Simplifying:
\[
\text{Percentage difference} = \left(\frac{848 - 600}{848}\right) \times 100\% = \left(\frac{248}{848}\right) \times 100\%
\]
\[
\text{Percentage difference} \approx 29.25\%
\]

Thus, Karan's speed is approximately **29.25%** lower than Arjun's.

Would you like more details on any step or have any questions?

Here are 5 related questions for further practice:
1. If both Karan and Arjun run at the same speed, how many rounds of the outer track would Arjun complete in the time Karan completes 8 rounds of the inner track?
2. How much time does each person take to complete one round of their respective tracks?
3. What is the ratio of the radii of the two tracks?
4. If the inner track's radius were increased by 10%, how would that affect the percentage difference in their speeds?
5. If the number of rounds for both Karan and Arjun were doubled, how would the percentage difference change?

**Tip:** When comparing speeds over different distances, always normalize the distances to a common reference if possible.

Explore how to calculate the speed difference between Karan and Arjun on concentric circular tracks. Detailed step-by-step solution involving circle geometry, distances covered, and speed calculations. Suitable for Grades 9-12.

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