Sure, let's go through the steps to find the distance the boy covered on foot.
1. Determine the Total Distance and Total Time:
- Total distance covered: \( 20 \) km
- Total time taken: \( 2 \frac{1}{2} \) hours (which is equivalent to \( 2 + \frac{1}{2} = 2.5 \) hours)
2. Define the Speeds:
- Speed on foot: \( 5 \) km/hour
- Speed on bicycle: \( 10 \) km/hour
3. Let the Distance Covered on Foot be \( x \) km:
- Hence, the distance covered on the bicycle will be \( 20 - x \) km.
4. Set Up the Time Equations for Each Mode of Travel:
- Time taken on foot = \( \frac{x}{5} \) hours
- Time taken on bicycle = \( \frac{20 - x}{10} \) hours
5. Set Up the Equation for Total Time:
- The sum of the time taken on foot and the time taken on the bike should equal the total time.
- Therefore, \( \frac{x}{5} + \frac{20 - x}{10} = 2.5 \)
6. Solve the Equation:
- To solve the equation, we'll combine the terms:
[tex]\[
\frac{x}{5} + \frac{20 - x}{10} = 2.5
\][/tex]
- To simplify, find a common denominator for the fractions (which is 10):
[tex]\[
\frac{2x}{10} + \frac{20 - x}{10} = 2.5
\][/tex]
[tex]\[
\frac{2x + 20 - x}{10} = 2.5
\][/tex]
[tex]\[
\frac{x + 20}{10} = 2.5
\][/tex]
- Multiply both sides of the equation by 10 to clear the denominator:
[tex]\[
x + 20 = 25
\][/tex]
- Subtract 20 from both sides to isolate \( x \):
[tex]\[
x = 25 - 20
\][/tex]
[tex]\[
x = 5
\][/tex]
7. Conclusion:
- The distance covered on foot is [tex]\( x = 5 \)[/tex] km.
