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A boy covers a distance of [tex]$20 \, \text{km}[tex]$[/tex] in [tex]$[/tex]2 \frac{1}{2} \, \text{hours}[tex]$[/tex], partly on foot at the rate of [tex]$[/tex]5 \, \text{km/hr}[tex]$[/tex] and partly on a bicycle at the rate of [tex]$[/tex]10 \, \text{km/hr}$[/tex]. Find the distance covered on foot.



Answer :

GinnyAnswer
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.

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