Certainly! Let's solve this step by step.
1. Define Variables for Speeds:
Let [tex]\( w \)[/tex] be the average walking speed of the boy in km/h. According to the question, his average cycling speed is 7 km/h faster than his walking speed. Hence, his average cycling speed is [tex]\( w + 7 \)[/tex] km/h.
2. Time Taken for Each Mode of Travel:
- Time taken to walk to the picnic spot: 6 hours.
- Time taken to cycle to the picnic spot: 2 hours.
3. Relationship Between Distance, Speed, and Time:
The distance traveled can be calculated using the formula:
[tex]\[
\text{Distance} = \text{Speed} \times \text{Time}
\][/tex]
4. Calculate the Distance Traveled for Walking and Cycling:
Since the distance to the picnic spot remains the same whether walking or cycling, we can set up the equations for distance.
- Walking distance:
[tex]\[
\text{Distance}_\text{walking} = w \times 6
\][/tex]
- Cycling distance:
[tex]\[
\text{Distance}_\text{cycling} = (w + 7) \times 2
\][/tex]
5. Set the Distances Equal:
Since the distance for walking and cycling are equal, we can write:
[tex]\[
w \times 6 = (w + 7) \times 2
\][/tex]
6. Solve for [tex]\( w \)[/tex]:
[tex]\[
6w = 2(w + 7)
\][/tex]
[tex]\[
6w = 2w + 14
\][/tex]
[tex]\[
6w - 2w = 14
\][/tex]
[tex]\[
4w = 14
\][/tex]
[tex]\[
w = \frac{14}{4} = \frac{7}{2} = 3.5 \text{ km/h}
\][/tex]
Therefore, the average walking speed [tex]\( w \)[/tex] is [tex]\( 3.5 \)[/tex] km/h.
7. Calculate the Average Cycling Speed:
The average cycling speed is 7 km/h faster than the walking speed:
[tex]\[
\text{Cycling speed} = w + 7 = 3.5 + 7 = 10.5 \text{ km/h}
\][/tex]
8. Summary:
- The boy's average walking speed is [tex]\( \frac{7}{2} \)[/tex] km/h or 3.5 km/h.
- His average cycling speed is [tex]\( \frac{21}{2} \)[/tex] km/h or 10.5 km/h.
Thus, the average walking speed is 3.5 km/h and the average cycling speed is 10.5 km/h.
