Certainly! Let's break the problem down step-by-step to find the boy's average walking speed and his average cycling speed.
### Step 1: Define Variables
- Let the boy's average walking speed be [tex]\( x \)[/tex] kilometers per hour (km/h).
- Given that his average cycling speed is 7 kilometers per hour faster than his walking speed, the average cycling speed is [tex]\( x + 7 \)[/tex] km/h.
### Step 2: Use the Time and Distance Relationship
We know the formula for distance is:
[tex]\[ \text{Distance} = \text{Speed} \times \text{Time} \][/tex]
Since the distance to the picnic spot is the same for both walking and cycling:
1. Distance when walking:
[tex]\[ \text{Distance} = x \times 6 \text{ hours} \][/tex]
2. Distance when cycling:
[tex]\[ \text{Distance} = (x + 7) \times 2 \text{ hours} \][/tex]
### Step 3: Set Up the Equation
Since both distances are the same, we can set these two expressions equal to each other:
[tex]\[ x \times 6 = (x + 7) \times 2 \][/tex]
### Step 4: Solve the Equation
Let's solve for [tex]\( x \)[/tex]:
1. Expand the right-hand side:
[tex]\[ 6x = 2(x + 7) \][/tex]
2. Distribute the 2:
[tex]\[ 6x = 2x + 14 \][/tex]
3. Subtract [tex]\( 2x \)[/tex] from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ 6x - 2x = 14 \][/tex]
[tex]\[ 4x = 14 \][/tex]
4. Divide both sides by 4:
[tex]\[ x = \frac{14}{4} \][/tex]
[tex]\[ x = 3.5 \][/tex]
So, the average walking speed is 3.5 km/h.
### Step 5: Find the Average Cycling Speed
We know the average cycling speed is 7 km/h faster than the walking speed:
[tex]\[ \text{Cycling speed} = x + 7 \][/tex]
[tex]\[ \text{Cycling speed} = 3.5 + 7 \][/tex]
[tex]\[ \text{Cycling speed} = 10.5 \text{ km/h} \][/tex]
### Conclusion
- The boy's average walking speed is 3.5 kilometers per hour.
- The boy's average cycling speed is 10.5 kilometers per hour.
