A cyclist rides into the country at an average speed of 10 miles per hour. When his bicycle gets a flat tire, he walks it back at an average speed of 3 miles per hour. If he returns home 6 and a half hours after he starts, how far into the country does he go?
The Correct Answer and Explanation is :
To solve this problem, let’s define some variables and set up an equation based on the total time the cyclist spends.
Step 1: Define Variables
- Let ( d ) represent the distance (in miles) the cyclist rides into the country before the flat tire occurs.
- The cyclist’s speed while riding is 10 miles per hour, and his speed while walking is 3 miles per hour.
- The total time taken for the trip is 6.5 hours.
Step 2: Set up Equations for Time
We can break the trip into two parts:
- The time spent riding into the country (outbound trip).
- The time spent walking the bike back home (return trip).
Time Riding into the Country:
The time to travel a distance ( d ) at 10 miles per hour is:
[
\text{Time riding} = \frac{d}{10}
]
Time Walking Back:
The time to walk the same distance ( d ) back at 3 miles per hour is:
[
\text{Time walking} = \frac{d}{3}
]
Step 3: Set up the Total Time Equation
The total time for the entire trip is the sum of the time riding and the time walking. We are told the total time is 6.5 hours, so we set up the equation:
[
\frac{d}{10} + \frac{d}{3} = 6.5
]
Step 4: Solve the Equation
To solve for ( d ), first find a common denominator for the fractions. The least common denominator of 10 and 3 is 30, so rewrite the equation:
[
\frac{3d}{30} + \frac{10d}{30} = 6.5
]
Now, combine the terms on the left side:
[
\frac{13d}{30} = 6.5
]
Multiply both sides by 30 to eliminate the denominator:
[
13d = 6.5 \times 30
]
[
13d = 195
]
Now divide both sides by 13:
[
d = \frac{195}{13}
]
[
d = 15
]
Step 5: Interpret the Result
The cyclist travels 15 miles into the country before getting the flat tire.
Verification
Now, let’s check the solution by calculating the total time:
- Time riding out: ( \frac{15}{10} = 1.5 ) hours
- Time walking back: ( \frac{15}{3} = 5 ) hours
- Total time: ( 1.5 + 5 = 6.5 ) hours, which matches the given total time.
Thus, the correct distance the cyclist rides into the country is 15 miles.