Question
A cyclist traveling 2km/h takes 15 minutes longer than his friend traveling at 32km/h. Calculate the distances covered
Answers
GPT-4o mini
To solve the problem, we can first convert the time difference of 15 minutes into hours, as the speeds are given in kilometers per hour (km/h).
15 minutes is equal to \( \frac{15}{60} = \frac{1}{4} \) hours.
Let \( d \) be the distance covered by both cyclists. We can set up equations based on the relationship between distance, speed, and time, which is given by the formula:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]
**For the slower cyclist (2 km/h):**
\[
\text{Time}_{\text{slow}} = \frac{d}{2}
\]
**For the faster cyclist (32 km/h):**
\[
\text{Time}_{\text{fast}} = \frac{d}{32}
\]
According to the problem, the time taken by the slower cyclist is 15 minutes longer than that taken by the faster cyclist. Thus, we can write the equation:
\[
\frac{d}{2} = \frac{d}{32} + \frac{1}{4}
\]
Next, we need a common denominator to combine the fractions. The common denominator for 2 and 32 is 32. Thus, we can rewrite the equation:
\[
\frac{16d}{32} = \frac{d}{32} + \frac{1}{4}
\]
Now, since \( \frac{1}{4} \) needs to be expressed with a denominator of 32, we have:
\[
\frac{1}{4} = \frac{8}{32}
\]
Now substitute this back into the equation:
\[
\frac{16d}{32} = \frac{d}{32} + \frac{8}{32}
\]
Next, multiply through by 32 to eliminate the denominator:
\[
16d = d + 8
\]
Now, subtract \( d \) from both sides:
\[
16d - d = 8
\]
\[
15d = 8
\]
Now, solve for \( d \):
\[
d = \frac{8}{15} \text{ km}
\]
Now we can verify the time taken by both cyclists:
1. **Time taken by the slower cyclist:**
\[
\text{Time}_{\text{slow}} = \frac{d}{2} = \frac{8/15}{2} = \frac{8}{30} = \frac{4}{15} \text{ hours}
\]
2. **Time taken by the faster cyclist:**
\[
\text{Time}_{\text{fast}} = \frac{d}{32} = \frac{8/15}{32} = \frac{8}{480} = \frac{1}{60} \text{ hours}
\]
3. **Time difference calculation:**
\[
\text{Difference} = \text{Time}_{\text{slow}} - \text{Time}_{\text{fast}} = \frac{4}{15} - \frac{1}{60}
\]
To subtract these fractions, we need a common denominator. The least common multiple of 15 and 60 is 60. Thus:
\[
\frac{4}{15} = \frac{16}{60}
\]
Therefore:
\[
\text{Difference} = \frac{16}{60} - \frac{1}{60} = \frac{15}{60} = \frac{1}{4} \text{ hours} = 15 \text{ minutes}
\]
This confirms that our calculations are correct.
Thus, the distance covered by both cyclists is:
\[
\boxed{\frac{8}{15} \text{ km}}
\]
Both cyclists travel the same distance of \( \frac{8}{15} \) km.
15 minutes is equal to \( \frac{15}{60} = \frac{1}{4} \) hours.
Let \( d \) be the distance covered by both cyclists. We can set up equations based on the relationship between distance, speed, and time, which is given by the formula:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]
**For the slower cyclist (2 km/h):**
\[
\text{Time}_{\text{slow}} = \frac{d}{2}
\]
**For the faster cyclist (32 km/h):**
\[
\text{Time}_{\text{fast}} = \frac{d}{32}
\]
According to the problem, the time taken by the slower cyclist is 15 minutes longer than that taken by the faster cyclist. Thus, we can write the equation:
\[
\frac{d}{2} = \frac{d}{32} + \frac{1}{4}
\]
Next, we need a common denominator to combine the fractions. The common denominator for 2 and 32 is 32. Thus, we can rewrite the equation:
\[
\frac{16d}{32} = \frac{d}{32} + \frac{1}{4}
\]
Now, since \( \frac{1}{4} \) needs to be expressed with a denominator of 32, we have:
\[
\frac{1}{4} = \frac{8}{32}
\]
Now substitute this back into the equation:
\[
\frac{16d}{32} = \frac{d}{32} + \frac{8}{32}
\]
Next, multiply through by 32 to eliminate the denominator:
\[
16d = d + 8
\]
Now, subtract \( d \) from both sides:
\[
16d - d = 8
\]
\[
15d = 8
\]
Now, solve for \( d \):
\[
d = \frac{8}{15} \text{ km}
\]
Now we can verify the time taken by both cyclists:
1. **Time taken by the slower cyclist:**
\[
\text{Time}_{\text{slow}} = \frac{d}{2} = \frac{8/15}{2} = \frac{8}{30} = \frac{4}{15} \text{ hours}
\]
2. **Time taken by the faster cyclist:**
\[
\text{Time}_{\text{fast}} = \frac{d}{32} = \frac{8/15}{32} = \frac{8}{480} = \frac{1}{60} \text{ hours}
\]
3. **Time difference calculation:**
\[
\text{Difference} = \text{Time}_{\text{slow}} - \text{Time}_{\text{fast}} = \frac{4}{15} - \frac{1}{60}
\]
To subtract these fractions, we need a common denominator. The least common multiple of 15 and 60 is 60. Thus:
\[
\frac{4}{15} = \frac{16}{60}
\]
Therefore:
\[
\text{Difference} = \frac{16}{60} - \frac{1}{60} = \frac{15}{60} = \frac{1}{4} \text{ hours} = 15 \text{ minutes}
\]
This confirms that our calculations are correct.
Thus, the distance covered by both cyclists is:
\[
\boxed{\frac{8}{15} \text{ km}}
\]
Both cyclists travel the same distance of \( \frac{8}{15} \) km.