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08 Jan 2025, 05:44
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Clair bikes from home to school to drop off her daughter at an average (arithmetic mean) speed of 15 miles per hour. She then immediately bikes back home along the same route at a different average speed. What was her average speed for the entire round trip?
(1) The journey from school to home took her 40% less time than the journey from home to school.
(2) The ratio of the time she spent biking from home to school to the time she spent biking from school to home was 5 to 3.
(1) The journey from school to home took her 40% less time than the journey from home to school.
(2) The ratio of the time she spent biking from home to school to the time she spent biking from school to home was 5 to 3.
08 Jan 2025, 05:44
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Official Solution:
Clair bikes from home to school to drop off her daughter at an average (arithmetic mean) speed of 15 miles per hour. She then immediately bikes back home along the same route at a different average speed. What was her average speed for the entire round trip?
(1) The journey from school to home took her 40% less time than the journey from home to school.
This implies that the (time to school) : (time from school) = 1 : 0.6 = 5 : 3. If times were in the ratio of 5 to 3, the speeds must have been in the ratio of 3 to 5. Given the speed to school was 15 miles per hour, the speed from school must have been 25 miles per hour. The average speed for the entire journey, assuming one leg was \(d\) miles long, would then be \(\frac{\text{total distance}}{\text{total time}} = \frac{2d}{\frac{d}{15} + \frac{d}{25}}\). The value of \(d\) cancels out, and we can calculate the average speed. Sufficient.
(2) The ratio of the time she spent biking from home to school to the time she spent biking from school to home was 5 to 3.
This statement gives the same information as the first one. Since the first statement is sufficient, this one is also sufficient.
Answer: D
Clair bikes from home to school to drop off her daughter at an average (arithmetic mean) speed of 15 miles per hour. She then immediately bikes back home along the same route at a different average speed. What was her average speed for the entire round trip?
(1) The journey from school to home took her 40% less time than the journey from home to school.
This implies that the (time to school) : (time from school) = 1 : 0.6 = 5 : 3. If times were in the ratio of 5 to 3, the speeds must have been in the ratio of 3 to 5. Given the speed to school was 15 miles per hour, the speed from school must have been 25 miles per hour. The average speed for the entire journey, assuming one leg was \(d\) miles long, would then be \(\frac{\text{total distance}}{\text{total time}} = \frac{2d}{\frac{d}{15} + \frac{d}{25}}\). The value of \(d\) cancels out, and we can calculate the average speed. Sufficient.
(2) The ratio of the time she spent biking from home to school to the time she spent biking from school to home was 5 to 3.
This statement gives the same information as the first one. Since the first statement is sufficient, this one is also sufficient.
Answer: D
