Problem:
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?
Answer Choices:
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Solution:
Let be Yan's walking speed, and let and be the distances from Yan to his home and to the stadium, respectively. The time required for Yan to walk to the stadium is , and the time required for him to walk home is . Because he rides his bicycle at a speed of , the time required for him to ride his bicycle from his home to the stadium is . Thus
As a consequence, , so . The required ratio is .
{OR}
Because we are interested only in the ratio of the distances, we may assume that the distance from Yan's home to the stadium is 1 mile. Let be his present distance from his home. Imagine that Yan has a twin, Nay. While Yan walks to the stadium, Nay walks to their home and continues of a mile past their home. Because walking of a mile requires the same amount of time as riding 1 mile, Yan and Nay will complete their trips at the same time. Yan has walked miles while Nay has walked miles, so . Thus , , and the required ratio is .
The problems on this page are the property of the MAA's American Mathematics Competitions