Question

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?

$\dfrac{2}{3}$

$\dfrac{3}{4}$

$\dfrac{4}{5}$

$\dfrac{5}{6}$

$\dfrac{6}{7}$

Solution:

Let $w$ be the walking speed and let $x$ and $y$ be Yan's distances from home and from the stadium.

Walking to the stadium takes $\dfrac{y}{w}.$ Walking home then biking takes $\dfrac{x}{w}+\dfrac{x+y}{7w}=\dfrac{8x+y}{7w}.$

Setting these equal gives $7y=8x+y,$ so $8x=6y$ and $\dfrac{x}{y}=\dfrac{3}{4}.$

Thus, the correct answer is B.

2007 AMC 12A Problem 9

Solution:

Problem 9 in Other Years