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

Answer 1

3/4

Explanation:

The relationship between time, speed, and distance can be used together with the travel time relations to find the desired ratio.

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setup

Let H represent the distance from Yan's location to home. Let S represent the distance to the stadium, and let w represent Yan's walking speed.

The time required for Yan to walk home is ...

time = distance/speed = H/w

The time required for Yan to walk to the stadium is ...

time = S/w

The time required for Yan to ride his bike from home to the stadium is ...

time = (H +S)/(7w) . . . . . his riding speed is 7 times his walking speed.

In order for the travel times to be equal, we must have ...

time to walk home + time to bike to stadium = time to walk to stadium

H/w + (H +S)/(7w) = S/w

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solution

Multiplying by 7w gives ...

7H +(H +S) = 7S

8H = 6S . . . . . . . . . subtract S

H/S = 6/8 = 3/4 . . . . divide by 8S

The ratio of Yan's distance from home to his distance from the stadium is 3/4.

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Additional comment

Another way to think about this is ...

The stadium is farther away than home by a distance that Yan can walk in the same time he can bike the entire distance from home to the stadium. That is, the difference in distances must be 1/7 of the total distance. Now, we have a "sum and difference" problem: S +H = 7, S -H = 1. The solution is S = (7+1)/2 = 4, H = (7-1)/2 = 3. H/S = 3/4.