SOLUTION: 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
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Question 1202967: 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? Found 2 solutions by math_tutor2020, greenestamps:Answer by math_tutor2020(3817) (Show Source):
Draw out a number line to plot these points.
S = stadium location
Y = Yan's start position
H = Yan's home
Make sure point Y is between S and H.
Define these distances.
m = distance from S to Y
n = distance from Y to H
m+n = distance from S to H
And define these speeds.
w = walking speed
7w = biking speed
Yan has the option to walk directly to the stadium.
He must walk m miles and does so at w mph.
t1 = number of hours it takes Yan to walk to stadium
distance = rate*time
d = r*t
m = w*t1
t1 = m/w
We'll use this later.
The other option is to walk home to get his bike.
Yan walks n miles to his home at speed w mph.
t2 = number of hours it takes Yan to walk home
d = r*t
n = w*t2
t2 = n/w
Yan bikes m+n miles from home to the stadium at the speed 7w mph.
t3 = number of hours it takes Yan bike from home to stadium
d = r*t
m+n = 7w*t3
t3 = (m+n)/(7w)
Times t2 and t3 must add to t1 because of the phrasing "both choices require the same amount of time" mentioned in the instructions.
t1 = t2+t3
t1 = n/w + (m+n)/(7w)
t1 = (7n)/(7w) + (m+n)/(7w)
t1 = (7n+m+n)/(7w)
t1 = (m+8n)/(7w)
Set this equal to the previously stated t1
t1 = m/w
(m+8n)/(7w) = m/w
(m+8n)/7 = m
m+8n = 7m
8n = 7m-m
8n = 6m
Notice on the third step, the 'w's cancel.
This is because we can multiply both sides by w.
Therefore, the walking speed does not affect the final answer.
Now the goal is to form the fraction or ratio n/m
We will follow the cross multiplication process in reverse.
8n = 6m
8n/m = 6m/m
8n/m = 6
(8n/m)*(1/8) = 6*(1/8)
n/m = 6/8
n/m = 3/4
The ratio of the distance to his home to the stadium is 3:4
For instance, we could have n = 3 and m = 4.
Another option is n = 6 and m = 8.
There are infinitely many possibilities.
Let x be his distance from home and y be his distance from the stadium. Then x+y is the distance from his home to the stadium.
He rides his bicycle at a rate 7 times his walking rate. Since the question only asks for the ratio of x to y and doesn't involve any actual distances, we can use simply "1" for his walking rate and "7" for his cycling rate.
His time to return home a distance x at rate "1" and bicycle the whole distance x+y at rate "7" is the same as his time to walk the distance y at rate "1":
(