math problem. find the total trip distance?

Question

a biker travels downhill at 12mph, over flat land at 8mph, and uphill at 6mph. The biker rides from town A to town B in four hours. The return trip takes 4 hours 30 minutes. how far is town A from town B?

Answer 1

First, convert the speeds into minutes per mile instead of mpg. Downhill is 5 minutes per mile, uphill is 10 minutes per mile and flat is 7.5 minutes per mile. On the wat from A to B call the downhill distance d, uphill u and flat f. The travel time in minutes A to B is 240 minutes. The equation relating all this is:

10 u + 5 d + 7.5 f = 240

On the way back, whatever was uphill is now downhill. the return equation is:

5 u + 10 d + 7.5 f = 270

The distance to be solved is u + d + f.

Subtracting the 2nd equation from the first yields:

(10 u + 5 d + 7.5 f) -(5 u + 10 d + 7.5 f) = 240 - 270
5 u - 5d = -30

Dividing by 2 yields 2.5 u - 2.5 d = -15

Add this to the second equation:

(5 u + 10 d + 7.5 f) + (2.5 u - 2.5 d ) = 270 - 15
7.5 u + 7.5 d + 7.5 f = 255

u + d + f = 255/7.5 = 34

The distance A to B is 34 miles.

The interesting thing here is that the amount of flat, uphill, and downhill cannot be determined individually but the total distance can be determined.

Answer 2

While the previous solution is mathematically correct, you do not have to convert any units. Below is a solution using all of the original units. Although it appears lengthy, that's because of my detailed comments.

The basic formula is
t = d / v
t = time in hours
d = distance in miles
v = velocity in miles per hour

du = uphill distance
tu = uphill time
df = flat distance
tf = flat time
dd = uphill distance
td = downhill time

GOING FROM A to B
tu + tf + td = total time
(du / 12mph) + (df / 8mph) + (dd / 6mph) = 4 hours

GOING FROM B to A
uphill A to B distance is now downhill B to A distance and downhill A to B distance is now uphill B to A distance, so in the formula, either the distances for these two terms must be reversed or their corresponding speeds must be reversed. I will reverse their speeds. The flat speed and distance remains the same for both directions.
(du / 6mph) + (df / 8mph) + (dd / 12mph) = 4.5 hours

ADD BOTH EQUATIONS
(du / 12mph) + (df / 8mph) + (dd / 6mph) + (du / 6mph) + (df / 8mph) + (dd / 12mph) = 8.5 hours

CONVERT TO LEAST COMMON DENOMINATOR (24)
(2*du / 24mph) + (3*df / 24mph) + (4*dd / 24mph) + (4*du / 24mph) + (3*df / 24mph) + (2*dd / 24mph) = 8.5

COMBINE LIKE TERMS
(6*du / 24) + (6*df / 24) + (6*dd / 24) = 8.5

MULTIPLY BOTH SIDES BY 24 TO ELIMINATE DENOMINATOR
6du + 6df + 6dd = 204

DIVIDE BOTH SIDES BY 6
du + df + dd = 204 / 6 = 34 miles, total distance between A and B