Math Word Problems-Car Uphill downhill

A car travels downhill at 72 mph (miles per hour), on the level at 63 mph, and uphill at
only 56 mph. The car takes 4 hours to travel from town A to town B. The return trip
takes 4 hours and 40 minutes. Find the distance (in miles) between the two towns

Let the total distance travelled downhill, on the level, and uphill, on the outbound journey, be x, y, and z, respectively.
The time taken to travel a distance s at speed v is s/v.

Hence, for the outbound journey

x/72 + y/63 + z/56 = 4

While for the return journey, which we assume to be along the same roads

x/56 + y/63 + z/72 = 14/3

It may at first seem that we have too little information to solve the puzzle. After all, two equations in three unknowns do not have a unique solution. However, we are not asked for the values of x, y, and z, individually; but for the value of x + y + z.

Multiplying both equations by the least common multiple of denominators 56, 63, and 72, we obtain

7x + 8y + 9z = 4 · 7 · 8 · 9
9x + 8y + 7z = (14/3) · 7 · 8 · 9

Now it is clear that we should add the equations, yielding

16(x + y + z) = (26/3) · 7 · 8 · 9

Therefore x + y + z = 273; the distance between the two towns is 273 miles.

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