# Solution to puzzle 75: Car journey

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 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.

## Remarks

A unique solution is possible because the speeds are chosen so that a round trip over a sloping section of road takes the same time as that over a flat section of the same length.  Had we chosen to write down an equation for the round trip, the answer would have been immediately apparent.

Source: From A to B, on flooble :: perplexus