## I’ve been driving in my car, it’s not quite a Jaguar…

Posted by Karl Sharman on July 1, 2014 – 5:34 am

I went for a drive, and curiosly discovered that my car (Kevin the Capri) only has three speeds. It travels downhill at 72 mph (miles per hour), on the level at 63 mph, and uphill at only 56 mph Kevin the Capri takes 4 hours to travel from Tomsville in Tomshire to Tomerton in Tomerset. The return trip takes 4 hours and 40 minutes.

How far is it from Tomerton to Tomsville?

July 2nd, 2014 at 9:28 pm

273 miles.

July 3rd, 2014 at 6:41 am

Let u, f and d be the distances up, down and flat when going outwards.

The outward time = f / 63 + u / 56 + d / 72 = 4 hours.

The return time = f / 63 + u / 72 + d / 56 = 4 + 2 / 3 hours.

Adding => 2f / 63 + (u + d)(1 / 56 + 1 / 72) = 8 + 2 / 3

=> 2f / 63 + (u + d)(2 / 63) = 8 + 2 / 3.

Multiplying by 63 / 2 => f + u + d = 273

July 5th, 2014 at 3:10 am

A quicker solution:

The round trip takes 8 hours and 40 minutes. The uphills and downhills will cancel out. For the same distance the average of 56 mph uphill and 72 mph downhill is 63 mph.

[Proof: Speed = Distance (d) / Time (t)

So the time for equivalent up and down hill sections of distance d = (d/56 + d/72) = 2d/63]

So we can use the average speed of 63 mph for the whole round trip. 8 hours and 40 minutes at 63 mph gives us 546 miles, or 273 miles each way.

July 5th, 2014 at 8:40 pm

Hi Wiz. I like your answer.

July 10th, 2014 at 1:46 am

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 x 7 x 8 x 9

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

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

16(x + y + z) = (26/3) x 7 x 8 x 9

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

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.

Or, what Wiz said….