Distance Ken Is From Uncle Frank

Here is a problem by J.A.H. Hunter.

'How far's Guelph?' asked Ken, reading the postmark on a letter. His father laughed. 'Depends on your route.' The boy nodded. Of course, but is it near Uncle Frank's farm?' 'Seventeen miles on from there,' replied his father, 'but if we drove that way it would be twice as far for us as the way past your Aunt Kate's place.' 'I know she lives 23 miles from us,' commented Ken, 'and you once told me she is 45 miles nearer to Guelph than Uncle Frank is to us. So I guess I'll have to figure out how far we are from Uncle Frank.' What do you make of it?

Please give the answer and your method to get it.

Peter Heichelheim

We just need to define a couple of variables, then turn all these words into a couple of equations, and then solve them.

I'll start.

Let's suppose Aunt Kate is x miles from Guelph, and we are y miles from Uncle Frank. (Defining suitable variables is often the key to solving this sort of problem.)

'she is 45 miles nearer to Guelph than Uncle Frank is to us' will yield one equation in x and y

'but if we drove that way [i.e. via Uncle Frank's farm] it would be twice as far for us as the way past your Aunt Kate's place' will yield another.

I hope this is enough for you to solve it.

Spoiler follows

61 miles

I used a pencil and the back of an envelope.

Oh, all right, I don't know why that request irritates me so much.

If the distance Aunt-Kate - Guelph is a, and the distance here - Uncle Frank is b, we have two equations 17+b = 2(23+a) a+45 = b Solving them, we get a=16 b=61.

Hi PCH,

According to the problem:

Uncle Frank to Us = x miles Uncle Frank to Guelph = 17 miles Aunt Kate to Us = 23 miles Aunt Kate to Guelph = ( x - 45) miles and Guelph to Us through Uncle Frank = 2 * (Guelph to Us through Aunt Kate)

then x + 17 = 2*[23 + (x - 45)]

x + 17 = 2*(x - 22)

x + 17 = 2x - 44

x = 61 miles

Very good. You and JJAthayde got the same answer as I did. However J.A.H. Hunter gave as his answer that Ken lives 29 miles from Uncle Frank. Anyone as any idea how he got this answer.

You would get that answer if, instead of 17+b = 2(23+a) you wrote 17+b = 2(23).

But it seems an odd kind of mistake to make, to set up a pair of simultaneous equations, and then omit a variable from one of them and treat it as a simple equation.

It is 61 miles to Uncle Frank's farm.

Let HF = distance from Home to Frank's. Then the distance from home to Guelph via Uncle Frank's farm is HF + 17.

The distance from Home to Aunt Kate's is 23 miles. Since the distance from Kate's to Guelph is 45 miles farther than the distance from Home to Frank's (given), then the distance from Home to Guelph via Aunt Kate's is 23 + (HF - 45)

And, given that the distance via Uncle Frank's is twice the distance via Aunt Kate's, we get the equation HF+17 = 2(23+(HF-45)) Solving for HF, we find the distance from Home to Uncle Frank's is 61 miles. Then, the distance from Uncle Frank's to Guelph is 17 miles, and the total trip is 78 miles. The distance from Home to Aunt Kate's is 23 miles, the distance from Aunt Kate's to Guelph is (61 - 45) or 16 miles, and the total trip is 39 miles.

Did nobody notice that the way past Aunt Kate and Guelph to Uncle Frank is only 56 miles long and therefor shorter than the 'direct' way.

Andreas