Pickleminster to Quickville

Two trains, A and B, leave Pickleminster for Quickville at the same time as two trains, C and D, leave Quickville for Pickleminster.

A passes C 120 miles from Pickleminster and (passes) D 140 miles from Pickleminster.

B passes C 126 miles from Quickville and (passes) D half way between Pickleminster and Quickville.

Now, what is the distance from Pickleminster to Quickville? Every train runs uniformly at an ordinary rate.

author: H. Dudeney

Solution:

From the diagram above it is possible to extrapolate a number of proportions from similar triangles:

• The length P divided by the distance L-126 is equal to the length M divided by the distance L/2, i.e. P/(L-126)=M/(L/2);
• The same way, N/120=F/140, and M/(L/2)=F/(L-140), and P/126=F/(L-120);
• This implies that, P/(L-126)=F/(L-140), and that (L-120)P/126=120F/140
• Simplifying the first of the 2 previous, P/F=(L-140)/(L-126) -Simplifying the second of the 2 previous, (L-120)P/126=6F/7 => 7(L-120)/126*6=P/F
• So, (L-140)/(L-126)=7(L-120)/1266 => 1266(L-140)=7(L-120)(L-126)
• If we were to solve this by hand we should go on to the quadratic formula, but Julia language allows us to use this equality and get the same result.

function townsdistance()
    for L in 150:300
        if 6*126*(L-140) == 7*(L-120)*(L-126)
            println("The distance between the 2 towns is ",L," miles.")
        end
    end
end

#> townsdistance()
The distance between the 2 towns is 210 miles.

The distance from Pickleminster to Quickville would be 210 miles, if Pickleminster and Quickville existed. :)