Ratios of Line Segments

Question

If the points A, B, C, D, and E are placed on a line in alphabetical order, and line segment AB:CD = 2:3, BC:DE = 4:5, and AB:BC = 5:8. Then if it is given that AE = 250, what is the length of CD?

Answer 1

In this problem, there are five lengths that remain in ratio to each other:

$AB:BC:CD:DE:AE$

We are given three partial pieces of information.

Statement 1: $AB:BC:CD:DE:AE = 2:?:3:?:?$

Statement 2: $AB:BC:CD:DE:AE = ?:4:?:5:?$

Statement 3: $AB:BC:CD:DE:AE = 5:8:?:?:?$

The nature of ratio statements is that they can be scaled up and down - think equivalent fractions. It would be helpful to alter Statement 2 so that it has a matching entry to Statement 3. This can be achieved by multiplying Statement 2 by $2$ to give:

Statement 1: $AB:BC:CD:DE:AE = 2:?:3:?:?$

Statement 4: $AB:BC:CD:DE:AE = ?:8:?:10:?$

Statement 3: $AB:BC:CD:DE:AE = 5:8:?:?:?$

We can now combine Statements 4 and 3 to give:

Statement 1: $AB:BC:CD:DE:AE = 2:?:3:?:?$

Statement 5: $AB:BC:CD:DE:AE = 5:8:?:10:?$

We now want to change both Statements so that they have a matching entry. This can be achieved by multiplying Statement 1 by $5$ and Statement 5 by $2$:

Statement 6: $AB:BC:CD:DE:AE = 10:?:15:?:?$

Statement 7: $AB:BC:CD:DE:AE = 10:16:?:20:?$

These can be combined to give:

Statement 8: $AB:BC:CD:DE:AE = 10:16:15:20:?$

Now we move to the fact that $AB+BC+CD+DE=AE$, giving us:

Statement 9: $AB:BC:CD:DE:AE = 10:16:15:20:61$

Now all that remains is to alter Staement 9 so that $AE=250$

Answer 2

I'll give you a hint:

let AB = u; BC = x; CD = y; DE = z;

u + x + y + z = 250

u/y = 2/3; x/z = 4/5; u/x = 5/8;

The question is asking you to find y.

Extra hint: if you can find (u + x + y + z) / y = [a number] you can solve for y.