On the number line shown, the distance between \(0\) and \(a,\) \(a\) and \(b,\) and \(a\) and \(c\) is in the ratio of

[Gmat_mission]

On the number line shown, the distance between \(0\) and \(a,\) \(a\) and \(b,\) and \(a\) and \(c\) is in the ratio of \(1:2:3.\) If the distance of point \(b\) from \(15\) is twice the distance of point a from \(15,\) what is the value of \(|c|?\)

A. 9
B. 27
C. 36
D. 45
E. 54

Answer: C

Source: e-GMAT

[Scott@TargetTestPrep]

01_01.png

On the number line shown, the distance between \(0\) and \(a,\) \(a\) and \(b,\) and \(a\) and \(c\) is in the ratio of \(1:2:3.\) If the distance of point \(b\) from \(15\) is twice the distance of point a from \(15,\) what is the value of \(|c|?\)

A. 9
B. 27
C. 36
D. 45
E. 54

Answer: C

Solution:

Since a < b, we see that if b < 15, then b would be closer to 15 than a to 15. Since the distance between b and 15 is twice that between a and 15, we see that b can’t be less than 15. If a > 15, then the distance between b and 15 would be more than twice that between a and 15 since the distance between b and 0 is twice that between a and 0. We see that a can’t be greater than 15. Therefore, 15 must be between a and b. Now, using the distance relationships between the points a and b and between either one and 15, we can create the equations.

(b - a)/a = 2/1

and

(b - 15)/(15 - a) = 2

Simplifying the first equation, we have b = 3a. Substituting this into the second equation, we have:

(3a - 15)/(15 - a) = 2

3a - 15 = 30 - 2a

5a = 45

a = 9

Therefore, b = 3(9) = 27. Since (c - b)/(b - a) = 3/2, we have:

(c - 27)/(27 - 9) = 3/2

2(c - 27) = 3(27 - 9)

2c - 54 = 54

2c = 108

c = 54

Since c is positive, |c| = c = 54.

Answer: E