Five points are located on a line. When the ten distances between pairs of points are listed from smallest to largest, the list reads: $2,4,5,7,8, k, 13,15,17,19$. What is the value of $k$ ?
(A) 11
(B) 9
(C) 13
(D) 10
(E) 12
Five points are located on a line. When the ten distances between pairs of points are listed from smallest to largest, the list reads: $2,4,5,7,8, k, 13,15,17,19$. What is the value of $k$ ?
(A) 11
(B) 9
(C) 13
(D) 10
(E) 12

## Solution

The solution to this problem is difficult to write up because it really comes down to systematic trial and error. We'll start by drawing a number line and placing two points at 0 and 19 . Since the first required distance is 2 , we'll place our first three points as shown.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-224.jpg?height=121&width=1277&top_left_y=796&top_left_x=435)

Since we need to have a distance of 7, we'll place a point at 7 on the number line. This gives us distances which are not inconsistent with the required.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-224.jpg?height=122&width=1277&top_left_y=1142&top_left_x=435)

Distances so far are, 2, 7, 19, 5, 17 and 12.

And finally, if we place a point at 15 on the number line, we will have the following.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-224.jpg?height=124&width=1277&top_left_y=1491&top_left_x=435)

If we place the point anywhere else, we will have inconsistencies with the required. If the points are placed as shown we will have the distances, $\{2,7,15,19,5,13,17,8,(12), 4\}$.

Thus the required distance is 12 .

ANSWER: (E)