On a standard number line, points A, B, C, and D have distinct integer coordinates a, b, c… — ACT

On a standard number line, points $A$ , $B$ , $C$ , and $D$ have distinct integer coordinates $a$ , $b$ , $c$ , and $d$ , respectively, such that $a < b < c < d$ . The distance between $A$ and $B$ is equal to the distance between $C$ and $D$ . The distance between $B$ and $C$ is $\frac{2}{3}$ of the distance between $A$ and $B$ . If the average (arithmetic mean) of the four coordinates is $0$ and $|a - d| = 24$ , what is the coordinate of $B$ ?

Answer: -3

Answer

The coordinate of

B

-3

The coordinate of

B

is found by setting the segment lengths to

3k

2k

, and

3k

based on the given ratio. Using the absolute value distance

|a - d| = 24

, we find

k = 3

, meaning the segments are

9

6

, and

9

. Using the coordinate sum of

0

, we get

4a + 48 = 0 \implies a = -12

. Substituting this back gives the coordinate of

B

-3

Step-by-Step Solution

Define segment lengths using a variable

k

based on the given ratio.

Let the distance

AB = CD = 3k

and

BC = 2k

This allows us to write all distances as integer multiples of a single variable since the ratio of

BC

AB

\frac{2}{3}

Set up an equation for the total distance from

A

D

using the absolute value

|a - d| = 24

Since

a < d

d - a = |a - d| = 24

. The sum of the segments is

3k + 2k + 3k = 8k

. Solving

8k = 24

yields

k = 3

The absolute value of the difference between the outermost points represents the total length of the number line segment containing all four points.

Express the coordinates of

b

c

, and

d

in terms of

a

using the calculated segment lengths.

b = a + 9

c = a + 15

, and

d = a + 24

Since the points are in order

a < b < c < d

, we add the segment lengths successively to find the coordinates.

Apply the average condition to solve for the coordinate

a

The sum of the coordinates is

4 \times 0 = 0

, so

a + (a + 9) + (a + 15) + (a + 24) = 4a + 48 = 0

, giving

a = -12

An average of

0

for four numbers means their sum must be

0

Find the coordinate of

B

by substituting the value of

a

into the expression for

b

b = -12 + 9 = -3

This yields the specific coordinate requested by the question.

Key Concept

Using absolute value as distance on a number line and expressing relationships between ordered coordinates algebraically.

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