On a standard number line, point A is located at -24 and point B is located at 6. Point C … — ACT

On a standard number line, point $A$ is located at $-24$ and point $B$ is located at $6$ . Point $C$ is located between point $A$ and point $B$ such that the distance from $A$ to $C$ is $\frac{2}{3}$ of the distance from $C$ to $B$ . What is the coordinate of point $C$ ?

Answer: -12

Answer

The coordinate of point

C

-12

Because point

C

with coordinate

c

lies between

-24

and

6

, the distance from

A

C

c - (-24) = c + 24

, and the distance from

C

B

6 - c

. Setting the distance from

A

C

\frac{2}{3}

of the distance from

C

B

gives the equation

c + 24 = \frac{2}{3}(6 - c)

. Multiplying both sides by

3

to clear the fraction results in

3c + 72 = 12 - 2c

. Collecting like terms yields

5c = -60

, which simplifies to

c = -12

Step-by-Step Solution

Express the distances between the points on the number line using their coordinates.

The distance from

A

C

c + 24

, and the distance from

C

B

6 - c

Since point

C

is positioned between points

A

and

B

, the inequality

-24 < c < 6

holds. This allows the absolute value distance expressions

|c - (-24)|

and

|6 - c|

to simplify directly to positive expressions without absolute value bars.

Formulate an equation based on the specified ratio of distances.

c + 24 = \frac{2}{3}(6 - c)

The problem states that the distance from

A

C

\frac{2}{3}

of the distance from

C

B

Solve the linear equation for the coordinate

c

c = -12

Multiplying both sides by

3

gives

3(c + 24) = 2(6 - c)

, which expands to

3c + 72 = 12 - 2c

. Rearranging the terms by adding

2c

to both sides and subtracting

72

from both sides results in

5c = -60

. Dividing by

5

gives

c = -12

Key Concept

Using absolute value properties to express distances on a number line and solving partitioning coordinate problems.

Estimated Time:1m 15s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic