On a standard number line, point A has coordinate -15 and point B has coordinate 17. Point… — ACT

On a standard number line, point $A$ has coordinate $-15$ and point $B$ has coordinate $17$ . Point $C$ is located to the right of point $B$ such that the distance between $A$ and $C$ is exactly $3$ times the distance between $B$ and $C$ . What is the coordinate of point $C$ ?

Answer: 33

Answer

The coordinate of point

C

is 33.

The correct coordinate is found by setting up the distance equation for point

C

(with coordinate

c > 17

) relative to

A(-15)

and

B(17)

. The distance

AC

c - (-15) = c + 15

, and the distance

BC

c - 17

. Setting

c + 15 = 3(c - 17)

and solving yields

c = 33

, which is to the right of

B

Step-by-Step Solution

Define the variable for the coordinate of point C and write the expressions for distances.

Let

c

be the coordinate of point

C

. The distance between

A

and

C

|c - (-15)| = |c + 15|

, and the distance between

B

and

C

|c - 17|

. Since point

C

is to the right of point

B

(which is at

17

), we know

c > 17

, so

|c + 15| = c + 15

and

|c - 17| = c - 17

To set up an algebraic equation representing the physical distance relations on the number line.

Set up the equation using the given relationship.

The equation is

c + 15 = 3(c - 17)

The problem states the distance between

A

and

C

3

times the distance between

B

and

C

Solve the equation for

c

c + 15 = 3c - 51 \implies 15 + 51 = 3c - c \implies 66 = 2c \implies c = 33

To find the coordinate of point

C

Key Concept

Calculating distances between points on a number line using absolute value and solving the resulting equations.

Alternative Method

Use geometric visualization: The distance from

A(-15)

B(17)

17 - (-15) = 32

units. Since point

C

lies to the right of

B

, the distance

AC

is the sum of

AB

and

BC

. Therefore,

AC = 32 + BC

. We are given that

AC = 3 \times BC

. Substituting this gives

32 + BC = 3 \times BC \implies 2 \times BC = 32 \implies BC = 16

. Since

C

is 16 units to the right of

B(17)

, its coordinate is

17 + 16 = 33

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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