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Point A lies on a number line at 6. Point [tex][tex]$C$[/tex][/tex] lies between points [tex][tex]$B$[/tex][/tex] and [tex][tex]$A$[/tex][/tex] at 1.875. The ratio [tex][tex]$AC : CB = 3 : 5$[/tex][/tex]. What is the length of [tex][tex]$\overline{AB}$[/tex][/tex]?

A. [tex][tex]$AB = 15$[/tex][/tex] units
B. [tex][tex]$AB = 8.75$[/tex][/tex] units
C. [tex][tex]$AB = 11$[/tex][/tex] units
D. [tex][tex]$AB = 6.875$[/tex][/tex] units



Answer :

GinnyAnswer
Sure! Let's solve the problem step-by-step.

We are given:
- Point [tex]\( A \)[/tex] at 6 on the number line.
- Point [tex]\( C \)[/tex] at 1.875 on the number line.
- The ratio [tex]\( AC : CB = 3 : 5 \)[/tex].

First, we need to calculate the length of [tex]\( AC \)[/tex]:
[tex]\[ AC = |6 - 1.875| = 4.125 \][/tex]

Let [tex]\( x \)[/tex] be the length of [tex]\( CB \)[/tex].

Since [tex]\( AC \)[/tex] and [tex]\( CB \)[/tex] are in the ratio [tex]\( 3:5 \)[/tex], we can write:
[tex]\[ \frac{AC}{CB} = \frac{3}{5} \][/tex]

In this ratio, let's use the known values:
[tex]\[ \frac{4.125}{x} = \frac{3}{5} \][/tex]

To find [tex]\( x \)[/tex], we solve for [tex]\( x \)[/tex]:
[tex]\[ 4.125 \cdot 5 = 3 \cdot x \][/tex]
[tex]\[ 20.625 = 3x \][/tex]
[tex]\[ x = \frac{20.625}{3} \][/tex]
[tex]\[ x = 6.875 \][/tex]

So, the length of [tex]\( CB \)[/tex] is 6.875 units.

Next, we need to calculate the total length of [tex]\( \overline{AB} \)[/tex]:
[tex]\[ AB = AC + CB \][/tex]
[tex]\[ AB = 4.125 + 6.875 \][/tex]
[tex]\[ AB = 11 \][/tex]

Therefore, the length of [tex]\( \overline{AB} \)[/tex] is 11 units.

The correct answer is:
C. [tex]\( AB = 11 \)[/tex] units

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