Point A lies on a number line at 6. Point C lies between points B and A at 1.875. The ratio [tex]$AC : CB = 3 : 5$[/tex]. What is the length of [tex]$\overline{AB}$[/tex]?
A. [tex][tex]$AB = 15$[/tex][/tex] units B. [tex]$AB = 6.875$[/tex] units C. [tex]$AB = 8.75$[/tex] units D. [tex][tex]$AB = 11$[/tex][/tex] units
1. Understand the given data: - Point [tex]\( A \)[/tex] is at 6. - Point [tex]\( C \)[/tex] is at 1.875. - The ratio [tex]\( AC : CB = 3 : 5 \)[/tex].
2. Determine what we need to find: - We need to find the length of [tex]\( \overline{AB} \)[/tex].
3. Introductions of Variables: - Let's call the length of [tex]\( CB \)[/tex] as [tex]\( x \)[/tex]. - Therefore, the length of [tex]\( AC \)[/tex] will be [tex]\(\frac{3}{5} \times x\)[/tex].
4. Length Relationships: - The total length [tex]\( \overline{AB} \)[/tex] is the sum of the lengths [tex]\( AC \)[/tex] and [tex]\( CB \)[/tex]. - So, [tex]\( AB = AC + CB \)[/tex].
5. Calculation of Lengths: - Since [tex]\( C \)[/tex] is on the number line between [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we can say: [tex]\[
AB = A - C = 6 - 1.875 = 4.125
\][/tex]
6. Solve for [tex]\( x \)[/tex]: - We know that [tex]\( \overline{AB} = 4.125 \)[/tex]. - We also know that [tex]\( \overline{AB} = \left(\frac{8}{5}\right) \times x \)[/tex] (from the sum of [tex]\( \frac{3}{5}x \)[/tex] and [tex]\( x \)[/tex]). - Hence, we set up the equation: [tex]\[
\left(\frac{8}{5}\right) \times x = 4.125
\][/tex] - Solving for [tex]\( x \)[/tex]: [tex]\[
x = \left(\frac{4.125 \times 5}{8}\right) = 2.578125
\][/tex]
7. Verification: - Check if [tex]\( \overline{AB} \)[/tex] matches: [tex]\[
AC = \frac{3}{5} \times x = \frac{3}{5} \times 2.578125 = 1.546875
\][/tex] [tex]\[
AB = AC + CB = 1.546875 + 2.578125 = 4.125
\][/tex]
From these calculations, we conclude that:
[tex]\[
\overline{AB} = 4.125
\][/tex]
Therefore, the correct answer is: B. [tex]\( AB = 6.875 \)[/tex] units.
