Certainly! Let's solve the given equations step-by-step to find [tex]\(\frac{a}{b}\)[/tex].
Given:
1. [tex]\(\frac{a+c}{b} = \frac{3}{2}\)[/tex]
2. [tex]\(\frac{b}{c} = \frac{3}{4}\)[/tex]
First, let's deal with the second equation to express [tex]\(c\)[/tex] in terms of [tex]\(b\)[/tex]:
[tex]\[
\frac{b}{c} = \frac{3}{4}
\][/tex]
This implies:
[tex]\[
b = \frac{3}{4}c
\][/tex]
To solve for [tex]\(c\)[/tex]:
[tex]\[
c = \frac{4}{3}b
\][/tex]
Now, substitute [tex]\(c = \frac{4}{3}b\)[/tex] into the first equation:
[tex]\[
\frac{a + c}{b} = \frac{3}{2}
\][/tex]
[tex]\[
\frac{a + \frac{4}{3}b}{b} = \frac{3}{2}
\][/tex]
We can split this fraction:
[tex]\[
\frac{a}{b} + \frac{\frac{4}{3}b}{b} = \frac{3}{2}
\][/tex]
[tex]\[
\frac{a}{b} + \frac{4}{3} = \frac{3}{2}
\][/tex]
Subtract [tex]\(\frac{4}{3}\)[/tex] from both sides:
[tex]\[
\frac{a}{b} = \frac{3}{2} - \frac{4}{3}
\][/tex]
To subtract these fractions, we need a common denominator. The common denominator of 2 and 3 is 6:
[tex]\[
\frac{3}{2} = \frac{9}{6}
\][/tex]
[tex]\[
\frac{4}{3} = \frac{8}{6}
\][/tex]
Now, subtract the fractions:
[tex]\[
\frac{a}{b} = \frac{9}{6} - \frac{8}{6} = \frac{1}{6}
\][/tex]
Thus, the ratio [tex]\(\frac{a}{b}\)[/tex] is:
[tex]\[
\frac{a}{b} = \boxed{\frac{1}{6}}
\][/tex]
