To find the ratio [tex]\(a: b: c\)[/tex] given [tex]\(\frac{a}{b} = \frac{2}{3}\)[/tex] and [tex]\(\frac{b}{c} = \frac{3}{5}\)[/tex], follow these steps:
1. Represent the given ratios in fractional form:
[tex]\[
\frac{a}{b} = \frac{2}{3} \quad \text{and} \quad \frac{b}{c} = \frac{3}{5}
\][/tex]
2. Assign a common variable to [tex]\(b\)[/tex]. We'll let [tex]\(b = 3k\)[/tex], where [tex]\(k\)[/tex] is a constant.
3. Determine [tex]\(a\)[/tex] using [tex]\(\frac{a}{b} = \frac{2}{3}\)[/tex]:
[tex]\[
\frac{a}{3k} = \frac{2}{3} \implies a = 2k
\][/tex]
4. Determine [tex]\(c\)[/tex] using [tex]\(\frac{b}{c} = \frac{3}{5}\)[/tex]:
[tex]\[
\frac{3k}{c} = \frac{3}{5} \implies c = 5k
\][/tex]
5. Now we have the values in terms of [tex]\(k\)[/tex]:
[tex]\[
a = 2k, \quad b = 3k, \quad c = 5k
\][/tex]
6. The ratio [tex]\(a: b: c\)[/tex] is:
[tex]\[
a: b: c = 2k : 3k : 5k
\][/tex]
7. To simplify this ratio, we can divide each term by the common factor [tex]\(k\)[/tex]:
[tex]\[
a: b: c = \frac{2k}{k} : \frac{3k}{k} : \frac{5k}{k} = 2 : 3 : 5
\][/tex]
So, the simplest form of the ratio [tex]\(a : b : c\)[/tex] is [tex]\(\boxed{2 : 3 : 5}\)[/tex].
