To determine the simplified ratio [tex]\( a : b : c \)[/tex] given the ratios [tex]\(\frac{a}{b} = \frac{2}{9}\)[/tex] and [tex]\(\frac{b}{c} = \frac{9}{11}\)[/tex], follow these steps:
1. Express [tex]\(a\)[/tex] and [tex]\(c\)[/tex] in terms of [tex]\(b\)[/tex]:
From the given ratio [tex]\(\frac{a}{b} = \frac{2}{9}\)[/tex], we can write:
[tex]\[
a = \frac{2}{9}b
\][/tex]
And from the ratio [tex]\(\frac{b}{c} = \frac{9}{11}\)[/tex], we can write:
[tex]\[
c = \frac{11}{9}b
\][/tex]
2. Determine the value of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in the same terms:
Choose a convenient value for [tex]\(b\)[/tex] to make calculations easy. Let’s work with a representative value where [tex]\(b\)[/tex] can be easily expressed in these terms. Notice that [tex]\(b\)[/tex] appears as [tex]\(9\)[/tex] in the given ratios, which suggest we can simply let [tex]\(b = 9\)[/tex]:
[tex]\[
a = \frac{2}{9} \times 9 = 2
\][/tex]
[tex]\[
c = \frac{11}{9} \times 9 = 11
\][/tex]
3. Collect the values and write the ratio:
Now, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] values are:
[tex]\[
a = 2, \quad b = 9, \quad c = 11
\][/tex]
Therefore, the ratio [tex]\(a : b : c\)[/tex] is:
[tex]\[
2 : 9 : 11
\][/tex]
4. Check for the simplest form:
The values are already in their simplest integer form. There is no common factor among 2, 9, and 11 other than 1, so the ratio does not require any further simplification.
Thus, the answer to [tex]\(a : b : c\)[/tex] in its simplest form is:
[tex]\[
a : b : c = 2 : 9 : 11
\][/tex]
