To determine which of the given ratios is equivalent to [tex]\(7 : 9\)[/tex], we'll follow these steps:
1. Simplify each given ratio to its simplest form:
- A ratio is in its simplest form when the greatest common divisor (GCD) of both numbers in the ratio is 1.
2. Compare the simplified form of each ratio to [tex]\(7 : 9\)[/tex]:
- If the simplified form of any ratio matches [tex]\(7 : 9\)[/tex], then that ratio is equivalent to [tex]\(7 : 9\)[/tex].
### Step-by-Step Solution
#### Simplifying [tex]\(5 : 15\)[/tex]:
1. Find the GCD of 5 and 15. The GCD of 5 and 15 is 5.
2. Divide both terms of the ratio by their GCD:
[tex]\[
\frac{5}{5} : \frac{15}{5} = 1 : 3
\][/tex]
3. The simplified form of [tex]\(5 : 15\)[/tex] is [tex]\(1 : 3\)[/tex].
#### Simplifying [tex]\(84 : 108\)[/tex]:
1. Find the GCD of 84 and 108. The GCD of 84 and 108 is 12.
2. Divide both terms of the ratio by their GCD:
[tex]\[
\frac{84}{12} : \frac{108}{12} = 7 : 9
\][/tex]
3. The simplified form of [tex]\(84 : 108\)[/tex] is [tex]\(7 : 9\)[/tex].
#### Simplifying [tex]\(27 : 54\)[/tex]:
1. Find the GCD of 27 and 54. The GCD of 27 and 54 is 27.
2. Divide both terms of the ratio by their GCD:
[tex]\[
\frac{27}{27} : \frac{54}{27} = 1 : 2
\][/tex]
3. The simplified form of [tex]\(27 : 54\)[/tex] is [tex]\(1 : 2\)[/tex].
### Comparing Simplified Ratios:
- The simplified form of [tex]\(5 : 15\)[/tex] is [tex]\(1 : 3\)[/tex], which is not equivalent to [tex]\(7 : 9\)[/tex].
- The simplified form of [tex]\(84 : 108\)[/tex] is [tex]\(7 : 9\)[/tex], which is equivalent to [tex]\(7 : 9\)[/tex].
- The simplified form of [tex]\(27 : 54\)[/tex] is [tex]\(1 : 2\)[/tex], which is not equivalent to [tex]\(7 : 9\)[/tex].
### Conclusion
Among the given ratios, only [tex]\(84 : 108\)[/tex] is equivalent to [tex]\(7 : 9\)[/tex]. Thus, the ratio equivalent to [tex]\(7 : 9\)[/tex] is:
[tex]\[
84 : 108
\][/tex]
