To determine which ratio is equivalent to [tex]\(\frac{10}{9}\)[/tex], let's compare each of the given options with [tex]\(\frac{10}{9}\)[/tex].
1. Option A: [tex]\(\frac{50}{36}\)[/tex]
[tex]\[
\frac{50}{36} = \frac{50 \div 2}{36 \div 2} = \frac{25}{18}
\][/tex]
Now, let's see if [tex]\(\frac{25}{18}\)[/tex] is equivalent to [tex]\(\frac{10}{9}\)[/tex].
A quick way to compare [tex]\(\frac{10}{9}\)[/tex] and [tex]\(\frac{25}{18}\)[/tex] is to see if their cross-products are equal.
[tex]\[
10 \times 18 = 180 \quad \text{and} \quad 9 \times 25 = 225
\][/tex]
Since [tex]\(180 \neq 225\)[/tex], the ratios are not equivalent.
2. Option B: [tex]\(\frac{50}{45}\)[/tex]
[tex]\[
\frac{50}{45} = \frac{50 \div 5}{45 \div 5} = \frac{10}{9}
\][/tex]
This simplifies directly to [tex]\(\frac{10}{9}\)[/tex], which is exactly the ratio we are given.
Therefore, [tex]\(\frac{50}{45}\)[/tex] is equivalent to [tex]\(\frac{10}{9}\)[/tex].
3. Option C: [tex]\(\frac{15}{14}\)[/tex]
To check if [tex]\(\frac{15}{14}\)[/tex] is equivalent to [tex]\(\frac{10}{9}\)[/tex], we can also use the cross-products method:
[tex]\[
10 \times 14 = 140 \quad \text{and} \quad 9 \times 15 = 135
\][/tex]
Since [tex]\(140 \neq 135\)[/tex], the ratios are not equivalent.
Among the given options, only [tex]\(\frac{50}{45}\)[/tex] simplifies to [tex]\(\frac{10}{9}\)[/tex].
Thus, the ratio that is equivalent to [tex]\(\frac{10}{9}\)[/tex] is:
Option B: [tex]\(\frac{50}{45}\)[/tex]
