To determine whether the ratios [tex]\(\frac{10}{17}\)[/tex] and [tex]\(\frac{30}{68}\)[/tex] are equivalent, we need to simplify each fraction and then compare the simplified forms.
### Step-by-step solution:
1. Simplify [tex]\(\frac{10}{17}\)[/tex]:
The fraction [tex]\(\frac{10}{17}\)[/tex] is already in its simplest form since the greatest common divisor (GCD) of 10 and 17 is 1. Hence, the simplified form of [tex]\(\frac{10}{17}\)[/tex] is [tex]\(\frac{10}{17}\)[/tex].
2. Simplify [tex]\(\frac{30}{68}\)[/tex]:
To simplify [tex]\(\frac{30}{68}\)[/tex], we need to find the GCD of 30 and 68. The GCD of 30 and 68 is 2.
Next, divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{30 \div 2}{68 \div 2} = \frac{15}{34}
\][/tex]
Therefore, the simplified form of [tex]\(\frac{30}{68}\)[/tex] is [tex]\(\frac{15}{34}\)[/tex].
3. Compare the simplified forms:
Now, compare the simplified fractions:
[tex]\[
\frac{10}{17} \quad \text{and} \quad \frac{15}{34}
\][/tex]
Since [tex]\(\frac{10}{17} \neq \frac{15}{34}\)[/tex] (their numerators and denominators are not equal), the ratios are not equivalent.
### Conclusion:
The simplified form of [tex]\(\frac{10}{17}\)[/tex] is [tex]\(\frac{10}{17}\)[/tex], and the simplified form of [tex]\(\frac{30}{68}\)[/tex] is [tex]\(\frac{15}{34}\)[/tex]. Since these simplified fractions are not equal, the original ratios are not equivalent.
