Absolutely, Sophia is correct. Let's carefully walk through the problem step-by-step.
1. Original Ratio: We start with the given ratio, which is 60:36.
2. Multiplying by [tex]\(\frac{1}{6}\)[/tex]: Sophia suggests multiplying both numbers in the original ratio by [tex]\(\frac{1}{6}\)[/tex]. Let's do that:
[tex]\[
60 \times \frac{1}{6} = 10
\][/tex]
[tex]\[
36 \times \frac{1}{6} = 6
\][/tex]
3. New Ratio: After performing the multiplication, the new numbers we get are 10 and 6. This gives us a new ratio of 10:6.
4. Simplifying the New Ratio: We need to check if the new ratio (10:6) is equivalent to the original ratio (60:36). For that, let's simplify both ratios to their lowest terms:
- For 60:36:
- The greatest common divisor (GCD) of 60 and 36 is 12.
- Dividing both parts by 12, we get:
[tex]\[
\frac{60}{12} : \frac{36}{12} = 5 : 3
\][/tex]
- For 10:6:
- The greatest common divisor (GCD) of 10 and 6 is 2.
- Dividing both parts by 2, we get:
[tex]\[
\frac{10}{2} : \frac{6}{2} = 5 : 3
\][/tex]
5. Verification: Both ratios simplify to the same lowest terms of 5:3. Hence, the new ratio (10:6) maintains the same proportion as the original ratio (60:36).
6. Ratio Comparison: We can also confirm by comparing the division of both ratios:
[tex]\[
\frac{60}{36} = \frac{5}{3}
\][/tex]
[tex]\[
\frac{10}{6} = \frac{5}{3}
\][/tex]
Both yield the same value, confirming they are indeed equivalent.
Thus, Sophia's method of multiplying both quantities in the ratio 60:36 by [tex]\(\frac{1}{6}\)[/tex] is correct, as it maintains the proportional relationship, rendering the ratios equivalent. This confirms that the new ratio (10:6) is correctly scaled down from the original ratio (60:36).
