Grant Newton
06/17/2024 · Elementary School
(vi) \( 3 \frac{1}{2}: 2 \frac{1}{4} \)
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Tutor-Verified Answer
Step-by-step Solution
To simplify the ratio \( 3 \frac{1}{2} : 2 \frac{1}{4} \), we first convert the mixed numbers into improper fractions.
1. Convert \( 3 \frac{1}{2} \):
\[
3 \frac{1}{2} = 3 \times 2 + 1 = 6 + 1 = 7 \quad \text{(so, } 3 \frac{1}{2} = \frac{7}{2}\text{)}
\]
2. Convert \( 2 \frac{1}{4} \):
\[
2 \frac{1}{4} = 2 \times 4 + 1 = 8 + 1 = 9 \quad \text{(so, } 2 \frac{1}{4} = \frac{9}{4}\text{)}
\]
Now we can rewrite the ratio:
\[
3 \frac{1}{2} : 2 \frac{1}{4} = \frac{7}{2} : \frac{9}{4}
\]
To simplify the ratio of two fractions, we can multiply by the reciprocal of the second fraction:
\[
\frac{7}{2} : \frac{9}{4} = \frac{7}{2} \times \frac{4}{9} = \frac{7 \times 4}{2 \times 9} = \frac{28}{18}
\]
Now, we simplify \( \frac{28}{18} \):
\[
\frac{28}{18} = \frac{14}{9} \quad \text{(dividing both numerator and denominator by 2)}
\]
Thus, the simplified ratio is:
\[
\boxed{14 : 9}
\]
Quick Answer
The simplified ratio is 14:9.
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