Certainly! Let's simplify each given ratio so that the second quantity in each ratio is 1.
### Ratio 1: [tex]\(\frac{14}{5} : \frac{7}{5}\)[/tex]
To simplify [tex]\(\frac{14}{5}: \frac{7}{5}\)[/tex], we want the second quantity to be 1.
- We can do this by dividing both parts of the ratio by [tex]\(\frac{7}{5}\)[/tex]:
[tex]\[ \left(\frac{14}{5}\right) \div \left(\frac{7}{5}\right) = \frac{\frac{14}{5}}{\frac{7}{5}} = \frac{14}{5} \times \frac{5}{7} = \frac{14 \times 5}{5 \times 7} = \frac{14}{7} = 2 \][/tex]
Thus, the simplified ratio [tex]\(\frac{14}{5} : \frac{7}{5}\)[/tex] is equivalent to [tex]\(2 : 1\)[/tex].
### Ratio 2: [tex]\(3 : \frac{1}{4}\)[/tex]
To simplify [tex]\(3 : \frac{1}{4}\)[/tex], we want the second quantity to be 1.
- We can do this by dividing both parts of the ratio by [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[ 3 \div \left(\frac{1}{4}\right) = 3 \times 4 = 12 \][/tex]
Thus, the simplified ratio [tex]\(3 : \frac{1}{4}\)[/tex] is equivalent to [tex]\(12 : 1\)[/tex].
### Ratio 3: [tex]\(\frac{1}{2} : \frac{1}{6}\)[/tex]
To simplify [tex]\(\frac{1}{2} : \frac{1}{6}\)[/tex], we want the second quantity to be 1.
- We can do this by dividing both parts of the ratio by [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[ \left(\frac{1}{2}\right) \div \left(\frac{1}{6}\right) = \frac{\frac{1}{2}}{\frac{1}{6}} = \frac{1}{2} \times 6 = 3 \][/tex]
Thus, the simplified ratio [tex]\(\frac{1}{2} : \frac{1}{6}\)[/tex] is equivalent to [tex]\(3 : 1\)[/tex].
### Summary
- [tex]\(\frac{14}{5} : \frac{7}{5}\)[/tex] simplifies to [tex]\(2 : 1\)[/tex]
- [tex]\(3 : \frac{1}{4}\)[/tex] simplifies to [tex]\(12 : 1\)[/tex]
- [tex]\(\frac{1}{2} : \frac{1}{6}\)[/tex] simplifies to [tex]\(3 : 1\)[/tex]
So the final results matching each ratio with their corresponding simplified ratio are:
- [tex]\(\frac{14}{5} : \frac{7}{5}\)[/tex] matches with [tex]\(2 : 1\)[/tex]
- [tex]\(3 : \frac{1}{4}\)[/tex] matches with [tex]\(12 : 1\)[/tex]
- [tex]\(\frac{1}{2} : \frac{1}{6}\)[/tex] matches with [tex]\(3 : 1\)[/tex]
