Certainly! Let's solve each fraction equivalence step-by-step:
### Given Fractions:
1. [tex]\(\frac{1}{5} = \frac{3}{2}\)[/tex]
This fraction equality is already complete, so there's no need to solve further.
2. [tex]\(\frac{1}{10} = \frac{x}{20}\)[/tex]
To find [tex]\(x\)[/tex], we set up the proportion:
[tex]\[
\frac{1}{10} = \frac{x}{20}
\][/tex]
Cross-multiplying to solve for [tex]\(x\)[/tex]:
[tex]\[
1 \cdot 20 = 10 \cdot x \implies 20 = 10x \implies x = \frac{20}{10} \implies x = 2
\][/tex]
So, [tex]\(\frac{1}{10} = \frac{2}{20}\)[/tex].
3. [tex]\(\frac{3}{4} = \frac{9}{9}\)[/tex]
This fraction equality is already complete, so there's no need to solve further.
4. [tex]\(\frac{1}{2} = \frac{9}{y}\)[/tex]
To find [tex]\(y\)[/tex], we set up the proportion:
[tex]\[
\frac{1}{2} = \frac{9}{y}
\][/tex]
Cross-multiplying to solve for [tex]\(y\)[/tex]:
[tex]\[
1 \cdot y = 2 \cdot 9 \implies y = 18
\][/tex]
So, [tex]\(\frac{1}{2} = \frac{9}{18}\)[/tex].
5. [tex]\(\frac{1}{3} = \frac{x}{12}\)[/tex]
To find [tex]\(x\)[/tex], we set up the proportion:
[tex]\[
\frac{1}{3} = \frac{x}{12}
\][/tex]
Cross-multiplying to solve for [tex]\(x\)[/tex]:
[tex]\[
1 \cdot 12 = 3 \cdot x \implies 12 = 3x \implies x = \frac{12}{3} \implies x = 4
\][/tex]
So, [tex]\(\frac{1}{3} = \frac{4}{12}\)[/tex].
6. [tex]\(\frac{2}{4} = \frac{8}{y}\)[/tex]
To find [tex]\(y\)[/tex], we set up the proportion:
[tex]\[
\frac{2}{4} = \frac{8}{y}
\][/tex]
Cross-multiplying to solve for [tex]\(y\)[/tex]:
[tex]\[
2 \cdot y = 4 \cdot 8 \implies 2y = 32 \implies y = \frac{32}{2} \implies y = 16
\][/tex]
So, [tex]\(\frac{2}{4} = \frac{8}{16}\)[/tex].
7. [tex]\(\frac{1}{12} = \frac{2}{y}\)[/tex]
To find [tex]\(y\)[/tex], we set up the proportion:
[tex]\[
\frac{1}{12} = \frac{2}{y}
\][/tex]
Cross-multiplying to solve for [tex]\(y\)[/tex]:
[tex]\[
1 \cdot y = 12 \cdot 2 \implies y = 24
\][/tex]
So, [tex]\(\frac{1}{12} = \frac{2}{24}\)[/tex].
8. [tex]\(\frac{2}{9} = \frac{x}{18}\)[/tex]
To find [tex]\(x\)[/tex], we set up the proportion:
[tex]\[
\frac{2}{9} = \frac{x}{18}
\][/tex]
Cross-multiplying to solve for [tex]\(x\)[/tex]:
[tex]\[
2 \cdot 18 = 9 \cdot x \implies 36 = 9x \implies x = \frac{36}{9} \implies x = 4
\][/tex]
So, [tex]\(\frac{2}{9} = \frac{4}{18}\)[/tex].
### Final Results:
1. [tex]\(\frac{1}{5} = \frac{3}{2}\)[/tex]
2. [tex]\(\frac{1}{10} = \frac{2}{20}\)[/tex]
3. [tex]\(\frac{3}{4} = \frac{9}{9}\)[/tex]
4. [tex]\(\frac{1}{2} = \frac{9}{18}\)[/tex]
5. [tex]\(\frac{1}{3} = \frac{4}{12}\)[/tex]
6. [tex]\(\frac{2}{4} = \frac{8}{16}\)[/tex]
7. [tex]\(\frac{1}{12} = \frac{2}{24}\)[/tex]
8. [tex]\(\frac{2}{9} = \frac{4}{18}\)[/tex]
These are the equivalent fractions found for each given pair.
