Answer :
To solve whether the given pairs of fractions are equivalent, we first need to simplify each fraction and compare them. Let's go through each step.
### Step-by-Step Solution:
1. List the fractions and numbers:
- Fractions: [tex]\( 9 \)[/tex], [tex]\( \frac{18}{10} \)[/tex], [tex]\( \frac{20}{2} \)[/tex], [tex]\( \frac{34}{85} \)[/tex]
- Pairs of fractions to compare:
- [tex]\( \left(\frac{8}{9}, \frac{17}{18}\right) \)[/tex]
- [tex]\( (10, 6) \)[/tex] (Note: This entry does not seem to be a fraction pair but rather two numbers.)
2. Simplify each fraction:
- For [tex]\( 9 \)[/tex], it stays as [tex]\( 9 \)[/tex].
- For [tex]\( \frac{18}{10} \)[/tex]: Simplify by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 2:
[tex]\[
\frac{18}{10} = \frac{18 \div 2}{10 \div 2} = \frac{9}{5}
\][/tex]
- For [tex]\( \frac{20}{2} \)[/tex]: Simplify by dividing the numerator and the denominator by their GCD, which is 2:
[tex]\[
\frac{20}{2} = \frac{20 \div 2}{2 \div 2} = 10
\][/tex]
- For [tex]\( \frac{34}{85} \)[/tex]: Simplify by dividing the numerator and the denominator by their GCD, which is 17:
[tex]\[
\frac{34}{85} = \frac{34 \div 17}{85 \div 17} = \frac{2}{5}
\][/tex]
3. Results of simplification:
[tex]\[
[9, \frac{9}{5}, 10, \frac{2}{5}]
\][/tex]
#### Compare the given pairs of fractions:
4. First pair: [tex]\( \left(\frac{8}{9}, \frac{17}{18}\right) \)[/tex]:
- Simplify [tex]\( \frac{8}{9} \)[/tex] (it is already in simplest form).
- Simplify [tex]\( \frac{17}{18} \)[/tex] (it is already in simplest form).
- Compare [tex]\( \frac{8}{9} \)[/tex] and [tex]\( \frac{17}{18} \)[/tex]:
[tex]\[
\frac{8}{9} \neq \frac{17}{18}
\][/tex]
-> This pair is not equivalent.
5. Second pair: [tex]\( (10, 6) \)[/tex]:
- Compare the numbers [tex]\( 10 \)[/tex] and [tex]\( 6 \)[/tex]:
- They are not the same.
-> This pair is not equivalent.
### Final Answer:
The simplified fractions are: [tex]\[ [9, \frac{9}{5}, 10, \frac{2}{5}] \][/tex]
The pairs of fractions: [tex]\[ [(\frac{8}{9}, \frac{17}{18}), (10, 6)] \][/tex] are not equivalent.
### Step-by-Step Solution:
1. List the fractions and numbers:
- Fractions: [tex]\( 9 \)[/tex], [tex]\( \frac{18}{10} \)[/tex], [tex]\( \frac{20}{2} \)[/tex], [tex]\( \frac{34}{85} \)[/tex]
- Pairs of fractions to compare:
- [tex]\( \left(\frac{8}{9}, \frac{17}{18}\right) \)[/tex]
- [tex]\( (10, 6) \)[/tex] (Note: This entry does not seem to be a fraction pair but rather two numbers.)
2. Simplify each fraction:
- For [tex]\( 9 \)[/tex], it stays as [tex]\( 9 \)[/tex].
- For [tex]\( \frac{18}{10} \)[/tex]: Simplify by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 2:
[tex]\[
\frac{18}{10} = \frac{18 \div 2}{10 \div 2} = \frac{9}{5}
\][/tex]
- For [tex]\( \frac{20}{2} \)[/tex]: Simplify by dividing the numerator and the denominator by their GCD, which is 2:
[tex]\[
\frac{20}{2} = \frac{20 \div 2}{2 \div 2} = 10
\][/tex]
- For [tex]\( \frac{34}{85} \)[/tex]: Simplify by dividing the numerator and the denominator by their GCD, which is 17:
[tex]\[
\frac{34}{85} = \frac{34 \div 17}{85 \div 17} = \frac{2}{5}
\][/tex]
3. Results of simplification:
[tex]\[
[9, \frac{9}{5}, 10, \frac{2}{5}]
\][/tex]
#### Compare the given pairs of fractions:
4. First pair: [tex]\( \left(\frac{8}{9}, \frac{17}{18}\right) \)[/tex]:
- Simplify [tex]\( \frac{8}{9} \)[/tex] (it is already in simplest form).
- Simplify [tex]\( \frac{17}{18} \)[/tex] (it is already in simplest form).
- Compare [tex]\( \frac{8}{9} \)[/tex] and [tex]\( \frac{17}{18} \)[/tex]:
[tex]\[
\frac{8}{9} \neq \frac{17}{18}
\][/tex]
-> This pair is not equivalent.
5. Second pair: [tex]\( (10, 6) \)[/tex]:
- Compare the numbers [tex]\( 10 \)[/tex] and [tex]\( 6 \)[/tex]:
- They are not the same.
-> This pair is not equivalent.
### Final Answer:
The simplified fractions are: [tex]\[ [9, \frac{9}{5}, 10, \frac{2}{5}] \][/tex]
The pairs of fractions: [tex]\[ [(\frac{8}{9}, \frac{17}{18}), (10, 6)] \][/tex] are not equivalent.
