To determine if two fractions are equivalent, we need to check if they simplify to the same fraction. Let's evaluate each pair of fractions step-by-step:
1. Checking [tex]\( \frac{5}{6} \)[/tex] and [tex]\( \frac{20}{36} \)[/tex]:
- Simplify [tex]\( \frac{20}{36} \)[/tex]:
- The greatest common divisor (GCD) of 20 and 36 is 4.
- [tex]\( \frac{20}{36} = \frac{20 \div 4}{36 \div 4} = \frac{5}{9} \)[/tex].
- Since [tex]\( \frac{5}{6} \)[/tex] and [tex]\( \frac{5}{9} \)[/tex] are not the same, these fractions are not equivalent.
2. Checking [tex]\( \frac{4}{5} \)[/tex] and [tex]\( \frac{8}{20} \)[/tex]:
- Simplify [tex]\( \frac{8}{20} \)[/tex]:
- The GCD of 8 and 20 is 4.
- [tex]\( \frac{8}{20} = \frac{8 \div 4}{20 \div 4} = \frac{2}{5} \)[/tex].
- Since [tex]\( \frac{4}{5} \)[/tex] and [tex]\( \frac{2}{5} \)[/tex] are not the same, these fractions are not equivalent.
3. Checking [tex]\( \frac{3}{5} \)[/tex] and [tex]\( \frac{15}{30} \)[/tex]:
- Simplify [tex]\( \frac{15}{30} \)[/tex]:
- The GCD of 15 and 30 is 15.
- [tex]\( \frac{15}{30} = \frac{15 \div 15}{30 \div 15} = \frac{1}{2} \)[/tex].
- Since [tex]\( \frac{3}{5} \)[/tex] and [tex]\( \frac{1}{2} \)[/tex] are not the same, these fractions are not equivalent.
4. Checking [tex]\( \frac{6}{10} \)[/tex] and [tex]\( \frac{18}{30} \)[/tex]:
- Simplify [tex]\( \frac{18}{30} \)[/tex]:
- The GCD of 18 and 30 is 6.
- [tex]\( \frac{18}{30} = \frac{18 \div 6}{30 \div 6} = \frac{3}{5} \)[/tex].
- Simplify [tex]\( \frac{6}{10} \)[/tex]:
- The GCD of 6 and 10 is 2.
- [tex]\( \frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5} \)[/tex].
- Since [tex]\( \frac{6}{10} \)[/tex] and [tex]\( \frac{3}{5} \)[/tex] simplify to the same fraction, these fractions are equivalent.
In summary, among the given pairs of fractions, only [tex]\( \frac{6}{10} \)[/tex] and [tex]\( \frac{18}{30} \)[/tex] represent equivalent fractions. Therefore, the result is:
[tex]\[ \left[ \frac{5}{6} = \frac{20}{36} \, (\text{False}), \, \frac{4}{5} = \frac{8}{20} \, (\text{False}), \, \frac{3}{5} = \frac{15}{30} \, (\text{False}), \, \frac{6}{10} = \frac{18}{30} \, (\text{True}) \right]. \][/tex]
