Answer :
To determine which pair of fractions is equivalent, we'll check each pair to see if they can be simplified to the same fraction.
Let's go through each pair step-by-step:
1. First Pair: [tex]\(\frac{12}{35}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex]
- Both fractions have the same denominator (35), but their numerators are different (12 and 14), so they are not equivalent.
2. Second Pair: [tex]\(\frac{14}{21}\)[/tex] and [tex]\(\frac{8}{20}\)[/tex]
- Simplify [tex]\(\frac{14}{21}\)[/tex]:
- The greatest common divisor (GCD) of 14 and 21 is 7.
- [tex]\(\frac{14}{21} = \frac{14 \div 7}{21 \div 7} = \frac{2}{3}\)[/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].
- The simplified fractions are [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex], which are not equivalent.
3. Third Pair: [tex]\(\frac{15}{25}\)[/tex] and [tex]\(\frac{24}{30}\)[/tex]
- Simplify [tex]\(\frac{15}{25}\)[/tex]:
- The GCD of 15 and 25 is 5.
- [tex]\(\frac{15}{25} = \frac{15 \div 5}{25 \div 5} = \frac{3}{5}\)[/tex].
- Simplify [tex]\(\frac{24}{30}\)[/tex]:
- The GCD of 24 and 30 is 6.
- [tex]\(\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}\)[/tex].
- The simplified fractions are [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex], which are not equivalent.
4. Fourth Pair: [tex]\(\frac{18}{45}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex]
- Simplify [tex]\(\frac{18}{45}\)[/tex]:
- The GCD of 18 and 45 is 9.
- [tex]\(\frac{18}{45} = \frac{18 \div 9}{45 \div 9} = \frac{2}{5}\)[/tex].
- Simplify [tex]\(\frac{14}{35}\)[/tex]:
- The GCD of 14 and 35 is 7.
- [tex]\(\frac{14}{35} = \frac{14 \div 7}{35 \div 7} = \frac{2}{5}\)[/tex].
- The simplified fractions are [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex], which are equivalent.
Therefore, the fraction pair [tex]\(\frac{18}{45}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex] is equivalent.
Let's go through each pair step-by-step:
1. First Pair: [tex]\(\frac{12}{35}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex]
- Both fractions have the same denominator (35), but their numerators are different (12 and 14), so they are not equivalent.
2. Second Pair: [tex]\(\frac{14}{21}\)[/tex] and [tex]\(\frac{8}{20}\)[/tex]
- Simplify [tex]\(\frac{14}{21}\)[/tex]:
- The greatest common divisor (GCD) of 14 and 21 is 7.
- [tex]\(\frac{14}{21} = \frac{14 \div 7}{21 \div 7} = \frac{2}{3}\)[/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].
- The simplified fractions are [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex], which are not equivalent.
3. Third Pair: [tex]\(\frac{15}{25}\)[/tex] and [tex]\(\frac{24}{30}\)[/tex]
- Simplify [tex]\(\frac{15}{25}\)[/tex]:
- The GCD of 15 and 25 is 5.
- [tex]\(\frac{15}{25} = \frac{15 \div 5}{25 \div 5} = \frac{3}{5}\)[/tex].
- Simplify [tex]\(\frac{24}{30}\)[/tex]:
- The GCD of 24 and 30 is 6.
- [tex]\(\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}\)[/tex].
- The simplified fractions are [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex], which are not equivalent.
4. Fourth Pair: [tex]\(\frac{18}{45}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex]
- Simplify [tex]\(\frac{18}{45}\)[/tex]:
- The GCD of 18 and 45 is 9.
- [tex]\(\frac{18}{45} = \frac{18 \div 9}{45 \div 9} = \frac{2}{5}\)[/tex].
- Simplify [tex]\(\frac{14}{35}\)[/tex]:
- The GCD of 14 and 35 is 7.
- [tex]\(\frac{14}{35} = \frac{14 \div 7}{35 \div 7} = \frac{2}{5}\)[/tex].
- The simplified fractions are [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex], which are equivalent.
Therefore, the fraction pair [tex]\(\frac{18}{45}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex] is equivalent.
