Answer :
To determine which pair of fractions is equivalent, we'll simplify each fraction and compare their simplest forms. Two fractions are equivalent if they have the same simplest form.
Let's go through each pair of fractions:
1. First 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.
- Divide both the numerator and the denominator by 7:
[tex]\(\frac{14 \div 7}{21 \div 7} = \frac{2}{3}\)[/tex].
- Simplify [tex]\(\frac{8}{20}\)[/tex]:
- The GCD of 8 and 20 is 4.
- Divide both the numerator and the denominator by 4:
[tex]\(\frac{8 \div 4}{20 \div 4} = \frac{2}{5}\)[/tex].
These fractions simplify to [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex] respectively, which are not equivalent.
2. Second 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.
- Divide both the numerator and the denominator by 9:
[tex]\(\frac{18 \div 9}{45 \div 9} = \frac{2}{5}\)[/tex].
- Simplify [tex]\(\frac{14}{35}\)[/tex]:
- The GCD of 14 and 35 is 7.
- Divide both the numerator and the denominator by 7:
[tex]\(\frac{14 \div 7}{35 \div 7} = \frac{2}{5}\)[/tex].
Both fractions simplify to [tex]\(\frac{2}{5}\)[/tex], so they are 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.
- Divide both the numerator and the denominator by 5:
[tex]\(\frac{15 \div 5}{25 \div 5} = \frac{3}{5}\)[/tex].
- Simplify [tex]\(\frac{24}{30}\)[/tex]:
- The GCD of 24 and 30 is 6.
- Divide both the numerator and the denominator by 6:
[tex]\(\frac{24 \div 6}{30 \div 6} = \frac{4}{5}\)[/tex].
These fractions simplify to [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex] respectively, which are not equivalent.
4. Fourth Pair: [tex]\(\frac{12}{35}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex]
- These fractions have the same denominator (35), but different numerators (12 and 14).
- Without simplification, they clearly aren't equivalent.
The only pair of equivalent fractions is the second pair: [tex]\(\frac{18}{45}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex].
Let's go through each pair of fractions:
1. First 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.
- Divide both the numerator and the denominator by 7:
[tex]\(\frac{14 \div 7}{21 \div 7} = \frac{2}{3}\)[/tex].
- Simplify [tex]\(\frac{8}{20}\)[/tex]:
- The GCD of 8 and 20 is 4.
- Divide both the numerator and the denominator by 4:
[tex]\(\frac{8 \div 4}{20 \div 4} = \frac{2}{5}\)[/tex].
These fractions simplify to [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex] respectively, which are not equivalent.
2. Second 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.
- Divide both the numerator and the denominator by 9:
[tex]\(\frac{18 \div 9}{45 \div 9} = \frac{2}{5}\)[/tex].
- Simplify [tex]\(\frac{14}{35}\)[/tex]:
- The GCD of 14 and 35 is 7.
- Divide both the numerator and the denominator by 7:
[tex]\(\frac{14 \div 7}{35 \div 7} = \frac{2}{5}\)[/tex].
Both fractions simplify to [tex]\(\frac{2}{5}\)[/tex], so they are 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.
- Divide both the numerator and the denominator by 5:
[tex]\(\frac{15 \div 5}{25 \div 5} = \frac{3}{5}\)[/tex].
- Simplify [tex]\(\frac{24}{30}\)[/tex]:
- The GCD of 24 and 30 is 6.
- Divide both the numerator and the denominator by 6:
[tex]\(\frac{24 \div 6}{30 \div 6} = \frac{4}{5}\)[/tex].
These fractions simplify to [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex] respectively, which are not equivalent.
4. Fourth Pair: [tex]\(\frac{12}{35}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex]
- These fractions have the same denominator (35), but different numerators (12 and 14).
- Without simplification, they clearly aren't equivalent.
The only pair of equivalent fractions is the second pair: [tex]\(\frac{18}{45}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex].
