Answer :
To determine which of the given fraction pairs are equivalent, we need to simplify each fraction to its lowest terms and then compare the simplified forms.
### First Fraction Pair: [tex]\(\frac{18}{45}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex]
Simplifying [tex]\(\frac{18}{45}\)[/tex]:
1. Determine the greatest common divisor (GCD) of 18 and 45. The GCD is 9.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{18 \div 9}{45 \div 9} = \frac{2}{5}
\][/tex]
Simplifying [tex]\(\frac{14}{35}\)[/tex]:
1. Determine the GCD of 14 and 35. The GCD is 7.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{14 \div 7}{35 \div 7} = \frac{2}{5}
\][/tex]
Since both fractions simplify to [tex]\(\frac{2}{5}\)[/tex], the fraction pair [tex]\(\frac{18}{45}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex] is equivalent.
### Second Fraction Pair: [tex]\(\frac{12}{35}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex]
Simplifying [tex]\(\frac{12}{35}\)[/tex]:
1. The GCD of 12 and 35 is 1.
2. The fraction is already in its simplest form:
[tex]\[
\frac{12}{35}
\][/tex]
Simplifying [tex]\(\frac{14}{35}\)[/tex]:
1. Determine the GCD of 14 and 35. The GCD is 7.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{14 \div 7}{35 \div 7} = \frac{2}{5}
\][/tex]
Since [tex]\(\frac{12}{35}\)[/tex] does not simplify to [tex]\(\frac{2}{5}\)[/tex], the fraction pair [tex]\(\frac{12}{35}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex] is not equivalent.
### Third Fraction Pair: [tex]\(\frac{14}{21}\)[/tex] and [tex]\(\frac{8}{20}\)[/tex]
Simplifying [tex]\(\frac{14}{21}\)[/tex]:
1. Determine the GCD of 14 and 21. The GCD is 7.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{14 \div 7}{21 \div 7} = \frac{2}{3}
\][/tex]
Simplifying [tex]\(\frac{8}{20}\)[/tex]:
1. Determine the GCD of 8 and 20. The GCD is 4.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{8 \div 4}{20 \div 4} = \frac{2}{5}
\][/tex]
Since [tex]\(\frac{2}{3}\)[/tex] is not equal to [tex]\(\frac{2}{5}\)[/tex], the fraction pair [tex]\(\frac{14}{21}\)[/tex] and [tex]\(\frac{8}{20}\)[/tex] is not equivalent.
### Fourth Fraction Pair: [tex]\(\frac{15}{25}\)[/tex] and [tex]\(\frac{24}{30}\)[/tex]
Simplifying [tex]\(\frac{15}{25}\)[/tex]:
1. Determine the GCD of 15 and 25. The GCD is 5.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{15 \div 5}{25 \div 5} = \frac{3}{5}
\][/tex]
Simplifying [tex]\(\frac{24}{30}\)[/tex]:
1. Determine the GCD of 24 and 30. The GCD is 6.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{24 \div 6}{30 \div 6} = \frac{4}{5}
\][/tex]
Since [tex]\(\frac{3}{5}\)[/tex] is not equal to [tex]\(\frac{4}{5}\)[/tex], the fraction pair [tex]\(\frac{15}{25}\)[/tex] and [tex]\(\frac{24}{30}\)[/tex] is not equivalent.
### Conclusion:
The only fraction pair that is equivalent is:
[tex]\[
\frac{18}{45} \text{ and } \frac{14}{35}
\][/tex]
### First Fraction Pair: [tex]\(\frac{18}{45}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex]
Simplifying [tex]\(\frac{18}{45}\)[/tex]:
1. Determine the greatest common divisor (GCD) of 18 and 45. The GCD is 9.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{18 \div 9}{45 \div 9} = \frac{2}{5}
\][/tex]
Simplifying [tex]\(\frac{14}{35}\)[/tex]:
1. Determine the GCD of 14 and 35. The GCD is 7.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{14 \div 7}{35 \div 7} = \frac{2}{5}
\][/tex]
Since both fractions simplify to [tex]\(\frac{2}{5}\)[/tex], the fraction pair [tex]\(\frac{18}{45}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex] is equivalent.
### Second Fraction Pair: [tex]\(\frac{12}{35}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex]
Simplifying [tex]\(\frac{12}{35}\)[/tex]:
1. The GCD of 12 and 35 is 1.
2. The fraction is already in its simplest form:
[tex]\[
\frac{12}{35}
\][/tex]
Simplifying [tex]\(\frac{14}{35}\)[/tex]:
1. Determine the GCD of 14 and 35. The GCD is 7.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{14 \div 7}{35 \div 7} = \frac{2}{5}
\][/tex]
Since [tex]\(\frac{12}{35}\)[/tex] does not simplify to [tex]\(\frac{2}{5}\)[/tex], the fraction pair [tex]\(\frac{12}{35}\)[/tex] and [tex]\(\frac{14}{35}\)[/tex] is not equivalent.
### Third Fraction Pair: [tex]\(\frac{14}{21}\)[/tex] and [tex]\(\frac{8}{20}\)[/tex]
Simplifying [tex]\(\frac{14}{21}\)[/tex]:
1. Determine the GCD of 14 and 21. The GCD is 7.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{14 \div 7}{21 \div 7} = \frac{2}{3}
\][/tex]
Simplifying [tex]\(\frac{8}{20}\)[/tex]:
1. Determine the GCD of 8 and 20. The GCD is 4.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{8 \div 4}{20 \div 4} = \frac{2}{5}
\][/tex]
Since [tex]\(\frac{2}{3}\)[/tex] is not equal to [tex]\(\frac{2}{5}\)[/tex], the fraction pair [tex]\(\frac{14}{21}\)[/tex] and [tex]\(\frac{8}{20}\)[/tex] is not equivalent.
### Fourth Fraction Pair: [tex]\(\frac{15}{25}\)[/tex] and [tex]\(\frac{24}{30}\)[/tex]
Simplifying [tex]\(\frac{15}{25}\)[/tex]:
1. Determine the GCD of 15 and 25. The GCD is 5.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{15 \div 5}{25 \div 5} = \frac{3}{5}
\][/tex]
Simplifying [tex]\(\frac{24}{30}\)[/tex]:
1. Determine the GCD of 24 and 30. The GCD is 6.
2. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{24 \div 6}{30 \div 6} = \frac{4}{5}
\][/tex]
Since [tex]\(\frac{3}{5}\)[/tex] is not equal to [tex]\(\frac{4}{5}\)[/tex], the fraction pair [tex]\(\frac{15}{25}\)[/tex] and [tex]\(\frac{24}{30}\)[/tex] is not equivalent.
### Conclusion:
The only fraction pair that is equivalent is:
[tex]\[
\frac{18}{45} \text{ and } \frac{14}{35}
\][/tex]
