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To determine which fraction pair is equivalent, we need to simplify each fraction and compare them with each other. Let's go through each pair step-by-step:
1. Pair: [tex]\(\frac{16}{64}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]
- Simplify [tex]\(\frac{16}{64}\)[/tex]:
[tex]\[ \text{Numerator: } 16, \quad \text{Denominator: } 64 \][/tex]
The greatest common divisor (GCD) of 16 and 64 is 16.
[tex]\[ \frac{16 \div 16}{64 \div 16} = \frac{1}{4} \][/tex]
- Simplify [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[ \frac{1}{4} \quad \text{is already in simplest form}. \][/tex]
- Both fractions simplify to [tex]\(\frac{1}{4}\)[/tex], so these fractions are equivalent.
2. Pair: [tex]\(\frac{8}{9}\)[/tex] and [tex]\(\frac{9}{8}\)[/tex]
- Simplify [tex]\(\frac{8}{9}\)[/tex]:
[tex]\[ \text{Numerator: } 8, \quad \text{Denominator: } 9 \][/tex]
The greatest common divisor (GCD) of 8 and 9 is 1.
[tex]\[ \frac{8 \div 1}{9 \div 1} = \frac{8}{9} \][/tex]
- Simplify [tex]\(\frac{9}{8}\)[/tex]:
[tex]\[ \text{Numerator: } 9, \quad \text{Denominator: } 8 \][/tex]
The greatest common divisor (GCD) of 9 and 8 is 1.
[tex]\[ \frac{9 \div 1}{8 \div 1} = \frac{9}{8} \][/tex]
- These fractions are not equivalent because [tex]\(\frac{8}{9} \ne \frac{9}{8}\)[/tex].
3. Pair: [tex]\(\frac{3}{6}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex]
- Simplify [tex]\(\frac{3}{6}\)[/tex]:
[tex]\[ \text{Numerator: } 3, \quad \text{Denominator: } 6 \][/tex]
The greatest common divisor (GCD) of 3 and 6 is 3.
[tex]\[ \frac{3 \div 3}{6 \div 3} = \frac{1}{2} \][/tex]
- Simplify [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \frac{1}{3} \quad \text{is already in simplest form}. \][/tex]
- These fractions are not equivalent because [tex]\(\frac{1}{2} \ne \frac{1}{3}\)[/tex].
4. Pair: [tex]\(\frac{5}{8}\)[/tex] and [tex]\(\frac{25}{48}\)[/tex]
- Simplify [tex]\(\frac{5}{8}\)[/tex]:
[tex]\[ \text{Numerator: } 5, \quad \text{Denominator: } 8 \][/tex]
The greatest common divisor (GCD) of 5 and 8 is 1.
[tex]\[ \frac{5 \div 1}{8 \div 1} = \frac{5}{8} \][/tex]
- Simplify [tex]\(\frac{25}{48}\)[/tex]:
[tex]\[ \text{Numerator: } 25, \quad \text{Denominator: } 48 \][/tex]
The greatest common divisor (GCD) of 25 and 48 is 1.
[tex]\[ \frac{25 \div 1}{48 \div 1} = \frac{25}{48} \][/tex]
- These fractions are not equivalent because [tex]\(\frac{5}{8} \ne \frac{25}{48}\)[/tex].
After simplifying and comparing each pair of fractions, we can see that the equivalent pair is [tex]\(\frac{16}{64}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex].
1. Pair: [tex]\(\frac{16}{64}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]
- Simplify [tex]\(\frac{16}{64}\)[/tex]:
[tex]\[ \text{Numerator: } 16, \quad \text{Denominator: } 64 \][/tex]
The greatest common divisor (GCD) of 16 and 64 is 16.
[tex]\[ \frac{16 \div 16}{64 \div 16} = \frac{1}{4} \][/tex]
- Simplify [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[ \frac{1}{4} \quad \text{is already in simplest form}. \][/tex]
- Both fractions simplify to [tex]\(\frac{1}{4}\)[/tex], so these fractions are equivalent.
2. Pair: [tex]\(\frac{8}{9}\)[/tex] and [tex]\(\frac{9}{8}\)[/tex]
- Simplify [tex]\(\frac{8}{9}\)[/tex]:
[tex]\[ \text{Numerator: } 8, \quad \text{Denominator: } 9 \][/tex]
The greatest common divisor (GCD) of 8 and 9 is 1.
[tex]\[ \frac{8 \div 1}{9 \div 1} = \frac{8}{9} \][/tex]
- Simplify [tex]\(\frac{9}{8}\)[/tex]:
[tex]\[ \text{Numerator: } 9, \quad \text{Denominator: } 8 \][/tex]
The greatest common divisor (GCD) of 9 and 8 is 1.
[tex]\[ \frac{9 \div 1}{8 \div 1} = \frac{9}{8} \][/tex]
- These fractions are not equivalent because [tex]\(\frac{8}{9} \ne \frac{9}{8}\)[/tex].
3. Pair: [tex]\(\frac{3}{6}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex]
- Simplify [tex]\(\frac{3}{6}\)[/tex]:
[tex]\[ \text{Numerator: } 3, \quad \text{Denominator: } 6 \][/tex]
The greatest common divisor (GCD) of 3 and 6 is 3.
[tex]\[ \frac{3 \div 3}{6 \div 3} = \frac{1}{2} \][/tex]
- Simplify [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \frac{1}{3} \quad \text{is already in simplest form}. \][/tex]
- These fractions are not equivalent because [tex]\(\frac{1}{2} \ne \frac{1}{3}\)[/tex].
4. Pair: [tex]\(\frac{5}{8}\)[/tex] and [tex]\(\frac{25}{48}\)[/tex]
- Simplify [tex]\(\frac{5}{8}\)[/tex]:
[tex]\[ \text{Numerator: } 5, \quad \text{Denominator: } 8 \][/tex]
The greatest common divisor (GCD) of 5 and 8 is 1.
[tex]\[ \frac{5 \div 1}{8 \div 1} = \frac{5}{8} \][/tex]
- Simplify [tex]\(\frac{25}{48}\)[/tex]:
[tex]\[ \text{Numerator: } 25, \quad \text{Denominator: } 48 \][/tex]
The greatest common divisor (GCD) of 25 and 48 is 1.
[tex]\[ \frac{25 \div 1}{48 \div 1} = \frac{25}{48} \][/tex]
- These fractions are not equivalent because [tex]\(\frac{5}{8} \ne \frac{25}{48}\)[/tex].
After simplifying and comparing each pair of fractions, we can see that the equivalent pair is [tex]\(\frac{16}{64}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex].
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