Let's evaluate each fraction pair to determine if they are equivalent.
To determine if two fractions are equivalent, we need to see if they simplify to the same fraction or denominator:
a. \( \frac{1}{4} \) and \( \frac{8}{32} \)
- Simplify \( \frac{8}{32} \): Divide both the numerator and the denominator by their greatest common divisor, which is 8.
[tex]\[
\frac{8}{32} = \frac{8 \div 8}{32 \div 8} = \frac{1}{4}
\][/tex]
- Since \( \frac{1}{4} = \frac{1}{4} \), these fractions are equivalent.
b. \( \frac{1}{4} \) and \( \frac{9}{32} \)
- Simplify \( \frac{9}{32} \): The greatest common divisor of 9 and 32 is 1, so \( \frac{9}{32} \) cannot be simplified further.
[tex]\[
\frac{9}{32} \neq \frac{1}{4}
\][/tex]
- Since \( \frac{1}{4} \neq \frac{9}{32} \), these fractions are not equivalent.
c. \( \frac{1}{4} \) and \( \frac{3}{8} \)
- To compare these fractions, let's find a common denominator: The least common multiple of 4 and 8 is 8.
[tex]\[
\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}
\][/tex]
[tex]\[
\frac{3}{8} \text{ remains } \frac{3}{8}
\][/tex]
- Since \( \frac{2}{8} \neq \frac{3}{8} \), these fractions are not equivalent.
d. \( \frac{1}{4} \) and \( \frac{42}{128} \)
- Simplify \( \frac{42}{128} \): Divide both the numerator and the denominator by their greatest common divisor, which is 2.
[tex]\[
\frac{42}{128} = \frac{42 \div 2}{128 \div 2} = \frac{21}{64}
\][/tex]
- Simplifying further, we see \( \frac{21}{64} \) can't be simplified any more because 21 and 64 do not have any common factors.
[tex]\[
\frac{21}{64} \neq \frac{1}{4}
\][/tex]
- Since \( \frac{1}{4} \neq \frac{21}{64} \), these fractions are not equivalent.
So, the fraction pair that is equivalent is:
a. \( \frac{1}{4} \) and \( \frac{8}{32} \)
The correct answer is:
a. [tex]\( \frac{1}{4} \)[/tex] and [tex]\( 8 / 32 \)[/tex].
