To determine which of the given fractions are equivalent to [tex]\(\frac{28}{32}\)[/tex], we need to compare them individually. We do this by simplifying [tex]\(\frac{28}{32}\)[/tex] and then checking it against each given fraction.
First, let's simplify [tex]\(\frac{28}{32}\)[/tex].
1. Find the greatest common divisor (GCD) of 28 and 32.
- The factors of 28 are: 1, 2, 4, 7, 14, 28
- The factors of 32 are: 1, 2, 4, 8, 16, 32
- The GCD of 28 and 32 is 4.
2. Divide the numerator and the denominator by the GCD.
[tex]\[
\frac{28 \div 4}{32 \div 4} = \frac{7}{8}
\][/tex]
So, [tex]\(\frac{28}{32}\)[/tex] simplifies to [tex]\(\frac{7}{8}\)[/tex].
Next, we need to compare this simplified fraction [tex]\(\frac{7}{8}\)[/tex] with the given fractions:
A. [tex]\(\frac{7}{8}\)[/tex]
- [tex]\(\frac{28}{32} = \frac{7}{8}\)[/tex]
- Therefore, [tex]\(\frac{7}{8}\)[/tex] is equivalent to [tex]\(\frac{28}{32}\)[/tex]. This checks out as True.
B. [tex]\(\frac{13}{16}\)[/tex]
- Compare [tex]\(\frac{7}{8}\)[/tex] with [tex]\(\frac{13}{16}\)[/tex].
- To compare, find a common denominator. The least common denominator (LCD) of 8 and 16 is 16.
- Convert [tex]\(\frac{7}{8}\)[/tex] to an equivalent fraction with a denominator of 16:
[tex]\[
\frac{7}{8} = \frac{7 \times 2}{8 \times 2} = \frac{14}{16}
\][/tex]
- Compare [tex]\(\frac{14}{16}\)[/tex] with [tex]\(\frac{13}{16}\)[/tex].
- [tex]\(\frac{14}{16} \neq \frac{13}{16}\)[/tex]
- Therefore, [tex]\(\frac{13}{16}\)[/tex] is not equivalent to [tex]\(\frac{28}{32}\)[/tex]. This checks out as False.
C. [tex]\(\frac{14}{16}\)[/tex]
- To compare [tex]\(\frac{7}{8}\)[/tex] with [tex]\(\frac{14}{16}\)[/tex], use the previous result.
- [tex]\(\frac{7}{8} = \frac{14}{16}\)[/tex]
- Therefore, [tex]\(\frac{14}{16}\)[/tex] is equivalent to [tex]\(\frac{28}{32}\)[/tex]. This checks out as True.
D. [tex]\(\frac{7}{12}\)[/tex]
- Compare [tex]\(\frac{7}{8}\)[/tex] with [tex]\(\frac{7}{12}\)[/tex]. To do this, find a common denominator. The least common multiple (LCM) of 8 and 12 is 24.
- Convert both fractions to equivalent fractions with denominator 24:
[tex]\[
\frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}
\][/tex]
[tex]\[
\frac{7}{12} = \frac{7 \times 2}{12 \times 2} = \frac{14}{24}
\][/tex]
- Compare [tex]\(\frac{21}{24}\)[/tex] with [tex]\(\frac{14}{24}\)[/tex].
- [tex]\(\frac{21}{24} \neq \frac{14}{24}\)[/tex]
- Therefore, [tex]\(\frac{7}{12}\)[/tex] is not equivalent to [tex]\(\frac{28}{32}\)[/tex]. This checks out as False.
In conclusion, the fractions equivalent to [tex]\(\frac{28}{32}\)[/tex] are:
A. [tex]\(\frac{7}{8}\)[/tex]
C. [tex]\(\frac{14}{16}\)[/tex]
