To determine which two of the given fractions multiply together to yield [tex]\(\frac{9}{28}\)[/tex], we need to check the product of each possible pair of fractions.
Given fractions:
[tex]\[
\frac{1}{4}, \frac{6}{23}, \frac{2}{7}, \frac{3}{5}, \frac{9}{8}
\][/tex]
We need to find pairs whose product equals [tex]\(\frac{9}{28}\)[/tex].
First, let's define the multiplication of two fractions:
[tex]\[
\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}
\][/tex]
We will proceed by checking each possible combination:
1. [tex]\(\frac{1}{4} \times \frac{6}{23}\)[/tex]:
[tex]\[
\frac{1 \cdot 6}{4 \cdot 23} = \frac{6}{92} = \frac{3}{46} \quad \text{(not equal to) } \frac{9}{28}
\][/tex]
2. [tex]\(\frac{1}{4} \times \frac{2}{7}\)[/tex]:
[tex]\[
\frac{1 \cdot 2}{4 \cdot 7} = \frac{2}{28} = \frac{1}{14} \quad \text{(not equal to) } \frac{9}{28}
\][/tex]
3. [tex]\(\frac{1}{4} \times \frac{3}{5}\)[/tex]:
[tex]\[
\frac{1 \cdot 3}{4 \cdot 5} = \frac{3}{20} \quad \text{(not equal to) } \frac{9}{28}
\][/tex]
4. [tex]\(\frac{1}{4} \times \frac{9}{8}\)[/tex]:
[tex]\[
\frac{1 \cdot 9}{4 \cdot 8} = \frac{9}{32} \quad \text{(not equal to) } \frac{9}{28}
\][/tex]
5. [tex]\(\frac{6}{23} \times \frac{2}{7}\)[/tex]:
[tex]\[
\frac{6 \cdot 2}{23 \cdot 7} = \frac{12}{161} \quad \text{(not equal to) } \frac{9}{28}
\][/tex]
6. [tex]\(\frac{6}{23} \times \frac{3}{5}\)[/tex]:
[tex]\[
\frac{6 \cdot 3}{23 \cdot 5} = \frac{18}{115} \quad \text{(not equal to) } \frac{9}{28}
\][/tex]
7. [tex]\(\frac{6}{23} \times \frac{9}{8}\)[/tex]:
[tex]\[
\frac{6 \cdot 9}{23 \cdot 8} = \frac{54}{184} = \frac{27}{92} \quad \text{(not equal to) } \frac{9}{28}
\][/tex]
8. [tex]\(\frac{2}{7} \times \frac{3}{5}\)[/tex]:
[tex]\[
\frac{2 \cdot 3}{7 \cdot 5} = \frac{6}{35} \quad \text{(not equal to) } \frac{9}{28}
\][/tex]
9. [tex]\(\frac{2}{7} \times \frac{9}{8}\)[/tex]:
[tex]\[
\frac{2 \cdot 9}{\7 \cdot 8} = \frac{18}{56} = \frac{9}{28} \quad \text{(equal to) } \frac{9}{28}
\][/tex]
10. [tex]\(\frac{3}{5} \times \frac{9}{8}\)[/tex]:
[tex]\[
\frac{3 \cdot 9}{5 \cdot 8} = \frac{27}{40} \quad \text{(not equal to) } \frac{9}{28}
\][/tex]
After checking all pairs, we find that the fractions [tex]\(\frac{2}{7}\)[/tex] and [tex]\(\frac{9}{8}\)[/tex] multiply together to give [tex]\(\frac{9}{28}\)[/tex].
So, the two fractions are:
[tex]\[
\frac{2}{7} \quad \text{and} \quad \frac{9}{8}
\][/tex]
Therefore, [tex]\(\boxed{\frac{2}{7} \text{ and } \frac{9}{8}}\)[/tex].
