Let's solve the problem step-by-step to determine which number belongs in the box to make the equation true.
First, we need to rewrite the mixed numbers as improper fractions:
1. Convert [tex]\( 3 \frac{1}{2} \)[/tex]:
[tex]\[
3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2}
\][/tex]
2. Convert [tex]\( 5 \frac{2}{3} \)[/tex]:
[tex]\[
5 \frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{17}{3}
\][/tex]
Now, we need to divide these two improper fractions:
[tex]\[
\frac{7}{2} \div \frac{17}{3}
\][/tex]
To divide fractions, we multiply by the reciprocal of the divisor:
[tex]\[
\frac{7}{2} \div \frac{17}{3} = \frac{7}{2} \times \frac{3}{17}
\][/tex]
Calculate this multiplication:
[tex]\[
\frac{7}{2} \times \frac{3}{17} = \frac{7 \times 3}{2 \times 17} = \frac{21}{34}
\][/tex]
We need to find the number that, when multiplied by [tex]\(\frac{7}{2}\)[/tex], gives us [tex]\(\frac{21}{34}\)[/tex]. Let [tex]\( x \)[/tex] represent this number, so we have:
[tex]\[
\frac{7}{2} \times x = \frac{21}{34}
\][/tex]
To solve for [tex]\( x \)[/tex], we divide both sides by [tex]\(\frac{7}{2}\)[/tex]:
[tex]\[
x = \frac{\frac{21}{34}}{\frac{7}{2}} = \frac{21}{34} \times \frac{2}{7}
\][/tex]
Calculate this expression:
[tex]\[
\frac{21}{34} \times \frac{2}{7} = \frac{21 \times 2}{34 \times 7} = \frac{42}{238}
\][/tex]
Simplify [tex]\(\frac{42}{238}\)[/tex]:
[tex]\[
\frac{42}{238} = \frac{21}{119} = \frac{3}{17}
\][/tex]
Therefore, the number that belongs in the box is:
[tex]\[
\boxed{\frac{3}{17}}
\][/tex]
So, option A ([tex]\(\frac{3}{17}\)[/tex]) is correct.
