What number belongs in the box to make the equation true?

[tex]\[ 3 \frac{1}{2} \div 5 \frac{2}{3} = \frac{7}{2} \times \][/tex]

A. [tex]\(\frac{3}{17}\)[/tex]

B. [tex]\(\frac{3}{2}\)[/tex]

C. [tex]\(\frac{17}{3}\)[/tex]

D. [tex]\(\frac{13}{2}\)[/tex]



Answer :

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.