To find the missing factor in the equation [tex]\(\left(\frac{6x^2}{5y}\right)(?) = \frac{3xy^2}{10}\)[/tex], let's denote the missing factor as [tex]\(k\)[/tex]. Then, the equation becomes:
[tex]\[\left(\frac{6x^2}{5y}\right) \cdot k = \frac{3xy^2}{10}\][/tex]
We need to solve for \(k\). To do this, we can multiply both sides by the reciprocal of \(\frac{6x^2}{5y}\):
[tex]\[k = \frac{\frac{3xy^2}{10}}{\frac{6x^2}{5y}}\][/tex]
Simplify the division of fractions by multiplying by the reciprocal:
[tex]\[k = \frac{3xy^2}{10} \cdot \frac{5y}{6x^2}\][/tex]
Multiply the numerators together and the denominators together:
[tex]\[k = \frac{3xy^2 \cdot 5y}{10 \cdot 6x^2}\][/tex]
Simplify the expression:
[tex]\[k = \frac{15xy^3}{60x^2}\][/tex]
Simplify the fraction by canceling common factors:
[tex]\[k = \frac{15}{60} \cdot \frac{xy^3}{x^2} = \frac{1}{4} \cdot y\][/tex]
So, the missing factor \(k\) is:
[tex]\[k = \frac{y}{4}\][/tex]
Therefore, the missing factor in the equation is [tex]\(\frac{y}{4}\)[/tex].