Question 19

5/108

To solve the equation \( \frac{4}{9} \left( \frac{3}{16} - \frac{1}{12} \right) \), we first simplify the expression inside the parentheses. The correct answer is \( \frac{5}{108} \).

A) 5/108

This option is correct. First, we calculate \( \frac{3}{16} - \frac{1}{12} \) by finding a common denominator, which is 48. Thus, \( \frac{3}{16} = \frac{9}{48} \) and \( \frac{1}{12} = \frac{4}{48} \). Therefore, \( \frac{3}{16} - \frac{1}{12} = \frac{9}{48} - \frac{4}{48} = \frac{5}{48} \). Next, we multiply it by \( \frac{4}{9} \):

\[ \frac{4}{9} \times \frac{5}{48} = \frac{20}{432} = \frac{5}{108} \] after simplifying.

B) 5/48

This option is incorrect. While \( \frac{5}{48} \) is the result of \( \frac{3}{16} - \frac{1}{12} \), it is not the final answer, as we need to multiply this result by \( \frac{4}{9} \) to find the value of the original expression.

C) 2/9

This choice is incorrect as well. The value \( \frac{2}{9} \) does not result from the operations performed in the equation. It does not correspond to any step in the calculations, showing a misunderstanding of the multiplication involved.

D) 20/48

This option is also incorrect. While \( 20/48 \) may appear to stem from multiplying \( 4/9 \) by \( 5/48 \), it has not been simplified. The final answer must be fully simplified to \( \frac{5}{108} \), making \( 20/48 \) an improper representation of the final result.

Conclusion

The correct answer is \( \frac{5}{108} \) because it accurately reflects the final result of the calculations required by the equation. Options B, C, and D fail to represent the correct final value, either presenting intermediate results or incorrect simplifications. Thus, only option A provides the definitive correct solution.