$4$ mat-weavers can weave $4$ mats in $4$ days. At the same rate, how many mats would be woven by $8$ mat-weavers in $8$ days?

### Concept & Work Equivalence Formula The fundamental chain rule equation relates men ($M$), days ($D$), and work ($W$): $$ \frac{M_1 \times D_1}{W_1} = \frac{M_2 \times D_2}{W_2} $$ ### Step-by-Step Solution * **Given**: $M_1 = 4$, $W_1 = 4$, $D_1 = 4$. For the second scenario: $M_2 = 8$, $D_2 = 8$. Let the required work be $W_2$. * **Calculation**: Substitute the given values into the formula: $$ \frac{4 \times 4}{4} = \frac{8 \times 8}{W_2} $$ $$ 4 = \frac{64}{W_2} $$ $$ W_2 = \frac{64}{4} = 16 $$ ### Exam Strategy & Shortcut Double the workers and double the time means the output is quadrupled. $2 \text{ (from workers)} \times 2 \text{ (from days)} = 4$. The original work was $4$ mats, so the new work is $4 \times 4 = 16$ mats. ### Common Pitfall Assuming that because $4$ weavers in $4$ days make $4$ mats, $8$ weavers in $8$ days will make $8$ mats. This ignores that both the workforce and the duration have doubled. ### Final Answer Therefore, the correct answer is **16**.