64 persons can dig a trench 50 m long, 2 m wide and 2 m deep in 5 days, working 12 hours daily. In how many days, working 8 hours daily, will 80 persons dig another trench 75 m long, 4 m wide and 3 m deep?

### Concept & 4-Variable Chain Rule This is a comprehensive chain rule problem utilizing all variables: Men, Days, Hours, and Work (calculated as a 3D volume: Length $\times$ Width $\times$ Depth). $$ \frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2} $$ ### Step-by-Step Solution * Scenario 1 parameters: $M_1 = 64$, $D_1 = 5$, $H_1 = 12$. The work $W_1$ is the volume: $50 \times 2 \times 2 = 200 \text{ m}^3$. * Scenario 2 parameters: $M_2 = 80$, $D_2 = x$, $H_2 = 8$. The work $W_2$ is the volume: $75 \times 4 \times 3 = 900 \text{ m}^3$. * Plug these values into the master formula: $$ \frac{64 \times 5 \times 12}{200} = \frac{80 \times x \times 8}{900} $$ * Simplify the denominators by cancelling two zeros across the equation (multiply both sides by 100): $$ \frac{64 \times 5 \times 12}{2} = \frac{80 \times x \times 8}{9} $$ * Simplify the left side by computing the math: $\frac{64}{2} = 32$. Then $32 \times 5 \times 12 = 160 \times 12 = 1920$. * The equation is now: $1920 = \frac{640 \times x}{9}$. * Notice that 1920 is exactly 3 times 640 ($640 \times 3 = 1920$). Divide both sides by 640: $$ 3 = \frac{x}{9} $$ * Multiply by 9 to find $x$: $x = 27$. ### Exam Strategy & Shortcut Always arrange the formula and ruthlessly cancel numbers on the top and bottom before multiplying. Write it as: $x = \frac{64 \times 5 \times 12 \times 900}{80 \times 8 \times 200}$. Cancel $900$ and $200$ to $9$ and $2$. Cancel $80 \times 8$ ($640$) with $64 \times 10$ from the top. The math will beautifully collapse to $3 \times 9 = 27$ in seconds without ever dealing with a four-digit number. ### Common Pitfall Calculating the total product of the numerator and denominator before dividing ($3840/200 = 640x/900 \rightarrow 19.2 = 0.711x$). Introducing decimals into chain rule problems makes the math significantly harder to track and solve without a calculator. ### Final Answer Therefore, the correct answer is **27**.