1. **State the problem:** We need to find how many days 16 men and 20 boys will take to finish the same job given the work rates of men and boys. 2. **Define variables:** Let the work rate of one man be $m$ jobs/day and one boy be $b$ jobs/day. 3. **Set up equations from given data:** - 8 men and 16 boys finish the job in 6 days, so total work is 1 job: $$6 \times (8m + 16b) = 1$$ Simplifying: $$8m + 16b = \frac{1}{6}$$ - 12 men and 24 boys finish in 8 days: $$8 \times (12m + 24b) = 1$$ Simplifying: $$12m + 24b = \frac{1}{8}$$ 4. **Simplify the equations:** - From the first: $$8m + 16b = \frac{1}{6}$$ - From the second: $$12m + 24b = \frac{1}{8}$$ 5. **Divide the first equation by 8:** $$m + 2b = \frac{1}{48}$$ 6. **Divide the second equation by 12:** $$m + 2b = \frac{1}{96}$$ 7. **Notice inconsistency:** The values for $m + 2b$ differ. Check carefully the initial simplification: Reevaluate step 3: - For the first equation, total work is 1, so work done per day is $\frac{1}{6}$. Thus: $$8m + 16b = \frac{1}{6}$$ - Similarly, for the second: $$12m + 24b = \frac{1}{8}$$ 8. **Divide first equation by 8:** $$m + 2b = \frac{1}{48}$$ 9. **Divide second equation by 12:** $$m + 2b = \frac{1}{96}$$ 10. The discrepancy implies an error in division; re-calculate step 8 and 9. Wait, $\frac{1}{6} ÷ 8 = \frac{1}{48}$ and $\frac{1}{8} ÷ 12 = \frac{1}{96}$. The contradiction implies men and boys' work rates cannot be linearly combined like this, or an error exists. 11. **Use simultaneous equations:** From the two equations: $$8m + 16b = \frac{1}{6}$$ $$12m + 24b = \frac{1}{8}$$ Multiply first equation by 3: $$24m + 48b = \frac{3}{6} = \frac{1}{2}$$ Multiply second equation by 2: $$24m + 48b = \frac{2}{8} = \frac{1}{4}$$ Again inconsistency: left sides equal but right sides differ. This suggests a potential misinterpretation: perhaps the equation should be work rate times days equals 1: Given 8 men and 16 boys can finish in 6 days, total work per day: $$W = 8m + 16b$$ Total work done in 6 days: $$6W = 1$$ Similarly, for 12 men and 24 boys and 8 days: $$8(12m + 24b) = 1$$ Check again: Equation 1: $$6(8m + 16b) = 1 \implies 48m + 96b = 1$$ Equation 2: $$8(12m + 24b) = 1 \implies 96m + 192b = 1$$ Divide equation 2 by 2: $$48m + 96b = \frac{1}{2}$$ Compare with equation 1: $$48m + 96b = 1$$ and $$48m + 96b = \frac{1}{2}$$ contradiction again. 12. **Re-express carefully:** Probably total job per day = work rate; total job done in days equals 1 job: Equation 1: $$6 \times (8m + 16b) = 1$$ $$48m + 96b = 1$$ Equation 2: $$8 \times (12m + 24b) = 1$$ $$96m + 192b = 1$$ 13. **Solve the system:** Equation 1: $$48m + 96b = 1$$ Equation 2: $$96m + 192b = 1$$ Divide equation 2 by 2: $$48m + 96b = \frac{1}{2}$$ Again contradiction with equation 1! 14. **Conclusion:** The problem statement might be inconsistent or contains a typo because 8 men and 16 boys finishing in 6 days conflicts with 12 men and 24 boys finishing in 8 days. 15. Assuming the problem meant: - 8 men and 16 boys can finish the job in 6 days - 12 men and 24 boys can finish in 4 days (instead of 8) Then proceed: Equation 1: $$6 \times (8m + 16b) = 1 \implies 48m + 96b = 1$$ Equation 2: $$4 \times (12m + 24b) = 1 \implies 48m + 96b = 1$$ Both equations become same; no unique solution. 16. **Alternate approach:** Express work rates per day: From equation 1: $$8m + 16b = \frac{1}{6}$$ From equation 2: $$12m + 24b = \frac{1}{8}$$ Divide first by 8: $$m + 2b = \frac{1}{48}$$ Divide second by 12: $$m + 2b = \frac{1}{96}$$ Again inconsistent. 17. **Therefore, there is an error in problem data as given.** Assuming problem states: - 8 men and 16 boys finish in 6 days - 12 men and 24 boys finish in 8 days Work rates contradict. 18. **If instead the days are:** - 8 men and 16 boys in 6 days - 12 men and 24 boys in 4 days Set equations: $$8m + 16b = \frac{1}{6}$$ $$12m + 24b = \frac{1}{4}$$ Multiply first equation by 3: $$24m + 48b = \frac{1}{2}$$ Multiply second equation by 2: $$24m + 48b = \frac{1}{2}$$ Consistent, proceed with one equation. 19. **Solve for $m$ and $b$:** From $$8m + 16b = \frac{1}{6}$$ Divide by 8: $$m + 2b = \frac{1}{48}$$ 20. **Find days for 16 men and 20 boys:** Work rate: $$16m + 20b$$ Express $m$ from above: $$m = \frac{1}{48} - 2b$$ Substitute: $$16 \left(\frac{1}{48} - 2b \right) + 20b = 16 \times \frac{1}{48} - 32b + 20b = \frac{1}{3} - 12b$$ 21. Since $b$ unknown, assume $b = x$ and solve for $x$ using second equation: $$12m + 24b = \frac{1}{4}$$ Substitute $m$: $$12 \left(\frac{1}{48} - 2b \right) + 24b = \frac{1}{4}$$ $$\frac{12}{48} - 24b + 24b = \frac{1}{4}$$ $$\frac{1}{4} = \frac{1}{4}$$ 22. Equation satisfied for any $b$, so infinite solutions, cannot find unique number of days. 23. **Hence, problem data inconsistent or insufficient.** --- **Final answer:** Cannot determine the number of days for 16 men and 20 boys with the given inconsistent data.