Sixteen men can complete a work in twelve days. Twenty-four children can complete the same work in eighteen days. Twelve men and eight children started working and after eight days three more children joined them. How many days will they now take to complete the remaining work ?

### Concept & Fractional Daily Work When the equivalence ratio between two groups (like men and children) results in a fraction, it is often safer to calculate the 1-day work fraction for each individual type of worker and build the daily equation from there. $$\text{Work of 1 person in 1 day} = \frac{1}{\text{Total Persons} \times \text{Total Days}}$$ ### Step-by-Step Solution * **Find 1-day work values:** 16 men take 12 days; 24 children take 18 days. * 1 man's 1-day work = $\frac{1}{16 \times 12} = \frac{1}{192}$. * 1 child's 1-day work = $\frac{1}{24 \times 18} = \frac{1}{432}$. * **Calculate initial team's daily work:** 12 men and 8 children start the work. * Team's 1-day work = $12 \times \frac{1}{192} + 8 \times \frac{1}{432} = \frac{1}{16} + \frac{1}{54}$. * Find common denominator for 16 and 54, which is 432. * $\frac{27}{432} + \frac{8}{432} = \frac{35}{432}$. * **Calculate work done in 8 days:** * Work completed = $8 \times \frac{35}{432} = \frac{35}{54}$. * **Calculate remaining work:** * Remaining work = $1 - \frac{35}{54} = \frac{19}{54}$. * **Calculate new team's daily work:** 3 more children join, making it 12 men and 11 children. * New 1-day work = $12 \times \frac{1}{192} + 11 \times \frac{1}{432} = \frac{27}{432} + \frac{11}{432} = \frac{38}{432} = \frac{19}{216}$. * **Find final days required:** Time = Remaining Work / New Daily Work. * Time = $\frac{19}{54} \div \frac{19}{216} = \frac{19}{54} \times \frac{216}{19} = \frac{216}{54} = 4$ days. ### Exam Strategy & Shortcut Instead of fractions, use total work equivalence: $16m \times 12 = 24c \times 18 \Rightarrow 192m = 432c \Rightarrow 4m = 9c$. Total work = $192m$. Initial workforce: $12m + 8c = 12m + 8(4/9)m = (140/9)m$. In 8 days, work done = $8 \times (140/9)m = (1120/9)m$. Remaining work = $192m - (1120/9)m = (1728 - 1120)/9 = (608/9)m$. New workforce: $12m + 11c = 12m + 11(4/9)m = (152/9)m$. Days = $(608/9) / (152/9) = 608 / 152 = 4$. Both methods yield 4 swiftly! ### Common Pitfall Miscalculating the Least Common Multiple (LCM) when adding fractions like $1/16$ and $1/54$. Always prime factorize to ensure accurate LCM generation ($16 = 2^4$, $54 = 2 \times 3^3$, LCM = $2^4 \times 3^3 = 432$). ### Final Answer Therefore, the correct answer is **4 days**.