Two men undertake to do a piece of work for ₹ 1400. The first man alone can do this work in 7 days while the second man alone can do this work in 8 days. If they working together complete this work in 3 days with the help of a boy, how should the money be divided? (M.A.T., 2007)

### Concept & Work Segments When multiple people work together for a defined period, their total wages must be divided strictly based on the proportion of the total work each individual completed during that time. ### Step-by-Step Solution * **Man 1's Work:** In 1 day, he completes $\frac{1}{7}$ of the work. In 3 days, he completes $\frac{3}{7}$. * **Man 2's Work:** In 1 day, he completes $\frac{1}{8}$ of the work. In 3 days, he completes $\frac{3}{8}$. * **Combined Men's Work:** Work done by both men in 3 days = $\frac{3}{7} + \frac{3}{8}$ $$ \frac{3}{7} + \frac{3}{8} = \frac{24 + 21}{56} = \frac{45}{56} $$ * **Boy's Work:** The remaining work was done by the boy in those 3 days. $$ 1 - \frac{45}{56} = \frac{11}{56} $$ * **Wage Ratio:** The money is divided according to the ratio of work done: $\frac{24}{56} : \frac{21}{56} : \frac{11}{56} = 24 : 21 : 11$. * **Man 1's Share:** $\frac{24}{56} \times 1400 = 600$ * **Man 2's Share:** $\frac{21}{56} \times 1400 = 525$ * **Boy's Share:** $\frac{11}{56} \times 1400 = 275$ ### Exam Strategy & Shortcut Notice the relationship between the denominator of the work fractions (56) and the total payment (1400). Since $1400 \div 56 = 25$, you can simply multiply the numerator of each person's fractional work by 25 to find their exact payout ($24 \times 25 = 600$, $21 \times 25 = 525$, $11 \times 25 = 275$). ### Common Pitfall A common mistake is assuming the boy's 3-day efficiency is equal to one of the men's, or attempting to divide the money by the inverse of their individual days to completion without accounting for the actual work fractions completed. ### Final Answer Therefore, the correct answer is **₹ 600, ₹ 525, ₹ 275**.