UPSC CDS 2 2022 Math Paper

Concept:The total work done is directly proportional to the number of workers, the number of days, and the hours worked per day.Explanation:Let $y$ be the number of additional persons needed.From the first part: 480 persons work 10 hours/day for 10 days to complete $\frac{1}{4}$ of the work.So, $M_1 = 480$, $D_1 = 10$, $H_1 = 10$, $W_1 = \frac{1}{4}$.The remaining work is $1 - \frac{1}{4} = \frac{3}{4}$.This remaining work must be done in 20 days with 8 hours/day.So, $M_2 = 480 + y$, $D_2 = 20$, $H_2 = 8$, $W_2 = \frac{3}{4}$.Use the work equivalence formula: $\frac{M_1 \times D_1 \times H_1}{W_1} = \frac{M_2 \times D_2 \times H_2}{W_2}$.Substitute the values: $\frac{480 \times 10 \times 10}{\frac{1}{4}} = \frac{(480 + y) \times 20 \times 8}{\frac{3}{4}}$.Simplify each side. Left: $\frac{48000}{\frac{1}{4}} = 48000 \times 4 = 192000$.Right: $\frac{(480 + y) \times 160}{\frac{3}{4}} = (480 + y) \times 160 \times \frac{4}{3} = (480 + y) \times \frac{640}{3}$.Thus, $192000 = (480 + y) \times \frac{640}{3}$.Multiply both sides by 3: $576000 = (480 + y) \times 640$.Divide by 640: $900 = 480 + y$.So, $y = 900 - 480 = 420$.Answer:420 additional persons are required.