Question 1

8 men can complete a piece of work in 20 days. 8 women can complete the same work in 32 days. In how many days will 5 men and 8 women together complete the same work? (Bank P.O., 2010)

Accepted Answer

Concept & Work Rates: When different groups finish the same work in different times, we can calculate their individual 1-day fractional work to find combined rates. 1-day work = (1/Total man-days) Step-by-Step Solution: • Calculate 1-Day Work for Men: 8 men complete the work in 20 days. Total man-days = 8 × 20 = 160. Therefore, 1 man's 1-day work = 1/160. • Calculate 1-Day Work for Women: 8 women complete the work in 32 days. Total woman-days = 8 × 32 = 256. Therefore, 1 woman's 1-day work = 1/256. • Calculate Target Group's 1-Day Work: The target group is 5 men and 8 women. Their combined 1-day work = 5 × (1/160) + 8 × (1/256) = 5/160 + 8/256 Simplify the fractions: = 1/32 + 1/32 = 2/32 = 1/16 • Calculate Total Time: Since their combined 1-day work is 1/16, they will take 16 days to complete the full work. Exam Strategy & Shortcut: Use the equivalence method directly. 8 men take 20 days, so total work = 160 M-days. 8 women take 32 days, so total work = 256 W-days. Equating them: 160 M = 256 W \Rightarrow 5 M = 8 W. The question asks for the time taken by "5 men + 8 women". Substitute 5 men with 8 women: (8 W) + 8 W = 16 W. We already know 8 women take 32 days. So, 16 women (double the workforce) will take half the time: 32 / 2 = 16 days. Common Pitfall: A common trap is attempting to average the days or arbitrarily combining the numbers without establishing a fundamental equivalence between the work rate of a man and a woman. Final Answer: Therefore, the correct answer is 16 days.

Question 2

If 12 men and 16 boys can do a piece of work in 5 days; 13 men and 24 boys can do it in 4 days, then the ratio of the daily work done by a man to that of a boy is

Accepted Answer

Concept & Equating Total Work: When two different groups complete the same piece of work in different numbers of days, the total work (Workforce × Days) for both groups is equal. Equating these allows us to find the efficiency relationship between the individual worker types. M_1 × D_1 = M_2 × D_2 Step-by-Step Solution: • Define variables: Let the daily work of a man be m and a boy be b. • Calculate total work from the first group: 12 men and 16 boys finish the work in 5 days. • Total Work = 5 × (12m + 16b). • Calculate total work from the second group: 13 men and 24 boys finish the work in 4 days. • Total Work = 4 × (13m + 24b). • Equate and solve: Since both groups are doing the same work, set the equations equal to each other. • 5(12m + 16b) = 4(13m + 24b). • 60m + 80b = 52m + 96b. • Move m terms to one side and b terms to the other. • 60m - 52m = 96b - 80b. • 8m = 16b. • Simplify by dividing by 8. • 1m = 2b. • Determine ratio: The ratio of a man's daily work to a boy's is m/b = 2/1. Exam Strategy & Shortcut: Quickly cross-multiply the days and the workforce numbers mentally: (12 × 5)m = 60m vs (13 × 4)m = 52m, difference is 8m. For boys: (16 × 5)b = 80b vs (24 × 4)b = 96b, difference is 16b. Thus, 8m = 16b \Rightarrow m = 2b. The ratio is 2:1. Common Pitfall: Miscalculating the distribution when multiplying the days across the workforce numbers (e.g., forgetting to multiply the boys by the days). Final Answer: Therefore, the correct answer is 2 : 1.

Question 3

If 5 men and 3 women can reap 18 acre of crop in 4 days; 3 men and 2 women can reap 22 acre of crop in 8 days, then how many men are required to join 21 women to reap 54 acre of crop in 6 days?

Accepted Answer

Concept & Formula: When different groups of people perform different amounts of work in a given time, we use the chain rule formula linking Men (M), Days (D), and Work (W): (M_1 × D_1/W_1) = (M_2 × D_2/W_2) We must first find the efficiency ratio between a man and a woman to convert all workers into a single unit. Step-by-Step Solution: 1. Find the Efficiency Ratio: Let the daily work of 1 man be m and 1 woman be w. Case 1: (5m + 3w) reap 18 acres in 4 days. Case 2: (3m + 2w) reap 22 acres in 8 days. Using the formula: ((5m + 3w) × 4/18) = ((3m + 2w) × 8/22) Simplify the denominators and numerators: (2(5m + 3w)/9) = (4(3m + 2w)/11) (5m + 3w/9) = (2(3m + 2w)/11) Cross-multiplying: 11(5m + 3w) = 18(3m + 2w) 55m + 33w = 54m + 36w m = 3w This means 1 man is as efficient as 3 women. 2. Calculate the Required Men: Let x men join 21 women. Total workforce in terms of women = x(3w) + 21w = (3x + 21)w. Convert the first group to women: 5m + 3w = 5(3w) + 3w = 18w. Now, 18 women reap 18 acres in 4 days. We need (3x + 21) women to reap 54 acres in 6 days. (18 × 4/18) = ((3x + 21) × 6/54) 4 = ((3x + 21) × 1/9) 36 = 3x + 21 3x = 15 \implies x = 5 Exam Strategy & Shortcut: Always convert variables (m, w) to a single unit as early as possible to minimize complex fractional calculations. Identifying that m = 3w simplifies the equation dramatically. Common Pitfall: Forgetting to include the work (W) in the denominator when setting up the chain rule equation. Work done (acres reaped) is not constant here. Final Answer: Therefore, the correct answer is 5.

Question 4

X can do a piece of work in 24 days. When he had worked for 4 days, Y joined him. If complete work was finished in 16 days, Y can alone finish that work in

Accepted Answer

Concept & Logic: To find the time taken by an individual working alone, determine what fraction of the total work they completed and how long they worked on it. If someone works from the start to the end, calculate their total contribution first. Total Work = Work done by X + Work done by Y Step-by-Step Solution: 1. Analyze X's Work: X can finish the whole work in 24 days, meaning X's 1-day work is (1/24). The complete work finished in 16 days. Since X worked from the beginning to the end, X worked for all 16 days. Work done by X in 16 days = 16 × (1/24) = (16/24) = (2/3). 2. Analyze Y's Work: Remaining work = 1 - (2/3) = (1/3). This remaining (1/3) of the work was done by Y. Y joined after 4 days and worked until the end (day 16). Therefore, Y worked for 16 - 4 = 12 days. 3. Calculate Y's Total Time: If Y can complete (1/3) of the work in 12 days, then the time Y will take to complete the whole work alone is 12 × 3 = 36 days. Exam Strategy & Shortcut: Instead of tracking work day-by-day (before and after Y joined), just look at the total duration each person worked. X worked for 16 days ((2/3) of X's total required time), leaving exactly (1/3) for Y. Since Y worked for 12 days to do that (1/3), Y's total time is simply 12 × 3. Common Pitfall: Assuming Y worked for 16 days alongside X. Pay close attention to the fact that Y joined *after* 4 days, meaning Y only worked for 12 days. Final Answer: Therefore, the correct answer is 36 days.

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