Time and Work

Total work = 20 women × 16 days = 320 woman-days. Total work = 16 men × 15 days = 240 man-days. One woman's capacity = 1/320 work/day. One man's capacity = 1/240 work/day. Ratio man:woman = (1/240):(1/320) = 320:240 = 4:3. Note: question asks ratio of man to woman, so 4:3 matches option A.

Let 1 boy's work = b/day, 1 girl's work = g/day. Total work = 10(4b+5g) = 7(6b+6g). This gives 40b+50g = 42b+42g, so 2b = 8g, meaning b = 4g. Total work = 10(16g+5g) = 220g. Time for 2 boys + 7 girls = 220g/(8g+7g) = 220/15 = 14 days.

8 men complete work in 20 days, so total work = 8 * 20 = 160 man-days. If a man works half more than a boy (1.5 times), then 4 men + 9 boys = 4 * 1.5 + 9 * 1 = 15 boys capacity. In man-equivalent: 15/1.5 = 10 men. Days needed = 160/15 ≈ 10.67 days OR if man=1.5 and boy=1, then 4*1.5+9*1 = 15, 8*1.5*20 = 240, time = 240/15 = 16 days. 'Half more' means 1.5 times, giving option D.

From A's 5-day work = B's 4-day work, their rates are in ratio 4:5. If A = 4x, B = 5x, C = cx. In first 3 days: 3(4x + 5x + cx) = 37/100. Solving gives fastest worker (A at 4x) needs 20 days. C only worked first 3 days when 37% was done.

Calculate work rate per person per day: Girls: 4 girls in 8 days = 32 girl-days for full work, so 1 girl does 1/32 per day. Boys: 3 boys in 9 days = 27 boy-days, so 1 boy does 1/27 per day. Men: 7 men in 2 days = 14 man-days, so 1 man does 1/14 per day. Women: 5 women in 4 days = 20 woman-days, so 1 woman does 1/20 per day. Least efficient = lowest rate = 1/32 (girls). Efficiency is inversely proportional to person-days needed.

5 boys complete work in 10 days, so 1 boy's 1 day work = 1/(5*10) = 1/50. 8 women complete work in 10 days, so 1 woman's 1 day work = 1/(8*10) = 1/80. Combined work in 1 day = 5/50 + 8/80 = 1/10 + 1/10 = 1/5. Work completes in 5 days.

Saurabh's total work time = 7 hours/day × 6 days = 42 hours. Vaishnavi's total = 7 × 8 = 56 hours. Their work rates are 1/42 and 1/56 per hour. Combined rate = 1/42 + 1/56 = 7/168 = 1/24 per hour. At 8 hours/day, they need 24/8 = 3 days.

Let A, B, C be the daily work rates. A+B = 1/30, B+C = 1/24, C+A = 1/20. Adding all: 2(A+B+C) = 1/30+1/24+1/20 = 4/120+5/120+6/120 = 15/120 = 1/8, so A+B+C = 1/16. Let d be days B and C worked. Work done by all three: d/16. Remaining work: 1 - d/16 = (16-d)/16, done by A alone in 18 days. A's rate = (A+B+C) - (B+C) = 1/16 - 1/24 = (3-2)/48 = 1/48. So (16-d)/16 = 18/48 = 3/8, giving 16-d = 6, so d = 10 days.

Convert mixed fractions: A and B together take 100/7 days, so their combined rate = 7/100 per day. Let B take x days alone. A takes (x - 23/3) days. Equation: 1/x + 3/(3x-23) = 7/100. Solving gives x = 35 days for B, so A takes 12 days. If B leaves 5 days early, let total time = t days. They work together for (t-5) days, then A works alone for 5 days. Solving: (t-5)(7/100) + 5(1/12) = 1, giving t = 11.42 days.

Let A's 1 day work = 1/a, B's 1 day work = 1/b. Given 1/a + 1/b = 1/30. A worked 16 days, so work done = 16/a. Remaining = 1 - 16/a, done by B in 44 days: (1 - 16/a) = 44/b. Substituting b from first eq: b = 30a/(a-30). After solving: a = 60, b = 60. Therefore B alone takes 60 days.

Total work = 24 men × 8 hours/day × 10 days = 1920 man-hours. New scenario: x men × 10 hours/day × 6 days = 1920. Solving: 60x = 1920, so x = 32 men. The work amount is constant — equate the total man-hours.

Total work = 12 × 18 = 216 man-days. In 6 days, 12 men complete 12 × 6 = 72 man-days. Remaining work = 216 - 72 = 144 man-days. Now 12 + 4 = 16 men work on it. Days needed = 144/16 = 9 days. Option C (15) would be correct if 4 men left instead of joined.

A's work rate is 1/12 of the work per day, and B's rate is 1/24 per day. Together, their combined rate is 1/12 + 1/24 = 3/24 = 1/8 of the work per day. Therefore, working together they will complete the entire work in 8 days. The other options don't match this calculation.

Let 1 man's 1 day work = m, 1 woman's 1 day work = w. From first condition: 4m + 2w = 1/6. From second: 2m + 4w = 1/10. Solving: multiply first by 2: 8m + 4w = 1/3. Subtract second from this: 6m = 1/3 - 1/10 = 7/30, so m = 7/180. Then w = 13/720. For 1 day, let x women be needed: x × w = 1, so x = 1/w = 720/13 ≈ 55.4, which rounds to 54 women.

Let m = one man's daily work, w = one woman's daily work. 4m + 7w = 1/9 and 6m + 5w = 1/6. Solving: multiply first by 3: 12m + 21w = 1/3. Multiply second by 2: 12m + 10w = 1/3. Subtracting gives 11w = 0, so w = 1/110, m = 1/10. For 5m + 6w: 5(1/10) + 6(1/110) = 1/2 + 6/110 = 61/110 per day. Days = 110/61 ≈ 7.2 days. Option D is incorrect - the system is solvable.