Question 1

Finding an Individual's Time from a Trio's Combined Time A can do the work in 24 days, B in 30 days, and together with C they finish in 12 days. How long would C take to complete the work alone?

Accepted Answer

Introduction / Context:We are given two individual times and a combined time with a third worker. Convert to rates and solve for C's rate by subtraction, then invert to get C's solo time. Given Data / Assumptions: A: 24 days ⇒ 1/24 per day. B: 30 days ⇒ 1/30 per day. A + B + C: 12 days ⇒ 1/12 per day. Concept / Approach:r_C = r_total − (r_A + r_B). Then time_C = 1/r_C. Step-by-Step Solution:r_A + r_B = 1/24 + 1/30 = (5 + 4)/120 = 9/120 = 3/40.r_total = 1/12 = 10/120.r_C = 10/120 − 9/120 = 1/120.C alone time = 1 / (1/120) = 120 days. Verification / Alternative check:Rates add up: 1/24 + 1/30 + 1/120 = 1/12, confirming consistency. Why Other Options Are Wrong:100, 125, 72, or 90 days do not match the rate difference result. Common Pitfalls:Arithmetic slips when adding fractions; always use a common denominator. Final Answer:120 days

Question 2

Efficiency vs. Time — Inverse Relationship A is thrice as efficient as B, and C is twice as efficient as B. What is the ratio of the number of days taken by A, B, and C respectively when working alone?

Accepted Answer

Introduction / Context:Efficiency (rate) and time are inversely proportional: higher efficiency means less time for the same work. Converting relative efficiencies to time ratios is a common aptitude step. Given Data / Assumptions: B's efficiency = x. A's efficiency = 3x. C's efficiency = 2x. Concept / Approach:Time ∝ 1/efficiency. Therefore, days_A : days_B : days_C = 1/(3x) : 1/x : 1/(2x) = 1/3 : 1 : 1/2. Multiply by 6 to clear fractions. Step-by-Step Solution:Days ratio = 1/3 : 1 : 1/2Multiply by 6 ⇒ 2 : 6 : 3 Verification / Alternative check:Check inverses against efficiencies: A (3x) gets smallest time; B (x) largest; C (2x) middle — the 2 : 6 : 3 order matches. Why Other Options Are Wrong:Any ordering not reflecting the inverse relationship (A shortest, B longest) is incorrect. Common Pitfalls:Confusing the direct vs. inverse relation between time and efficiency. Final Answer:2 : 6 : 3

Question 3

Relative Speeds and Combined Duration A takes only 4/5 as many days as B takes to do the same work. Working together, they finish the job in 20/3 days. What is B’s efficiency (as a percentage of the whole job per day)?

Accepted Answer

Introduction / Context:“A takes 4/5 as many days as B” means A is faster. We use the relationship between times and rates to find B’s per-day efficiency, expressed as a percent of one full job. Given Data / Assumptions: T_A = 4/5 * T_B. Working together time = 20/3 days ⇒ combined rate = 3/20 job/day. Concept / Approach:Rates: r_A = 1/T_A and r_B = 1/T_B. Then r_A + r_B = 3/20. Substitute T_A = (4/5)T_B to solve for T_B and r_B, then convert to percent. Step-by-Step Solution:1/((4/5)T_B) + 1/T_B = 3/20(5/4)*(1/T_B) + 1/T_B = (9/4)*(1/T_B) = 3/201/T_B = (3/20)*(4/9) = 12/180 = 1/15 job/dayB’s efficiency = (1/15)*100% = 6 2/3%. Verification / Alternative check:Then A’s rate is 1/T_A = 1/((4/5)T_B) = 5/(4T_B) = 5/60 = 1/12; sum 1/12 + 1/15 = 3/20, consistent. Why Other Options Are Wrong:16%, 10%, 8 1/3%, and 5.55% do not match the exact 1/15 per day. Common Pitfalls:Interpreting “4/5 as many days” as 4/5 the rate; it refers to time, not rate. Final Answer:6 2/3%

Question 4

Comparative Efficiency with Time Difference A is thrice as efficient as B. Together they complete the work in 3 days. If B takes 8 days more than A to do the work alone, how many days does A take alone?

Accepted Answer

Introduction / Context:This problem couples a relative-efficiency statement (“A is thrice as efficient as B”) with a time difference between their solo durations. Use the inverse relation between efficiency and time to form equations and solve. Given Data / Assumptions: A is 3 times as efficient as B. They finish together in 3 days. B’s solo time is 8 days more than A’s solo time. Concept / Approach:If A is 3 times as efficient as B, then T_A = T_B/3. Also given T_B = T_A + 8. Solve these to find T_A. (The “together in 3 days” statement is a cross-check.) Step-by-Step Solution:From efficiencies: T_A = T_B/3.From time difference: T_B = T_A + 8.Substitute: T_A = (T_A + 8)/3 ⇒ 3T_A = T_A + 8 ⇒ 2T_A = 8 ⇒ T_A = 4 days.Check together rate: 1/4 + 1/12 = 1/3 ⇒ 3 days total, consistent. Verification / Alternative check:With T_A = 4, T_B = 12 (8 more). The together time is exactly 3 days. Why Other Options Are Wrong:2, 6, 12, and 16 days don’t satisfy both the efficiency and the time-difference constraints. Common Pitfalls:Confusing “times as efficient” with “times as long”; they are inversely related. Final Answer:4

Question 5

Pairwise Work Times Given — Find C Alone A and B can do a job in 12 days, B and C in 15 days, and C and A in 20 days. How many days will C alone take to complete the job?

Accepted Answer

Introduction / Context:Given three pairwise completion times, we can find the sum of individual rates and then isolate the rate for a single worker by subtracting an appropriate pair rate. This is a classic system-of-rates problem. Given Data / Assumptions: (A + B) time = 12 ⇒ rate = 1/12. (B + C) time = 15 ⇒ rate = 1/15. (C + A) time = 20 ⇒ rate = 1/20. Concept / Approach:Sum: 2(A + B + C) rate = 1/12 + 1/15 + 1/20. Then (A + B + C) rate = half of that. C's rate = (A + B + C) − (A + B). Step-by-Step Solution:1/12 + 1/15 + 1/20 = (5 + 4 + 3)/60 = 12/60 = 1/5.(A + B + C) rate = (1/5)/2 = 1/10.C's rate = 1/10 − 1/12 = (6 − 5)/60 = 1/60.C alone time = 60 days. Verification / Alternative check:Compute A and B similarly if desired; all rates are consistent. Why Other Options Are Wrong:Any time other than 60 contradicts the derived rate 1/60. Common Pitfalls:Forgetting to divide by 2 after summing the three pairwise rates. Final Answer:60

Question 6

Three Pairwise Times — Find All Together A and B can finish the work in 8 days, B and C in 12 days, and A and C in 8 days. How many days will A, B, and C together take to finish the work?

Accepted Answer

Introduction / Context:Given three pairwise completion times, we can find the sum of all three rates by adding the pairwise rates and halving, since each person's rate is counted twice in the sum of pairs. Given Data / Assumptions: (A + B) rate = 1/8. (B + C) rate = 1/12. (A + C) rate = 1/8. Concept / Approach:(A + B + C) rate = ( (A + B) + (B + C) + (A + C) ) / 2. Then total time = 1 / (A + B + C rate). Step-by-Step Solution:Sum of pair rates = 1/8 + 1/12 + 1/8 = (3 + 2 + 3)/24 = 8/24 = 1/3.(A + B + C) rate = (1/3)/2 = 1/6 job/day.All together time = 1 / (1/6) = 6 days. Verification / Alternative check:The symmetry of two pairs at 8 days supports a moderate all-together time of 6 days. Why Other Options Are Wrong:4, 9, 12, or 5 days would not match the derived combined rate of 1/6 per day. Common Pitfalls:Forgetting to divide by 2 after summing pairwise rates or attempting to average the days directly. Final Answer:6

Question 7

Different Individual Times in a Factory Assignment A, B, and C can do the same assignment in 24, 6, and 12 days respectively. If they all work together from the beginning, in how many days will the assignment be completed?

Accepted Answer

Introduction / Context:We must sum three rates and invert to get total time. These numbers are chosen so the result is a tidy fraction in simplest form, a common testing style. Given Data / Assumptions: A: 24 days ⇒ 1/24. B: 6 days ⇒ 1/6. C: 12 days ⇒ 1/12. Concept / Approach:Combined rate r = 1/24 + 1/6 + 1/12. Total time = 1/r. Step-by-Step Solution:Common denominator 24: r = 1/24 + 4/24 + 2/24 = 7/24.Time = 1 / (7/24) = 24/7 days = 3 3/7 days = 33/7 days. Verification / Alternative check:Decimal: 24/7 ≈ 3.4286, between B’s 6 and C’s 12 when combined with A’s slower rate. Why Other Options Are Wrong:43/7, 4, 3, or 24/7 (if misread) don’t match the exact 24/7. Common Pitfalls:Arithmetic errors in fraction addition or misinterpreting 24/7 as an option separate from 33/7. Final Answer:33/7 days

Question 8

Mother–Daughter Team vs. Mother Alone A woman can weave a basket in 6 days. With the help of her daughter, they finish in 4 days. How long would the daughter alone take to weave the basket?

Accepted Answer

Introduction / Context:This is a basic “helper” problem: compare the solo time of one worker to the combined time with a helper, then solve for the helper’s solo time via rate subtraction. Given Data / Assumptions: Woman alone: 6 days ⇒ 1/6 per day. Together: 4 days ⇒ 1/4 per day. Rates add linearly. Concept / Approach:Daughter’s rate = together rate − mother’s rate = 1/4 − 1/6. Daughter’s time = reciprocal of her rate. Step-by-Step Solution:Daughter rate = 1/4 − 1/6 = (3 − 2)/12 = 1/12 per day.Daughter alone time = 1 / (1/12) = 12 days. Verification / Alternative check:Check: 1/6 + 1/12 = 1/4; correct combined rate. Why Other Options Are Wrong:16, 10, 8, or 14 days do not match the calculated helper rate. Common Pitfalls:Subtracting times instead of rates; remember to subtract 1/6 from 1/4, not 6 from 4. Final Answer:12 days

Question 9

Three Pairwise Times — Find All Three Together P and Q can complete a job in 6 days, Q and R in 10 days, and P and R in 5 days. If all three work together, how long will they take to finish the job?

Accepted Answer

Introduction / Context:Using pairwise completion times, we can find the sum of individual rates. The total rate of all three equals half the sum of pairwise rates (because each person is counted in two pairs). Given Data / Assumptions: (P + Q) rate = 1/6. (Q + R) rate = 1/10. (P + R) rate = 1/5. Concept / Approach:(P + Q + R) rate = ( (P + Q) + (Q + R) + (P + R) ) / 2. Then total time = 1 / (P + Q + R rate). Step-by-Step Solution:Sum of pair rates = 1/6 + 1/10 + 1/5 = 5/30 + 3/30 + 6/30 = 14/30 = 7/15.All-three rate = (7/15)/2 = 7/30 job/day.Time = 1 / (7/30) = 30/7 days ≈ 4.2857 days. Verification / Alternative check:Each pair's time is reasonable, and the all-together time is less than the fastest pair time (5 days), as expected. Why Other Options Are Wrong:42/7, 43/7, 44/7, and 45/7 do not equal the exact 30/7 derived from rates. Common Pitfalls:Forgetting to divide the sum of pairwise rates by 2 or adding days rather than rates. Final Answer:30/7 days

Question 10

Comparative Efficiency with Time Difference (Hours) Romil is twice as efficient as Sanjay and finishes the work 4 hours sooner than Sanjay. If both work together from the start, in how many hours will they finish the work?

Accepted Answer

Introduction / Context:This problem links relative efficiency (twice as efficient) with a time difference. Solve for each person’s solo time first, then add rates to get the combined finishing time. Given Data / Assumptions: Romil's efficiency = 2 * Sanjay's efficiency. Romil finishes 4 hours earlier than Sanjay. Constant rates throughout. Concept / Approach:If Sanjay's time is t hours, Romil's time is t/2 (twice as efficient ⇒ half the time). Given t/2 = t − 4 allows solving for t. Then compute the combined rate. Step-by-Step Solution:t/2 = t − 4 ⇒ t − t/2 = 4 ⇒ t/2 = 4 ⇒ t = 8 hours (Sanjay).Romil's time = t/2 = 4 hours.Combined rate = 1/8 + 1/4 = 3/8 job/hour.Together time = 1 / (3/8) = 8/3 hours ≈ 2.667 hours. Verification / Alternative check:Romil being faster and both working together should yield a time less than 4 hours, which 8/3 satisfies. Why Other Options Are Wrong:3 h or 4/3 h do not match the combined rate; 8 h is Sanjay’s solo time; 5 h is arbitrary. Common Pitfalls:Misreading “twice as efficient” as “2 hours less” or subtracting times rather than adding rates to combine efforts. Final Answer:8 / 3 h

Question 11

Wage calculation with overtime: A worker is paid ₹ 5 per hour for regular time and ₹ 8 per hour for overtime. A regular working day has 8 hours and a regular week has 5 working days. Over 4 weeks he earned a total of ₹ 920 and he was present on all …

Accepted Answer

Introduction / Context: This is a classic wages-and-time problem that mixes a fixed schedule of regular hours with higher-paid overtime hours. The goal is to separate regular pay from overtime pay and then recover the total number of hours from the given total earnings. Given Data / Assumptions: Regular wage = ₹ 5 per hour. Overtime wage = ₹ 8 per hour. Regular working day = 8 hours; regular week = 5 days. He worked all regular days for 4 weeks. Total earnings across 4 weeks = ₹ 920. Concept / Approach: First compute the guaranteed regular hours and the corresponding regular-pay amount. Subtract this from the total earnings to isolate how much was paid at the overtime rate. Convert overtime earnings to overtime hours, then add regular hours to obtain the total hours worked. Step-by-Step Solution: Regular hours per week = 5 days * 8 h = 40 h.Regular hours in 4 weeks = 4 * 40 = 160 h.Regular pay = 160 h * ₹ 5/h = ₹ 800.Overtime earnings = Total − Regular pay = 920 − 800 = ₹ 120.Overtime hours = 120 / 8 = 15 h.Total hours = 160 + 15 = 175 h. Verification / Alternative check: Rebuild the wages from hours: 160 h at ₹ 5 is ₹ 800 and 15 h at ₹ 8 is ₹ 120, total ₹ 920. This matches the given earnings, so the hour count is consistent. Why Other Options Are Wrong: 160 h ignores overtime; 180 h and 200 h imply overtime earnings that would exceed ₹ 120; 168 h does not match the computed overtime component. Common Pitfalls: Forgetting that regular hours are fixed by the schedule, or accidentally applying the overtime rate to regular hours. Always separate earnings by rate and then recombine hours at the end. Final Answer: 175 h

Question 12

Three pipes with filling and emptying: Pipes A and B can fill a tank in 20 h and 30 h respectively, while pipe C can empty a full tank in 40 h. If all three are opened together on an empty tank, in how many hours will the tank become full?

Accepted Answer

Introduction / Context: Multiple pipe problems rely on adding or subtracting rates. Filling pipes contribute positively to the net rate, while an emptying pipe contributes negatively. Once the net hourly fraction of the tank filled is known, the total time is found by taking the reciprocal. Given Data / Assumptions: A fills in 20 h ⇒ rate = 1/20 tank per hour. B fills in 30 h ⇒ rate = 1/30 tank per hour. C empties in 40 h ⇒ rate = −1/40 tank per hour. Tank starts empty; all three are opened together continuously. Concept / Approach: Net rate = 1/20 + 1/30 − 1/40. Convert to a common denominator to add/subtract correctly. Time to fill = 1 / (net rate). Mixed fraction form improves readability of the final time. Step-by-Step Solution: Compute with denominator 120: 1/20 = 6/120; 1/30 = 4/120; 1/40 = 3/120.Net rate = (6 + 4 − 3)/120 = 7/120 tank per hour.Time to fill = 1 / (7/120) = 120/7 h = 17 1/7 h. Verification / Alternative check: Decimal check: 120/7 ≈ 17.1429 h. A quick sanity check: if the emptying pipe were closed, time would be less than with it open; 120/7 is indeed greater than the combined time of A and B alone (which would be 12 h). Why Other Options Are Wrong: 16 1/7 h or 18 1/7 h result from arithmetic slips in summing the fractions; 19 1/7 h and 15 h are inconsistent with the computed net rate. Common Pitfalls: Adding the emptying rate instead of subtracting it, or using incorrect common denominators for fractional rates. Always standardize denominators before combining rates. Final Answer: 17 1/7 h

Question 13

Team up with different efficiencies: A man, a woman, and a boy can complete a job in 3 days, 4 days, and 12 days respectively when working alone. How many boys must assist one man and one woman so that the job is finished in 1.5 days?

Accepted Answer

Introduction / Context: Work-rate problems convert “days to finish” into unit rates (work per day). When multiple workers cooperate, their rates add. We then set the combined rate equal to the required rate for the targeted completion time and solve for the number of assisting workers needed. Given Data / Assumptions: Man alone: 3 days ⇒ rate = 1/3 work/day. Woman alone: 4 days ⇒ rate = 1/4 work/day. Boy alone: 12 days ⇒ rate = 1/12 work/day. Target completion time = 1.5 days ⇒ required combined rate = 1 / 1.5 = 2/3 work/day. Assume the job size is 1 unit of work. Concept / Approach: Let n be the number of boys. Then the combined rate with one man and one woman is 1/3 + 1/4 + n*(1/12). Set this equal to 2/3 and solve for n. Because work rates are linear, this is a straightforward single-variable equation. Step-by-Step Solution: 1/3 + 1/4 = (4 + 3)/12 = 7/12.Add boys: 7/12 + n/12 = 2/3 = 8/12.n/12 = 1/12 ⇒ n = 1. Verification / Alternative check: With one boy: total rate = 1/3 + 1/4 + 1/12 = 8/12 = 2/3 work/day. Time = 1 / (2/3) = 1.5 days, matching the requirement. Why Other Options Are Wrong: Larger values (such as 3, 6, or 12) would overshoot the required rate and finish earlier than 1.5 days, which does not match the condition. Common Pitfalls: Confusing rate (work/day) with time (days/work), or forgetting to convert 1.5 days to the required rate of 2/3 work/day before setting up the equation. Final Answer: 1

Question 14

Leak and inlet together determine capacity: A leak at the bottom of a tank can empty the full tank in 6 h. An inlet pipe can fill water at 4 L per minute. With the tank initially full, the inlet is opened and, due to the leak, the tank becomes empty…

Accepted Answer

Introduction / Context: When a leak and an inlet operate simultaneously, the net rate decides whether the tank level rises or falls. With the tank starting full and eventually emptying, the leak dominates the combined effect. We can model both in consistent “tank-per-hour” units using the unknown capacity to link the inlet’s liters per hour to a fraction of the tank per hour. Given Data / Assumptions: Leak empties a full tank in 6 h ⇒ leak rate L = 1/6 tank/h (outflow). Inlet flow = 4 L/min = 240 L/h. Starting full, after opening inlet, the tank empties in 8 h ⇒ net rate = −1/8 tank/h. Capacity = V liters (unknown). Concept / Approach: Express the inlet as a fraction of a tank per hour: inlet rate = 240/V tank/h. The net rate is inlet minus leak, and equals −1/8 tank/h. Solve for V from 240/V − 1/6 = −1/8. Step-by-Step Solution: 240/V − 1/6 = −1/8240/V = 1/6 − 1/8 = (4 − 3)/24 = 1/24V = 240 * 24 = 5760 L. Verification / Alternative check: Inlet as fraction: 240/5760 = 1/24 tank/h. Net = 1/24 − 1/6 = 1/24 − 4/24 = −3/24 = −1/8 tank/h ⇒ the full tank empties in 8 h, consistent. Why Other Options Are Wrong: Values like 5670 L or 5700 L would not satisfy the precise balance 240/V = 1/24, leading to a different emptying time. Common Pitfalls: Mixing liters per hour and tank fractions without introducing the tank capacity V, or assuming the tank must fill because the inlet is open. The sign of the net rate determines the direction. Final Answer: 5760 L

Question 15

Pairwise work times for three workers: A and B can complete a job in 12 days, B and C in 15 days, and C and A in 20 days. In how many days will A alone complete the work?

Accepted Answer

Introduction / Context: When pairwise completion times are given, summing pair rates provides the total combined rate of all three workers. From that, each individual rate can be extracted by subtracting the appropriate pair rate. This method avoids solving simultaneous equations explicitly. Given Data / Assumptions: (A + B) finish in 12 days ⇒ rate = 1/12. (B + C) finish in 15 days ⇒ rate = 1/15. (C + A) finish in 20 days ⇒ rate = 1/20. Job size assumed as 1 unit of work. Concept / Approach: Add the three pair rates: (A+B) + (B+C) + (C+A) = 2(A+B+C). Hence, total triple rate is half of that sum. Then, A’s rate = (A+B+C) − (B+C). Finally, the time for A alone is 1 divided by A’s rate. Step-by-Step Solution: Compute sum: 1/12 + 1/15 + 1/20 = (5 + 4 + 3)/60 = 12/60 = 1/5.Thus 2(A+B+C) = 1/5 ⇒ (A+B+C) = 1/10.A’s rate = (A+B+C) − (B+C) = 1/10 − 1/15 = (3 − 2)/30 = 1/30.Therefore A alone takes 30 days. Verification / Alternative check: With A = 1/30, B + C = 1/15. Also A + B = 1/12, so B = 1/12 − 1/30 = 1/20; then C = 1/15 − B = 1/15 − 1/20 = 1/60. Checks with C + A = 1/60 + 1/30 = 1/20, consistent. Why Other Options Are Wrong: 20 or 45 days contradict the derived individual rates; 60 days would halve A’s rate and mismatch pair totals. Common Pitfalls: Forgetting to divide the sum of pair rates by 2, or mixing up which pair to subtract to isolate A’s rate. Final Answer: 30 days

Question 16

Different efficiencies, combined time known: Two friends A and B working together finish a job in 26 days. Their efficiencies (work rates) are in the ratio 8 : 5. If B works alone, how many days will he take to complete the job?

Accepted Answer

Introduction / Context: When two workers have rates in a given ratio and their combined time is known, we can express their individual rates as proportional parts of the combined rate. From there, the solo completion time for either worker follows by taking the reciprocal of the individual rate. Given Data / Assumptions: Together time = 26 days ⇒ combined rate = 1/26 work/day. Rate ratio A : B = 8 : 5. Let A’s rate = 8k, B’s rate = 5k (work/day). Concept / Approach: Since 8k + 5k = 13k equals the combined rate 1/26, we can solve for k. Then B’s rate is 5k, and B’s required time is 1 divided by that rate. Keep units consistent as work/day. Step-by-Step Solution: 13k = 1/26 ⇒ k = 1/338.B’s rate = 5k = 5/338 work/day.B’s time = 1 ÷ (5/338) = 338/5 days = 67 3/5 days. Verification / Alternative check: A’s rate would be 8/338; combined with B’s 5/338 gives 13/338 = 1/26, matching the given teamwork time, so the proportioning is correct. Why Other Options Are Wrong: Values like 105 1/4 or 112 3/5 arise from arithmetic slips or wrong reciprocals; 214 2/3 and 52 days are inconsistent with the proportional split of the 1/26 rate. Common Pitfalls: Conflating time ratios with rate ratios. Remember, rates add; times do not. Use the combined rate to determine the constant of proportionality. Final Answer: 67 3/5 days

Question 17

Separating individual time from combined time: A and B together can complete a work in 35 days, while A alone can do it in 60 days. In how many days will B alone complete the same work?

Accepted Answer

Introduction / Context: This is a standard decomposition: with A’s solo time and the team time known, we can compute B’s solo rate by subtracting A’s rate from the joint rate. The solo time for B is then the reciprocal of that rate. Keep all rates in work/day for consistency. Given Data / Assumptions: (A + B) time = 35 days ⇒ rate(A+B) = 1/35. A time = 60 days ⇒ rate(A) = 1/60. Job size = 1 unit of work. Concept / Approach: rate(B) = rate(A+B) − rate(A) = 1/35 − 1/60. Find a common denominator and compute. Then time(B) = 1 / rate(B). Step-by-Step Solution: 1/35 − 1/60 = (60 − 35)/(35*60) = 25/2100 = 1/84.Therefore, B alone takes 84 days. Verification / Alternative check: Check the combined rate: 1/60 + 1/84 = (7 + 5)/420 = 12/420 = 1/35, matching the given team time. Why Other Options Are Wrong: 80, 88, 92, and 72 days do not satisfy the exact rate subtraction and will not reconstruct the 35-day joint time when combined with A’s rate. Common Pitfalls: Subtracting times instead of rates, or using incorrect denominators when combining fractions. Final Answer: 84 days

Question 18

Find A’s solo time from three pairwise times: A and B can do a job in 72 days, B and C in 120 days, and A and C in 90 days. How many days would A alone take to complete the work?

Accepted Answer

Introduction / Context: This mirrors the earlier pairwise method: sum the three pair rates to obtain twice the combined rate of all three. Then subtract the relevant pair to isolate the individual. The approach is algebraically neat and avoids a system of three unknowns directly. Given Data / Assumptions: (A + B) = 1/72 work/day. (B + C) = 1/120 work/day. (A + C) = 1/90 work/day. Concept / Approach: Sum: (A+B) + (B+C) + (A+C) = 2(A+B+C). Divide by 2 to get the combined rate of A, B, and C. Then A’s rate = (A+B+C) − (B+C). Invert to get A’s solo time in days. Step-by-Step Solution: Sum = 1/72 + 1/120 + 1/90.Using denominator 360: (5 + 3 + 4)/360 = 12/360 = 1/30.Hence (A+B+C) = (1/30)/2 = 1/60.A’s rate = 1/60 − 1/120 = 1/120 ⇒ A alone takes 120 days. Verification / Alternative check: Back-check: A = 1/120; then C = (A+C) − A = 1/90 − 1/120 = 1/360; and B = (A+B) − A = 1/72 − 1/120 = 1/180. Verify B + C = 1/180 + 1/360 = 1/120, correct. Why Other Options Are Wrong: 60 or 90 days contradict the derived individual rates; 115 and 100 days are not supported by the rate sums given. Common Pitfalls: Forgetting to halve the sum of pair rates; mixing up which pair to subtract to isolate A’s rate. Final Answer: 120 days

Question 19

Alternating workers by day: Ram finishes a job in 7 days, Rahim in 8 days, and Robert in 6 days. They work one after another in a three-day cycle: Day 1 Ram, Day 2 Rahim, Day 3 Robert, then repeat. Who is working on the last day when the work finish…

Accepted Answer

Introduction / Context: In cyclic work schedules, the total work done per full cycle helps estimate how many cycles are required. Then compare the remaining work with the contribution of the next worker in sequence to determine who completes the job on the final day. Given Data / Assumptions: Ram’s rate = 1/7 per day. Rahim’s rate = 1/8 per day. Robert’s rate = 1/6 per day. Cycle order: Ram → Rahim → Robert → repeat. Concept / Approach: Compute the three-day cycle contribution S = 1/7 + 1/8 + 1/6. Multiply by the number of full cycles and then evaluate the remainder relative to the next person’s daily rate. The person whose day reaches or exceeds the remaining work is the finisher. Step-by-Step Solution: S = 1/7 + 1/8 + 1/6 = (24 + 21 + 28)/168 = 73/168 per 3 days.After 2 cycles (6 days): work done = 2 * 73/168 = 146/168.Remaining = 1 − 146/168 = 22/168.Next is Ram (Day 7) with rate 1/7 = 24/168 ≥ 22/168 ⇒ Ram finishes. Verification / Alternative check: If Ram’s final-day effort slightly exceeds remaining work, it simply means he completes it that day. There is no need for the others to work further. Why Other Options Are Wrong: Neither Rahim nor Robert is needed after Ram’s final contribution covers the remaining 22/168 of the work. Common Pitfalls: Misordering the cycle or assuming equal daily contributions. Always compare the exact remaining fraction with the next worker’s full-day contribution. Final Answer: Ram

Question 20

Men and boys with different efficiencies: 12 men and 16 boys can finish a job in 5 days, while 13 men and 24 boys can finish it in 4 days. In how many days will 5 men and 10 boys complete the same job?

Accepted Answer

Introduction / Context: Multi-category workforce problems use linear equations in unknown per-day efficiencies. By forming two equations from two scenarios, we can deduce the relative efficiencies of men and boys, then compute the time for a different combination of workers to finish the same job. Given Data / Assumptions: Let a man’s daily work = m, a boy’s daily work = b. 12m + 16b completes in 5 days ⇒ total work W = 5(12m + 16b). 13m + 24b completes in 4 days ⇒ W = 4(13m + 24b). Concept / Approach: Equate the two expressions for W and solve for the ratio m : b. Then compute W explicitly and use the daily work rate of 5 men + 10 boys to find the required days = W / daily rate. Step-by-Step Solution: 5(12m + 16b) = 4(13m + 24b)60m + 80b = 52m + 96b ⇒ 8m = 16b ⇒ m = 2b.W = 5(12*2b + 16b) = 5(24b + 16b) = 200b.Daily rate for 5 men + 10 boys = 5m + 10b = 5(2b) + 10b = 20b.Days required = W / rate = 200b / 20b = 10 days. Verification / Alternative check: Using m = 2b, check the second scenario: 13m + 24b = 26b + 24b = 50b; time = W / (50b) = 200b / 50b = 4 days, consistent. Why Other Options Are Wrong: 12, 9, 15, or 8 days conflict with the deduced efficiencies and do not satisfy both given scenarios simultaneously. Common Pitfalls: Treating men and boys as equally efficient or failing to equate total work across scenarios. Always solve for relative efficiencies first. Final Answer: 10 days

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