Question 1

Man-days held constant, compressing duration: 10 men can build a wall in 8 days. If the job must be completed in half a day (0.5 day), how many men are required, assuming everyone works at the same constant rate?

Accepted Answer

Introduction / Context:This is a pure man-days calculation. For the same job, (men * days) remains constant if per-man productivity is constant. To reduce the number of days, the number of men must increase proportionally. Given Data / Assumptions: Initial plan: 10 men * 8 days = 80 man-days of work. New plan: finish in 0.5 day. All workers have equal, constant productivity; no setup overheads. Concept / Approach:Let M be the required number of men for half a day. Since total work is unchanged, M * 0.5 = 80. Solve for M. Step-by-Step Solution:Work W = 80 man-days.Let M men work for D = 0.5 day: M * 0.5 = 80.M = 80 / 0.5 = 160 men. Verification / Alternative check:Cross-check with proportionality: If time is reduced by a factor of 16 (from 8 days to 0.5 day), the men required increase by the same factor: 10 * 16 = 160. Why Other Options Are Wrong: 80, 100, 120 are too few; they would not supply the necessary 80 man-days within 0.5 day. Common Pitfalls: Reading “half days” ambiguously; here we explicitly interpret as “half a day” to match the viable option and standard man-day logic. Final Answer:160

Question 2

Inverse proportionality with number of workers: 6 boys can complete a job in 16 hours. Assuming equal productivity for each boy, in how many hours will 8 boys complete the same job?

Accepted Answer

Introduction / Context:When more identical workers are added, the time decreases inversely with the number of workers, provided productivity per worker remains the same and there are no constraints like crowding or setup times. Given Data / Assumptions: 6 boys finish in 16 hours. All boys have equal constant productivity. We need the time with 8 boys. Concept / Approach:Total work W is fixed. Work = (number of boys) * (hours). Therefore, time for 8 boys = (6 * 16) / 8 hours. Step-by-Step Solution:W = 6 * 16 = 96 boy-hours.With 8 boys, time = 96 / 8 = 12 hours. Verification / Alternative check:Proportional check: Going from 6 to 8 workers is a factor of 6/8 = 0.75 in time. 0.75 * 16 = 12 hours, consistent. Why Other Options Are Wrong: 10, 8, 14 do not match the invariant W = 96 boy-hours with 8 workers. Common Pitfalls: Assuming time decreases linearly with an absolute difference in workers (it is inverse proportional, not additive). Final Answer:12

Question 3

Stocks for students: direct inverse proportion: A hostel has food stock for 45 days for 120 students. If 30 more students join (total 150), in how many days will the stock be exhausted, assuming equal consumption per student?

Accepted Answer

Introduction / Context:Food stock problems are classic inverse proportion questions. If total stock is fixed and per-head daily consumption is constant, days available vary inversely with the number of consumers. Given Data / Assumptions: Initial: 120 students for 45 days. New total: 150 students (120 + 30). Equal daily consumption per student; no wastage. Concept / Approach:Total stock S = students * days (in “student-days”). After joining, new days = S / new_students. Step-by-Step Solution:S = 120 * 45 = 5400 student-days.New days = 5400 / 150 = 36 days. Verification / Alternative check:Proportional check: Increasing students by factor 150/120 = 1.25 shrinks days by factor 1/1.25 = 0.8; 0.8 * 45 = 36 days. Why Other Options Are Wrong: 38, 40, 32 do not reflect the exact inverse proportion result. Common Pitfalls: Subtracting or adding days directly rather than using the product students * days constant. Final Answer:36

Question 4

Work-rate from a given case, then scale to a new target: 5 boys take 7 hours to pack 35 toys. How many boys are required to pack 65 toys in 3 hours, assuming all work at the same constant rate?

Accepted Answer

Introduction / Context:Here we infer the group productivity from a known case and then scale it to meet a different production target within a different time window. The method is a straightforward application of proportion based on (boys * hours) = toys / productivity per boy-hour. Given Data / Assumptions: 5 boys * 7 hours produce 35 toys. Assume linear productivity and no setup overheads; all boys identical. Goal: 65 toys in 3 hours. Concept / Approach:First compute toys per boy-hour from the given case, then compute required boy-hours for 65 toys, and finally divide by 3 hours to get boys needed. Round up to the next whole number since a fraction of a boy is not possible in headcount terms. Step-by-Step Solution:From the given: (5 * 7) boy-hours produce 35 toys ⇒ 35 / 35 = 1 toy per boy-hour.Required boy-hours for 65 toys = 65 / 1 = 65 boy-hours.In 3 hours, boys needed = 65 / 3 ≈ 21.666…, i.e., 22 boys. Verification / Alternative check:If 22 boys work 3 hours, total boy-hours = 66, and toys at 1 toy per boy-hour = 66 ≥ 65. With 21 boys, only 63 toys result, which is insufficient. Why Other Options Are Wrong: 26, 39, 45 greatly exceed the minimum required 22 and are not the least feasible integer headcount. Common Pitfalls: Treating 21.666… as 21 instead of rounding up to ensure the target is met. Final Answer:None of these

Question 5

Man-days constant for fixed work: A stock of food is sufficient for 240 men for 48 days. For how many days will the same stock last for 160 men, assuming equal daily consumption per man?

Accepted Answer

Introduction / Context:Food-stock questions rely on the constancy of total consumption measured in man-days. If the number of men changes, the duration adjusts inversely so that (men * days) remains constant, provided per-man consumption is unchanged. Given Data / Assumptions: Initial: 240 men for 48 days. All men consume equal amounts per day; no wastage. New team size: 160 men. Concept / Approach:Total stock S (in man-days) = men * days. New days = S / new_men. Step-by-Step Solution:S = 240 * 48 = 11520 man-days.For 160 men, days = 11520 / 160 = 72 days. Verification / Alternative check:Proportionality: Reducing men from 240 to 160 is multiplying by 160/240 = 2/3; days should increase by 3/2: 48 * 1.5 = 72 days. Why Other Options Are Wrong: 54, 60, 64 do not satisfy S = 11520 when multiplied by 160. Common Pitfalls: Adding/subtracting days linearly instead of using the product men * days. Final Answer:72 days

Question 6

Joint work time from individual rates: Worker A can complete the job in 4 days, while worker B can complete the same job in 12 days. If both A and B work together at their constant daily rates from start to finish, how many days will they take to co…

Accepted Answer

Introduction / Context:This problem reinforces the core Time-and-Work principle that individual work rates add when people work together. We convert each person’s completion time into a daily rate (jobs per day), add the rates, and take the reciprocal to get the total time. Given Data / Assumptions: A alone finishes in 4 days ⇒ rate_A = 1/4 job per day. B alone finishes in 12 days ⇒ rate_B = 1/12 job per day. They work together continuously at constant rates. Concept / Approach:For one complete job, total time T equals 1 divided by the combined rate R. That is, T = 1 / (rate_A + rate_B). Always add rates, not times, when tasks are done simultaneously. Step-by-Step Solution:rate_A = 1/4rate_B = 1/12Combined rate R = 1/4 + 1/12 = 3/12 + 1/12 = 4/12 = 1/3 job/dayTime T = 1 / (1/3) = 3 days Verification / Alternative check:In 3 days, A would complete 3 * (1/4) = 3/4 of the job and B would complete 3 * (1/12) = 1/4 of the job. Total = 3/4 + 1/4 = 1 job, confirming correctness. Why Other Options Are Wrong: 2 days: too short; at rate 1/3 per day, 2 days produce only 2/3 of the job. 4 days and 5 days: too long; they exceed the exact reciprocal of the combined rate. Common Pitfalls: Adding the times 4 and 12 directly, instead of adding rates 1/4 and 1/12. Arithmetic slip in finding the common denominator and simplifying 1/4 + 1/12. Final Answer:3 days

Question 7

Isolating a third worker’s rate from total: Workers A, B, and C together complete a job in 2 hours. A alone would take 6 hours and B alone would take 5 hours. How long would C alone take to finish the job if working at a constant rate?

Accepted Answer

Introduction / Context:When three workers together finish a job in a given time, we can subtract known individual rates from the total joint rate to discover the unknown third worker’s rate. This is a standard pipes-and-cisterns/time-and-work technique based on linear additivity of rates. Given Data / Assumptions: Total time (A+B+C) = 2 h ⇒ combined rate = 1/2 job per hour. A alone: 6 h ⇒ rate_A = 1/6 job per hour. B alone: 5 h ⇒ rate_B = 1/5 job per hour. Concept / Approach:Let rate_C be C’s solo rate. Since rates add: rate_A + rate_B + rate_C = 1/2. Hence, rate_C = 1/2 − 1/6 − 1/5. Once rate_C is known, C’s time alone is the reciprocal, 1 / rate_C. Step-by-Step Solution:rate_A = 1/6, rate_B = 1/5, total rate = 1/2rate_C = 1/2 − 1/6 − 1/5Use denominator 30: 1/2 = 15/30, 1/6 = 5/30, 1/5 = 6/30rate_C = 15/30 − 5/30 − 6/30 = 4/30 = 2/15 job per hourTime_C = 1 / (2/15) = 15/2 hours = 7 1/2 h Verification / Alternative check:Re-sum rates: 1/6 + 1/5 + 2/15 = 5/30 + 6/30 + 4/30 = 15/30 = 1/2 job per hour → total time 2 h, consistent. Why Other Options Are Wrong: 5 1/2 h and 4 1/2 h: imply rates greater than 2/15, which would overshoot the total when summed with A and B. 9 h: implies a rate lower than 2/15 and fails to reach the total combined rate of 1/2. Common Pitfalls: Subtracting times instead of rates. Arithmetic mistakes in handling fractions with different denominators. Final Answer:7 1/2 h

Question 8

Inferring individual rates from joint output: A can copy 75 pages in 25 hours. Working together, A and B can copy 135 pages in 27 hours. Assuming constant rates, how long will B alone take to copy 42 pages?

Accepted Answer

Introduction / Context:This question requires deducing the individual rates of two workers from given production data. Once A’s solo rate and the A+B combined rate are known, B’s solo rate is the difference. Finally, time for B to complete a specified number of pages is pages divided by B’s rate. Given Data / Assumptions: A copies 75 pages in 25 hours ⇒ rate_A = 75/25 = 3 pages/hour. A and B together copy 135 pages in 27 hours ⇒ rate_{A+B} = 135/27 = 5 pages/hour. Rates are constant; no setup or idle time. Concept / Approach:Since the combined rate equals the sum of individual rates, rate_B = rate_{A+B} − rate_A. Then compute B’s time for a 42-page job using T = pages / rate_B. Step-by-Step Solution:rate_A = 3 pages/hourrate_{A+B} = 5 pages/hourrate_B = 5 − 3 = 2 pages/hourTime for B to copy 42 pages = 42 / 2 = 21 hours Verification / Alternative check:Check with proportions: In 21 hours, B would copy 42 pages, and A would copy 63 pages in the same time at 3 pages/hour; together they would produce 105 pages in 21 hours, which scales to 135 pages in 27 hours at 5 pages/hour, consistent with the given joint data. Why Other Options Are Wrong: 18 hours: implies B’s rate 42/18 = 2.333… pages/hour, which contradicts the deduced rate 2 pages/hour. 24 hours: implies B’s rate 1.75 pages/hour, too slow. 15 hours: implies B’s rate 2.8 pages/hour, too fast given the combined rate constraint. Common Pitfalls: Dividing 135 by 27 incorrectly; ensure rate_{A+B} = 5 pages/hour. Assuming the difference of times instead of the difference of rates. Final Answer:21 hours

Question 9

Portion of work remaining after partial progress: Mohan can mow his entire lawn in x hours at a constant rate. After he works for 2 hours, it begins to rain and he stops. What fraction of the lawn remains unmowed at that moment?

Accepted Answer

Introduction / Context:This is a straightforward fraction-of-work problem. If a job takes x hours at a steady rate, then each hour completes an equal fraction of the job. After a certain number of hours, we can compute the completed fraction and subtract from 1 to find what remains. Given Data / Assumptions: Total time to mow entire lawn = x hours. Work rate is constant over time. Mohan works for 2 hours before stopping. Concept / Approach:The fraction of the job completed in t hours at constant speed is t/x. Therefore, the remaining fraction is 1 − t/x. Substitute t = 2 to get the specific fraction left unmowed. Step-by-Step Solution:Completed fraction after 2 hours = 2/xRemaining fraction = 1 − 2/x = (x − 2)/x Verification / Alternative check:Edge cases: If x = 2 hours (he needs exactly 2 hours to mow all), then remaining = (2−2)/2 = 0, meaning nothing remains—correct. If x is very large, the remaining is close to 1, which is sensible because only a tiny fraction would be done in 2 hours. Why Other Options Are Wrong: 2/x: this is the completed fraction, not the remaining part. (2 − x)/2 and x/2: both have wrong structure and units for a fraction of the whole and do not simplify to a value between 0 and 1 for all x > 2. Common Pitfalls: Confusing completed fraction with remaining fraction. Not expressing both parts over the common denominator x. Final Answer:(x - 2)/x

Question 10

Rates and time are inversely proportional: If the working rates of A and B are in the ratio 3 : 4, then the numbers of days taken by A and B to finish the same job (working individually) will be in what ratio?

Accepted Answer

Introduction / Context:Time-and-Work questions often pivot on the inverse relation between rate and time for a fixed amount of work. If one worker is faster, they take less time, and the ratios invert perfectly when comparing identical jobs. Given Data / Assumptions: rate_A : rate_B = 3 : 4. They work individually on the same one-job task. Concept / Approach:For one complete job, time = 1 / rate. Therefore, the ratio of times is the reciprocal of the ratio of rates. If rate_A : rate_B = 3 : 4, then time_A : time_B = 1/3 : 1/4 = 4 : 3 after clearing denominators. Step-by-Step Solution:rate_A : rate_B = 3 : 4time_A : time_B = (1/3) : (1/4)Multiply both terms by 12 to clear denominators ⇒ 4 : 3 Verification / Alternative check:Pick concrete rates: let rate_A = 3 units/day and rate_B = 4 units/day. For 12 units of work: time_A = 12/3 = 4 days; time_B = 12/4 = 3 days; thus time_A : time_B = 4 : 3, confirming the inverse relationship. Why Other Options Are Wrong: 3 : 4 is the rate ratio, not the time ratio. 9 : 16 is the square of the rate ratio; time does not scale with the square of rates here. None of these: incorrect because a valid ratio (4 : 3) is available. Common Pitfalls: Confusing the rate ratio with the time ratio. Forgetting to invert the ratio when moving from rates to times. Final Answer:4 : 3

Question 11

Time and work comparison: A does half as much work as B in three-fourths of B's time. If together A and B complete the whole job in 18 days, how many days would B alone require to finish the entire work?

Accepted Answer

Introduction / Context: This time-and-work problem compares efficiencies. The statement “A does half as much work as B in three-fourths of the time” encodes a relationship between their daily work rates. Using that, we combine their rates to match the given combined time and then isolate B’s solo time. Given Data / Assumptions: Together time to finish 1 job = 18 days. In 3/4 unit of time, A does half as much work as B does in 1 unit of time. Work is uniform; rates are constant; 1 job is the total work. Concept / Approach: Translate the wording into a rate ratio. If B’s rate is r_B (job/day), the comparison gives A’s rate r_A as a multiple of r_B. Then r_A + r_B equals the together rate. Finally, invert B’s rate to get B’s time alone. Step-by-Step Solution: Interpretation: In 3/4 time, A does 1/2 of what B does in 1 time ⇒ r_A = (1/2) / (3/4) * r_B = (2/3) * r_B. Together rate r_A + r_B = (2/3)r_B + r_B = (5/3)r_B. Given together time = 18 ⇒ (5/3)r_B = 1/18 ⇒ r_B = 3/(5*18) = 1/30 job/day. Therefore, B alone takes 30 days. Verification / Alternative check: Together rate computed from B’s rate: (5/3)*(1/30) = 1/18 job/day, matching the given 18 days for the whole work. Why Other Options Are Wrong: 40 days and 35 days contradict the derived rate relation and the 18-day team time. “None of these” is invalid because 30 days is consistent with all conditions. Common Pitfalls: Misreading “half as much work in three-fourths of the time” and setting r_A = 1/2 r_B (ignoring the time scaling). Always convert comparative statements to rates carefully. Final Answer: 30 days

Question 12

Work allocation with partial progress: A and B together can complete a job in 24 days. B alone completes one-third of the job in 12 days. How long will A alone take to complete the remaining two-thirds of the job?

Accepted Answer

Introduction / Context: This problem blends combined work with a partial-completion scenario. We are told B's performance for one-third of the job and the joint time for the full job. From these, we compute A’s rate and then the time A takes to finish the remaining portion alone. Given Data / Assumptions: Together time (A + B) = 24 days ⇒ joint rate = 1/24 job/day. B does 1/3 of the job in 12 days ⇒ B’s rate = (1/3) / 12 = 1/36 job/day. Uniform work rates; total work = 1 job. Concept / Approach: Compute A’s rate as (joint rate − B’s rate). The remaining work after B’s one-third is two-thirds. Time for A to finish that amount is (remaining work) / (A’s rate). Step-by-Step Solution: Joint rate = 1/24; B’s rate = 1/36. A’s rate = 1/24 − 1/36 = (3 − 2)/72 = 1/72 job/day. Remaining work after B’s 1/3 = 2/3. Time for A = (2/3) / (1/72) = (2/3)*72 = 48 days. Verification / Alternative check: A alone for the whole job would take 72 days (since rate 1/72). Doing only two-thirds thus requires 48 days, which matches the computation. Why Other Options Are Wrong: 36 or 24 days underestimate A’s slower solo rate; 72 days corresponds to the full job by A, not just the remainder. Common Pitfalls: Treating 12 days as B's whole-job time (it is only for one-third). Always convert to per-day rates before combining or subtracting. Final Answer: 48 days

Question 13

Man-days method: If 15 men complete a work in 16 days, in how many days will 24 men complete the same work (assume identical efficiency for all men)?

Accepted Answer

Introduction / Context: This question uses conservation of man-days (work = workers * days) under the assumption that all workers are equally efficient. Increasing the number of workers should reduce the time proportionally for the same total amount of work. Given Data / Assumptions: 15 men finish the job in 16 days. All men have equal efficiency; work scale is linear. We need time for 24 men to do the same job. Concept / Approach: Compute the total man-days from the first scenario, then divide by the new workforce to get the required time. Formula: days_2 = (men_1 * days_1) / men_2. Step-by-Step Solution: Total man-days = 15 * 16 = 240. With 24 men, required days = 240 / 24 = 10. Verification / Alternative check: Ratio method: Time varies inversely with men. 24/15 = 8/5 ⇒ time factor = 5/8 ⇒ 16 * 5/8 = 10 days. Why Other Options Are Wrong: 7, 8, or 12 days do not match the linear inverse proportion implied by identical efficiencies. Common Pitfalls: Forgetting that the product men * days remains constant for the same total work when efficiency is unchanged. Final Answer: 10 days

Question 14

Combined daily work fraction: A can complete a piece of work in 18 days and B can complete the same work in half of A’s time (i.e., 9 days). Working together, what fraction of the work do they complete in one day?

Accepted Answer

Introduction / Context: When two workers collaborate, their daily work fractions add. Given individual completion times, we can compute each daily rate and sum them to obtain the combined one-day fraction of the job completed. Given Data / Assumptions: A’s time = 18 days ⇒ A’s rate = 1/18 job/day. B’s time = 9 days ⇒ B’s rate = 1/9 job/day. Total work = 1 job; rates are constant. Concept / Approach: Combined rate = (1/18 + 1/9). This directly gives the fraction of the work done in a single day when they work together. Step-by-Step Solution: 1/18 + 1/9 = 1/18 + 2/18 = 3/18 = 1/6. So, in one day together they complete 1/6 of the job. Verification / Alternative check: Time together would be 1 ÷ (1/6) = 6 days, which lies between 9 and 18 days, as expected for combined work. Why Other Options Are Wrong: 1/9 is A’s rate alone if B were slower (not the case). 2/5 and 2/7 are unrelated to these exact reciprocals. Common Pitfalls: Averaging times instead of adding rates. Always combine work problems via rates, not raw days. Final Answer: 1/6

Question 15

Find A’s solo time: A and B together can finish a piece of work in 12 days, while B alone can do it in 30 days. In how many days can A alone complete the entire work?

Accepted Answer

Introduction / Context: We are given the combined time and one individual time. Subtract B’s rate from the joint rate to get A’s rate, then invert to obtain A’s solo time to finish the full work. Given Data / Assumptions: Together time (A + B) = 12 days ⇒ joint rate = 1/12. B alone time = 30 days ⇒ B’s rate = 1/30. Work is uniform; total work = 1 job. Concept / Approach: A’s rate = (joint rate − B’s rate). A’s time = 1 / (A’s rate). Step-by-Step Solution: Joint rate = 1/12; B’s rate = 1/30. A’s rate = 1/12 − 1/30 = (5 − 2)/60 = 3/60 = 1/20. Hence, A’s time = 20 days. Verification / Alternative check: Combined check: 1/20 + 1/30 = (3 + 2)/60 = 5/60 = 1/12, which is consistent with the given joint time. Why Other Options Are Wrong: 15, 18, 25 days do not satisfy the rate equation when combined with 30 days for B. Common Pitfalls: Averaging 12 and 30 to estimate A’s time. The correct method is to use rates and subtraction. Final Answer: 20 days

Question 16

Scaling identical work: Aarti completes one job in 6 days. In how many days will she complete three jobs of the same type (assuming identical size and rate)?

Accepted Answer

Introduction / Context: When identical jobs are repeated and the worker’s rate remains constant, total time scales linearly with the number of jobs. Here, “three jobs” means three times the original amount of work at the same pace. Given Data / Assumptions: Aarti’s time per job = 6 days. Number of identical jobs = 3. Rate is constant; no setup or downtime is introduced. Concept / Approach: Total time for k identical jobs at the same rate = k * (time for one job). Thus, multiply the single-job duration by 3. Step-by-Step Solution: Time for 1 job = 6 days. Time for 3 jobs = 3 * 6 = 18 days. Verification / Alternative check: Rate-based view: Aarti’s rate = 1/6 job/day. To complete 3 jobs, needed days = 3 / (1/6) = 18 days. Why Other Options Are Wrong: 21 days assumes slower rate; 3 and 6 days ignore the scaling by three. Common Pitfalls: Confusing “three times the work” with “three workers.” The scenario keeps the same worker and rate, so time multiplies by 3. Final Answer: 18 days

Question 17

Combine two workers: A and B can complete a piece of work in 6 days and 12 days respectively. If both work together at their constant rates, in how many days will the job be finished?

Accepted Answer

Introduction / Context: Classic combined-work calculation: sum individual daily rates to get the team rate, then invert to find the total time needed when both work simultaneously without interruptions or interference. Given Data / Assumptions: A’s time = 6 days ⇒ rate = 1/6 job/day. B’s time = 12 days ⇒ rate = 1/12 job/day. Work is 1 job; rates add when working together. Concept / Approach: Team rate = 1/6 + 1/12 = 1/4 job/day. Total time = 1 ÷ team rate = 4 days. Step-by-Step Solution: 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4. Time = 1 / (1/4) = 4 days. Verification / Alternative check: As a reasonableness check, the combined time (4) is less than the smaller of 6 and 12, which is expected. Why Other Options Are Wrong: 6 or 9 days exceed the correct joint time; 18 is far too large for a team that includes a faster worker. Common Pitfalls: Averaging 6 and 12 to get 9 instead of adding rates. Always add rates for combined work. Final Answer: 4 days

Question 18

Proportional workforce scaling: If 3 men and 4 boys can complete a piece of work in 8 days, then how many days are required for 6 men and 8 boys to complete the same work (assume same individual efficiencies)?

Accepted Answer

Introduction / Context: When the entire workforce is doubled (both men and boys in the same proportions), and all efficiencies are constant, the time required halves. This uses the linearity of work with respect to total effective workers and time. Given Data / Assumptions: (3 men + 4 boys) finish in 8 days. (6 men + 8 boys) is exactly double the effective workforce. Efficiencies are constant; work remains the same 1 job. Concept / Approach: Work = (effective workers) * days. Doubling workers halves the required days for the same work. Step-by-Step Solution: Original man-equivalents: W = (3M + 4B) * 8. New team is 2*(3M + 4B), so days = 8 / 2 = 4. Verification / Alternative check: If you define rates m and b, original daily rate = 3m + 4b; doubled rate = 6m + 8b = 2*(3m + 4b); the time divides by 2. Why Other Options Are Wrong: 2 days would require a quadrupling, not doubling. 6 and 16 days contradict proportionality. Common Pitfalls: Changing only men or only boys in comparisons; here both categories are doubled, preserving ratio. Final Answer: 4 days

Question 19

Rates with a parameter: A can do a piece of work in x days and B can do the same work in 3x days. Working together, they finish it in 12 days. Find the value of x.

Accepted Answer

Introduction / Context: This parameterized work problem asks you to form and solve an equation in x using combined rates. The key is to express both A’s and B’s rates in terms of x and match their sum to the given joint time of 12 days for the full job. Given Data / Assumptions: A’s time = x days ⇒ rate = 1/x. B’s time = 3x days ⇒ rate = 1/(3x). Together time = 12 days ⇒ joint rate = 1/12. Concept / Approach: Add rates: 1/x + 1/(3x) = (4/3)*(1/x). Set equal to 1/12 and solve for x. Then select the correct numerical value from the options. Step-by-Step Solution: 1/x + 1/(3x) = (4/3)*(1/x). Set (4/3)*(1/x) = 1/12 ⇒ 1/x = (1/12)*(3/4) = 1/16. Therefore, x = 16. Verification / Alternative check: Rates: A = 1/16, B = 1/48; Sum = 1/12, giving 12 days jointly, consistent with the data. Why Other Options Are Wrong: 8, 10, 12 do not satisfy the rate equation when substituted back. Common Pitfalls: Averaging times (x and 3x) or averaging numbers (1 and 3) instead of adding reciprocal rates and equating to 1/12. Final Answer: 16

Question 20

Two-worker collaboration: X alone completes a job in 12 days. Together, X and Y complete the job in 6 2/3 days. In how many days can Y alone finish the job?

Accepted Answer

Introduction / Context: Here we know one person's solo time and the pair's time. We first find the joint rate, subtract X’s rate to get Y’s rate, and finally invert to determine Y’s solo completion time. Given Data / Assumptions: X alone = 12 days ⇒ rate_X = 1/12. X + Y together = 6 2/3 days = 20/3 days ⇒ joint rate = 3/20. Total work = 1 job; constant rates. Concept / Approach: rate_Y = joint rate − rate_X. Then Y’s time = 1 / rate_Y. Keep mixed numbers consistent by converting to improper fractions where needed. Step-by-Step Solution: Joint rate = 3/20; X’s rate = 1/12. rate_Y = 3/20 − 1/12 = (9 − 5)/60 = 4/60 = 1/15. Hence, Y alone takes 15 days. Verification / Alternative check: Check: 1/12 + 1/15 = (5 + 4)/60 = 9/60 = 3/20 ⇒ 20/3 days together, as given. Why Other Options Are Wrong: 10 or 12 days are too fast for Y given the combined time; 18 days would make the pair slower than stated. Common Pitfalls: Mishandling the mixed fraction 6 2/3; always convert to an improper fraction to avoid arithmetic slips. Final Answer: 15 days

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