Question 1

A and B together can complete a certain job in 20 days. They work together for 15 days, after which B leaves and A alone finishes the remaining work in another 10 days. In how many days can A alone complete the entire work if he works from the begin…

Accepted Answer

Introduction / Context: This problem is a classic Time and Work question involving two workers, A and B, starting together and then one of them, B, leaving after some days. We are asked to deduce the individual time A would take to finish the entire job alone, using information about their combined time and the extra days A works alone. Given Data / Assumptions: A and B together complete the job in 20 days. They work together for 15 days. After 15 days, B leaves and A alone finishes the remaining work in 10 more days. Total work is taken as 1 unit. Both A and B work at constant rates. Concept / Approach: We use the idea that rate of work multiplied by time equals total work. First, we find the combined rate of A and B by using the fact that they together finish in 20 days. Next, we calculate how much work they complete together in 15 days and how much work is left. Since A alone finishes the remaining work in 10 days, his individual daily rate can be determined, and thus his total time for the whole job can be calculated. Step-by-Step Solution: Let the rates of A and B be a and b units per day respectively. Total work W = 1 unit. A and B together finish in 20 days, so (a + b) * 20 = 1. Thus, a + b = 1 / 20. They work together for 15 days, so work done in 15 days = 15 * (a + b) = 15 * (1 / 20) = 3 / 4. Remaining work after 15 days = 1 - 3 / 4 = 1 / 4. A alone finishes this 1 / 4 work in 10 days, so a * 10 = 1 / 4. Therefore, a = (1 / 4) / 10 = 1 / 40 units per day. Time taken by A alone to finish entire work = total work / rate = 1 / (1 / 40) = 40 days. Verification / Alternative check: If A takes 40 days alone, then his daily rate is 1 / 40. We know a + b = 1 / 20, so b = 1 / 20 - 1 / 40 = 1 / 40. Thus, A and B have equal rates, each doing 1 / 40 of the work per day. Together they do 1 / 20 per day, so in 20 days they finish the job, which matches the given information. The calculations are therefore consistent. Why Other Options Are Wrong: Option A (30 days) would give a rate of 1 / 30, which combined with some b cannot satisfy both the 20 days total and 10 days remaining conditions simultaneously. Option C (50 days) and Option D (60 days) similarly yield inconsistent rates when substituted back into the combined-work equation and the remaining-work condition. Only 40 days satisfies all constraints in the problem statement. Common Pitfalls: Students often try to average times instead of working with rates, leading to incorrect answers. Another mistake is to assume that since they worked 15 out of 20 days together, the remaining work must be exactly 1 / 4 without checking the rate; in this case, it fortunately is 1 / 4, but that comes from the proper calculation. Always convert time to rate (1 / time), compute work done, and then determine unknown quantities. Final Answer: A alone can complete the entire job in 40 days if he works from the beginning by himself.

Question 2

Ten women can complete a certain piece of work in 6 days, while six men can complete the same work in 5 days. What is the ratio of the efficiency (daily work rate) of a man to that of a woman?

Accepted Answer

Introduction / Context: This question is a straightforward application of the Time and Work concept, focusing on comparing efficiencies of men and women when both groups complete the same job in different times and numbers. Efficiency here means the amount of work a person can do per day. Since the total work is the same, we can compare the total man-days and woman-days to obtain the ratio of their individual capacities. Given Data / Assumptions: 10 women complete the work in 6 days. 6 men complete the same work in 5 days. All women have equal efficiency. All men have equal efficiency. Total work is taken as 1 unit. Concept / Approach: The central idea is that Work = (Number of workers) * (Efficiency per worker) * (Time). Since both groups complete the same work, their total worker-days represent the same total work. By equating these and simplifying, we can find the ratio of the efficiency of one man to that of one woman. We will treat the woman-day and man-day as units of work produced and then derive efficiencies from them. Step-by-Step Solution: Let the total work be W = 1 unit. Efficiency of one woman = Ew units per day. Efficiency of one man = Em units per day. Work done by 10 women in 6 days = 10 * Ew * 6 = 60Ew. Since this completes the job, 60Ew = 1. So, Ew = 1 / 60 units per day. Work done by 6 men in 5 days = 6 * Em * 5 = 30Em. This also completes the job, so 30Em = 1. Thus, Em = 1 / 30 units per day. Ratio of efficiency of one man to one woman = Em : Ew = (1 / 30) : (1 / 60). = (1 / 30) * (60 / 1) = 2 : 1. Verification / Alternative check: To verify, note that if one woman does 1 / 60 of the work in a day, 10 women do 10 / 60 = 1 / 6 per day, and hence finish in 6 days. Similarly, if one man does 1 / 30 of the work in a day, 6 men do 6 / 30 = 1 / 5 per day, which completes the work in 5 days. This confirms the consistency of our computed efficiencies and the ratio 2 : 1. Why Other Options Are Wrong: Option A (1 : 2) reverses the correct ratio, implying women are twice as efficient as men, which contradicts our calculations. Option C (2 : 3) and Option D (3 : 2) are arbitrary ratios that do not satisfy the given conditions when tested. Only 2 : 1 correctly reflects that a man's daily work is twice that of a woman under the data provided. Common Pitfalls: A common mistake is to compare times directly rather than computing efficiencies. For example, students might incorrectly think that since women take 6 days and men take 5 days, the ratio should simply be 5 : 6 or similar. However, the number of workers is different, so both time and number of workers must be considered. Always use Work = rate * time and then solve for individual rates before forming the ratio. Final Answer: The ratio of the efficiency of a man to that of a woman is 2 : 1.

Question 3

A shopkeeper must print a fixed number of documents and has three machines P, Q and R for this job. Machine P can complete the entire job in 3 days, machine Q in 4 days and machine R in 6 days. If all three machines are used simultaneously from the …

Accepted Answer

Introduction / Context: This question focuses on combined work rates when multiple machines are working together. Machines P, Q and R can each complete the entire job individually in different amounts of time. We are required to find the total time taken when all three operate simultaneously. Such problems are very common in aptitude tests in the Time and Work chapter, especially in scenarios involving machines, taps or workers. Given Data / Assumptions: Machine P completes the job in 3 days. Machine Q completes the job in 4 days. Machine R completes the job in 6 days. All three work together from the beginning. Total work is considered as 1 complete job. Each machine works at a constant rate. Concept / Approach: We convert the individual times into daily work rates. If a machine finishes a job in T days, its rate is 1 / T of the job per day. The combined rate is the sum of individual rates since they work simultaneously. Once we know the combined daily rate, the total time taken is the reciprocal of that combined rate, because time = work / rate and total work is 1 unit. Step-by-Step Solution: Let total work W = 1 unit (one complete printing job). Rate of P = 1 / 3 job per day. Rate of Q = 1 / 4 job per day. Rate of R = 1 / 6 job per day. Combined rate when P, Q and R work together = 1 / 3 + 1 / 4 + 1 / 6. Take LCM of 3, 4, 6 which is 12. 1 / 3 = 4 / 12, 1 / 4 = 3 / 12, 1 / 6 = 2 / 12. Combined rate = (4 + 3 + 2) / 12 = 9 / 12 = 3 / 4 job per day. Time taken when all three work together = total work / combined rate = 1 / (3 / 4) = 4 / 3 days. Verification / Alternative check: In 4 / 3 days, the total work done should equal 1 job. Check: (3 / 4) * (4 / 3) = 1, so the calculation is consistent. Alternatively, you can imagine 12 days as a convenient common multiple: in 12 days, P does 4 jobs, Q does 3 jobs and R does 2 jobs, totaling 9 jobs. Therefore, 1 job takes 12 / 9 = 4 / 3 days. Why Other Options Are Wrong: Option B (2 days) would imply a combined rate of 1 / 2 job per day, which is slower than P alone, so it cannot be correct. Option C (3/2 days) corresponds to a combined rate of 2 / 3 job per day, which does not match 1 / 3 + 1 / 4 + 1 / 6. Option D (4 days) is even slower and clearly not possible since each machine alone can finish in 3 to 6 days, and working together must be faster than the fastest individual time. Common Pitfalls: Many students make mistakes while adding fractions with different denominators. It is important to find the correct LCM before converting each rate into a common denominator. Another pitfall is to compute some sort of average of the times (3, 4 and 6 days) instead of using rates; averaging times does not work in work-rate problems and leads to incorrect answers. Final Answer: When machines P, Q and R all work together from the start, the printing job will be completed in 4/3 days (that is, 1 day and 8 hours).

Question 4

Person A can write 32 pages in 6 hours, while person B can write 40 pages in 5 hours. If they both write together at their respective constant speeds, in how many hours will they together be able to write a total of 110 pages?

Accepted Answer

Introduction / Context: This problem belongs to the Time and Work category, framed in terms of writing pages. We are given the individual writing speeds of A and B, and we must find the total time taken when they work together to produce a specific number of pages. It involves converting total pages written in a given time into rates (pages per hour), adding those rates, and then applying the formula time = work / rate. Given Data / Assumptions: A writes 32 pages in 6 hours. B writes 40 pages in 5 hours. We need time to write 110 pages when A and B work together. Both A and B write at constant speeds. They start and work together without any break. Concept / Approach: First, we calculate the individual rates of A and B in pages per hour. Next, we add these rates to find the combined rate when they write together. Finally, we use the basic relation: Time = Required pages / Combined rate. The answer then needs to be converted from hours into hours and minutes so that it matches one of the given options exactly. Step-by-Step Solution: Rate of A = 32 pages / 6 hours = 32 / 6 = 16 / 3 pages per hour. Rate of B = 40 pages / 5 hours = 8 pages per hour. Combined rate of A and B = 16 / 3 + 8. Express 8 as 24 / 3, so combined rate = 16 / 3 + 24 / 3 = 40 / 3 pages per hour. Required work = 110 pages. Time taken together = 110 / (40 / 3) hours. This is equal to 110 * 3 / 40 = 330 / 40 = 33 / 4 hours. 33 / 4 hours = 8.25 hours. 0.25 hours = 0.25 * 60 minutes = 15 minutes. Therefore, total time = 8 hours 15 minutes. Verification / Alternative check: We can verify by computing how many pages they write in 8 hours and then in an additional 15 minutes. In 8 hours, pages written = (40 / 3) * 8 = 320 / 3 ≈ 106.67 pages. In 0.25 hours (15 minutes), they write (40 / 3) * 0.25 = 10 / 3 ≈ 3.33 pages. Adding these gives approximately 110 pages, confirming the correctness of the result. Why Other Options Are Wrong: Option A (7 hours), Option B (6 hours 10 minutes) and Option C (5 hours 25 minutes) all correspond to lesser times. If we multiply the combined rate of 40 / 3 pages per hour by any of these times, the total pages produced will be less than 110 pages. Only 8 hours 15 minutes gives very close to 110 pages, exactly matching the required total when calculated precisely with fractions. Common Pitfalls: A common mistake is to average the times 6 hours and 5 hours instead of working with pages per hour. Another frequent error is in converting mixed fractions like 33 / 4 into hours and minutes, or forgetting to divide by the combined rate correctly. Always remember that rates add directly, and use fractions accurately to avoid rounding errors. Final Answer: Working together, A and B will write 110 pages in 8 hours 15 minutes.

Question 5

A can complete a piece of work in 12 days when working 8 hours per day. B can complete the same work in 8 days when working 10 hours per day. If both A and B work together, each working 8 hours per day, in how many days will they complete the entire…

Accepted Answer

Introduction / Context: This Time and Work problem introduces an additional twist where A and B initially work for different hours per day, but later they are asked to work the same number of hours per day. We must convert their given information into hourly work rates so that we can compare them on a common basis and then compute the time required when both work together for 8 hours each day. Given Data / Assumptions: A finishes the work in 12 days working 8 hours per day. B finishes the same work in 8 days working 10 hours per day. When they work together, both work 8 hours per day. Total work is considered as 1 job. Each person's hourly rate is constant, regardless of the schedule. Concept / Approach: We first convert the given information into total person-hours needed to complete the job for A and B individually. From that we determine their hourly work rates. Once we know their hourly rates, we can sum them to get the combined hourly rate when they work together. Since they work 8 hours a day, we convert this combined hourly rate into a daily rate, and finally obtain the total days required to finish the task together. Step-by-Step Solution: Let total work W = 1 unit. Total hours A takes = 12 days * 8 hours/day = 96 hours. So, A's hourly rate = 1 / 96 units per hour. Total hours B takes = 8 days * 10 hours/day = 80 hours. So, B's hourly rate = 1 / 80 units per hour. When both work together, combined hourly rate = 1 / 96 + 1 / 80. Compute: 1 / 96 + 1 / 80 = (5 + 6) / 480 = 11 / 480 units per hour. They work 8 hours per day together, so combined daily rate = 8 * (11 / 480) = 88 / 480 = 11 / 60 units per day. Time required = total work / daily rate = 1 / (11 / 60) = 60 / 11 days. Verification / Alternative check: As a check, note that in 60 / 11 days, the combined work done is (11 / 60) * (60 / 11) = 1 unit, which perfectly matches the total work. Also, A alone would have taken 12 days and B alone 8 days in their original schedules, so a joint time of a little more than 5 days (60 / 11 ≈ 5.45 days) is reasonable because working together at 8 hours per day should be faster than either working alone at their respective patterns. Why Other Options Are Wrong: Option A (51/24 days) is approximately 2.125 days, which is unrealistically small compared to the individual times of A and B. Option B (87/5 days) and Option C (57/12 days) either overestimate or underestimate the time such that the total work computed using the combined daily rate does not equal 1. Only 60 / 11 days maintains consistency with the derived hourly rates and the total work equation. Common Pitfalls: A frequent error is to treat 12 days and 8 days as if A and B worked the same hours per day, directly using 1 / 12 and 1 / 8 as daily rates. This ignores the difference in their daily working hours and leads to incorrect results. It is crucial to convert everything to a common measure, such as person-hours, before summing or comparing rates. Another mistake is mishandling fraction arithmetic when adding 1 / 96 and 1 / 80. Final Answer: Working together for 8 hours each day, A and B can complete the job in 60/11 days (approximately 5.45 days).

Question 6

A takes twice as much time as B to finish a piece of work. Working together, they can complete the work in 2 hours. In how many hours can B alone complete the same work if he works by himself from the beginning?

Accepted Answer

Introduction / Context: This is a classic Time and Work question involving two workers, A and B, whose times are in a simple ratio. We are told that A is slower, taking twice as much time as B to finish the entire work. When they work together, they complete the job in 2 hours. Using this information, we must determine the time B alone would need to finish the same work. Given Data / Assumptions: A takes twice as much time as B to complete the work. Working together, A and B finish the work in 2 hours. Both work at constant rates throughout. Total work is considered as 1 unit. Concept / Approach: When time taken is known, efficiency (rate of work) is the reciprocal of time. If A takes twice as long as B, then A's efficiency is half of B's. We let B's time be x hours, so A's time is 2x hours, and then we convert these times to rates 1 / x and 1 / (2x). Their combined rate is the sum of individual rates. Since their combined time is 2 hours, the combined rate must be 1 / 2 of the work per hour. We set up an equation and solve for x, which gives B's time alone. Step-by-Step Solution: Let B alone take x hours to complete the work. Then A alone takes 2x hours (as given). Rate of B = 1 / x units per hour. Rate of A = 1 / (2x) units per hour. Combined rate of A and B = 1 / x + 1 / (2x). = (2 + 1) / (2x) = 3 / (2x) units per hour. They complete the work together in 2 hours, so combined rate = 1 / 2 unit per hour. Thus, 3 / (2x) = 1 / 2. Cross-multiplying: 3 * 2 = 2x * 1 ⇒ 6 = 2x. So, x = 3 hours. Therefore, B alone can complete the work in 3 hours. Verification / Alternative check: If B takes 3 hours alone, then A takes 2 * 3 = 6 hours alone. That means B's rate is 1 / 3 and A's rate is 1 / 6. Combined rate = 1 / 3 + 1 / 6 = 1 / 2. Time taken together = 1 / (1 / 2) = 2 hours, which exactly matches the given condition. This confirms the correctness of the result x = 3 hours. Why Other Options Are Wrong: If B took 6 hours alone (Option B), A would take 12 hours, giving rates 1 / 6 and 1 / 12; their combined rate would be 1 / 4, implying 4 hours together, not 2 hours. Similarly, with 5 hours (Option D), A would need 10 hours, giving combined time different from 2 hours. Option C (7 hours) is also inconsistent in the same way. Only 3 hours gives a combined time of exactly 2 hours, as required in the problem statement. Common Pitfalls: Some students mistakenly assume that since A takes twice as long as B, B's time must simply be 1 hour if the combined time is 2 hours, which is incorrect because work rates do not combine linearly in that way. Another common error is to forget that rates add, not times. Always convert times to rates before summing them, and then use the combined rate to find the total time. Final Answer: B alone can complete the work in 3 hours.

Question 7

A piece of work could originally be finished in 11 days by a certain number of men. However, 4 more men joined from the beginning, and as a result the work was completed 4 days earlier. How many men were originally employed to do the work?

Accepted Answer

Introduction / Context: This question is about how adding more workers affects the completion time of a job. We are told that there was an original plan to finish a job in 11 days, but after increasing the workforce from the start by 4 men, the job finished 4 days earlier. We must determine the original number of men. This is a standard Time and Work scenario that uses the idea that total work is proportional to (number of men) * (time). Given Data / Assumptions: Originally, some men working together could finish a job in 11 days. From the very beginning, 4 extra men joined the original group. With this increased workforce, the job finished 4 days earlier, i.e., in 7 days. All men have equal and constant efficiency. Total work is considered as 1 unit. Concept / Approach: Let the original number of men be m and each man's daily work rate be r units per day. Then the total work is m * r * 11. After 4 additional men join from the start, the new workforce is (m + 4) men, and they finish the same work in 7 days, so total work is also (m + 4) * r * 7. Since total work is unchanged, these two expressions must be equal. From this equation, we can solve for m. The rate r cancels out, leaving a simple algebraic equation in m only. Step-by-Step Solution: Let m be the original number of men. Let each man's daily work rate be r units. Total work W = m * r * 11 (original plan). With 4 more men from the start, number of men = m + 4. New time taken = 11 - 4 = 7 days. Total work W is also equal to (m + 4) * r * 7. Since both expressions equal the same work: m * r * 11 = (m + 4) * r * 7. Cancel r from both sides: 11m = 7(m + 4). 11m = 7m + 28. 11m - 7m = 28 ⇒ 4m = 28. m = 28 / 4 = 7. Thus, originally there were 7 men employed. Verification / Alternative check: Originally, 7 men would finish the work in 11 days, so total work = 7 * 11 = 77 man-days. With 4 extra men, total men = 11. Time taken with 11 men = total work / men = 77 / 11 = 7 days, which matches the statement that the work was finished 4 days earlier (11 - 7 = 4 days). Hence our value m = 7 is consistent with all the data given. Why Other Options Are Wrong: If m were 8, total work would be 8 * 11 = 88 man-days, and with 12 men the time would be 88 / 12 ≈ 7.33 days, not an exact 4-day reduction. Similarly, with m = 9 or 10, the resulting times do not provide exactly 4 days of savings and do not satisfy the equality between the two work expressions. Only m = 7 gives a precise match to the problem conditions. Common Pitfalls: A typical mistake is to misinterpret the phrase "finished 4 days earlier" as meaning that the extra men joined sometime in the middle of the project instead of from the beginning. In that case, more unknown variables arise, and the problem becomes ambiguous. However, the usual interpretation in such aptitude questions is that the extra workers are counted from day one. Another error is forgetting to cancel the common rate r and thus overcomplicating the algebra. Final Answer: The number of men originally employed to do the work was 7 men.

Question 8

A labourer is engaged for 30 days on the condition that he will receive ₹25 for each day he works and will be fined ₹7.50 for each day he is absent. If his total earnings at the end of the period are ₹425, for how many days was he absent from work?

Accepted Answer

Introduction / Context: This question blends Time and Work ideas with simple arithmetic about wages and penalties. The labourer is paid a fixed amount per working day and fined a smaller amount per day of absence. Given the total number of days in the contract and his final earnings, we are asked to determine the number of days he was absent. Such problems commonly appear in aptitude tests under the category of wages, work and simple equations. Given Data / Assumptions: Total contract period = 30 days. Earning per working day = ₹25. Fine per absent day = ₹7.50. Total money received by the labourer = ₹425. Every day is either a working day or an absent day (no partial days). Concept / Approach: Let the number of working days be w and the number of absent days be a. We know that w + a = 30. His total earnings are calculated as (w * wage per day) minus (a * fine per day). Using these two relationships, we form two linear equations in w and a, then solve to find the value of a, which gives the number of days of absence. Step-by-Step Solution: Let w = number of days he actually worked. Let a = number of days he was absent. Total days: w + a = 30. ...(1) Earnings on working days = 25w. Fine on absent days = 7.5a. Net amount received = earnings - fines = 25w - 7.5a. Given that 25w - 7.5a = 425. ...(2) From equation (1), w = 30 - a. Substitute into (2): 25(30 - a) - 7.5a = 425. This gives 750 - 25a - 7.5a = 425. Combine like terms: 750 - 32.5a = 425. Rearrange: 32.5a = 750 - 425 = 325. a = 325 / 32.5 = 10. Therefore, the labourer was absent for 10 days. Verification / Alternative check: If a = 10 days, then w = 30 - 10 = 20 working days. Earnings without fines would be 20 * 25 = ₹500. Fines for 10 absent days total 10 * 7.5 = ₹75. Net amount = 500 - 75 = ₹425, which perfectly matches the given total earnings. This confirms that 10 days of absence is correct. Why Other Options Are Wrong: If he were absent 9 days, he would have worked 21 days. Net earnings would be 21 * 25 - 9 * 7.5 = 525 - 67.5 = ₹457.50, not ₹425. Similarly, for 11 or 14 days of absence, the computed net earnings do not match ₹425. Only 10 days of absence yields the correct final amount. Common Pitfalls: One frequent mistake is to add the fine instead of subtracting it, leading to a higher net amount. Another is to mis-handle the decimal 7.5 when solving the equation. To avoid confusion, some students prefer to multiply the entire equation by 2 or 10 to eliminate decimals before solving. Also, forgetting that the total number of days is fixed at 30 can lead to incorrect systems of equations. Final Answer: The labourer was absent from work for 10 days.

Question 9

Sixty men can stitch 150 shirts in 30 days while working 8 hours each day. In how many days can 35 men stitch 200 shirts if they work 6 hours per day, assuming everyone works at the same constant rate?

Accepted Answer

Introduction / Context: This Time and Work question uses the concept of man-hours to compare two different working scenarios. In the first scenario, we know how many men, shirts, days and hours per day are involved. In the second scenario, we change the number of men, the number of shirts, and the working hours per day, and need to calculate the number of days required. The key is to compute the total man-hours per shirt from the first case and then apply it to the second case. Given Data / Assumptions: 60 men stitch 150 shirts in 30 days at 8 hours per day. We must find the days needed for 35 men to stitch 200 shirts at 6 hours per day. All men have equal efficiency and work at a constant rate. Efficiency does not change with time or number of shirts. Concept / Approach: First, we calculate the total man-hours required to stitch 150 shirts in the first scenario. From this, we obtain the man-hours needed per shirt. Then we multiply this per-shirt requirement by 200 shirts to find the total man-hours needed in the new scenario. Finally, we divide this total by the man-hours produced per day in the new scenario (35 men working 6 hours per day) to find the number of days required. Step-by-Step Solution: Total man-hours used in first scenario = number of men * days * hours per day. = 60 * 30 * 8 = 14400 man-hours. These 14400 man-hours produce 150 shirts. Man-hours required per shirt = 14400 / 150 = 96 man-hours per shirt. Total man-hours needed for 200 shirts = 200 * 96 = 19200 man-hours. In the new scenario, 35 men work 6 hours per day. Daily man-hours produced = 35 * 6 = 210 man-hours per day. Number of days required = total man-hours / daily man-hours. = 19200 / 210. Simplify: 19200 / 210 ≈ 91.42857 days. This is approximately 91.43 days, which is closest to 91.5 days among the options. Verification / Alternative check: To verify, we can check the approximate shirts per day in the second case. One day of work by 35 men at 6 hours gives 210 man-hours, which equals 210 / 96 ≈ 2.1875 shirts per day. Over about 91.5 days, total shirts ≈ 2.1875 * 91.5 ≈ 200 shirts. This confirms that 91.5 days is the best approximation and aligns well with the exact fractional result 19200 / 210 ≈ 91.4286. Why Other Options Are Wrong: Option A (87.5 days), Option B (89 days) and Option D (93.5 days) do not match the exact calculation 19200 / 210. For example, at 87.5 days, total shirts would be significantly less than 200, while at 93.5 days, total shirts would exceed 200. Only 91.5 days is close enough to the exact result and matches the reasoning from man-hour calculations. Common Pitfalls: Common mistakes include ignoring the working hours per day and comparing only men and days, or directly proportioning shirts and men without considering the product of men, hours and days. Another frequent error is performing incorrect division when calculating man-hours per shirt. Always keep track of units (man-hours per shirt) carefully and carry out the arithmetic step by step. Final Answer: The required time for 35 men working 6 hours per day to stitch 200 shirts is approximately 91.5 days.

Question 10

Two persons K and L can complete a piece of work in 30 days and 45 days respectively when working alone. If they work together, what fraction of the work will be completed in 3 days?

Accepted Answer

Introduction / Context: This is a direct Time and Work question about two people working together for a given number of days. We are given the individual times taken by K and L to complete the work alone, and asked to find the fraction of the work done when they work together for 3 days. The key is to convert the times into daily work rates and then scale by the number of days worked together. Given Data / Assumptions: K alone can finish the work in 30 days. L alone can finish the work in 45 days. They work together for 3 days. Total work is considered as 1 unit. Both K and L have constant daily work rates. Concept / Approach: If a person takes T days to complete a job alone, their daily work rate is 1 / T of the job per day. For K and L, we find 1 / 30 and 1 / 45 as their rates. When they work together, their combined rate is the sum of these individual rates. Multiplying this combined rate by 3 (the number of days worked) gives the fraction of the job completed in those 3 days. Step-by-Step Solution: Let total work W = 1 unit. K's daily rate = 1 / 30 units per day. L's daily rate = 1 / 45 units per day. Combined rate of K and L = 1 / 30 + 1 / 45. LCM of 30 and 45 is 90. 1 / 30 = 3 / 90 and 1 / 45 = 2 / 90. Combined rate = (3 + 2) / 90 = 5 / 90 = 1 / 18 units per day. In 3 days, work done = 3 * (1 / 18) = 3 / 18 = 1 / 6. Therefore, they complete 1 / 6 of the work in 3 days. Verification / Alternative check: You can also think in terms of how long it would take them together to complete the entire job: time together T satisfies (1 / 30 + 1 / 45) * T = 1. Since the combined rate is 1 / 18, T = 18 days. In 3 days, the fraction completed is 3 / 18 = 1 / 6, exactly matching our earlier calculation. Why Other Options Are Wrong: Option B (1/3) would imply they complete 1/3 of the work in 3 days, meaning their daily rate would be 1/9, which contradicts the actual combined rate of 1/18. Option C (2/3) and Option D (1/18) are also inconsistent with the rates derived from 30 and 45 days. Only 1/6 is supported by the rate and time calculations. Common Pitfalls: Students sometimes wrongly average the times 30 and 45, or they forget to compute the LCM correctly when adding fractions. Another error is to multiply individual times instead of converting times to rates. Always remember: work rate is 1 / time, and combined work rate is the sum of individual rates, not the average of times. Final Answer: Together, K and L will complete 1/6 of the work in 3 days.

Question 11

A and B together can complete a piece of work in 9 days. In any given time interval, A does thrice as much work as B. In how many days will A alone finish the entire work if he works by himself from the beginning?

Accepted Answer

Introduction / Context: This problem is based on relative efficiencies where one person does a multiple of the work done by another in the same time. We are told that A and B together complete the work in 9 days and that A does three times as much work as B in any given time. Using this relationship, we can find the individual rates of A and B and from that deduce how long A alone would take to complete the entire job. Given Data / Assumptions: A and B together complete the work in 9 days. A does thrice the work of B in the same time (efficiency ratio A : B = 3 : 1). Both work at constant rates. Total work is treated as 1 unit. Concept / Approach: Let B's daily work rate be b units. Then A's rate is 3b units because A is three times as efficient as B. The combined rate of A and B is therefore 4b units per day. Since they can finish the work in 9 days together, we know that 4b * 9 = 1. From this, we find b and then determine A's rate as 3b. The time taken by A alone is simply 1 divided by his daily rate. Step-by-Step Solution: Let B's daily rate = b units per day. Then A's daily rate = 3b units per day (since A does thrice the work of B). Combined daily rate of A and B = 3b + b = 4b. They complete the work in 9 days, so total work W = 4b * 9 = 36b. Set W = 1 unit, so 36b = 1. Thus, b = 1 / 36 units per day. A's daily rate = 3b = 3 * (1 / 36) = 1 / 12 units per day. Time taken by A alone = total work / A's rate = 1 / (1 / 12) = 12 days. Verification / Alternative check: In 12 days, A alone would complete the whole work. B's time would be 1 / b = 36 days. As a cross-check, if A and B work together, their combined rate is 1 / 12 + 1 / 36 = (3 + 1) / 36 = 4 / 36 = 1 / 9. This means they together take 9 days to finish the work, which perfectly matches the given condition in the question. Why Other Options Are Wrong: Option A (4 days), Option B (6 days) and Option C (8 days) all correspond to incorrect values of A's rate that do not satisfy the condition that A and B together finish in 9 days while A does thrice the work of B. Only 12 days leads to a consistent set of rates (1 / 12 and 1 / 36) that satisfy all given relationships. Common Pitfalls: Students sometimes reverse the ratio and treat B as three times more efficient than A, which leads to incorrect answers. Another typical error is to try averaging times instead of converting to rates. Remember that time is inversely proportional to efficiency; if A is three times as efficient as B, his time alone should be one-third of B's time, not vice versa. Final Answer: A alone will finish the entire work in 12 days.

Question 12

Ram and Shyam undertake a piece of work for a total payment of ₹5600. Ram alone can finish the work in 5 days, while Shyam alone can finish the same work in 9 days. If they work together and share the payment in proportion to the work done by each, …

Accepted Answer

Introduction / Context: This is a Time and Work problem combined with profit sharing. Ram and Shyam are paid a fixed total amount for a job, and they share this amount in proportion to the work each has actually done. Since we are given the times taken by each to complete the job alone, we can derive their work rates and use these to determine the ratio in which they should share the payment. Finally, we compute the difference between their shares. Given Data / Assumptions: Ram alone completes the work in 5 days. Shyam alone completes the work in 9 days. Total payment for the work = ₹5600. They work together for the entire duration and share payment proportional to their work done (i.e., their efficiencies). Total work is taken as 1 unit. Concept / Approach: The key is that when two people work together, their contributions to the work are proportional to their individual rates of work. If a person can do the work in T days, their rate is 1 / T of the work per day. The ratio of Ram's rate to Shyam's rate is therefore (1 / 5) : (1 / 9), which simplifies to 9 : 5. This is the ratio in which they should share the total money. We then find each share and compute the difference between them. Step-by-Step Solution: Let total work W = 1 unit. Ram's daily work rate = 1 / 5 units. Shyam's daily work rate = 1 / 9 units. Ratio of their rates (Ram : Shyam) = (1 / 5) : (1 / 9) = 9 : 5. Total payment = ₹5600. Sum of ratio parts = 9 + 5 = 14 parts. Value of one part = 5600 / 14 = ₹400. Ram's share = 9 * 400 = ₹3600. Shyam's share = 5 * 400 = ₹2000. Difference between their shares = 3600 - 2000 = ₹1600. Verification / Alternative check: You can also check using actual work done per unit time. Suppose they work together for T days and finish the work: (1 / 5 + 1 / 9) * T = 1. Their individual contributions are (1 / 5) * T and (1 / 9) * T. The ratio of these contributions is (T / 5) : (T / 9) = 9 : 5, confirming that our ratio is correct and independent of T. Therefore, the payment must be shared in the ratio 9 : 5, and the resulting difference in money must be ₹1600. Why Other Options Are Wrong: Option A (₹1800), Option B (₹2400) and Option D (₹2200) correspond to incorrect share ratios. If any of these were correct, the total payment would not be consistently shared in the 9 : 5 ratio. By explicitly calculating the shares based on the correct ratio and comparing them with the total amount, we see that only a difference of ₹1600 matches the given total payment of ₹5600. Common Pitfalls: A common mistake is to assume that the payment should be divided according to the times they would take to complete the work alone (5 and 9) instead of their rates (1 / 5 and 1 / 9). Remember that contribution depends on how much work each does per unit time, not on how long they would individually take. Always convert times to rates and then compare rates to determine the correct sharing ratio. Final Answer: The difference between the amounts received by Ram and Shyam is ₹1600.

Question 13

Raman can complete a certain work in 5 days, Jatin can complete it in 7 days and Sachin can complete it in 9 days. If all three work together and earn a total of ₹2860 for completing the work, what is Raman's share (in rupees) of the total payment?

Accepted Answer

Introduction / Context: This question is another example of work and wages. Three persons, Raman, Jatin and Sachin, can complete the same work in different times. They do the work together and are paid a lump sum. The payment is to be divided in proportion to the work done by each, which is directly proportional to their individual efficiencies (rates of work). We are asked to find Raman's share of the total amount ₹2860. Given Data / Assumptions: Raman alone finishes the work in 5 days. Jatin alone finishes the work in 7 days. Sachin alone finishes the work in 9 days. Total payment = ₹2860. All three work together and share the payment in proportion to their work done (i.e., their rates). Total work is treated as 1 unit. Concept / Approach: We first compute the work rates for each person: 1 / 5, 1 / 7 and 1 / 9 of the job per day for Raman, Jatin and Sachin respectively. Their shares in the money must be in the ratio of these rates. We then find the sum of these rates and multiply the total amount by Raman's fraction of the total rate to determine his exact monetary share. Step-by-Step Solution: Let total work W = 1 unit. Raman's daily rate = 1 / 5 units. Jatin's daily rate = 1 / 7 units. Sachin's daily rate = 1 / 9 units. Total combined rate = 1 / 5 + 1 / 7 + 1 / 9. Compute using fractions: LCM of 5, 7 and 9 is 315. 1 / 5 = 63 / 315, 1 / 7 = 45 / 315, 1 / 9 = 35 / 315. Combined rate = (63 + 45 + 35) / 315 = 143 / 315. Raman's share of total rate = (1 / 5) / (1 / 5 + 1 / 7 + 1 / 9) = (1 / 5) / (143 / 315). (1 / 5) = 63 / 315, so Raman's fraction = 63 / 315 ÷ 143 / 315 = 63 / 143. Raman's share of money = (63 / 143) * 2860. Compute: 2860 * 63 / 143 = 1260. Therefore, Raman receives ₹1260. Verification / Alternative check: As a check, we can also compute Jatin's and Sachin's shares and see if they sum correctly to ₹2860. Jatin's fraction = (1 / 7) / (1 / 5 + 1 / 7 + 1 / 9) = 45 / 143; Sachin's fraction = 35 / 143. Using these, Jatin's share = 2860 * 45 / 143 = 900, Sachin's share = 2860 * 35 / 143 = 700. Summing 1260 + 900 + 700 = 2860, which matches the total payment, confirming the calculation. Why Other Options Are Wrong: Options B (₹700), C (₹900) and D (₹870) correspond to the shares of Sachin, Jatin or some incorrect fraction, not to Raman's share. Only ₹1260 agrees with the ratio of work rates and the total payment. Common Pitfalls: Students sometimes divide the payment inversely proportional to the number of days (5, 7, 9) without converting to rates, but luckily in this case using 1 / 5 : 1 / 7 : 1 / 9 or equivalently 63 : 45 : 35 leads to the same ratio. However, arithmetic with large LCMs like 315 can cause mistakes. Carefully handling each fraction and simplifying ratios before multiplying with the total money helps avoid errors. Final Answer: Raman's share of the total payment is ₹1260.

Question 14

Ten women can complete a piece of work in 6 days, six men can complete the same work in 5 days and eight children can complete it in 10 days. What is the ratio of the efficiencies of a woman, a man and a child respectively?

Accepted Answer

Introduction / Context: In this Time and Work problem, we are comparing the efficiencies of three different categories of workers: women, men and children. Each group, working alone, can finish the same job in different times with different numbers of workers. By interpreting total work as constant and using worker-days, we can determine how much work one woman, one man and one child can do per day, and then express these capacities as a ratio. Given Data / Assumptions: 10 women complete the work in 6 days. 6 men complete the work in 5 days. 8 children complete the work in 10 days. All women have equal efficiency, all men have equal efficiency, and all children have equal efficiency within their groups. Total work is treated as 1 unit. Concept / Approach: If a group of workers completes a job in D days, the total work is equal to (number of workers) * (individual efficiency) * D. For each group, we can write an expression for total work, set each equal to 1 unit, and deduce individual efficiencies. Once we know the individual daily work done by a woman, a man and a child, we express these in a simple ratio form to get the required answer. Step-by-Step Solution: Let Ew, Em and Ec be the daily efficiencies of one woman, one man and one child respectively. Total work W = 1 unit. For 10 women in 6 days: 10 * Ew * 6 = 1 ⇒ 60Ew = 1 ⇒ Ew = 1 / 60. For 6 men in 5 days: 6 * Em * 5 = 1 ⇒ 30Em = 1 ⇒ Em = 1 / 30. For 8 children in 10 days: 8 * Ec * 10 = 1 ⇒ 80Ec = 1 ⇒ Ec = 1 / 80. Now we want the ratio Ew : Em : Ec. Substitute values: Ew : Em : Ec = 1 / 60 : 1 / 30 : 1 / 80. To clear denominators, multiply by the LCM of 60, 30 and 80 which is 240. Ew * 240 = 240 / 60 = 4. Em * 240 = 240 / 30 = 8. Ec * 240 = 240 / 80 = 3. Hence, Ew : Em : Ec = 4 : 8 : 3. Verification / Alternative check: We can reverse-check by assuming that 1 woman does 4 units of some arbitrary work measure, 1 man does 8 units and 1 child does 3 units each day, with total work scaled accordingly. Then 10 women for 6 days would do 10 * 4 * 6 = 240 units, 6 men for 5 days would do 6 * 8 * 5 = 240 units, and 8 children for 10 days would do 8 * 3 * 10 = 240 units. Since all groups complete the same 240 units of work, the ratio 4 : 8 : 3 is consistent across all three relationships. Why Other Options Are Wrong: Option A (4 : 6 : 3) would imply that a man is only 1.5 times as efficient as a woman, which does not satisfy the condition that 6 men finish in 5 days compared to 10 women in 6 days. Option B (4 : 5 : 3) and Option C (2 : 4 : 3) similarly fail when we examine the total work done by each group. Only 4 : 8 : 3 allows all three given scenarios to represent the same total work. Common Pitfalls: Students often get confused by the presence of three different groups and attempt to combine all data in a single equation. The correct approach is to treat each group separately, compute individual efficiencies and then form the ratio. Another frequent source of error is failing to take a proper LCM or mishandling the fraction simplification when converting 1 / 60, 1 / 30 and 1 / 80 into a simple ratio. Final Answer: The ratio of the efficiencies of a woman, a man and a child is 4 : 8 : 3 respectively.

Question 15

B would take 10 hours more than A to complete a certain task if each of them worked alone. Working together, A and B can complete the entire task in 12 hours. How many hours would B take to complete 50% of the task if he works alone?

Accepted Answer

Introduction / Context: This Time and Work problem involves two workers, A and B, whose individual times are related by a fixed difference. We are told that B takes 10 hours longer than A to complete a task alone, and that together they complete the task in 12 hours. We must first determine B's individual time for the full task and then find how long B alone would take to complete half of the work. This requires setting up and solving a quadratic equation in terms of A's time or B's time. Given Data / Assumptions: A's time to complete the task alone = x hours (unknown). B's time to complete the task alone = x + 10 hours. Working together, A and B finish the whole task in 12 hours. Both A and B work at constant rates. Total work is considered as 1 unit. Concept / Approach: Since time and efficiency are inversely related, we express A's rate as 1 / x and B's rate as 1 / (x + 10). The combined rate of A and B is thus 1 / x + 1 / (x + 10). Given that they complete the task in 12 hours, their combined rate must equal 1 / 12. Solving this equation gives the value of x (A's time). From x, we obtain B's time as x + 10 hours. To find the time B takes for 50% of the work, we divide half the work by B's rate. Step-by-Step Solution: Let A alone take x hours. Then B alone takes x + 10 hours. A's rate = 1 / x units per hour. B's rate = 1 / (x + 10) units per hour. Combined rate = 1 / x + 1 / (x + 10). Given that together they complete the work in 12 hours, so combined rate = 1 / 12. Thus, 1 / x + 1 / (x + 10) = 1 / 12. Combine the left side: (x + 10 + x) / [x(x + 10)] = 1 / 12. (2x + 10) / [x(x + 10)] = 1 / 12. Cross-multiplying: 12(2x + 10) = x(x + 10). 24x + 120 = x^2 + 10x. Rearrange: x^2 + 10x - 24x - 120 = 0 ⇒ x^2 - 14x - 120 = 0. Solve the quadratic: x^2 - 14x - 120 = 0. This factors as (x - 20)(x + 6) = 0. Since time cannot be negative, x = 20 hours. So, A alone takes 20 hours and B alone takes 20 + 10 = 30 hours. B's rate = 1 / 30 units per hour. To complete 50% (i.e., 1 / 2) of the work, time taken by B = (1 / 2) / (1 / 30) = 15 hours. Verification / Alternative check: Check the combined time: A's rate is 1 / 20, B's rate is 1 / 30. Combined rate = 1 / 20 + 1 / 30 = (3 + 2) / 60 = 5 / 60 = 1 / 12. Therefore, together they take 12 hours, which matches the given condition. For half the work, B at 1 / 30 units per hour needs 15 hours, since in 15 hours he does (15 / 30) = 1 / 2 of the job. Everything is consistent. Why Other Options Are Wrong: Option A (30 hours) is the time B takes to finish the entire task, not just 50% of it. Option C (20 hours) or Option D (10 hours) do not correspond to half the work at B's actual rate of 1 / 30; they yield different fractions of the total work (2/3 and 1/3 respectively). Only 15 hours gives exactly half the job completed by B alone. Common Pitfalls: One common mistake is to assume that if together they take 12 hours and B is slower, B must automatically take 24 hours for the whole work, which is not correct because of the specific 10-hour difference from A. Another error is to forget that half the work means dividing by the rate while using 1 / 2 as the numerator, not confusing it with doubling the time incorrectly. Careful handling of the quadratic equation and the half-work calculation is essential. Final Answer: B would take 15 hours to complete 50% of the task working alone.

Question 16

K is 4 times as fast as L in doing a certain job. Working together, they can complete the work in 24 days. In how many days can L alone complete the entire work if he works by himself from the beginning?

Accepted Answer

Introduction / Context: This Time and Work problem involves two workers, K and L, whose efficiencies are given in a simple ratio. K is 4 times as fast as L, meaning K does four times as much work as L in the same time. We are also told how long they take together to complete the work. From this, we need to determine how long L alone would take to complete the same work. Given Data / Assumptions: K is 4 times as fast as L (efficiency ratio K : L = 4 : 1). Working together, K and L complete the work in 24 days. Both work at constant rates. Total work is regarded as 1 unit. Concept / Approach: If K is 4 times as fast as L, then K's time to complete the work alone is one-fourth of L's time. Let L's time be T days, making K's time T / 4 days. Their daily work rates are therefore 1 / T and 4 / T respectively. The combined rate is the sum of these rates, and since together they take 24 days, the combined rate must be 1 / 24. We set up an equation in T and solve to get L's time. Step-by-Step Solution: Let L alone take T days to complete the work. Then K alone, being 4 times as fast, takes T / 4 days to complete the work. L's daily rate = 1 / T units per day. K's daily rate = 1 / (T / 4) = 4 / T units per day. Combined daily rate = 1 / T + 4 / T = 5 / T units per day. Working together, they complete the work in 24 days, so combined rate = 1 / 24 units per day. Thus, 5 / T = 1 / 24. Cross-multiplying: 5 * 24 = T * 1 ⇒ T = 120 days. Therefore, L alone takes 120 days to finish the work. Verification / Alternative check: If L takes 120 days alone, his rate is 1 / 120 units per day. K, being 4 times as fast, has time 120 / 4 = 30 days, so his rate is 1 / 30 units per day, which equals 4 / 120 units per day. Combined rate = 1 / 120 + 1 / 30 = 1 / 120 + 4 / 120 = 5 / 120 = 1 / 24. Thus, together they take 24 days, which matches the given condition and confirms our answer. Why Other Options Are Wrong: If L took 30 or 40 or 80 days, then K's corresponding times would not yield a combined rate of 1 / 24 when added. For example, if L took 80 days, K would take 20 days, and combined rate would be 1 / 80 + 1 / 20 = 5 / 80 = 1 / 16, implying a joint time of 16 days instead of 24. Only 120 days for L gives a consistent solution. Common Pitfalls: A common mistake is to misinterpret "K is 4 times as fast as L" as meaning K takes 4 times as much time as L, which is the opposite of the correct relationship. Always remember that if someone is faster, they take less time, and efficiencies are inversely proportional to time. Another error is to forget to convert K's time correctly to T / 4 before computing the rate 4 / T. Final Answer: L alone will complete the work in 120 days.

Question 17

K, L and M can complete a piece of work in 20 days, 30 days and 60 days respectively when working alone. If K works every day and is assisted by L and M only on every third day, in how many days will the entire work be completed?

Accepted Answer

Introduction / Context: This is a Time and Work question involving a repeating pattern of work over days. Three workers, K, L and M, have different individual times to complete a job. K works every day, but L and M assist him only on every third day. We must compute the effective work done in each 3-day cycle and then determine how many such cycles are needed to complete the entire job. This type of problem tests understanding of periodic work patterns and combined work rates. Given Data / Assumptions: K alone can finish the work in 20 days. L alone can finish the work in 30 days. M alone can finish the work in 60 days. K works every day. On every third day, L and M also work with K; on the first and second day of each cycle, only K works. Total work is considered as 1 unit. All three work at constant rates. Concept / Approach: We first compute the individual daily work rates by taking reciprocals of the times. Then we analyze a 3-day cycle: on day 1 and day 2, only K works; on day 3, K, L and M work together. We compute the total work done in one full 3-day cycle. Knowing how much of the work is completed per cycle, we find how many such cycles are required to reach 1 full unit of work. If the work is completed exactly in an integer number of cycles, the total days is the number of cycles multiplied by 3. Step-by-Step Solution: Let total work W = 1 unit. K's daily rate = 1 / 20 units per day. L's daily rate = 1 / 30 units per day. M's daily rate = 1 / 60 units per day. On day 1: only K works ⇒ work done = 1 / 20. On day 2: only K works ⇒ work done = 1 / 20. On day 3: K, L and M all work ⇒ work done = 1 / 20 + 1 / 30 + 1 / 60. Compute: 1 / 20 + 1 / 30 + 1 / 60 = (3 + 2 + 1) / 60 = 6 / 60 = 1 / 10. Total work in one 3-day cycle = 1 / 20 + 1 / 20 + 1 / 10. = 1 / 20 + 1 / 20 + 2 / 20 = 4 / 20 = 1 / 5. So, every 3 days, they complete 1 / 5 of the job. To complete the full work (1 unit), number of such cycles required = 1 / (1 / 5) = 5 cycles. Total days taken = 5 cycles * 3 days per cycle = 15 days. Verification / Alternative check: After 5 cycles, total work done = 5 * (1 / 5) = 1 unit, so the entire job is completed exactly at the end of a cycle. There is no partial cycle needed. It is also reasonable that the total days (15) lies between K's individual time (20 days) and the combined time of all working every day, which would be faster than 15 days. Since L and M only assist on every third day, the overall time is longer than if all three had worked daily, but shorter than K alone; 15 days fits that expectation. Why Other Options Are Wrong: Option B (13 days), Option C (12 days) and Option D (10 days) all imply faster completion, which would require more daily work than the computed pattern of 1 / 5 per 3 days allows. If we divide 1 unit of work by the effective average rate over the 3-day pattern, we obtain exactly 15 days, not any of the smaller values. Thus, only 15 days satisfies the periodic working pattern precisely. Common Pitfalls: One common error is to assume that K, L and M work together every day and then simply use their combined rate, ignoring the given condition that L and M only help on every third day. Another mistake is to average the individual times (20, 30, 60) directly instead of computing work per cycle. The correct approach is to focus on the repeating pattern and calculate work done in one complete cycle before scaling up to the total work. Final Answer: The entire work will be completed in 15 days when K works every day and is assisted by L and M on every third day.

Question 18

A is three times as fast as B at doing a piece of work. A alone finishes the entire work in 32 days less than B alone. In how many days will A and B, working together at their respective constant rates, complete the whole work?

Accepted Answer

Introduction / Context: This is a classic time and work aptitude question where two workers, A and B, have different efficiencies. We are told that A is three times as fast as B and that A takes 32 days less than B to finish the same job alone. The goal is to determine how long they will take to complete the work when they both work together at their constant individual rates. Given Data / Assumptions: A is three times as fast as B. A alone finishes the work 32 days faster than B. Both A and B work at constant speeds with no breaks. The total work is treated as one complete job. Concept / Approach: In time and work problems, efficiency is measured as work done per day. If a person takes T days to do a job alone, the daily work rate is 1 / T. The phrase three times as fast means the rate of A is three times the rate of B. Using the relationship between their times and the 32 day difference, we can find their individual times, then compute the combined rate and finally the total time when they work together. Step-by-Step Solution: Let the time taken by B alone be TB days, so B's rate is 1 / TB.A is three times as fast as B, so A's rate is 3 * (1 / TB) = 3 / TB.Thus A's time TA = 1 / (3 / TB) = TB / 3.Given that A finishes 32 days sooner: TA = TB - 32, so TB / 3 = TB - 32.Multiply both sides by 3: TB = 3TB - 96, so 2TB = 96 and TB = 48 days.Then TA = TB / 3 = 48 / 3 = 16 days.Combined rate of A and B together = 1 / 16 + 1 / 48 = (3 + 1) / 48 = 4 / 48 = 1 / 12.Time taken together = 1 / (1 / 12) = 12 days. Verification / Alternative check: If B takes 48 days, B's rate is 1 / 48. A is three times as fast, so A's rate is 3 / 48 = 1 / 16, matching the earlier time of 16 days. The difference in times is 48 - 16 = 32 days, which agrees with the condition. Their combined rate 1 / 16 + 1 / 48 equals 1 / 12, giving 12 days for the joint work. All conditions fit consistently, so the solution is verified. Why Other Options Are Wrong: 16 days would be the time taken by A alone, not the time taken together. Twenty four days and 32 days are both larger than the time of the faster worker A, so they cannot be the result of combining two positive work rates. Any combined effort must finish sooner than the fastest worker alone, which immediately rules out those larger values. Common Pitfalls: A common mistake is to confuse three times as fast with taking three times as many days, which is actually the opposite. Another typical error is to subtract rates instead of using the 32 day difference in time. Some learners also forget that when two people work together, their rates add, which must always give a completion time less than that of the faster individual worker. Final Answer: The required time for A and B working together to finish the entire work is 12 days.

Question 19

In a 5 day working week with 8 hours of regular work per day, a man is paid Rs. 2.40 per hour for regular work and Rs. 3.20 per hour for overtime. Over 4 such weeks he earns a total of Rs. 432. How many hours in total, including both regular hours a…

Accepted Answer

Introduction / Context: This problem tests the link between time, wage rate, and total earnings in a typical working week. We are given different hourly rates for regular work and overtime, along with the total earnings over several weeks. The task is to calculate the total number of hours worked, combining both regular and overtime hours, based on the pay structure and total income. Given Data / Assumptions: There are 5 working days per week. Each working day has 8 regular working hours. Regular work is paid at Rs. 2.40 per hour. Overtime work is paid at Rs. 3.20 per hour. Total time considered is 4 weeks. Total earnings in 4 weeks are Rs. 432. Concept / Approach: First, we compute the total regular hours over 4 weeks and the corresponding regular wages. The difference between the total earnings and the regular wages must be the overtime pay. Using the overtime rate, we find the number of overtime hours. Finally, we add regular and overtime hours to obtain the total hours of work across the 4 week period. Step-by-Step Solution: Regular hours per week = 5 days * 8 hours per day = 40 hours.Regular hours in 4 weeks = 40 * 4 = 160 hours.Regular wages in 4 weeks = 160 * 2.40 = Rs. 384.Total earnings are Rs. 432, so overtime earnings = 432 - 384 = Rs. 48.Overtime rate is Rs. 3.20 per hour, so overtime hours = 48 / 3.20 = 15 hours.Total hours worked = regular hours + overtime hours = 160 + 15 = 175 hours. Verification / Alternative check: We can verify by recomputing earnings from the obtained hours. Regular wages: 160 hours * Rs. 2.40 = Rs. 384. Overtime wages: 15 hours * Rs. 3.20 = Rs. 48. Combined, 384 + 48 = Rs. 432, which matches the total income given in the question. Therefore, the calculated total time of 175 hours is correct and consistent with the data. Why Other Options Are Wrong: 145 hours and 165 hours would imply smaller overtime hours and would not produce total earnings of Rs. 432 at the given pay rates. One hundred thirty five hours is even further from the required figure and leads to much less than Rs. 432. Only 175 hours leads to a correct split between regular earnings and overtime earnings that exactly matches the given total salary. Common Pitfalls: Candidates often forget to multiply the weekly hours by the number of weeks or may mistakenly assume that all hours are paid at the overtime rate. Another common error is to divide the total earnings directly by one of the hourly rates and ignore the fact that there are two different pay rates. Correctly separating regular pay and overtime pay is essential to obtaining the right answer. Final Answer: The man works for a total of 175 hours over the 4 week period.

Question 20

A sum of money is sufficient to pay the wages of worker K alone for 10 days or worker L alone for 15 days. If both K and L are paid from this same amount at the same time, for how many days will the money last?

Accepted Answer

Introduction / Context: This question involves the concept of total available money being used to pay wages at fixed daily rates. The money can pay one worker for a certain number of days or another worker for a different number of days. We are asked to find how long that same amount of money will last if both workers are paid simultaneously from it. Given Data / Assumptions: The money can pay worker K alone for 10 days. The same money can pay worker L alone for 15 days. Daily wages of each worker remain constant over time. Both K and L start receiving wages on the same day from the same fund. The entire amount is used only for their wages. Concept / Approach: Let the daily wage of K be k, and that of L be l. The total money available is the same in both equivalent descriptions. From this we can express the fund in two ways: 10k and 15l. Equating these expressions lets us find a relationship between k and l. Once we know the combined daily wage k + l, we can find the number of days the money will last by dividing the total fund by the total daily payout. Step-by-Step Solution: Let the total money be M.If it pays K alone for 10 days, M = 10k.If it pays L alone for 15 days, M = 15l.Equate the two expressions: 10k = 15l.So k / l = 15 / 10 = 3 / 2, hence l = (2 / 3)k.Combined daily wage when both work together = k + l = k + (2 / 3)k = (5 / 3)k.Total money M = 10k, so number of days with both = M / (k + l) = 10k / ((5 / 3)k) = 10 * (3 / 5) = 6 days. Verification / Alternative check: Assume a convenient value for K's daily wage, say k = Rs. 30. Then total money M = 10 * 30 = Rs. 300. From the earlier ratio, l = (2 / 3) * 30 = Rs. 20. If both are paid together, daily payout = 30 + 20 = Rs. 50. The number of days = 300 / 50 = 6 days, which matches our algebraic result. This confirms the correctness of the solution. Why Other Options Are Wrong: Nine days would require a much smaller combined daily wage and contradicts the fact that K alone uses up the fund in only 10 days. Twelve days and four days also do not satisfy the relationship between the wages derived from the given conditions. Only six days is consistent with both K and L's possible payment periods from the same fund. Common Pitfalls: A frequent mistake is to average the two given days as (10 + 15) / 2 = 12.5, which has no basis in the actual wage structure. Another error is to treat wages as equal without using the ratio implied by the total fund. Always translate such problems into algebra using a symbol for the daily wages and then equate the two expressions for the same money. Final Answer: The money will last for 6 days when both K and L are paid simultaneously from it.

Time and Work Questions