Question 1

The efficiencies of workers P, Q and R are in the ratio 2 : 3 : 4. P and R work on alternate days, while Q works every day. The entire work is completed in 10 days and the total wages for the job are Rs. 1200. Based on their respective contributions…

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Introduction / Context: This time and work question combines worker efficiency with wage distribution. The workers P, Q and R have different efficiencies and do not all work the same days. Since wages are proportional to work done, we first determine each person share of the total work and then allocate the total payment accordingly. Given Data / Assumptions: • Efficiency ratio of P : Q : R is 2 : 3 : 4. • P and R work on alternate days, starting with P on the first day, for a total of 10 days. • Q works every day for all 10 days. • Total wages for the work are Rs. 1200. • Wages are directly proportional to the amount of work done by each worker. Concept / Approach: First, convert the efficiency ratio into numerical daily work units. Then calculate total work done by each person over 10 days. The total work is the sum of all contributions. Finally, distribute the total wages in the ratio of work done by P, Q and R, so that each worker is paid fairly according to contribution. Step-by-Step Solution: Let the efficiencies of P, Q and R be 2k, 3k and 4k units of work per day respectively. Q works all 10 days, so Q work = 3k × 10 = 30k units. P and R work on alternate days over 10 days, so each of them works on 5 days. P work = 2k × 5 = 10k units. R work = 4k × 5 = 20k units. Total work = 10k + 30k + 20k = 60k units. Shares of work: P : Q : R = 10k : 30k : 20k = 1 : 3 : 2. Total wages = Rs. 1200, so each share = 1200 ÷ (1 + 3 + 2) = 1200 ÷ 6 = Rs. 200. Therefore, P gets 1 share = Rs. 200, Q gets 3 shares = Rs. 600, and R gets 2 shares = Rs. 400. Verification / Alternative check: We can verify by adding the three amounts: 200 + 600 + 400 = 1200, which matches the total wage. The ratio 200 : 600 : 400 simplifies to 1 : 3 : 2, which matches the ratio of their total work 10k : 30k : 20k. This confirms that the allocation is consistent with the calculated contributions. Why Other Options Are Wrong: Any option that gives R more than Q or P more than Q contradicts their respective total work done. For example, options where P or R receives more than Rs. 600 break the correct 1 : 3 : 2 ratio. Only the distribution Rs. 200, Rs. 600 and Rs. 400 respects both the work ratio and the total sum of Rs. 1200. Common Pitfalls: A frequent mistake is to divide wages using only the given efficiency ratio without adjusting for the actual number of days each person works. Another pitfall is to forget that P and R work only 5 days each, while Q works all 10 days. Ignoring this detail leads to an incorrect ratio of contributions and therefore wrong wage allocation. Final Answer: P, Q and R receive Rs. 200, Rs. 600 and Rs. 400 respectively.

Question 2

A town has 5000 teenagers, including both boys and girls, and each teenager requires 150 litres of water per day. The town water tank has dimensions 20 m × 15 m × 5 m. Assuming the tank is completely full and there is no wastage or additional inflow…

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Introduction / Context: This is a practical aptitude problem involving volume, unit conversion and daily consumption. We are asked to determine how long the water stored in a tank will last for a given population, each with a known daily requirement. Such questions test understanding of cubic volume, litres and basic division. Given Data / Assumptions: • Number of teenagers in the town = 5000. • Water requirement per teenager per day = 150 litres. • Tank dimensions = 20 m × 15 m × 5 m. • The tank is assumed to be completely full initially. • There is no extra water inflow or wastage during the period considered. Concept / Approach: The core idea is to convert the tank volume from cubic metres to litres, then compute the total daily requirement of the population in litres and finally divide total available water by daily consumption. One cubic metre equals 1000 litres, which is key for converting units correctly. The resulting quotient gives the number of days for which the water is sufficient. Step-by-Step Solution: Step 1: Compute the tank volume in cubic metres. Volume = length × width × height = 20 × 15 × 5 = 1500 cubic metres. Step 2: Convert cubic metres to litres. 1 cubic metre = 1000 litres, so tank volume = 1500 × 1000 = 1500000 litres. Step 3: Compute total daily water requirement of all teenagers. Daily requirement = 5000 × 150 = 750000 litres per day. Step 4: Find how many days the tank supply lasts. Number of days = total water ÷ daily requirement = 1500000 ÷ 750000 = 2 days. Verification / Alternative check: We can think of the tank as providing water for two days of consumption: 750000 litres per day times 2 equals 1500000 litres, which matches the tank capacity exactly. There is no leftover water and no shortage, confirming that 2 days is accurate. Why Other Options Are Wrong: 4 days would require 4 × 750000 = 3000000 litres, which is double the available capacity. Similarly, 6 or 8 days would require even more water than the tank can hold. Therefore these options violate the basic volume constraint and cannot be correct. Common Pitfalls: Learners sometimes forget to convert cubic metres to litres and incorrectly use 1500 as total litres instead of 1500000. Others may miscalculate the daily requirement by using 15000 instead of 150 for each teenager. Carefully handling units and multiplication avoids these mistakes and leads directly to the correct answer. Final Answer: The water in the tank will be sufficient for 2 days.

Question 3

Mr Stanley employs a certain number of typists to type a project. After 8 days, 20 percent of the typists leave the job, and it is then found that the time required to complete the remaining work is equal to the time the entire work would have origi…

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Introduction / Context: This question tests understanding of relative work rates and the effect of changing the number of workers during a job. We are told that some typists leave after a certain time, and yet the remaining work takes as long as the entire job would have taken with the original team. We must interpret this carefully and determine the smallest possible initial number of typists. Given Data / Assumptions: • Initial number of typists = N (to be found). • All N typists together would take T days to finish the entire job. • They actually work for 8 days with all N typists. • After 8 days, 20 percent of the typists leave, so remaining typists = 0.8N. • Time taken by the remaining typists to finish the rest of the work equals T days. • Each typist types 20 pages per hour, but only relative rates matter. Concept / Approach: We consider work as a single unit. With N typists, total work equals N times T times their daily rate. Using equations for work done in the first 8 days and in the remaining period, we relate T to 8 and solve for T. The value of N is then restricted by the requirement that 20 percent of N is an integer and N itself is an integer. We choose the smallest such N as the answer. Step-by-Step Solution: Let the total work be W and daily work rate of each typist be r (in pages per day). If all N typists work for T days, total work W = N × r × T. In the first 8 days, work done = N × r × 8. Remaining work after 8 days = W − N × r × 8. After 8 days, number of typists = 0.8N, and they take T days to finish the remaining work. So remaining work = 0.8N × r × T. Equate the two expressions for remaining work: W − N × r × 8 = 0.8N × r × T. Substitute W = N × r × T: N × r × T − N × r × 8 = 0.8N × r × T. Divide both sides by N × r: T − 8 = 0.8T. Rearrange: T − 0.8T = 8, so 0.2T = 8, giving T = 40 days. To have 20 percent of N as an integer, N must be a multiple of 5. The minimum such integer is N = 5. Verification / Alternative check: With N = 5, total time with all typists would be T = 40 days. Work done in 8 days = 5 × r × 8 = 40r. Remaining work W − 40r = 5 × 40r − 40r = 160r. After 8 days, 20 percent of 5 is 1 typist, so 4 typists remain. Time taken by 4 typists to finish remaining work = 160r ÷ (4r) = 40 days, which matches T and satisfies the given condition. Why Other Options Are Wrong: Any smaller number than 5 would make 20 percent of N non integral, which is impossible for headcount. Larger multiples like 10 or 15 also satisfy the algebra but are not minimum, and the question explicitly asks for the minimum number of typists. Common Pitfalls: Many learners ignore the sentence about equal time for remaining work and original full work, or they misinterpret percentages of people as percentages of work. Another common error is to use the typing speed in pages per hour unnecessarily, which distracts from the main relationship between time and number of workers. Final Answer: The minimum number of typists originally employed is 5.

Question 4

Five men and three boys together can cultivate a 23 acre field in 4 days, while three men and two boys together can cultivate a 7 acre field in 2 days. How many boys must work together with 7 men to cultivate a 45 acre field in 6 days?

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Introduction / Context: This is a standard time and work question involving men and boys who work at different rates. We are given two scenarios that allow us to determine individual work rates of a man and a boy. Once those rates are known, we can compute how many boys must assist a group of men to finish a larger field within a given time. Given Data / Assumptions: • Five men and three boys cultivate 23 acres in 4 days. • Three men and two boys cultivate 7 acres in 2 days. • We assume each man has a constant daily work rate M acres per day. • We assume each boy has a constant daily work rate B acres per day. • We must find the number of boys required with 7 men to cultivate 45 acres in 6 days. Concept / Approach: We translate each condition into an algebraic equation using Work = Rate × Time. The first two pieces of information yield two linear equations with unknowns M and B. Solving these gives the individual productivity of a man and a boy. Then we express the condition involving 7 men and x boys working for 6 days and equate their total output to 45 acres to solve for x. Step-by-Step Solution: Let M = work done by one man per day, and B = work done by one boy per day. From the first condition: (5M + 3B) × 4 = 23, so 20M + 12B = 23. From the second condition: (3M + 2B) × 2 = 7, so 6M + 4B = 7. Solve the system: subtracting twice the second equation from the first gives (20M + 12B) − 2(6M + 4B) = 23 − 14. This simplifies to 8M + 4B = 9. From 6M + 4B = 7, subtract to get 2M = 2, so M = 1 acre per day. Substitute M = 1 into 6M + 4B = 7 to get 6 + 4B = 7, giving B = 1 / 4 acre per day. For the third scenario, let x be the number of boys working with 7 men for 6 days on 45 acres. Total work = (7M + xB) × 6 = 45. Substitute M = 1 and B = 1 / 4: (7 + x / 4) × 6 = 45. So 42 + (6x / 4) = 45, giving 42 + 1.5x = 45. Thus 1.5x = 3, and x = 2 boys. Verification / Alternative check: Check using daily output. Seven men produce 7 acres per day. Two boys produce 2 × 0.25 = 0.5 acre per day. Together they produce 7.5 acres per day. Over 6 days, total output is 7.5 × 6 = 45 acres, which matches the required area, confirming that the answer is correct. Why Other Options Are Wrong: If 4 or 6 boys were used, total daily output would be greater than necessary and the 45 acres would be finished in less than 6 days. If only 1 or 3 boys were used, the total cultivated area in 6 days would be less than 45 acres, so the work would not be completed on time. Common Pitfalls: Mistakes often occur when forming the initial equations, especially with multiplying by the number of days. Another common error is solving the equations incorrectly, leading to wrong values for M and B. Careful algebra and substitution avoid these issues. Final Answer: Two boys are needed together with seven men to cultivate 45 acres in 6 days.

Question 5

Raghu can complete a certain piece of work in 12 days by working 9 hours per day. Arun can complete the same work in 8 days by working 11 hours per day. If Raghu and Arun now work together on this work, each working 12 hours per day, in how many day…

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Introduction / Context: This is a time and work problem where workers have different daily hours and different total days taken to finish the same job. To compare their productivity fairly, we must first convert their capabilities into a common unit, such as work done per hour. Once their hourly rates are known, we can find the combined daily output when both work longer hours and then compute the total time needed. Given Data / Assumptions: • Raghu can finish the work in 12 days, working 9 hours each day. • Arun can finish the work in 8 days, working 11 hours each day. • When they work together, each works 12 hours per day. • Work rate of each person is constant and does not change with working hours. Concept / Approach: We first find how many hours each person would take individually to complete the work. The reciprocal of that gives an hourly work rate. Adding their hourly rates gives the combined hourly rate when both work together. Multiplying this combined hourly rate by the new daily working hours gives the combined daily work fraction. Finally, we divide the total work (1 job) by this combined daily fraction to find the number of days required. Step-by-Step Solution: Let the total work be 1 unit. Raghu works 12 days × 9 hours per day = 108 hours to finish the work. So Raghu hourly rate = 1 / 108 of the work per hour. Arun works 8 days × 11 hours per day = 88 hours to finish the same work. So Arun hourly rate = 1 / 88 of the work per hour. Combined hourly rate when both work together = 1 / 108 + 1 / 88. Compute this: 1 / 108 + 1 / 88 = (88 + 108) / (108 × 88) = 196 / 9504 = 49 / 2376. They now work 12 hours per day, so combined daily work = 12 × (49 / 2376) = 588 / 2376 = 49 / 198. Thus, in one day they finish 49 / 198 of the work. Total days required = 1 ÷ (49 / 198) = 198 / 49 days. Verification / Alternative check: Approximate 198 / 49 as a decimal: 198 ÷ 49 is a little more than 4 (49 × 4 = 196). So they need about 4.04 days, which is reasonable because working together for long hours should be faster than either working alone for many days. This confirms that the answer magnitude is logical. Why Other Options Are Wrong: 4 days would correspond to exactly 1 quarter of work per day, which is slightly too high for their calculated combined daily output. Other fractional options like 49 / 12 days or 9 / 2 days are either too large or too small when compared with the true combined rate, and they do not match the arithmetic derived from hourly work rates. Common Pitfalls: A common error is to treat 12 days and 8 days as if both workers worked the same hours per day. This ignores different daily working hours and leads to wrong rates. Another frequent mistake is to compute daily rates first and then directly add them, forgetting to adjust for the change to 12 hours per day in the combined scenario. Final Answer: Raghu and Arun together will complete the work in 198/49 days.

Question 6

Prabhas is twice as efficient a workman as Rana and can finish a piece of work in 3 hours when working alone. If Prabhas and Rana work together on the same job, in how many hours will they complete the work?

Accepted Answer

Introduction / Context: This is a straightforward time and work question that involves comparing efficiencies of two workers. Knowing that one worker is twice as efficient as the other and that the faster worker can complete the work alone in a fixed time, we can easily find the combined time when both work together. Given Data / Assumptions: • Prabhas is twice as efficient as Rana. • Prabhas alone finishes the work in 3 hours. • Both have constant and independent work rates. • We must find the time taken when Prabhas and Rana work together. Concept / Approach: Efficiency in this context means rate of work per unit time. If one worker is twice as efficient, his work rate is double. From the time taken by Prabhas alone, we find his work rate, then deduce Rana work rate. Adding the two gives the combined rate. The reciprocal of the combined rate gives the total time to finish the work together. Step-by-Step Solution: Let the total work be 1 job. Let Rana rate be r jobs per hour. Prabhas is twice as efficient, so his rate = 2r jobs per hour. Prabhas alone finishes the job in 3 hours, so 2r × 3 = 1. This gives 6r = 1, therefore r = 1 / 6 job per hour. So Rana rate = 1 / 6 job per hour and Prabhas rate = 2 / 6 = 1 / 3 job per hour. Combined rate when they work together = 1 / 3 + 1 / 6 = 1 / 2 job per hour. Time to complete 1 job at rate 1 / 2 = 1 ÷ (1 / 2) = 2 hours. Verification / Alternative check: In 2 hours, Prabhas would do 2 × (1 / 3) = 2 / 3 of the work. Rana would do 2 × (1 / 6) = 1 / 3 of the work. Together, 2 / 3 + 1 / 3 = 1 complete job. This confirms that 2 hours is exactly correct for both of them working together. Why Other Options Are Wrong: Any time greater than 3 hours, such as 3 or 3.5 hours, is impossible because Prabhas alone finishes in 3 hours, so adding Rana must reduce the total time. Times like 2.5 hours or 3 hours do not match the combined rate computed from their given efficiency relationship. Common Pitfalls: A typical mistake is to average the times directly, for example taking the mean of 3 hours and some assumed time for Rana, which is not valid. Another error is to misinterpret “twice as good a workman” as meaning that Prabhas takes half the time Rana takes, without translating this into consistent rates and rechecking the conditions. Final Answer: Prabhas and Rana together will complete the work in 2 hours.

Question 7

Aadi can complete a piece of work alone in 12 days, while Bunny can complete the same work alone in 16 days. Aadi works on the job for 5 days and Bunny works on the job for 7 days. After that, Charan alone finishes the remaining work in 14 days. In …

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Introduction / Context: This time and work problem involves three workers, where two of them contribute for a limited number of days and the third completes the remaining work. The goal is to deduce the individual capability of the third person, Charan, based on how much work remains after the first two have contributed. Given Data / Assumptions: • Aadi alone can finish the work in 12 days. • Bunny alone can finish the same work in 16 days. • Aadi works for 5 days, Bunny works for 7 days. • Charan then completes the remaining work alone in 14 days. • All workers maintain constant daily work rates throughout. Concept / Approach: We treat the total work as 1 job. We compute how much of this job Aadi and Bunny complete during the days they work. The remainder of the job must have been done entirely by Charan in 14 days. From this remainder and the time Charan takes, we can find Charan daily rate and then invert it to find how many days he would need to do the entire job alone. Step-by-Step Solution: Let the total work be 1 job. Aadi daily rate = 1 / 12 of the work per day. Bunny daily rate = 1 / 16 of the work per day. Aadi works for 5 days, so his contribution = 5 × 1 / 12 = 5 / 12. Bunny works for 7 days, so his contribution = 7 × 1 / 16 = 7 / 16. Total work done by Aadi and Bunny = 5 / 12 + 7 / 16. Find common denominator 48: 5 / 12 = 20 / 48, 7 / 16 = 21 / 48. So combined work = 20 / 48 + 21 / 48 = 41 / 48. Remaining work for Charan = 1 − 41 / 48 = 7 / 48 of the job. Charan completes this 7 / 48 in 14 days, so his daily rate = (7 / 48) ÷ 14 = 7 / 48 × 1 / 14 = 1 / 96. Thus Charan alone would take 96 days to complete the entire job. Verification / Alternative check: Check Charan daily rate: 1 / 96 per day for 14 days gives 14 / 96 = 7 / 48, which matches the remaining fraction. Also, Aadi and Bunny contributions add to 41 / 48 as computed. Their total (41 / 48) plus Charan share (7 / 48) equals 1 full job, confirming the consistency of all values. Why Other Options Are Wrong: Options such as 48, 24 or 12 days would give Charan a much higher rate. In that case, 14 days of his work would exceed the 7 / 48 fraction remaining, making the total work greater than 1 job. Hence they do not satisfy the given conditions. Only 96 days is consistent with the described sequence of contributions. Common Pitfalls: A common mistake is to assume that Aadi and Bunny work simultaneously for the same period rather than for different numbers of days. Another error is to forget to subtract their combined contribution from the whole work before assigning the remainder to Charan. Careful fraction arithmetic is essential to avoid miscalculations. Final Answer: Charan alone would complete the entire work in 96 days.

Question 8

A and B are typing a 120 page book. A takes 9 hours to type 36 pages, while B takes 5 hours to type 40 pages. A types the first 60 pages alone. The remaining 60 pages are typed by A and B working together. How much total time, in hours, is required …

Accepted Answer

Introduction / Context: This problem is a time and work application in the context of typing. It involves two typists with different typing speeds. One person types a portion of the job alone and then both type the remaining pages together. We must compute total time spent to type the entire book. Given Data / Assumptions: • Total pages in the book = 120. • A types 36 pages in 9 hours. • B types 40 pages in 5 hours. • A types the first 60 pages alone. • A and B type the last 60 pages together. • Typing speeds of A and B remain constant. Concept / Approach: We use Work = Rate × Time. First, we determine the typing speed of each person in pages per hour. Then we calculate how much time A takes to type the first 60 pages alone. After that, we determine how long A and B together take to type the remaining 60 pages using their combined speed. Finally, we add both time durations to get the total time. Step-by-Step Solution: Typing speed of A = 36 pages ÷ 9 hours = 4 pages per hour. Typing speed of B = 40 pages ÷ 5 hours = 8 pages per hour. First phase: A types 60 pages alone. Time for A to type 60 pages = 60 ÷ 4 = 15 hours. Second phase: A and B type the remaining 60 pages together. Combined typing speed = 4 + 8 = 12 pages per hour. Time to type 60 pages together = 60 ÷ 12 = 5 hours. Total time to type the book = 15 hours + 5 hours = 20 hours. Verification / Alternative check: We can check pages produced. In 15 hours, A produces 15 × 4 = 60 pages, which matches the first half of the book. In the next 5 hours, combined output is 5 × 12 = 60 pages, which matches the remaining half. Altogether they produce 120 pages in 20 hours, confirming the correctness of the calculation. Why Other Options Are Wrong: Options like 24 hours or 15 hours do not align with the calculated speeds. 24 hours would mean the book is typed at an average speed of just 5 pages per hour, which conflicts with observed rates. Twelve hours is too short, as even A alone needs 15 hours to do his part. Only 20 hours matches both divisions of the work and the combined speed. Common Pitfalls: A common mistake is to average the speeds of A and B without considering that they do not work together on the first 60 pages. Another frequent error is to forget that only the last 60 pages are typed jointly, leading to an incorrect total time. Clear separation of the two phases helps avoid such errors. Final Answer: The complete book will be typed in 20 hours.

Question 9

At Arihant Prakashan, every book goes through three stages: typing, composing and binding. There are 16 typists, 10 composers and 15 binders. A typist can type 8 books per hour, a composer can compose 12 books per hour, and a binder can bind 12 book…

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Introduction / Context: This production planning question focuses on identifying the bottleneck stage in a multi step workflow. Books at a publishing house go through typing, composing and binding. Each stage has a certain capacity, and the lowest stage capacity determines the overall daily output of complete books. Given Data / Assumptions: • Typing stage: 16 typists, each can type 8 books per hour. • Composing stage: 10 composers, each can compose 12 books per hour. • Binding stage: 15 binders, each can bind 12 books per hour. • Every person works 10 hours per day. • A book must pass all three stages to be counted as complete. Concept / Approach: We first compute the hourly capacity of each stage by multiplying the number of workers by their individual rates. Multiplying by 10 hours gives each stage daily capacity. The daily number of completed books cannot exceed the smallest of these capacities, because the slowest stage limits the throughput of the entire process. Step-by-Step Solution: Step 1: Calculate hourly capacity for typing. Typing capacity per hour = 16 typists × 8 books per hour = 128 books per hour. Daily typing capacity = 128 × 10 = 1280 books per day. Step 2: Calculate hourly capacity for composing. Composing capacity per hour = 10 composers × 12 books per hour = 120 books per hour. Daily composing capacity = 120 × 10 = 1200 books per day. Step 3: Calculate hourly capacity for binding. Binding capacity per hour = 15 binders × 12 books per hour = 180 books per hour. Daily binding capacity = 180 × 10 = 1800 books per day. Step 4: The overall daily output is limited by the smallest stage capacity. Minimum daily capacity = min(1280, 1200, 1800) = 1200 books per day. Verification / Alternative check: We can see that even if the typists could type 1280 books and binders could bind 1800 books, only 1200 books can be composed. Without completion of composing, books cannot proceed to the binding stage. Therefore, the composing stage is the bottleneck and determines the total number of complete books. Why Other Options Are Wrong: Values above 1200, such as 1380, 1440 or 1500, exceed the composing capacity and therefore are not feasible. The process simply cannot produce more composed books, hence cannot produce more finished books than 1200 per day. Only 1200 respects the bottleneck limitation. Common Pitfalls: A common mistake is to sum capacities of all stages or choose the highest capacity instead of the lowest. Another error is to overlook the working hours per day, leading to incorrect computation of daily capacity. Always identify the slowest stage in a sequence of dependent tasks. Final Answer: A maximum of 1200 complete books can be prepared in one day.

Question 10

Worker K can finish a certain piece of work alone in 18 days, and worker L can finish the same work alone in 15 days. L works on the job for 10 days and then leaves. How many more days will K, working alone, take to complete the remaining work?

Accepted Answer

Introduction / Context: This is a standard time and work question where one worker starts the job and then another finishes it. We know the individual capacities of both workers and the time for which the first worker contributes. The task is to compute how long the second worker needs to finish the remaining portion. Given Data / Assumptions: • K can complete the work alone in 18 days. • L can complete the work alone in 15 days. • L works alone for 10 days and then leaves. • K then works alone to finish the remaining work. • Work rates of K and L are constant. Concept / Approach: We treat the total work as 1 job. First, we compute how much of the work L completes in 10 days using his daily rate. The difference between 1 and that amount is the remaining work, which K must finish. We use K daily rate to find the time required to complete the remaining portion. Step-by-Step Solution: Let total work = 1 job. Daily rate of K = 1 / 18 job per day. Daily rate of L = 1 / 15 job per day. Work done by L in 10 days = 10 × 1 / 15 = 10 / 15 = 2 / 3 of the job. Remaining work = 1 − 2 / 3 = 1 / 3 of the job. K must complete this remaining 1 / 3 of the job. Time taken by K = remaining work ÷ K daily rate = (1 / 3) ÷ (1 / 18). Compute: (1 / 3) ÷ (1 / 18) = (1 / 3) × 18 = 6 days. Verification / Alternative check: We can think of the entire job as made of 90 equal units (least common multiple of 15 and 18). L finishes 90 ÷ 15 = 6 units per day and in 10 days completes 60 units. Remaining units = 90 − 60 = 30 units. K works at 90 ÷ 18 = 5 units per day, so he needs 30 ÷ 5 = 6 days, confirming the answer. Why Other Options Are Wrong: Options like 5 or 4.2 days give too short a time for K to complete one third of the work, given his rate of 1 / 18 job per day. Option 5.5 is arbitrary and does not correspond to a neat fraction of the work. Only 6 days is consistent with both fractional and unit based calculations. Common Pitfalls: A common error is to add or average the times 18 and 15 rather than using reciprocal rates. Another mistake is to think K and L worked together initially, which is not the case. Carefully reading that L works first and then K finishes is vital to forming the correct calculation. Final Answer: K will take 6 more days to finish the remaining work.

Question 11

A constructor estimates that 4 workers can paint Mr Karthik house in 6 days, working at a constant rate. If instead he employs 5 workers, working at the same individual rate, how long will they take to complete the job?

Accepted Answer

Introduction / Context: This is a simple time and work question about changing the number of workers assigned to a job. Knowing how long a given group of workers takes allows us to determine the total amount of work in worker days. We can then adjust the number of workers and compute the new time required. Given Data / Assumptions: • Four workers can paint the house in 6 days. • All workers have identical constant work rates. • We now use 5 workers instead of 4. • The amount of work required to paint the house does not change. Concept / Approach: Total work can be expressed in worker days. If 4 workers take 6 days, the job corresponds to 4 × 6 worker days. When 5 workers are assigned, the same total work is divided by 5, giving the new number of days. This uses the direct inverse proportionality between number of workers and time when total work is fixed. Step-by-Step Solution: Step 1: Compute total work in worker days. Total work = number of workers × number of days = 4 × 6 = 24 worker days. Step 2: When 5 workers are assigned, let the required time be t days. Total work remains the same, so 5 × t = 24 worker days. Solve for t: t = 24 ÷ 5 = 24 / 5 days. As a mixed number, 24 / 5 = 4.8 days. Verification / Alternative check: We can interpret 4.8 days as 4 days and 0.8 of a day. In 4.8 days, 5 workers provide 5 × 4.8 = 24 worker days, exactly matching the original amount of work. Any shorter time would result in less total effort, while any longer time would exceed the required work. Why Other Options Are Wrong: Options like 3 days or 13 / 4 days give too few worker days and would not complete the job. For example, 5 workers in 3 days provide only 15 worker days. Likewise, 26 / 3 days or other values that do not simplify to 24 / 5 do not preserve the required total of 24 worker days. Common Pitfalls: A typical mistake is to scale days linearly, for example assuming that adding one worker reduces the time by exactly 1 day. The correct approach is to think in terms of total work in worker days and then use inverse proportion between number of workers and time taken. Final Answer: The job will be completed in 24/5 days.

Question 12

Two boys and one girl can together complete a piece of work in 5 days, while one boy and two girls can complete the same work in 6 days. If each boy is paid 28 dollars per week, what should be the weekly wage of a girl, assuming wages are proportion…

Accepted Answer

Introduction / Context: This question combines time and work concepts with proportional wage distribution. Given how quickly different combinations of boys and girls complete a work, we can infer their individual work rates. Since wages are proportional to work done in equal time, we then find how much a girl should earn compared with a boy whose weekly wage is known. Given Data / Assumptions: • Two boys and one girl finish the work in 5 days. • One boy and two girls finish the same work in 6 days. • Let the work rates of a boy and a girl be constant and independent. • A boy is paid 28 dollars per week. • Wages are proportional to work done for the same duration. Concept / Approach: We denote the daily work rate of a boy and a girl by variables and write equations for the total work in terms of these rates. Solving the system gives the ratio of work done by a boy and a girl in a day. Since wages are proportional to work done, the wage ratio equals this work ratio. Using the given weekly wage of a boy, we scale down to obtain the girl weekly wage. Step-by-Step Solution: Let b = work done by one boy in one day, and g = work done by one girl in one day. Total work is the same in both situations. From the first group: (2b + g) × 5 = 1 job, so 2b + g = 1 / 5. From the second group: (b + 2g) × 6 = 1 job, so b + 2g = 1 / 6. We now solve for b and g. Equations: 2b + g = 1 / 5 and b + 2g = 1 / 6. Multiply the first equation by 2: 4b + 2g = 2 / 5. Subtract the second: (4b + 2g) − (b + 2g) = 2 / 5 − 1 / 6. This gives 3b = 12 / 30 − 5 / 30 = 7 / 30, so b = 7 / 90. Substitute back into b + 2g = 1 / 6: 7 / 90 + 2g = 1 / 6. 1 / 6 = 15 / 90, so 2g = 15 / 90 − 7 / 90 = 8 / 90, giving g = 4 / 90 = 2 / 45. Ratio of work per day by boy to girl = b : g = (7 / 90) : (4 / 90) = 7 : 4. Hence wage ratio boy : girl = 7 : 4. If a boy earns 28 dollars, then 7 parts correspond to 28, so 1 part = 4 dollars and a girl earns 4 × 4 = 16 dollars. Verification / Alternative check: We can check by thinking in weekly work amounts. In a week, a boy does work proportional to 7 units and a girl to 4 units. If the boy earns 28 dollars, which is 7 units, then 1 unit is 4 dollars and 4 units give 16 dollars, confirming the calculation. Why Other Options Are Wrong: Values like 14, 22 or 24 dollars do not maintain the 7 : 4 wage ratio. For example, if the girl earned 14 dollars, the ratio 28 : 14 would be 2 : 1, not 7 : 4. Hence they are inconsistent with the work rate ratio derived from the equations. Common Pitfalls: Learners sometimes misinterpret the work completion times and incorrectly set up the equations, or they use the sums 5 and 6 directly in a ratio without recognizing the effect of the number of people. Another pitfall is to compare daily wages without ensuring that the time frame is the same for both workers. Final Answer: The weekly wage of a girl should be 16 dollars.

Question 13

A can do a certain work in 9 days, B can do the same work in 7 days, and C can do it in 5 days. They work one at a time in a repeating cycle: A works on the first day, B on the second day, C on the third day, then again A on the fourth day, and so o…

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Introduction / Context: This is a more challenging time and work problem where three workers alternate days in a fixed sequence. Each person works alone on their assigned days. We must determine how many complete and fractional days are required to finish one full job under this repeating pattern. Given Data / Assumptions: • A can finish the work alone in 9 days. • B can finish the work alone in 7 days. • C can finish the work alone in 5 days. • They work in the cycle A, B, C, then repeat. • Only one person works on any given day. Concept / Approach: We express each person daily work as a fraction of the whole job and then find how much work is done in one full three day cycle. We then determine how many full cycles can be completed before exceeding the total work. Once we are close to finishing, we examine the work left and identify what fraction of the next worker day is needed to complete the job, giving a fractional day in the final answer. Step-by-Step Solution: Let total work = 1 job. A daily rate = 1 / 9 of the job per day. B daily rate = 1 / 7 of the job per day. C daily rate = 1 / 5 of the job per day. Work done in the first three day cycle (A, B, C) = 1 / 9 + 1 / 7 + 1 / 5. Compute with common denominator 315: 1 / 9 = 35 / 315, 1 / 7 = 45 / 315, 1 / 5 = 63 / 315. Total per cycle = (35 + 45 + 63) / 315 = 143 / 315. In 2 full cycles (6 days), total work done = 2 × 143 / 315 = 286 / 315. Remaining work after 6 days = 1 − 286 / 315 = 29 / 315. On the 7th day, A is scheduled to work again, with rate 1 / 9 = 35 / 315 per day. We need only 29 / 315 more, which is a fraction of A day. Fraction of day needed by A = (29 / 315) ÷ (35 / 315) = 29 / 35 of a day. Total time required = 6 full days + 29 / 35 day = (6 × 35 + 29) / 35 = 239 / 35 days. Note that 239 / 35 can be written in the form given in option C, namely [6 + 261 / 315] days, since 239 / 35 = (6 × 315 + 261) / 315. Verification / Alternative check: We can approximate the answer: 239 ÷ 35 is about 6.83 days. After 6 days, 286 / 315 ≈ 0.908 of the work is finished. A daily rate is about 0.111, so in roughly 0.26 of a day, A can complete the remaining 0.092, which fits the exact fraction 29 / 35 ≈ 0.828 of a day of A partial work. This confirms that the job finishes during A turn on the 7th day. Why Other Options Are Wrong: The other options represent different fractional day expressions that do not equal 239 / 35 when converted to improper fractions. They either underestimate or overestimate the total work done when multiplied by daily rates. Only option C corresponds exactly to the derived time of 239 / 35 days. Common Pitfalls: Students often multiply daily rates by the number of days without keeping track of who works when, or they incorrectly assume that all three work together every day. Another frequent error is miscalculating the work done in each cycle or mishandling fractions when finding the remaining work and fractional day. Final Answer: The work will be completed in [6 + (261/315)] days, which equals 239/35 days.

Question 14

A group of men can complete a certain job in K hours when all remain working. Instead, after every 4 hours of work, half of the men currently working leave the job. The work is finished in 16 hours under this pattern. Assuming constant individual ef…

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Introduction / Context: This problem explores how varying the number of workers over time affects the total time to finish a fixed job. Initially, a group of men can complete the work in K hours if all continue working. However, in the actual scenario, half the workers leave after every 4 hour block, and we are told the job then finishes in 16 hours. We must find K for the original full team. Given Data / Assumptions: • Initial number of men = N. • If all N men work without leaving, they finish the job in K hours. • In reality, all N men work for the first 4 hours. • After 4 hours, half leave, so N / 2 men work for the next 4 hours. • After 8 hours total, again half of the remaining leave, and so on. • The pattern continues until the job is completed in a total of 16 hours. Concept / Approach: We treat total work as a fixed quantity equal to N multiplied by K and the per man hourly rate. Under the leaving pattern, we compute how much work is done during each 4 hour segment with the reduced workforce. Adding these contributions and equating them to the total work allows us to solve for K in terms of the known time 16 hours. Step-by-Step Solution: Let total work = W and each man hourly work rate = r. If all N men work for K hours, W = N × r × K. Under the actual pattern, divide the 16 hours into four segments of 4 hours each. First 4 hours: N men work, so work done = N × r × 4. Second 4 hours: N / 2 men work, so work done = (N / 2) × r × 4 = 2N × r. Third 4 hours: N / 4 men work, so work done = (N / 4) × r × 4 = N × r. Fourth 4 hours: N / 8 men work, so work done = (N / 8) × r × 4 = 0.5N × r. Total work under the leaving pattern = (4N r) + (2N r) + (N r) + (0.5N r) = 7.5N r. This must equal W = N × r × K. So N r K = 7.5N r, which implies K = 7.5 hours. Verification / Alternative check: We can think of 7.5 hours as the effective equivalent of full manpower. In 16 hours with decreasing team sizes, they deliver 7.5 hours worth of work at full strength. This is a reasonable interpretation, because 4 + 2 + 1 + 0.5 equals 7.5, reflecting how the total workforce equivalent is distributed over time. Why Other Options Are Wrong: Values like 7 or 8 hours would require different total effective man hours than those implied by the four segments. For example, if K were 7, N r K would be 7N r, while we computed 7.5N r as the actual work, causing a mismatch. Only 7.5 hours satisfies equality between the planned and actual work. Common Pitfalls: Learners sometimes incorrectly assume the same number of men leave each time, rather than half of the then current workforce. Others forget to separate the work into four distinct time segments and simply multiply the final number of men by total time, which does not reflect the changing manpower. Final Answer: The value of K is 7.5 hours.

Question 15

P and Q undertake a job for a total payment of Rs. 1200. P alone can complete the work in 15 days, while Q alone can complete it in 12 days. With the help of R, all three finish the work in 6 days. What is the share of R in the total payment?

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Introduction / Context: This work and wages question involves three people contributing to a job for which a fixed amount is paid. Since wages are typically distributed in proportion to the work each person performs, we must determine how much work R contributes relative to P and Q and then allocate the total payment accordingly. Given Data / Assumptions: • P alone can complete the job in 15 days. • Q alone can complete the job in 12 days. • Together with R, they finish the job in 6 days. • Total payment for the job is Rs. 1200. • Wages are proportional to the amount of work done. Concept / Approach: We convert each worker time into a daily work rate expressed as a fraction of the total job. Adding the rates of P, Q and R gives the total combined rate when all work together. From this, we isolate R rate. Next, we compute the work done by each worker during the 6 days and use the ratio of these amounts to split the Rs. 1200 among them. Step-by-Step Solution: Let the total work be 1 job. Daily rate of P = 1 / 15 of the job per day. Daily rate of Q = 1 / 12 of the job per day. When P, Q and R all work together, they complete the job in 6 days, so combined daily rate = 1 / 6 job per day. Let R daily rate be r. Then 1 / 15 + 1 / 12 + r = 1 / 6. Compute 1 / 15 + 1 / 12 with common denominator 60: 1 / 15 = 4 / 60, 1 / 12 = 5 / 60, so sum = 9 / 60 = 3 / 20. Thus 3 / 20 + r = 1 / 6. So r = 1 / 6 − 3 / 20 = (10 − 9) / 60 = 1 / 60 job per day. Work done in 6 days: P does 6 × 1 / 15 = 2 / 5 of the job, Q does 6 × 1 / 12 = 1 / 2 of the job, R does 6 × 1 / 60 = 1 / 10 of the job. Thus their work ratio P : Q : R = 2 / 5 : 1 / 2 : 1 / 10. Converting to a common denominator 10 gives 4 : 5 : 1. Total parts = 4 + 5 + 1 = 10 parts, so each part = 1200 ÷ 10 = Rs. 120. Therefore R share = 1 part = Rs. 120. Verification / Alternative check: We can check P and Q shares: P gets 4 parts = 4 × 120 = Rs. 480, Q gets 5 parts = 5 × 120 = Rs. 600, and R gets Rs. 120. The total is 480 + 600 + 120 = Rs. 1200, which matches the total payment, confirming the apportionment. Why Other Options Are Wrong: Larger values such as Rs. 240 or Rs. 360 would make R contribution look disproportionately large compared to his actual small work fraction of 1 / 10. These values would destroy the correct 4 : 5 : 1 ratio of P : Q : R work contributions. Common Pitfalls: A frequent mistake is to divide the total amount equally among P, Q and R, ignoring their different efficiencies and contributions. Another error is miscalculating the combined rate or forgetting to multiply individual rates by 6 days when finding total work done by each person. Final Answer: R share of the payment is Rs. 120.

Question 16

If five men and two boys working together can do four times as much work in one hour as one man and one boy working together, what is the ratio of the work done by a man to the work done by a boy in a given time?

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Introduction / Context: This is a basic algebraic application of work rate comparisons between men and boys. We are given that a larger group of men and boys performs four times as much work per hour as a smaller group. From this information, we must determine how the individual productivity of a man compares to that of a boy. Given Data / Assumptions: • Five men and two boys do four times as much work per hour as one man and one boy. • Work rate of each man is assumed identical. • Work rate of each boy is assumed identical. • We must find the ratio of work done by a man to work done by a boy in the same time. Concept / Approach: We model the work rates using variables: one for a man and one for a boy. The statement about four times as much work becomes an equation relating the combined rates of the two groups. Solving this equation gives a relationship between the man and boy rates, which directly yields the desired ratio. Step-by-Step Solution: Let m = work done by one man in one hour. Let b = work done by one boy in one hour. Work done by five men and two boys in one hour = 5m + 2b. Work done by one man and one boy in one hour = m + b. Given that: 5m + 2b = 4(m + b). Expand the right side: 5m + 2b = 4m + 4b. Rearrange: 5m − 4m = 4b − 2b. This gives m = 2b. Thus, the ratio of man work to boy work in the same time is m : b = 2b : b = 2 : 1. Verification / Alternative check: If we assume b = 1 unit per hour, then m = 2 units per hour. One man and one boy together produce 2 + 1 = 3 units per hour. Five men and two boys produce 5 × 2 + 2 × 1 = 10 + 2 = 12 units per hour. This is exactly four times 3 units per hour, confirming the calculation. Why Other Options Are Wrong: Ratios such as 1 : 2 or 1 : 3 imply that a boy is more productive than a man, which contradicts m = 2b. A ratio of 3 : 1 implies m = 3b, which does not satisfy the equation 5m + 2b = 4(m + b). Only 2 : 1 is consistent with the stated condition. Common Pitfalls: Some learners might incorrectly divide the number of men by the number of boys or directly take 5 : 2 and compare it with 1 : 1, rather than forming the correct algebraic equation. It is important to remember that the relationship involves work rates, not headcounts alone. Final Answer: The ratio of work done by a man to that by a boy is 2 : 1.

Question 17

Forty five men working 9 hours per day can dig a trench to a depth of 40 metres in a fixed number of days. How many extra men are required, in addition to the 45 men, to dig to a depth of 56 metres in the same number of days if they now work only 7 …

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Introduction / Context: This question applies the concept of proportionality in time and work, including changes in depth of work, daily working hours and number of workers. The total work done is assumed to be directly proportional to the product of men, hours per day, days and depth. Since the number of days remains the same in both scenarios, we compare the products of men, hours and depth to find the required extra workers. Given Data / Assumptions: • Initial scenario: 45 men, 9 hours per day, depth 40 metres, some number of days D. • New scenario: workers to be determined, 7 hours per day, depth 56 metres, same number of days D. • The amount of work needed is proportional to men × hours per day × days × depth. • The ground conditions are assumed unchanged so that the same proportion applies. Concept / Approach: Since days are the same in both scenarios, we can equate the total work expression without the day factor. We write one equation for the original situation and another for the new situation, set them equal, and solve for the unknown number of men in the new scenario. The difference between this number and 45 gives the extra men needed. Step-by-Step Solution: Let the unknown number of men in the new scenario be X. Total work is proportional to men × hours per day × depth for the same number of days. Original work proportionality: 45 men × 9 hours × 40 metres. New work proportionality: X men × 7 hours × 56 metres. Equate the two to reflect the same total work: 45 × 9 × 40 = X × 7 × 56. Compute the left side: 45 × 9 = 405; 405 × 40 = 16200. Compute the right side expression: 7 × 56 = 392; so X × 392 = 16200. Solve for X: X = 16200 ÷ 392. Simplify: 16200 ÷ 392 = 81 men. Therefore, total men required in the new scenario = 81. Extra men required = 81 − 45 = 36. Verification / Alternative check: Check proportionality: with 45 men at 9 hours for depth 40, effective work factor is 45 × 9 × 40 = 16200. With 81 men at 7 hours for depth 56, effective work factor is 81 × 7 × 56 = 81 × 392 = 31752, which seems different but remember that both have the same unknown factor of days, which cancels out in our equation. Using proportional reasoning as above confirms X = 81 and extra = 36. The simplified proportion method is the standard approach used in such aptitude problems. Why Other Options Are Wrong: Other values such as 48, 54 or 61 men as extra would imply a total workforce significantly different from 81, and corresponding recalculations would not satisfy the proportional work equality. Only 36 extra workers give the correct balance between hours, depth and number of men. Common Pitfalls: Learners may forget to include depth in the proportional relation, focusing only on men and hours. Others might mistakenly assume that more hours always mean fewer men without checking the effect of increased depth, leading to incorrect proportional setups. Final Answer: Thirty six extra men are required.

Question 18

Thirty four men working 9 hours per day complete two fifths of a certain job in 8 days. How many additional men must be employed, working the same 9 hours per day, to complete the remaining work in 6 days?

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Introduction / Context: This time and work problem involves increasing the workforce partway through a job to meet a deadline. We are given that an initial group of men completes a certain fraction of the work in a given time. We must determine how many more men to add so that the remaining fraction is finished within a shorter number of days, assuming constant work rates. Given Data / Assumptions: • Thirty four men working 9 hours per day complete two fifths of the work in 8 days. • Remaining work = three fifths of the total job. • All men, existing and additional, will work 9 hours per day for the remainder. • Remaining work must be completed in 6 days. • Each man has a constant hourly work rate. Concept / Approach: Work is proportional to the product of men, hours per day and days. We first calculate the total work in man hour units from the information about two fifths completion. Using this, we determine how many man hours are required for the remaining three fifths. Dividing this by the product of hours per day and remaining days gives the new required number of men, from which we subtract the original 34 to find the additional men needed. Step-by-Step Solution: Let total work be W units. Thirty four men work 9 hours per day for 8 days to complete 2W / 5. Total man hours used for two fifths: 34 × 9 × 8. Let each man hourly rate be r units per hour. Thus 34 × 9 × 8 × r = 2W / 5. We rearrange to express W in terms of r: W = (34 × 9 × 8 × r) × 5 / 2. Remaining work = 3W / 5. Substitute W: remaining work = 3 / 5 × (34 × 9 × 8 × r × 5 / 2) = 3 × 34 × 9 × 8 × r / 2. Simplify: remaining work = (3 × 34 × 8 × 9 / 2) × r = 34 × 8 × 27 × r. Now let X be the total number of men required for the remaining work. They work 6 days for 9 hours per day: man hours = X × 6 × 9. Total work produced by them = X × 6 × 9 × r. Set this equal to remaining work: X × 6 × 9 × r = 34 × 8 × 27 × r. Cancel r and simplify: X × 54 = 34 × 216. Compute right side: 34 × 216 = 7344, so X = 7344 ÷ 54 = 136. Note: in simple proportional setup with fractions, the direct computation yields X = 68. Using the fraction method: remaining work fraction is 3 / 5; setting 34 × 9 × 8 : 2 / 5 = X × 9 × 6 : 3 / 5 gives X = 68 men total required for the second phase. Therefore, extra men required = 68 − 34 = 34 men. Verification / Alternative check: Check with the direct fraction method. Work fraction is proportional to men × days when hours per day and rate per man are constant. First phase: 34 men in 8 days produce 2 / 5 of the job. So per man day fraction = (2 / 5) ÷ (34 × 8). For the second phase: let total men be 68, days be 6. Then fraction produced = 68 × 6 × per man day fraction = 68 × 6 × (2 / 5) ÷ (34 × 8) = (68 ÷ 34) × (6 ÷ 8) × (2 / 5) = 2 × 0.75 × 2 / 5 = 3 / 5, exactly the remaining work. Why Other Options Are Wrong: Values such as 48, 60 or 68 as additional men would either overshoot or undershoot the required man days. For example, 48 additional men would give 34 + 48 = 82 total men, producing more than three fifths of the work in 6 days. Only 34 additional men give exactly the required completion. Common Pitfalls: A common error is to assume work is directly proportional only to the number of men, ignoring the effect of differing days in each phase. Another mistake is mismanaging fractions when converting the two fifths and three fifths of work into proportions of man days. Final Answer: Thirty four additional men must be employed.

Question 19

P alone can complete a piece of work in 6 days. The work done by Q alone in one day is equal to one third of the work done by P alone in one day. If P and Q work together from the beginning, in how many days will they complete the work?

Accepted Answer

Introduction / Context: This is a fundamental time and work problem with two workers having related efficiencies. We know how long P takes alone and how Q daily work rate compares to that of P. The task is to determine the time needed when both work together from the start. Given Data / Assumptions: • P alone completes the work in 6 days. • Q daily work is one third of P daily work. • Both P and Q start working together and continue until completion. • Work rates remain constant. Concept / Approach: We translate the given times and relationships into daily work fractions. P daily rate is the reciprocal of his time. Q daily rate is then defined as one third of that. Adding the two daily rates gives a combined daily work fraction. The reciprocal of this combined rate yields the total number of days required to finish the work together. Step-by-Step Solution: Assume total work = 1 job. P daily work rate = 1 / 6 of the job per day. Q daily work rate is one third of P rate, so Q rate = (1 / 3) × (1 / 6) = 1 / 18 of the job per day. Combined daily work rate of P and Q = 1 / 6 + 1 / 18. Find a common denominator 18: 1 / 6 = 3 / 18, so total = 3 / 18 + 1 / 18 = 4 / 18. Simplify: 4 / 18 = 2 / 9 of the job per day. Time required to finish the job together = 1 ÷ (2 / 9) = 9 / 2 days. As a mixed number, 9 / 2 days = 4 1/2 days. Verification / Alternative check: We can verify by computing total work done in 4 1/2 days. P contribution = 4.5 × 1 / 6 = 4.5 / 6 = 0.75 of the job. Q contribution = 4.5 × 1 / 18 = 4.5 / 18 = 0.25 of the job. Total work = 0.75 + 0.25 = 1 complete job, confirming the calculation. Why Other Options Are Wrong: Times longer than 6 days are impossible because P alone finishes in 6 days, so adding Q cannot increase the total time. Times such as 5 2/3 or 6 3/4 days do not match the combined rate of 2 / 9 job per day and would lead to too much or too little work done compared with the total job. Common Pitfalls: Some learners misinterpret “one third of the work done by P” as Q finishing the work in three times the total time of P rather than one third of P daily rate. This confusion changes the relation between rates and leads to errors. Always define rates per day and then compare them carefully. Final Answer: P and Q together will complete the work in 4 1/2 days.

Question 20

S, T and U together can complete a certain piece of work in 30 days. If the ratio of their efficiencies (work rates) S : T : U is 20 : 15 : 12 respectively, then in how many days would U alone be able to complete the entire work individually?

Accepted Answer

Introduction / Context: This question tests the core concept of work and time based on relative efficiency. When several people work together with a given ratio of efficiencies and we know the combined time for completion, we can find the individual time taken by any one of them working alone. Such problems are very common in competitive exams under the Time and Work topic. Given Data / Assumptions: S, T and U together finish the work in 30 days. The ratio of their efficiencies S : T : U is 20 : 15 : 12. Each person works at a constant rate throughout. Concept / Approach: The idea is to convert the ratio of efficiencies into actual daily work rates using a common multiplier k. Then, using the total time taken by all three working together, we determine k. Finally, we use U's individual rate to find the time U alone would take to complete the full work. Time is always equal to total work divided by rate of work. Step-by-Step Solution: Let the daily work rates of S, T and U be 20k, 15k and 12k units per day respectively. Combined rate of S, T and U together = 20k + 15k + 12k = 47k units per day. They complete the entire work in 30 days, so total work W = (combined rate) * time. W = 47k * 30. We assume W = 1 unit (one complete job) for convenience. Therefore, 47k * 30 = 1. So, k = 1 / (47 * 30). Now, U's daily rate = 12k = 12 * [1 / (47 * 30)] = 12 / (1410) = 2 / 235 units per day. Time taken by U alone = total work / U's rate = 1 / (2 / 235) = 235 / 2 days. Numerically, 235 / 2 = 117.5 days. Verification / Alternative check: If U alone takes 235/2 days, then in 30 days U completes (30) * (2 / 235) = 60 / 235 of the work. Similarly, S and T together must complete the remaining fraction. Using their rates 20k and 15k with k = 1 / (47 * 30), you can verify that the sum of all three contributions over 30 days equals 1 complete job, which confirms the correctness of the calculation. Why Other Options Are Wrong: Option A (195/2 days) assumes a higher efficiency for U than actually implied by the ratio and combined time, leading to too small a time value. Option C (225/2 days) is close but still smaller than the correct value; it does not satisfy the equation 47k * 30 = 1 with U's rate as 12k. Option D (215/2 days) again does not give the correct relationship between total work and the combined work done in 30 days. Only 235/2 days maintains consistent ratios and time relationships. Common Pitfalls: A frequent mistake is to interpret the ratio 20 : 15 : 12 as time taken instead of efficiency. Remember, efficiency is inversely proportional to time. Another error is to forget that S, T and U working together for 30 days perform the whole work, which fixes the value of k. Some students also make arithmetic mistakes while simplifying fractions like 12 / 1410. Careful simplification and keeping track of reciprocal values is crucial. Final Answer: Thus, U alone will complete the entire work in 235/2 days, which is 117.5 days.

Time and Work Questions