Question 1

Workers A, B and C can complete a piece of work alone in 15 days, 16 days and 24 days respectively. They start working together on the job. A leaves after 3 days from the start, and B leaves after 4 days from the start. For how many total days, from…

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Introduction / Context:This time and work question involves three workers with different individual finishing times. They begin the work together but two of them leave at different stages. We must track the portions of work completed in different time intervals and finally determine the total duration from start to finish of the job. Given Data / Assumptions: A alone can complete the work in 15 days. B alone can complete the work in 16 days. C alone can complete the work in 24 days. All three start together. A leaves after 3 days from the beginning. B leaves after 4 days from the beginning. Work proceeds continuously with constant rates. Concept / Approach:We convert each worker's time into a daily work rate. Then we break the process into phases. In phase one, A, B, and C work together. In phase two, only B and C work. In phase three, only C works. For each phase, we compute the fraction of work completed. The sum of these fractions must total one complete job, and from that we can infer the total time. Step-by-Step Solution:Let total work = 1 unit.Rates: A = 1 / 15 per day, B = 1 / 16 per day, C = 1 / 24 per day.Phase 1 (days 1 to 3): all three together, rate = 1 / 15 + 1 / 16 + 1 / 24.Compute combined rate: 1 / 15 = 16 / 240, 1 / 16 = 15 / 240, 1 / 24 = 10 / 240, so total = 41 / 240.Work in first 3 days = 3 * 41 / 240 = 123 / 240 = 41 / 80.Phase 2 (day 4): A leaves, so only B and C work, rate = 1 / 16 + 1 / 24.B + C rate = 1 / 16 + 1 / 24 = 3 / 48 + 2 / 48 = 5 / 48.Work after 4 days = 41 / 80 + 5 / 48 = 37 / 60.Remaining work = 1 - 37 / 60 = 23 / 60, done by C alone.C's rate = 1 / 24, so time for remaining = (23 / 60) / (1 / 24) = 23 * 24 / 60 = 46 / 5 = 9.2 days.Total time = 4 days already passed + 46 / 5 days = (20 / 5 + 46 / 5) = 66 / 5 = 13.2 days = 13 1/5 days. Verification / Alternative check:We can check by computing the work done in each segment numerically. First 3 days produce 41 / 80 of the work. The next day adds 5 / 48, giving 37 / 60. The remaining 23 / 60 done by C at 1 / 24 per day corresponds to 23 / 60 * 24 = 46 / 5 days. When we add all phases, we get 3 + 1 + 9.2 = 13.2 days, exactly matching the converted mixed fraction of 13 1/5 days. Why Other Options Are Wrong:Ten and two thirds days, twelve and two thirds days, and eleven and five sevenths days are all less than the time that C alone needs to finish the remaining 23 / 60 of the work after the first four days. Since the last phase already requires more than nine days, any total under thirteen days is impossible. Therefore these options are inconsistent with the computed segment durations. Common Pitfalls:A common mistake is to misinterpret when each worker leaves, such as assuming A and B leave back to back without overlapping, or to forget the phase in which only C works. Another typical error is in adding fractions with different denominators, which can lead to an incorrect remaining work value and a wrong final time. Careful handling of each fractional part of the job is essential. Final Answer:The work is completed in a total of 13 1/5 days from the start.

Question 2

It takes exactly 1 minute to fill 2/7 of a vessel with a liquid at a constant rate. Assuming the filling rate does not change, how much time in minutes is required to completely fill the whole vessel?

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Introduction / Context:This is a simple ratio based time and work question involving the filling of a vessel. We know the fraction of the vessel filled in a given time at a constant rate. The aim is to use proportional reasoning to determine the total time required to fill the entire vessel completely at the same rate. Given Data / Assumptions: In 1 minute, 2/7 of the vessel is filled. The rate of filling is constant over time. The vessel starts empty. We want the time to fill 1 whole vessel (fraction 1). Concept / Approach:When rate is constant, the amount of work done is directly proportional to time. If a fraction of work is completed in a known time, we can scale the time up or down according to how many times larger the target fraction is. Here, 2/7 of the vessel corresponds to 1 minute. To get from 2/7 to the full vessel 1, we multiply by a factor that converts 2/7 into 1, and we apply the same factor to the time. Step-by-Step Solution:Fraction filled in 1 minute = 2 / 7.We want the time to fill 1, so we need a factor k such that (2 / 7) * k = 1.Solve for k: k = 1 / (2 / 7) = 7 / 2.Therefore, the required time = 1 minute * 7 / 2 = 7 / 2 minutes.Convert 7 / 2 minutes if desired: 7 / 2 = 3.5 minutes. Verification / Alternative check:You can verify by reversing the reasoning. If the whole vessel takes 7 / 2 minutes to fill, then the rate per minute is 1 / (7 / 2) = 2 / 7 of the vessel per minute, which matches the given condition. This symmetry confirms that our proportional scaling is correct and that 7 / 2 minutes is the correct time for the full vessel. Why Other Options Are Wrong:One seventh of a minute and five sevenths of a minute are both smaller than one minute, and therefore would correspond to less than 2 / 7 of the vessel, not the full amount. Seven fifths of a minute (which is 1.4 minutes) would also fill only (2 / 7) * 1.4 = 0.4 of the vessel. None of these match the requirement that all of the vessel be filled at the constant rate. Common Pitfalls:Many learners mistakenly multiply rather than divide by the given fraction, or they attempt to add extra steps instead of directly using proportional reasoning. Another common error is to assume that 1 minute corresponds to 1 / 7 of the vessel instead of 2 / 7, which leads to incorrect time calculations. Always base your ratio directly on the fraction explicitly provided in the question. Final Answer:The vessel will be completely filled in 7/2 minutes (that is, 3.5 minutes).

Question 3

Workers M, N and O can complete a piece of work alone in 18 days, 36 days and 54 days respectively. M works alone for 8 days, then N and O join him and they all work together for some more days. M leaves exactly one day before the work is finished, …

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Introduction / Context:This time and work question involves three workers with different individual completion times. The work is performed in stages, with some workers joining later and one worker leaving just before completion. The objective is to determine how long all three workers, M, N, and O, work together simultaneously before M leaves. Given Data / Assumptions: M alone can complete the work in 18 days. N alone can complete the work in 36 days. O alone can complete the work in 54 days. M works alone for the first 8 days. Then M, N and O work together for some days. M leaves one day before completion; N and O continue that last day. All rates are constant and the work is of fixed size. Concept / Approach:We convert each time into a daily work rate. Then we break the process into three phases. Phase one is M alone for 8 days. Phase two is M, N and O together for an unknown number of days t. Phase three is N and O together for the final 1 day. We express the total work as the sum of work done in these phases and set it equal to 1. Solving this equation gives t, the number of days all three work together. Step-by-Step Solution:Let total work = 1 unit.Rates: M = 1 / 18 per day, N = 1 / 36 per day, O = 1 / 54 per day.Phase 1: M alone for 8 days, work done = 8 * (1 / 18) = 8 / 18 = 4 / 9.Phase 3: last day, only N and O work, rate = 1 / 36 + 1 / 54.Compute N + O rate: 1 / 36 = 3 / 108, 1 / 54 = 2 / 108, total = 5 / 108.Work in last day = 5 / 108.Let t be the number of days in Phase 2 when all three work together.M + N + O combined rate = 1 / 18 + 1 / 36 + 1 / 54.Compute: 1 / 18 = 6 / 108, 1 / 36 = 3 / 108, 1 / 54 = 2 / 108, so total = 11 / 108.Total work equation: (4 / 9) + t * (11 / 108) + (5 / 108) = 1.Convert 4 / 9 to denominator 108: 4 / 9 = 48 / 108.So 48 / 108 + 5 / 108 = 53 / 108; remaining work = 1 - 53 / 108 = 55 / 108.Thus t * (11 / 108) = 55 / 108, so t = 55 / 108 * 108 / 11 = 5. Verification / Alternative check:We can check by explicitly summing all work. M alone in 8 days does 4 / 9. All three in 5 days do 5 * 11 / 108 = 55 / 108. N and O in the last day do 5 / 108. Total work is 4 / 9 + 55 / 108 + 5 / 108 = 48 / 108 + 55 / 108 + 5 / 108 = 108 / 108 = 1, which confirms that all work is completed and that the value t = 5 is consistent. Why Other Options Are Wrong:If t were 3 days or 4 days, the combined work of all phases would be less than one complete job and the work would remain unfinished. If t were 6 days, the total work done would exceed one and would imply that some of the work was undone, which is not possible under constant positive rates. Only 5 days gives an exact match of total work to one job. Common Pitfalls:One common mistake is to forget that M does not work during the last day and to treat it as if all three work throughout. Another error is in converting fractions to a common denominator when combining rates or total work. Careful algebra and structured phase wise thinking are necessary to correctly handle multi stage work problems like this. Final Answer:M, N and O work together for 5 days before M leaves.

Question 4

If a person earns Rs. 70,000 in one year, assuming 365 days in a year and 24 hours in a day, what is the equivalent earning per hour in rupees?

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Introduction / Context:This question is a unit conversion and rate problem framed in the context of annual income. A yearly salary is given, and we are asked to convert this into an hourly earning rate by using the total number of hours in a year. It tests understanding of basic time conversions and proportions. Given Data / Assumptions: Annual income = Rs. 70,000. We assume 365 days in a year. We assume 24 hours in each day. We want income per hour in rupees. Income is distributed uniformly across all hours in the year. Concept / Approach:The earning per hour is found by dividing the total yearly earnings by the total number of hours in a year. The total hours are computed by multiplying days per year by hours per day. Once we know the total number of hours, we perform a straightforward division to obtain the hourly rate. This is a direct application of the formula rate = total amount / total time. Step-by-Step Solution:Total days in a year = 365.Hours per day = 24.Total hours in a year = 365 * 24.Compute 365 * 24 = 8760 hours.Total income = Rs. 70,000.Hourly income = 70000 / 8760 rupees.70,000 / 8,760 is approximately equal to 7.99, which rounds to 8 rupees per hour. Verification / Alternative check:We can check by reversing the operation. If we assume the hourly rate is exactly Rs. 8, then yearly income would be 8 * 8,760 = 70,080 rupees. This is extremely close to 70,000 and consistent with rounding the hourly rate to the nearest integer. Hence, treating the answer as 8 rupees per hour is reasonable and matches the intent of the multiple choice options. Why Other Options Are Wrong:An hourly rate of 80 rupees would imply an annual income of 80 * 8,760 = 700,800, which is far higher than Rs. 70,000. An hourly rate of 0.8 rupee would lead to only 7,008 rupees per year. An hourly rate of 0.08 rupee would give only 700.8 rupees per year. None of these match the given yearly amount of Rs. 70,000. Common Pitfalls:Some learners may mistakenly divide by 365 or by 24 instead of by the full product 365 * 24, which mixes up hours, days, and years. Others may assume a working year of fewer hours without checking the assumption given in the question. Always pay attention to what time basis is specified and be systematic about counting total hours before performing the division. Final Answer:The equivalent earning rate is approximately 8 rupees per hour.

Question 5

Workers A and B together can complete a task in 1.5 days. In reality the task took 2 days because A left some time before completion and B finished the remaining work alone. If A alone can complete the entire task in 2.625 days, how many days before…

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Introduction / Context:This problem combines time and work with the idea of someone leaving before the completion of a task. A and B together can complete the work faster than either alone, but the actual duration is longer because A leaves early and B continues alone. We are asked to determine how long before completion A left, based on their individual and combined capabilities. Given Data / Assumptions: A and B together can complete the work in 1.5 days. The actual completion time of the work is 2 days. A alone can complete the work in 2.625 days. A leaves some time before completion and B works alone in the final part. Rates of A and B are constant throughout. Concept / Approach:We first convert the given times into daily work rates. Knowing A's solo time and the combined time, we can deduce B's solo time. Then we suppose that A works together with B for x days and that B works alone for the remaining (2 - x) days. We set up an equation for total work as the sum of the work done in both phases, and solve for x. The quantity 2 - x gives the number of days before completion when A left. Step-by-Step Solution:Let total work = 1 unit.A and B together finish in 1.5 days, so combined rate rAB = 1 / 1.5 = 2 / 3 per day.A alone finishes in 2.625 days. Convert 2.625 to an improper fraction: 2.625 = 21 / 8.So A's rate rA = 1 / (21 / 8) = 8 / 21 per day.Then B's rate rB = rAB - rA = 2 / 3 - 8 / 21 = 14 / 21 - 8 / 21 = 6 / 21 = 2 / 7 per day.So B alone would take 1 / (2 / 7) = 7 / 2 = 3.5 days.Let A and B work together for x days. Then B works alone for (2 - x) days.Total work equation: x * (2 / 3) + (2 - x) * (2 / 7) = 1.Solve: (2 / 3)x + (4 / 7) - (2 / 7)x = 1.Combine x terms: (2 / 3 - 2 / 7)x = 1 - 4 / 7 = 3 / 7.Compute 2 / 3 - 2 / 7 = 14 / 21 - 6 / 21 = 8 / 21.Thus (8 / 21)x = 3 / 7, so x = (3 / 7) * (21 / 8) = 9 / 8 = 1.125 days.Number of days before completion when A left = 2 - x = 2 - 1.125 = 0.875 days. Verification / Alternative check:Check the total work: A and B together for 1.125 days do 1.125 * 2 / 3 = 0.75 of the work. B alone for 0.875 days at 2 / 7 per day does 0.875 * 2 / 7 = 1.75 / 7 = 0.25 of the work. Total work = 0.75 + 0.25 = 1, confirming that the task is fully completed in 2 days with A leaving 0.875 days before the end. Why Other Options Are Wrong:If A had left 1.125 days before completion, that would mean he stopped very early and B alone would have to work for 1.125 days, which combined with the earlier joint work does not match the given total time and rates. Values like 0.625 or 0.375 days would result in different splits of the work that no longer satisfy the total work requirement when using the accurate rates of A and B. Only 0.875 days fits the equation and the conditions. Common Pitfalls:Learners often confuse whether x represents the time A works or the time B works alone, leading to misinterpretation of 2 - x. Another common error is to round decimals too early or mishandle fraction conversions, which can slightly distort the rates and upset the final equality. Careful algebra and clear identification of each phase are crucial in such layered time and work problems. Final Answer:A left the work 0.875 days before the task was completed.

Question 6

Worker A can complete a certain job in 12 days. Worker B is 60% more efficient than A. In how many days will B alone finish the same piece of work?

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Introduction / Context:Once again we have a comparison of efficiencies between two workers. A has a known completion time for a job, and B is described as being 60 percent more efficient. We need to convert this efficiency relationship into a time calculation and find B's solo completion time for the same job. Given Data / Assumptions: A alone completes the job in 12 days. B is 60% more efficient than A. Efficiency is measured as work done per day. Both workers operate at constant rates. Total work is one complete job. Concept / Approach:Efficiency and time are inversely related. If B is 60% more efficient, B's rate is 1.60 times A's rate. Given A's time, we find A's daily work rate as 1 / 12. Multiplying this by 1.6 gives B's daily rate. Then we invert this rate to find the time B alone needs to complete the job. This is a straight application of rate and time relationships in work problems. Step-by-Step Solution:Assume total work = 1 unit.A's time = 12 days, so A's rate = 1 / 12 work per day.B is 60% more efficient, so B's rate = 1.60 * (1 / 12) = 1.6 / 12.Simplify B's rate: 1.6 / 12 = 16 / 120 = 2 / 15 work per day.Time for B alone = 1 / (2 / 15) = 15 / 2 days = 7.5 days. Verification / Alternative check:We can verify with proportional reasoning. A takes 12 days. Since B is 1.6 times as fast, B's time must be 12 / 1.6. Compute 12 / 1.6 = 7.5, which confirms the rate based calculation. Both methods arrive at the same result, giving confidence in the answer. Why Other Options Are Wrong:Eight days would correspond to a smaller increase in efficiency than 60%. Six point four days or 5.5 days indicate significantly higher efficiencies than the stated 60% increase and do not match the specified relationship between A and B. Only 7.5 days is consistent with B being exactly 60% more efficient than A. Common Pitfalls:Students sometimes misinterpret 60% more efficient as meaning B takes 60% of A's time directly, which would imply a different efficiency factor. Another issue is performing decimal division incorrectly when computing 12 / 1.6. To avoid errors, it is often easier to work with fractions, for example, 12 / (8 / 5) = (12 * 5) / 8 = 60 / 8 = 7.5 days. Final Answer:B alone can complete the job in 7.5 days.

Question 7

Two men or three women can complete a job in 96 days. In how many days will a team consisting of six men and seven women complete the same job, assuming all men have equal efficiency and all women have equal efficiency?

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Introduction / Context:This time and work question involves converting between equivalent groups of workers. We are told that two men can do the same work as three women in a given time. Then a mixed group of six men and seven women is formed, and we must find how long this group will take to finish the same job. Given Data / Assumptions: Two men or three women can complete the job in 96 days. All men have the same rate of working. All women have the same rate of working. Six men and seven women work together on the same job. Rates are constant and additive. Concept / Approach:First, we translate the statement two men or three women into an equivalence of work rates. This tells us how many women correspond to one man. We then compute the total work in woman days or man days and use that to determine the time taken by the mixed group. Working in a single equivalent unit (either all man units or all woman units) makes the calculation straightforward. Step-by-Step Solution:Let total work = 1 job.Two men complete the job in 96 days, so total man days = 2 * 96 = 192 man days.Three women also complete the job in 96 days, so total woman days = 3 * 96 = 288 woman days.From this, 192 man days = 288 woman days, so 1 man day = 288 / 192 woman days = 3 / 2 woman days.Thus 1 man is equivalent to 1.5 women in daily work rate.Now consider the group of six men and seven women.Convert men to women equivalents: six men = 6 * 1.5 = 9 women equivalents.Total equivalent women in the team = 9 + 7 = 16 women equivalents.Total woman days required for the job = 288 woman days.Time taken by 16 women equivalents = 288 / 16 = 18 days. Verification / Alternative check:We can also work in man days. As found earlier, 1 woman is equivalent to 2 / 3 of a man. Seven women are therefore 7 * 2 / 3 = 14 / 3 men equivalents. Six men plus 14 / 3 men = (18 / 3 + 14 / 3) = 32 / 3 men equivalents. The total man days needed are 192. Hence time = 192 / (32 / 3) = 192 * 3 / 32 = 576 / 32 = 18 days, confirming the result. Why Other Options Are Wrong:Twenty seven days, twenty days, and twenty four days do not respect the total equivalent work required when converted to either man days or woman days. Any such value would imply that the mixed team produces either too much or too little work per day compared with the equivalence conditions given in the problem. Only 18 days correctly matches the necessary total work and team rate. Common Pitfalls:Some candidates directly average the times or numbers of men and women without setting up a proper equivalence. Others forget to convert all workers into a single comparable unit before calculating time, which leads to inconsistent units. Always express everything in either man days or woman days before solving mixed team work problems. Final Answer:The team of six men and seven women will complete the job in 18 days.

Question 8

Amar is twice as efficient a workman as Badal and therefore finishes a certain job in 15 days less time than Badal would take alone. In how many days will Amar and Badal together complete the job working at their constant individual rates?

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Introduction / Context:This problem combines relative efficiency and time difference. Amar is described as being twice as good a worker as Badal and, as a result, he finishes a job in 15 days less than Badal. We must use this information to find each worker's individual time and then compute the time if both work together on the same job. Given Data / Assumptions: Amar is twice as efficient as Badal. Amar takes 15 days less than Badal to finish the job. Each worker's efficiency is constant over time. The entire job is of fixed size. We need the time when Amar and Badal work together. Concept / Approach:If Amar is twice as efficient as Badal, then Amar's daily work rate is twice Badal's rate. Since time is inversely proportional to rate, Amar's time is half Badal's time. We set up an equation relating their times using the given 15 day difference. Once their individual times are found, we add their rates to get the combined rate and then invert this rate to find the time taken together. Step-by-Step Solution:Let Badal's time be TB days and Amar's time be TA days.Amar is twice as efficient as Badal, so TA = TB / 2.It is given that Amar finishes 15 days earlier: TB - TA = 15.Substitute TA = TB / 2 into the difference equation: TB - TB / 2 = 15.This simplifies to TB / 2 = 15, so TB = 30 days.Then TA = TB / 2 = 30 / 2 = 15 days.Rates: Amar = 1 / 15 per day, Badal = 1 / 30 per day.Combined rate = 1 / 15 + 1 / 30 = 2 / 30 + 1 / 30 = 3 / 30 = 1 / 10 per day.Time taken together = 1 / (1 / 10) = 10 days. Verification / Alternative check:Check the time difference: Amar needs 15 days, Badal needs 30 days, so the difference is indeed 30 - 15 = 15 days as stated. Together, in 10 days, Amar completes 10 / 15 = 2 / 3 of the job and Badal completes 10 / 30 = 1 / 3 of the job. The sum is 2 / 3 + 1 / 3 = 1 complete job, confirming that 10 days is accurate. Why Other Options Are Wrong:Thirty days is just Badal's solo time, not their combined time. Forty days is even longer and clearly impossible when combining two positive work rates. Twenty days would mean that two workers working together are slower than Amar alone, which contradicts basic logic. Only 10 days matches the derived combined rate correctly. Common Pitfalls:Some learners interpret twice as good incorrectly and treat Amar's time as twice Badal's instead of half. Others overlook the 15 day difference condition or set up the equation with the wrong sign. Carefully translating verbal descriptions of relative efficiency into correct algebraic relationships is critical in such problems. Final Answer:Amar and Badal together will complete the job in 10 days.

Question 9

Worker A completes 75% of a certain job in 25 days. He then calls in worker B, and together they finish the remaining 25% of the work in 5 days. How many days would B alone require to complete the entire job working at his own constant rate?

Accepted Answer

Introduction / Context:This time and work question involves one worker starting a job, then another worker joining later to help finish it. We are given how much of the job A completes alone and how long A and B together take to finish the remainder. Using this, we must deduce B's individual efficiency and then find how long B would take to do the entire job alone. Given Data / Assumptions: A completes 75% of the job in 25 days. The remaining 25% is finished by A and B working together in 5 days. Work rates remain constant throughout. The total work is one complete job. We are to find B's solo time for the entire job. Concept / Approach:We interpret the 75% completion in 25 days as A's rate. Then, since the remaining 25% is done by A and B together in 5 days, we interpret that as their combined rate. The difference between the combined rate and A's rate gives B's rate. Finally, we invert B's rate to find the time B would require to complete the entire job alone. Step-by-Step Solution:Let total work = 1 unit.A completes 75% = 3 / 4 of the job in 25 days.So A's rate = (3 / 4) / 25 = 3 / 100 work per day.Remaining work = 1 - 3 / 4 = 1 / 4.A and B together complete 1 / 4 of the job in 5 days.Combined rate of A and B = (1 / 4) / 5 = 1 / 20 work per day.So B's rate = (A + B rate) - A's rate = 1 / 20 - 3 / 100.Compute 1 / 20 = 5 / 100, so B's rate = 5 / 100 - 3 / 100 = 2 / 100 = 1 / 50 work per day.Therefore, time taken by B alone to do the entire job = 1 / (1 / 50) = 50 days. Verification / Alternative check:Check consistency: In 25 days, A does 25 * 3 / 100 = 75 / 100 = 3 / 4 of the work. There is 1 / 4 left. Together, A and B work for 5 days. A contributes 5 * 3 / 100 = 15 / 100, and B contributes 5 * 1 / 50 = 5 / 50 = 10 / 100. Total from this phase is 25 / 100 = 1 / 4. Summing 3 / 4 + 1 / 4 = 1, so the work is exactly completed and the rates are consistent. Why Other Options Are Wrong:Twenty four or 37.5 days would imply that B is significantly faster than the rate implied by the combined phase and would not maintain the correct division of work. Eighty days suggests B is much slower than A, which does not fit with the observed completion when they work together. Only 50 days matches the rate difference and the timing of the remaining work. Common Pitfalls:It is easy to confuse the fraction of work completed by A alone with the fraction completed jointly. Some learners incorrectly assume that A and B share the remaining work equally in the second phase. Others forget to convert percentages into fractions. Always express all portions of work as fractions of the whole and proceed step by step when combining rates. Final Answer:Worker B alone would complete the entire job in 50 days.

Question 10

Workers A and B together can finish a job in 24 days. Workers A, B and C together can finish the same job in 8 days. In how many days will worker C alone finish the job working at a constant rate?

Accepted Answer

Introduction / Context:This is a standard type of time and work question where various groups of workers complete a job in different times. We are given the time taken by A and B together, and by A, B and C together. The goal is to determine how long C alone would take to finish the same job, which we find by subtracting known combined rates. Given Data / Assumptions: A and B together finish the job in 24 days. A, B and C together finish the job in 8 days. All workers have constant individual rates of work. The job size is fixed and taken as one unit. Concept / Approach:We convert completion times to daily work rates. The rate of A and B together gives us one combined rate, and the rate of A, B and C together gives a larger combined rate. Subtracting the first from the second yields C's solo rate. Finally, we invert C's rate to obtain the number of days C alone needs to completed the job. Step-by-Step Solution:Let total work = 1 unit.A + B together finish in 24 days, so their combined rate rAB = 1 / 24 per day.A + B + C together finish in 8 days, so rABC = 1 / 8 per day.C's rate rC = rABC - rAB = 1 / 8 - 1 / 24.Compute: 1 / 8 = 3 / 24, so rC = 3 / 24 - 1 / 24 = 2 / 24 = 1 / 12 per day.Time taken by C alone = 1 / (1 / 12) = 12 days. Verification / Alternative check:We can check by recomputing group times from rates. If C's rate is 1 / 12, then A + B rate is 1 / 24. Combined A + B + C rate is 1 / 24 + 1 / 12 = 1 / 24 + 2 / 24 = 3 / 24 = 1 / 8, which matches the given 8 days for all three together. The relationships are consistent, confirming that 12 days is correct for C alone. Why Other Options Are Wrong:Fourteen, sixteen, and twenty four days correspond to smaller daily rates than 1 / 12. If C needed any of those longer times alone, then the combined rate A + B + C would be smaller than 1 / 8 and would not achieve completion in 8 days. Twelve days is the only duration that yields the correct rate differences. Common Pitfalls:Some learners incorrectly average the times instead of working with rates. Others may forget that rates add linearly while times do not. The correct process is always to convert times to rates, add or subtract the rates as appropriate, and then convert back to time at the end. Final Answer:Worker C alone will finish the job in 12 days.

Question 11

Twelve men can complete a certain work in 9 days. After they have worked together for 6 days, 6 more men join the group. How many additional days are required to complete the remaining work after the new men join?

Accepted Answer

Introduction / Context:This time and work question presents a situation where additional workers join partway through a job. We must calculate how much work remains at that point and then determine how quickly the larger team can finish it. The focus is on correctly handling total work and work done in different phases of the project. Given Data / Assumptions: Twelve men can complete the entire work in 9 days. These 12 men work together for the first 6 days. Then 6 more men join, giving a total of 18 men. All men have equal and constant efficiency. We need the number of extra days needed after the new men join. Concept / Approach:First, we compute the total amount of work in man days. Then we see how much work is done in the first 6 days by the initial 12 men. The difference between the total work and the completed portion is the remaining workload. Next, we compute the daily capacity of the larger group of 18 men and divide the remaining work by this daily capacity to find the extra days required. Step-by-Step Solution:Total work in man days = 12 men * 9 days = 108 man days.Work done in first 6 days by 12 men = 12 * 6 = 72 man days.Remaining work = 108 - 72 = 36 man days.After 6 days, 6 more men join, so total men = 12 + 6 = 18.Daily work capacity of 18 men = 18 man days per day.Time to finish remaining work = 36 man days / 18 man days per day = 2 days. Verification / Alternative check:We can also express the problem in terms of fractions of the job. Twelve men complete the job in 9 days, so the rate of 12 men is 1 / 9 job per day. In 6 days, they complete 6 * 1 / 9 = 6 / 9 = 2 / 3 of the job. Thus, 1 / 3 remains. Eighteen men are 1.5 times as many as 12 men, so their rate is 1.5 * 1 / 9 = 1 / 6 job per day. Time for 1 / 3 job at rate 1 / 6 is (1 / 3) / (1 / 6) = 2 days, confirming our answer. Why Other Options Are Wrong:Three, four, or five days would overestimate the time for the remaining 36 man days when 18 men are available. At 18 man days per day, even three days would represent 54 man days, which is more than the remaining work. Hence, only 2 days match both the man day and fractional analyses. Common Pitfalls:A common error is to forget that extra men join after 6 days, and to compute time based only on the original team. Another mistake is miscounting total man days or mixing up which portion of the work was completed before extra men joined. Writing down the full man day equation usually helps to avoid such confusion. Final Answer:The work will be completed in an additional 2 days after the six new men join.

Question 12

Worker A can do 75% of a job in 9 days, while worker B can complete 50% of the same job in 8 days. If A and B work together at their constant rates, how many days will they take to complete half of the job?

Accepted Answer

Introduction / Context:This problem involves two workers, A and B, whose rates are given in terms of fractions of a job completed in given times. We are asked how long they will take together to complete half the job. This requires translating each worker's partial completion information into daily rates and then combining those rates to find the required time for a specified fraction of the job. Given Data / Assumptions: A completes 75% (three quarters) of the job in 9 days. B completes 50% (one half) of the job in 8 days. We assume a fixed job size equal to one complete unit. A and B work simultaneously at their constant rates. We need the time taken to complete 50% of the job together. Concept / Approach:First, we find A's daily rate using the relation fraction of work divided by days. We do the same for B. The combined daily rate of A and B is the sum of their individual rates. Finally, we use that combined rate to calculate the time needed to accomplish 1/2 of the job by rearranging time = work / rate. Step-by-Step Solution:Take total work = 1 job.A does 75% = 3 / 4 of the job in 9 days, so A's rate = (3 / 4) / 9 = 3 / 36 = 1 / 12 per day.B does 50% = 1 / 2 of the job in 8 days, so B's rate = (1 / 2) / 8 = 1 / 16 per day.Combined rate of A and B = 1 / 12 + 1 / 16.Compute: 1 / 12 = 4 / 48, 1 / 16 = 3 / 48, so combined rate = 7 / 48 per day.We want time to complete 1 / 2 of the job at rate 7 / 48 per day.Time = (1 / 2) / (7 / 48) = (1 / 2) * (48 / 7) = 24 / 7 days. Verification / Alternative check:We can approximate 24 / 7 as about 3.4286 days. In that time, A would do 3.4286 * 1 / 12 ≈ 0.2857 of the job, and B would do 3.4286 * 1 / 16 ≈ 0.2143 of the job. Their combined contribution is about 0.2857 + 0.2143 ≈ 0.5, or exactly one half when calculated symbolically, confirming that 24 / 7 days is correct. Why Other Options Are Wrong:Forty over seven days would correspond to a much longer time and therefore more than half of the job. Seven over two and nine over two days both give smaller or larger contributions that do not total exactly 1 / 2 when multiplied by the combined rate of 7 / 48. Only 24 / 7 days provides the correct fraction of the job completed. Common Pitfalls:Some learners misinterpret 75% and 50% or convert them incorrectly into fractions. Others incorrectly average the times instead of working with rates. There can also be errors in adding the fractions 1 / 12 and 1 / 16 without using a common denominator. Careful attention to fraction operations is essential in such problems. Final Answer:A and B together will take 24/7 days to complete half of the job.

Question 13

Workers A, B and C together can finish a task in 7.5 days. Worker C is three times as productive as worker A, and worker B alone can finish the same task in 15 days. If B goes on leave, in how many days will workers A and C together complete the tas…

Accepted Answer

Introduction / Context:This question blends group work rates with a relationship between individual efficiencies. We know the time taken by all three workers together, we know C is three times as productive as A, and we know B's solo completion time. The objective is to deduce the individual rates, then find how long A and C together will take to complete the task once B is unavailable. Given Data / Assumptions: A, B and C together finish the task in 7.5 days. C is three times as productive as A. B alone finishes the task in 15 days. All workers have constant rates of work. We require the time for A and C together to finish the task. Concept / Approach:We convert all times into rates. B's rate is straightforward from the solo time. The combined rate of A, B and C is the reciprocal of 7.5 days. Using the relation C = 3A, we set up an equation for total rate in terms of A and B. By solving that equation we find A's rate, then C's rate from the productivity relation. Finally, we add A's and C's rates to get the combined rate and invert it to find the time required. Step-by-Step Solution:Let total work = 1 unit.Time for A, B and C together = 7.5 days = 15 / 2 days.So combined rate rABC = 1 / (15 / 2) = 2 / 15 per day.B alone finishes in 15 days, so B's rate rB = 1 / 15 per day.Let A's rate be a, so C's rate is 3a, because C is three times as productive as A.Then rABC = a + rB + 3a = 4a + 1 / 15.Set this equal to 2 / 15: 4a + 1 / 15 = 2 / 15.So 4a = 2 / 15 - 1 / 15 = 1 / 15, which gives a = 1 / 60.Therefore, A's rate is 1 / 60 and C's rate is 3 * 1 / 60 = 1 / 20.Combined rate of A and C together = 1 / 60 + 1 / 20 = 1 / 60 + 3 / 60 = 4 / 60 = 1 / 15.Thus A and C together take 1 / (1 / 15) = 15 days to complete the task. Verification / Alternative check:Check consistency: If A's rate is 1 / 60 and C's is 1 / 20, their total with B is 1 / 60 + 1 / 15 + 1 / 20. Convert to a common denominator 60: 1 / 60 + 4 / 60 + 3 / 60 = 8 / 60 = 2 / 15, which matches the combined 7.5 day time. Thus the derived rates are correct, and the resulting A + C time of 15 days is consistent with all given information. Why Other Options Are Wrong:Ten or twenty days would imply different combined rates for A and C that would not be compatible with the known A + B + C rate and B's solo time. Thirty days is too long and would correspond to a combined rate of 1 / 30, far smaller than the rates implied by the conditions. Only 15 days matches the precise arithmetic and relationships between the workers. Common Pitfalls:A common mistake is to misread C is three times as productive as A and treat it as C taking three times as long instead of having three times the rate. Another common error is to incorrectly subtract B's rate from the total combined rate. Always work with rates, not times, when combining groups of workers, and convert all quantities to consistent fractions before solving. Final Answer:If B goes on leave, workers A and C together will complete the task in 15 days.

Question 14

Workers A, B and C can complete a job alone in 9 days, 12 days and 36 days respectively. They all start working together, but A leaves after 3 days. For how many additional days must B and C work together to finish the remaining portion of the job?

Accepted Answer

Introduction / Context:This time and work problem describes three workers with different individual completion times. They start a job together, but one worker leaves after a certain period. We must work out how much of the job remains and then determine how long the remaining two workers need to finish the job working at their combined rate. Given Data / Assumptions: A alone completes the job in 9 days. B alone completes the job in 12 days. C alone completes the job in 36 days. A, B and C start together. A leaves after 3 days of joint work. B and C continue to work together thereafter at constant rates. Concept / Approach:We convert each solo completion time into a daily work rate. Then we calculate how much of the job is completed by all three workers during the first 3 days. The remaining fraction of the job is then completed by B and C together. We find the combined rate of B and C and divide the remaining work by this rate to determine the number of additional days required. Step-by-Step Solution:Assume total work = 1 job.Rates: A = 1 / 9 per day, B = 1 / 12 per day, C = 1 / 36 per day.Combined rate of A, B and C = 1 / 9 + 1 / 12 + 1 / 36.Convert to denominator 36: 1 / 9 = 4 / 36, 1 / 12 = 3 / 36, 1 / 36 = 1 / 36, so total = 8 / 36 = 2 / 9 per day.Work done in first 3 days by all three = 3 * 2 / 9 = 6 / 9 = 2 / 3.Remaining work = 1 - 2 / 3 = 1 / 3.After A leaves, only B and C work. Their combined rate = 1 / 12 + 1 / 36.Compute: 1 / 12 = 3 / 36, 1 / 36 = 1 / 36, so B + C rate = 4 / 36 = 1 / 9 per day.Time taken by B and C to finish remaining 1 / 3 = (1 / 3) / (1 / 9) = 3 days. Verification / Alternative check:Check the total work: first 3 days at 2 / 9 per day give 2 / 3 of the job. Next 3 days at 1 / 9 per day from B and C give 3 * 1 / 9 = 1 / 3. Adding these fractions, 2 / 3 + 1 / 3 = 1, which shows the job is exactly completed and confirms that B and C need 3 additional days after A leaves. Why Other Options Are Wrong:If B and C worked for 4, 5 or 6 extra days, they would complete more than the remaining 1 / 3 of the work, which is impossible under a constant positive rate. Each extra day adds another 1 / 9 of the job, so any time beyond 3 days overshoots the total work. Therefore, those options are inconsistent with the required amount of remaining work. Common Pitfalls:Some learners mistakenly think that the total time includes the first 3 days and then choose 6 instead of 3. However, the question specifically asks for the days after A leaves. Others miscalculate the combined rates by adding times instead of inverse times. Always remember that rates add, not raw times, when combining workers. Final Answer:B and C together will take an additional 3 days to complete the remaining portion of the job after A leaves.

Question 15

Six engines of a certain (former) type consume 24 metric tonnes of coal when each engine works for 8 hours per day. If two engines of this former type consume as much coal as three engines of a latter type, how many metric tonnes of coal will be req…

Accepted Answer

Introduction / Context:This question is a variation of a time and work type problem but with machines and fuel consumption instead of human workers and jobs. We are given how much coal a group of engines of one type consumes in a certain number of hours, and a relationship between the consumption of engines of two different types. The task is to determine the coal needed for a different number of engines of the second type working for a different number of hours. Given Data / Assumptions: Six engines of the former type consume 24 metric tonnes of coal when each works for 8 hours. Two engines of the former type consume as much as three engines of the latter type for the same time. We want coal required for 9 engines of the latter type, each running 13 hours. Consumption is directly proportional to the number of engines and to the hours they work. All engines of a given type have identical consumption rates. Concept / Approach:First, we find the coal consumption per engine per hour for the former type. Then we use the equivalence condition two former engines equal three latter engines to obtain the per engine per hour consumption for the latter type. Once we know that, we multiply by the number of latter type engines and the hours they run to find the total coal consumption required. Step-by-Step Solution:Total coal consumed by six former type engines in 8 hours = 24 metric tonnes.Total engine hours for former type = 6 engines * 8 hours = 48 engine hours.Consumption per engine hour for former type = 24 / 48 = 0.5 metric tonnes per engine hour.We are told that 2 former type engines consume as much coal as 3 latter type engines for the same time.Let cF be former per engine hour, cL be latter per engine hour. Then 2 * cF = 3 * cL.Substitute cF = 0.5: 2 * 0.5 = 3 * cL, so 1 = 3 * cL.Therefore, cL = 1 / 3 metric tonne per engine hour for the latter type.Now compute usage for 9 latter type engines running 13 hours.Total engine hours for latter type = 9 * 13 = 117 engine hours.Total coal required = 117 * (1 / 3) = 39 metric tonnes. Verification / Alternative check:We can cross check the equivalence condition. If cL = 1 / 3, then three latter engines in 1 hour use 3 * 1 / 3 = 1 metric tonne. Two former engines in 1 hour use 2 * 0.5 = 1 metric tonne. This exactly matches the given relationship. Thus the derived latter type rate is consistent, and multiplying by 117 engine hours to get 39 metric tonnes is correct. Why Other Options Are Wrong:Values like 45, 34 or 55 metric tonnes are not proportional to the correct per engine hour consumption and total engine hours. For example, 45 tonnes would correspond to a larger per engine hour rate than 1 / 3 and would violate the given equivalence between the two types of engines. Thirty four tonnes is too small and would imply a latter type engine is more efficient than implied by the relation. Only 39 metric tonnes satisfies all the constraints. Common Pitfalls:Some learners treat the 24 metric tonnes as a daily figure without computing per engine hour consumption. Others misinterpret the equivalence condition and set 2 former engines equal to 3 latter engines with inverse ratios. It is essential to first reduce everything to per engine hour consumption and then carefully apply proportional reasoning to the new situation. Final Answer:The coal required for 9 engines of the latter type running for 13 hours is 39 metric tonnes.

Question 16

A can complete a piece of work in 21 days and B can complete the same work in 28 days. They start working together, but B leaves after 4 days. In how many additional days can A alone finish the remaining work?

Accepted Answer

Introduction / Context: In this aptitude question on time and work, we are given the individual times taken by A and B to complete a job and are asked to determine how long A alone will take to finish the remaining work after both have worked together for some time. Such questions test understanding of work rates and the idea that work done is the product of rate and time. Given Data / Assumptions: - A can complete the entire work in 21 days. - B can complete the same work in 28 days. - A and B work together for the first 4 days. - After 4 days, B stops working and only A continues. - We assume they work at constant rates throughout. Concept / Approach: The key concept is that work rate is the reciprocal of time taken to finish one whole job. If someone finishes a job in T days, then the daily work rate is 1 / T. Total work is treated as 1 unit. When two people work together, their combined rate is the sum of their individual rates. First we find the work completed in the initial 4 days, then subtract from 1 to get the remaining work. Finally, we divide the remaining work by A's rate to find how many more days A will need to finish the job alone. Step-by-Step Solution: Step 1: Rate of A = 1 / 21 (job per day). Step 2: Rate of B = 1 / 28 (job per day). Step 3: Combined rate of A and B = 1 / 21 + 1 / 28. Step 4: Find a common denominator: 1 / 21 = 4 / 84 and 1 / 28 = 3 / 84, so combined rate = 7 / 84 = 1 / 12 job per day. Step 5: Work done in 4 days together = 4 * (1 / 12) = 4 / 12 = 1 / 3 of the total job. Step 6: Remaining work after 4 days = 1 - 1 / 3 = 2 / 3 of the job. Step 7: Now only A works, with rate 1 / 21 job per day. Step 8: Time taken by A to finish the remaining 2 / 3 job = (2 / 3) / (1 / 21) = (2 / 3) * 21 = 14 days. Verification / Alternative check: We can verify by computing the total work done explicitly. In 4 days together, A and B finish 1 / 3 of the work. A then works alone for 14 more days at a rate of 1 / 21 job per day, which yields 14 * (1 / 21) = 2 / 3 of the job. Adding 1 / 3 and 2 / 3 gives 1 full job. Thus, the answer 14 days is consistent and correct. Why Other Options Are Wrong: 18 days: This would yield more than the remaining 2 / 3 of the work and is inconsistent with the rates. 21 days: This would suggest A is doing more than the remaining work and ignores the initial contribution by B. 16 days: This also overshoots the exact required time; 16 * (1 / 21) = 16 / 21, which does not match 2 / 3. 12 days: This would give only 12 / 21 = 4 / 7 of the work, less than 2 / 3, so the job would not be completed. Common Pitfalls: A common mistake is to subtract days directly instead of working with rates and fractions of work. Some students also forget to account for the work already done in the first 4 days, or they may add times instead of rates. Always remember that when people work together, you add their rates, not the days. Only after finding the remaining fraction of work should you divide by the appropriate rate to get the remaining time. Final Answer: A alone will take 14 additional days to complete the remaining work.

Question 17

Ten boys can complete a piece of work in 7 days and ten girls can complete the same work in 14 days. If 5 boys and 10 girls work together, in how many days will they finish the entire work?

Accepted Answer

Introduction / Context: This time and work question compares the efficiencies of boys and girls working on the same job. We are given the time taken by groups of boys and girls to complete the work separately and must find the time when a mixed group works together. The problem tests understanding of individual work rates and how to combine them correctly. Given Data / Assumptions: - Ten boys can complete the full work in 7 days. - Ten girls can complete the same work in 14 days. - We want the time taken when 5 boys and 10 girls work together. - Work rate is constant for each boy and each girl. - Total work is treated as one complete unit. Concept / Approach: If a group finishes a job in T days, its daily work rate is 1 / T of the job. From the given group rates, we can find the rate of a single boy and a single girl. Then we calculate the combined rate of 5 boys and 10 girls. Finally, we find the total time by taking the reciprocal of that combined rate. This is a direct application of work rate and proportionality concepts commonly used in aptitude exams. Step-by-Step Solution: Step 1: Daily rate of 10 boys = 1 / 7 job per day. Step 2: Daily rate of 1 boy = (1 / 7) / 10 = 1 / 70 job per day. Step 3: Daily rate of 10 girls = 1 / 14 job per day. Step 4: Daily rate of 1 girl = (1 / 14) / 10 = 1 / 140 job per day. Step 5: Rate of 5 boys = 5 * (1 / 70) = 5 / 70 = 1 / 14 job per day. Step 6: Rate of 10 girls = 10 * (1 / 140) = 10 / 140 = 1 / 14 job per day. Step 7: Combined rate of 5 boys and 10 girls = 1 / 14 + 1 / 14 = 2 / 14 = 1 / 7 job per day. Step 8: Time to finish one full job = 1 / (1 / 7) = 7 days. Verification / Alternative check: Notice that 5 boys do the same work per day as 10 girls, since both groups have rate 1 / 14 job per day. So, 5 boys and 10 girls together are equivalent to two groups of 1 / 14 job per day, giving 1 / 7 job per day, exactly matching what 10 boys alone would do. Since 10 boys finish the job in 7 days, the mixed group must also take 7 days, confirming our result. Why Other Options Are Wrong: 13 days and 10 days: These are larger than 7 days and would imply a lower combined work rate than ten boys alone, which is illogical because we are using both boys and girls together. 5 days and 8 days: Five days is too short for this rate, and eight days does not match the calculated 1 / 7 job per day. Only 7 days is consistent with the combined rate. Common Pitfalls: A common mistake is to simply average the days (7 and 14) without considering rates, which gives a wrong value. Another error is to assume that 5 boys and 10 girls will finish in half the time of 10 boys, which is not correct unless the total rate doubles. Always convert times into rates first, add or compare rates, and then invert the final rate to get time. Final Answer: 5 boys and 10 girls together will complete the work in 7 days.

Question 18

Fifty men can complete a piece of work in 28 days. They all start together, but at the end of every 10th day, 10 men leave the job. In how many days will the entire work be completed?

Accepted Answer

Introduction / Context: This time and work problem involves changing manpower over time. The total number of men working on a job decreases periodically, so the overall speed of completing the work changes in stages. The question tests the ability to calculate work done in successive intervals with different workforce sizes and then to determine the total time taken to complete the job. Given Data / Assumptions: - Fifty men can complete the entire work in 28 days. - All 50 men start the work together. - At the end of every 10th day, 10 men leave the job. - Work rate of each man is constant over time. - We treat the total work as one complete unit. Concept / Approach: First we find the total amount of work in equivalent “man-days”. Total work = number of men * number of days when they alone could finish the job. Then we consider the work done in each 10 day block with a different number of men. At each stage, we subtract the work completed so far from the total work. Once the cumulative work equals the total required, we know the total number of days taken. This approach uses the idea that work = rate * time and that rates add linearly when multiple workers contribute. Step-by-Step Solution: Step 1: Total work in man-days = 50 men * 28 days = 1400 man-days. Step 2: Days 1 to 10: 50 men work for 10 days, work done = 50 * 10 = 500 man-days. Remaining = 1400 - 500 = 900 man-days. Step 3: Days 11 to 20: 10 men leave, so 40 men work for the next 10 days. Work done = 40 * 10 = 400 man-days. Remaining = 900 - 400 = 500 man-days. Step 4: Days 21 to 30: Again 10 men leave, so 30 men work for the next 10 days. Work done = 30 * 10 = 300 man-days. Remaining = 500 - 300 = 200 man-days. Step 5: Days 31 onward: Another 10 men leave, leaving 20 men to finish the remaining work. Step 6: Time needed for 20 men to finish 200 man-days of work = 200 / 20 = 10 days. Step 7: Total time = 10 + 10 + 10 + 10 = 40 days. Verification / Alternative check: We can check by adding all work segments: 500 + 400 + 300 + 200 = 1400 man-days, exactly equal to the total work required. The corresponding durations are 10, 10, 10 and 10 days, which add up to 40 days. Since all work is accounted for and there is no remaining work after 40 days, the answer is consistent. Why Other Options Are Wrong: 36 days or 38 days: These are too small; the total man-days contributed in those times would be less than 1400, so the work would remain incomplete. 45 days or 42 days: These are larger than necessary. By 40 days the entire work is already finished, so any extra days are not required. Hence these values do not match the calculated schedule. Common Pitfalls: Students sometimes assume that men leave every 10 working days from the start without carefully calculating the work done in each segment. Another common mistake is to average the total number of men over the entire duration, which does not correctly handle piecewise changes. Always convert to total work in man-days and then subtract work done interval by interval when workforce changes occur. Final Answer: The work will be completed in 40 days.

Question 19

Robin can complete a piece of work alone in 9 days, while Amit can complete the same work alone in 6 days. If they finish the work together and receive a total payment of Rs 2500, what is Robin's share (in rupees)?

Accepted Answer

Introduction / Context: This is a standard time and work plus wages distribution problem. Two workers, Robin and Amit, complete a job together and earn a total amount. Since payment is usually proportional to the amount of work each person does, we must compare their work rates and then split the total payment accordingly. The question specifically asks for Robin's share of the payment. Given Data / Assumptions: - Robin can complete the job alone in 9 days. - Amit can complete the same job alone in 6 days. - They work together to finish the entire job. - The total payment for the completed work is Rs 2500. - Wages are distributed in proportion to the work done by each person. Concept / Approach: If someone can complete a job in T days, their work rate is 1 / T of the job per day. When two people work together, the fraction of work each contributes is proportional to their respective rates. The ratio of their rates gives the ratio of their shares in the total payment. Therefore, we do not need the total time taken by them together; we only need the ratio of their individual rates to divide Rs 2500 between them correctly. Step-by-Step Solution: Step 1: Work rate of Robin = 1 / 9 job per day. Step 2: Work rate of Amit = 1 / 6 job per day. Step 3: Ratio of their rates = (1 / 9) : (1 / 6). Step 4: Simplify the ratio: (1 / 9) : (1 / 6) = 1 / 9 * 6 : 1 / 6 * 6 = 2 : 3. Step 5: Therefore, their payment share ratio is also Robin : Amit = 2 : 3. Step 6: Total shares = 2 + 3 = 5 parts. Step 7: Value of one part = 2500 / 5 = Rs 500. Step 8: Robin's share = 2 * 500 = Rs 1000. Verification / Alternative check: We can confirm that Amit's share would be 3 * 500 = Rs 1500. The sum 1000 + 1500 = 2500 matches the total payment, which verifies that the ratio and distribution are correct. Since the ratio of rates is logically consistent with their individual times (the one who works faster, Amit, gets the larger share), the result makes sense. Why Other Options Are Wrong: Rs 1700 or Rs 1200: These amounts do not follow the 2 : 3 ratio and would not sum to Rs 2500 when combined with Amit's share if calculated from the same ratio. Rs 1500: This would incorrectly assign the larger share to Robin, even though Amit works faster and should get a bigger share of the payment. Rs 900: This does not fit the 2 : 3 ratio and does not lead to a correct total when paired with Amit's corresponding share. Common Pitfalls: Students sometimes try to use the number of days directly instead of the reciprocal rates, mistakenly using 9 : 6 instead of 1 / 9 : 1 / 6. Another common error is to assume equal sharing of money without considering different efficiencies. Always convert times to rates and then form the correct ratio to divide wages fairly. Final Answer: Robin's share of the payment is Rs 1000.

Question 20

A does half as much work as B, and C does half as much work as A and B together. If C alone can finish the entire work in 40 days, in how many days will A, B and C together finish the work?

Accepted Answer

Introduction / Context: This time and work question involves comparative efficiencies. The problem describes how much work A does compared to B, and how much work C does compared to A and B together. We are also told how long C alone takes to complete the work. Using these relationships, we must determine how much time the three of them will take to finish the work together. This tests understanding of relative work rates and algebraic manipulation. Given Data / Assumptions: - A does half as much work as B. - C does half as much work as A and B together. - C alone can finish the work in 40 days. - Work rates remain constant throughout. - Total work is assumed to be one complete unit. Concept / Approach: We use variables to represent work rates. Let the daily work rate of B be b units. Then A's rate is b / 2. The combined rate of A and B is 3b / 2. Since C does half as much as A and B together, C's rate is (1 / 2) * (3b / 2) = 3b / 4. Using the information that C alone completes the work in 40 days, we find b and thus obtain the individual rates of A, B and C. Finally, we add these rates to get the combined rate of all three working together and invert it to find the time needed. Step-by-Step Solution: Step 1: Let B's work rate = b jobs per day. Step 2: A does half as much as B, so A's rate = b / 2 jobs per day. Step 3: A and B together have rate = b / 2 + b = 3b / 2 jobs per day. Step 4: C does half as much as A and B together, so C's rate = (1 / 2) * (3b / 2) = 3b / 4 jobs per day. Step 5: C alone takes 40 days to complete one job, so C's rate is 1 / 40 job per day. Step 6: Set 3b / 4 = 1 / 40, leading to b = (1 / 40) * (4 / 3) = 1 / 30 job per day. Step 7: Therefore, B's rate = 1 / 30, A's rate = 1 / 60, and C's rate = 1 / 40. Step 8: Combined rate of A, B and C = 1 / 60 + 1 / 30 + 1 / 40. Step 9: Use common denominator 120: 1 / 60 = 2 / 120, 1 / 30 = 4 / 120, 1 / 40 = 3 / 120, so total rate = 9 / 120 = 3 / 40 job per day. Step 10: Time to complete whole work together = 1 / (3 / 40) = 40 / 3 days. Verification / Alternative check: We can verify by checking the time C alone takes with the calculated rate: C's rate from our algebra is 1 / 40 job per day, which indeed corresponds to 40 days to finish the job, matching the given condition. The combined rate 3 / 40 job per day is greater than each individual rate, which makes sense because three people working together should be faster than any one of them alone. Thus, 40 / 3 days is a reasonable and verified answer. Why Other Options Are Wrong: 30 days or 16 days: These times are too small compared with C's 40 days alone and do not match the combined rate of 3 / 40 job per day. 15 days: This would correspond to a combined rate of 1 / 15 job per day, which is larger than the rate we computed using the given relationships and so does not fit the data. 20 days: This is still not equal to 40 / 3 and also contradicts the derived combined rate. Common Pitfalls: Many students misread “half as much work” and confuse it with “twice as much work”. Another frequent mistake is to treat the times directly instead of the rates, which leads to incorrect ratios. It is very important to translate verbal descriptions of efficiency into algebraic relations between rates, solve for the unknowns, and then sum the rates when workers collaborate on the same task. Final Answer: A, B and C together will complete the work in 40/3 days.

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