Question 1

S, T, and U can complete a piece of work in 40 days, 48 days, and 60 days respectively. They start the work together and receive Rs 10800 in total for completing it. T leaves 2 days before completion and U leaves 5 days before completion, so S alone…

Accepted Answer

Introduction / Context: This is a work and wages problem that involves three workers S, T, and U with different efficiencies. They all start the job together, but two of them leave before the work is completed, and the remaining person finishes the job alone. The total payment is given and must be divided among them in proportion to the work each person actually completed. This reflects real life situations where payment is linked to contribution. Given Data / Assumptions: S can complete the work alone in 40 days. T can complete the work alone in 48 days. U can complete the work alone in 60 days. All three start the job together. T leaves 2 days before the work is completed. U leaves 5 days before the work is completed. The total payment for the completed work is Rs 10800. Payment is proportional to the amount of work done by each person. Concept / Approach: First, we treat the total work as 1 job and compute the daily work rates of S, T, and U. Next, we let the total duration of the project be D days and use the fact that each worker contributes their rate multiplied by the number of days they actually work. Setting the sum of these contributions equal to 1 job allows us to solve for D. After that, we calculate how much work S performs and use the ratio of S work to total work to compute S share of the total payment. Step-by-Step Solution: Step 1: Let total work be 1 job. Step 2: Rate of S = 1/40 job per day. Step 3: Rate of T = 1/48 job per day. Step 4: Rate of U = 1/60 job per day. Step 5: Let the total duration of the project be D days. Step 6: S works for all D days, T works for (D - 2) days, and U works for (D - 5) days. Step 7: Total work equation: (D/40) + ((D - 2)/48) + ((D - 5)/60) = 1. Step 8: Solving this equation gives D = 18 days. Step 9: Work done by S = D/40 = 18/40 = 9/20 of the job. Step 10: Since total work is 1, S share of payment is (9/20) of Rs 10800. Step 11: S share = (9/20) * 10800 = 9 * 540 = Rs 4860. Verification / Alternative check: We can check the work proportions. T works (18 - 2) = 16 days at 1/48 job per day, so T contribution is 16/48 = 1/3 job. U works (18 - 5) = 13 days at 1/60 job per day, so U contribution is 13/60 job. Adding the contributions: S has 9/20 job, T has 1/3 job, and U has 13/60 job. Converting to a common denominator of 60 gives S = 27/60, T = 20/60, and U = 13/60. The sum is (27 + 20 + 13)/60 = 60/60 = 1 job, confirming the calculations and validating S share. Why Other Options Are Wrong: Rs 4320, Rs 4500, and Rs 4960 correspond to different and incorrect fractions of the total work; they do not match the actual 9/20 part that S performs. Rs 5400 would mean S does exactly half the job, which is not supported by the computed contributions. Only Rs 4860 correctly reflects S share based on his actual portion of the work done. Common Pitfalls: Students may assume that all workers share the money equally, ignoring differences in efficiency and time spent. Another error is miscalculating the total time D or incorrectly solving the fractional equation. Care must be taken when handling multiple time periods and different leaving times for the workers. Always compute each person contribution explicitly and then divide the total payment proportionally. Final Answer: The share of S from the total payment is Rs 4860.

Question 2

Ram and Hari together can cut 12 kilograms of nuts in 2 days. They work together for 5 days, after which Hari leaves and Ram alone takes 8 more days to cut the remaining nuts. If the total quantity of nuts cut is 58 kilograms, how much time (in days…

Accepted Answer

Introduction / Context: This question is a time and work scenario framed in terms of cutting nuts. Two workers, Ram and Hari, work together for some time and then one of them continues alone. We are given the total output and must deduce individual rates to then find how long Hari alone would take to cut a specific quantity of nuts. Such problems connect real life situations of shared work and are common in aptitude tests. Given Data / Assumptions: Ram and Hari together cut 12 kilograms of nuts in 2 days. They work together for 5 days initially. After 5 days, Hari leaves, and Ram alone works for 8 more days. The total quantity of nuts cut is 58 kilograms. We must find how long Hari alone would take to cut 10 kilograms of nuts. Both Ram and Hari work at constant daily rates. Concept / Approach: We treat the cutting of nuts as a work rate problem in kilograms per day. First we find the combined daily rate of Ram and Hari. Then, using the total production and additional days that Ram works alone, we calculate Ram individual rate. Subtracting Ram rate from the combined rate gives Hari rate. Finally, we use Hari rate to find the time required for him to cut 10 kilograms alone. This is a direct application of rate equations. Step-by-Step Solution: Step 1: Combined production of Ram and Hari is 12 kilograms in 2 days. Step 2: Combined daily rate of Ram and Hari = 12 / 2 = 6 kilograms per day. Step 3: For the first 5 days, they work together and produce 5 * 6 = 30 kilograms. Step 4: Total nuts cut is 58 kilograms, so remaining after 5 days = 58 - 30 = 28 kilograms. Step 5: Ram alone cuts these 28 kilograms in 8 days. Step 6: Ram daily rate = 28 / 8 = 3.5 kilograms per day. Step 7: Combined daily rate of Ram and Hari is 6 kilograms per day, so Hari daily rate = 6 - 3.5 = 2.5 kilograms per day. Step 8: Time taken by Hari alone to cut 10 kilograms = 10 / 2.5 = 4 days. Verification / Alternative check: We can verify by reconstructing the entire story using the found rates. On each day of the first 5 days, Ram cuts 3.5 kilograms and Hari cuts 2.5 kilograms, so together they cut 6 kilograms. In 5 days, they cut 30 kilograms. Then, Ram alone working 8 days at 3.5 kilograms per day cuts 28 kilograms. Thus, total nuts cut = 30 + 28 = 58 kilograms, which matches the given total. This confirms that the calculated rates and the final answer are correct. Why Other Options Are Wrong: If Hari took only 1, 2, or 3 days to cut 10 kilograms, his rate would be 10, 5, or 3.33 kilograms per day respectively, which are inconsistent with the combined rate and Ram rate. If he took 5 days, his rate would be only 2 kilograms per day, which would not yield a combined rate of 6 kilograms per day when added to Ram rate of 3.5 kilograms per day. Only 4 days gives Hari rate of 2.5 kilograms per day that fits all given conditions. Common Pitfalls: Some learners may assume that both workers are equally efficient and simply divide totals by two, which is not supported by the data. Others might forget to remove the contribution of the first phase when two people worked together and try to directly distribute the total among the individuals. The correct method is to find combined rate, then individual rate from the second phase, and finally deduce the other worker rate. Final Answer: Hari alone would take 4 days to cut 10 kilograms of nuts.

Question 3

Pannalal completes one-third of a job in 20 days. Saiprasad completes the remaining two-thirds of the same job in 10 days. In how many days can Pannalal and Saiprasad together complete the entire job if they work simultaneously from the start?

Accepted Answer

Introduction / Context: This problem involves two workers, Pannalal and Saiprasad, who complete different fractions of the same job in different times when working separately. The task is to compute how long they will take to complete the entire job if they work together from the beginning. This is a standard time and work problem that checks the understanding of fractional work and combined work rates. Given Data / Assumptions: Pannalal finishes 1/3 of the job in 20 days. Saiprasad finishes the remaining 2/3 of the job in 10 days. We must find the time taken if both work together from the start on the whole job. Total work is taken as 1 complete job. Both workers have constant work rates. Concept / Approach: We first convert the information about fractions of work completed into daily work rates. From the fact that Pannalal completes 1/3 of the job in 20 days, we can find his daily rate. Similarly, from Saiprasad completing 2/3 of the job in 10 days, we find his daily rate. Adding both rates gives the combined rate when they work together. Finally, time is computed as total work divided by combined rate. This direct approach is efficient and accurate. Step-by-Step Solution: Step 1: Let the total work be 1 job. Step 2: Pannalal completes 1/3 of the job in 20 days. Step 3: Therefore, Pannalal daily rate = (1/3) / 20 = 1/60 job per day. Step 4: Saiprasad completes 2/3 of the job in 10 days. Step 5: Saiprasad daily rate = (2/3) / 10 = 2/30 = 1/15 job per day. Step 6: Combined daily rate of Pannalal and Saiprasad = 1/60 + 1/15. Step 7: LCM of 60 and 15 is 60; 1/15 = 4/60. Step 8: Combined rate = (1/60 + 4/60) = 5/60 = 1/12 job per day. Step 9: Time taken to complete 1 job together = 1 / (1/12) = 12 days. Verification / Alternative check: To verify, in 12 days Pannalal would do 12 * (1/60) = 12/60 = 1/5 of the job. Saiprasad would do 12 * (1/15) = 12/15 = 4/5 of the job. Adding these gives 1/5 + 4/5 = 1 full job. This confirms that the combined time of 12 days is correct and consistent with their individual rates. Why Other Options Are Wrong: If they took 6 or 3 days, the total work done would be too small given their rates. If they took 24 or 15 days, the total work done would exceed one full job. Only 12 days results in a total of exactly one complete job when both work simultaneously. Common Pitfalls: A frequent mistake is to average the times 20 and 10 instead of converting to rates based on fractions of work. Another error is forgetting that Pannalal completed only one-third of the job in 20 days, not the entire job, which would incorrectly suggest a different daily rate. Always pay close attention to the fraction of work completed before calculating the rate. Final Answer: Pannalal and Saiprasad together will complete the entire job in 12 days.

Question 4

P and Q can complete a project in 6 days and 12 days respectively when working alone. If P and Q work together, in how many days will they complete 25% of the entire project?

Accepted Answer

Introduction / Context: This is another time and work question focused on partial completion of a project. Two workers P and Q can finish a project individually in different times. We are asked how long they will take to complete only 25 percent of the work when they collaborate. This type of problem frequently appears in quantitative aptitude sections and helps reinforce rate and fraction concepts. Given Data / Assumptions: P alone can complete the project in 6 days. Q alone can complete the project in 12 days. P and Q now work together on the same project. We must find how many days they need to complete 25% of the project. Total project work is considered as 1 full job. Concept / Approach: If a worker completes a job in N days, the worker daily rate is 1/N of the job per day. When two workers collaborate, their rates add. Once we determine the combined rate, we multiply by time to find the fraction of work completed. Here, we invert that idea to find the time required to complete a given fraction, namely 1/4 of the job. Therefore, time = required work / combined rate. Step-by-Step Solution: Step 1: Assume total work is 1 job. Step 2: P rate = 1/6 job per day. Step 3: Q rate = 1/12 job per day. Step 4: Combined rate of P and Q = 1/6 + 1/12. Step 5: LCM of 6 and 12 is 12, so 1/6 = 2/12 and 1/12 = 1/12. Step 6: Combined rate = (2/12 + 1/12) = 3/12 = 1/4 job per day. Step 7: Required fraction of the project = 25% = 1/4 of the job. Step 8: Time taken to complete 1/4 of the job at a rate of 1/4 job per day = (1/4) / (1/4) = 1 day. Verification / Alternative check: We can verify by checking how much work is done in exactly 1 day. At 1/4 job per day, P and Q together complete 1/4 of the project in 1 day, which is exactly 25 percent. This matches the requirement and confirms that the time needed is indeed 1 day. Why Other Options Are Wrong: If 2 days were taken, they would complete 2 * (1/4) = 1/2 of the project, which is too much. Similarly, 3 days, 4 days, or 8 days would lead to even more work, well beyond the required 25 percent. Only 1 day produces exactly one quarter of the job and therefore is the correct choice. Common Pitfalls: A common mistake is to compute the time needed for full completion and then simply divide by four, which in this case still gives 1.5 days, a different value. That approach is incorrect unless the combined rate is very carefully used. Another pitfall is mishandling percentages and fractions, so always convert 25 percent to 1/4 before proceeding with calculations. Final Answer: P and Q together will complete 25% of the project in 1 day.

Question 5

Ranjith can complete a task in 25 days while Anji can finish the same task in 20 days. They work together on the task for 5 days, after which Ranjith leaves. In how many additional days will Anji alone finish the remaining work?

Accepted Answer

Introduction / Context: This time and work problem considers two workers, Ranjith and Anji, who work together initially and then only one of them continues to complete the remaining work. We are asked to determine how long the second worker, Anji, will need to finish the task alone once the other leaves. This setup is typical in aptitude tests and helps build understanding of fractional completion and remaining work calculations. Given Data / Assumptions: Ranjith can complete the task alone in 25 days. Anji can complete the same task alone in 20 days. They work together for 5 days. Then Ranjith leaves, and Anji alone finishes the remainder. Total work is taken as 1 complete job. Work rates of both workers remain constant. Concept / Approach: We convert each worker time into a daily work rate, expressed as job per day. Then we compute how much work is completed during the initial period when both work together. Subtracting this from the total yields the remaining work. Finally, we divide the remaining work by Anji daily rate to find the number of additional days required for Anji to finish the task alone. This is a direct application of basic rate and time relationships. Step-by-Step Solution: Step 1: Let total work be 1 job. Step 2: Ranjith daily rate = 1/25 job per day. Step 3: Anji daily rate = 1/20 job per day. Step 4: Combined daily rate when both work together = 1/25 + 1/20. Step 5: LCM of 25 and 20 is 100, so 1/25 = 4/100 and 1/20 = 5/100. Step 6: Combined rate = (4/100 + 5/100) = 9/100 job per day. Step 7: Work done together in 5 days = 5 * (9/100) = 45/100 = 9/20 of the job. Step 8: Remaining work = 1 - 9/20 = 11/20 of the job. Step 9: Anji daily rate is 1/20, so time needed by Anji alone = (11/20) / (1/20) = 11 days. Verification / Alternative check: We can verify by checking total contributions. During the first 5 days, both together complete 9/20 of the task. If Anji then works for 11 more days at 1/20 job per day, she completes 11/20 of the task. Adding these gives 9/20 + 11/20 = 1 full job, confirming that the work is exactly completed with the calculated schedule. Thus, 11 days is the correct additional time for Anji alone. Why Other Options Are Wrong: If Anji took only 9 or 10 days, the total work done would add up to less than 1 job. If she took 12 or 15 days, the total work would exceed one full job. Therefore, these times would not match the exact completion of the task. Only 11 days gives a total of exactly one complete job when combined with the first 5 days of joint work. Common Pitfalls: A typical error is to treat the time for the joint work period as if it were done by one person alone, which leads to incorrect remaining work. Some learners also forget to use a common denominator while adding rates or computing fractional work. Proper handling of fractions and careful subtraction of completed work from the total are essential steps to avoid mistakes. Final Answer: Anji alone will take 11 more days to finish the remaining work after Ranjith leaves.

Question 6

Ganesh, Ram, and Sohan together can complete a piece of work in 16 days. Ganesh and Ram together can complete the same work in 24 days. How many days will Sohan alone take to finish the entire work?

Accepted Answer

Introduction / Context: This question asks us to find the time taken by one worker, Sohan, given the time taken by a group that includes him and the time taken by a subset of that group without him. Such problems are classic applications of time and work concepts and highlight how partial group times can be used to isolate the efficiency of one individual worker. Given Data / Assumptions: Ganesh, Ram, and Sohan together can complete the work in 16 days. Ganesh and Ram together can complete the work in 24 days. We must find how many days Sohan alone would take to complete the entire work. Total work is considered as 1 complete job. All workers have constant daily work rates. Concept / Approach: Let the daily work rates of Ganesh, Ram, and Sohan be G, R, and S respectively. From the given times, we can write equations for G + R + S and for G + R. Subtracting the second equation from the first isolates S, the daily rate of Sohan. Once we know S rate, we can obtain the time for Sohan alone by taking the reciprocal of that rate. This method is simple and very effective for combined work problems. Step-by-Step Solution: Step 1: Let total work be 1 job. Step 2: Since Ganesh, Ram, and Sohan together complete the job in 16 days, G + R + S = 1/16 job per day. Step 3: Since Ganesh and Ram together complete the job in 24 days, G + R = 1/24 job per day. Step 4: Subtract the second equation from the first to isolate Sohan rate. Step 5: S = (G + R + S) - (G + R) = 1/16 - 1/24. Step 6: Take LCM of 16 and 24, which is 48. Then 1/16 = 3/48 and 1/24 = 2/48. Step 7: S = (3/48 - 2/48) = 1/48 job per day. Step 8: Time taken by Sohan alone = 1 / (1/48) = 48 days. Verification / Alternative check: We can check consistency by reconstructing the combined rates. If Sohan rate is 1/48 and Ganesh plus Ram together have rate 1/24, then G + R + S = 1/24 + 1/48 = (2/48 + 1/48) = 3/48 = 1/16 job per day, which matches the given information. Thus, Sohan taking 48 days alone is consistent with both statements about the combined work times. Why Other Options Are Wrong: If Sohan took 32, 30, 40, or 24 days, his rate would be higher, leading to G + R + S being more than 1/16 job per day, which contradicts the given data. Only 48 days produces the correct combined rate and stays consistent with both given completion times for the two groups. Common Pitfalls: One mistake is to average the two time values instead of working with rates. Another is to miscalculate the subtraction of fractions, especially when not taking the right LCM. Always remember that adding or subtracting work times directly is incorrect; we must always add or subtract work rates instead. Final Answer: Sohan alone will take 48 days to complete the work.

Question 7

Nirmit can complete two-thirds of a job in 18 days. Kashish is twice as efficient as Nirmit. In how many days will Kashish alone complete the entire job?

Accepted Answer

Introduction / Context: This question compares the efficiencies of two workers, Nirmit and Kashish. We are told how long it takes Nirmit to complete a fraction of a job and that Kashish is twice as efficient as Nirmit. Based on this, we must determine how long Kashish would take to complete the full job alone. Questions of this type test the understanding of work rates and the effect of relative efficiency on completion time. Given Data / Assumptions: Nirmit completes two-thirds (2/3) of a job in 18 days. Kashish is twice as efficient as Nirmit. We must find the time Kashish alone needs to complete the entire job. Total work is taken as 1 full job. Efficiencies and work rates remain constant. Concept / Approach: First we calculate Nirmit daily work rate using the information about two-thirds of the job completed in 18 days. Once we know Nirmit rate, we find Kashish rate by doubling that value since Kashish is twice as efficient. To find how long Kashish alone takes to complete the entire job, we divide the total work, which is 1 job, by Kashish daily rate. This method is straightforward and relies on careful handling of fractions. Step-by-Step Solution: Step 1: Let total work be 1 job. Step 2: Nirmit completes 2/3 of the job in 18 days. Step 3: Nirmit daily rate = (2/3) / 18. Step 4: Simplify: (2/3) / 18 = (2/3) * (1/18) = 2 / 54 = 1/27 job per day. Step 5: Kashish is twice as efficient as Nirmit, so Kashish daily rate = 2 * (1/27) = 2/27 job per day. Step 6: Time taken by Kashish alone to complete 1 job = 1 / (2/27). Step 7: Simplify: 1 / (2/27) = 27/2 days. Step 8: Therefore, Kashish will complete the job in 27/2 days, which is 13.5 days. Verification / Alternative check: To check, in 13.5 days Kashish at rate 2/27 job per day completes 13.5 * (2/27) jobs. Converting 13.5 to a fraction gives 27/2. So work done = (27/2) * (2/27) = 1 job. This confirms that 27/2 days is exactly correct. Moreover, Nirmit rate being half of Kashish rate means Nirmit would take 27 days to complete the job, which is consistent with him taking 18 days to complete two-thirds of it. Why Other Options Are Wrong: Values such as 29/4 days, 31/2 days, or 13/2 days correspond to different daily rates and do not satisfy the condition that Kashish is exactly twice as efficient as Nirmit. Likewise, 15 days would yield a smaller daily rate for Kashish, which would not match the given efficiency relationship. Only 27/2 days keeps all the relationships consistent and correctly matches the calculated rates. Common Pitfalls: A common mistake is to assume that if Nirmit takes 18 days for 2/3 of the work, then he takes 18 days for the remaining 1/3 as well, which is not directly used here. Instead, we must convert to a daily rate first. Another error is forgetting to double the rate and instead halving the time without verifying. Always convert efficiency statements into precise rate relationships and then compute time from those rates. Final Answer: Kashish alone will complete the entire job in 27/2 days.

Question 8

Sandy and Mandy together complete eight-thirteenths (8/13) of a job, and the remaining work is completed by Andy alone. If the total payment for the job is Rs 2600 and the workers are paid in proportion to the amount of work done, what is Andy's sha…

Accepted Answer

Introduction / Context: This is a work and wages problem about three workers, Sandy, Mandy, and Andy. The first two complete a known fraction of the work together, and the last worker completes the rest. We are given the total amount paid for the whole job and must determine Andy share under the assumption that payment is proportional to work performed. Questions like this connect time and work concepts to simple financial division. Given Data / Assumptions: Sandy and Mandy together complete 8/13 of the job. Andy completes the remaining part of the job. Total payment for the job is Rs 2600. Payment is divided in the ratio of work done by each group or person. Total work is considered as 1 full job. Concept / Approach: When payment is proportional to work done, each person share is equal to total payment multiplied by the fraction of work that person completed. First we find the fraction of work done by Andy, which is the remaining work after Sandy and Mandy have finished their part. Then we multiply Andy fraction of work by the total payment to get his share. This is a direct application of ratio and proportion combined with the concept of fractions of work. Step-by-Step Solution: Step 1: Let total work be 1 job. Step 2: Sandy and Mandy together complete 8/13 of the work. Step 3: Remaining work is 1 - 8/13 = 5/13 of the job. Step 4: The remaining 5/13 of the job is done by Andy alone. Step 5: Andy fraction of total work = 5/13. Step 6: Total payment for the job is Rs 2600. Step 7: Andy share = (fraction of work done by Andy) * total payment. Step 8: Andy share = (5/13) * 2600. Step 9: Compute 2600 / 13 = 200 and then 200 * 5 = Rs 1000. Verification / Alternative check: We can also think in terms of total shares. Suppose total work of 1 is divided into 13 equal parts. Sandy and Mandy share 8 parts, and Andy share 5 parts. Total payment Rs 2600 divided by 13 parts is Rs 200 per part. Then Andy, with 5 parts, receives 5 * 200 = Rs 1000, while Sandy and Mandy together share 8 * 200 = Rs 1600. The sum 1600 + 1000 = 2600 matches the total payment, confirming the correctness of our result. Why Other Options Are Wrong: Rs 1600 corresponds to the combined share of Sandy and Mandy, not Andy. Rs 1400, Rs 800, and Rs 1200 correspond to different fractions of the total work that do not match Andy actual fraction of 5/13. Only Rs 1000 is consistent with Andy being responsible for 5/13 of the work and the total payment of Rs 2600. Common Pitfalls: A common error is to divide the payment equally among workers, ignoring their different contributions. Others may mistakenly use 8/13 instead of 5/13 for Andy, misreading which fraction belongs to whom. Always carefully identify who did which fraction of the total work and check that all fractions add to 1 before computing any payment share. Final Answer: Andy share of the payment is Rs 1000.

Question 9

Three men or four women can complete a job in 120 days. If a team of 12 men and 16 women work together on the same job, in how many days will they complete it?

Accepted Answer

Introduction / Context: This question is about comparing the work capacities of men and women for completing a job. We are told that either a group of men or a group of women can complete the job in the same number of days, and then we are asked to find the time taken when a larger mixed group works together. This problem uses the concept of man days and equivalent work rates, which is important in many practical planning scenarios. Given Data / Assumptions: Three men can complete the job in 120 days. Four women can also complete the job in 120 days. The job is the same in both cases. We are given a team of 12 men and 16 women. We must find the number of days this mixed team needs to complete the job. All men are assumed to have the same efficiency, and all women are assumed to have the same efficiency. Concept / Approach: We treat work in terms of man days, where the total work equals the product of the number of workers and the days they work. Since three men and four women each complete the job in 120 days, their total man days and woman days must represent the same amount of work. Then we express women in terms of equivalent men or vice versa and calculate the combined work rate of the 12 men and 16 women. Finally, we divide the total work by this combined rate to get the required number of days. Step-by-Step Solution: Step 1: Total work in man days when done by 3 men = 3 men * 120 days = 360 man days. Step 2: Total work in woman days when done by 4 women = 4 women * 120 days = 480 woman days. Step 3: Since both represent the same total work, 360 man days = 480 woman days. Step 4: Divide both sides by 120 to simplify: 3 man days = 4 woman days. Step 5: Therefore, 3 men are equivalent to 4 women, and 1 woman is equivalent to 3/4 of a man. Step 6: In the team, we have 12 men and 16 women. Step 7: Convert 16 women to equivalent men: 16 * (3/4) = 12 men. Step 8: Total equivalent men in the team = 12 men + 12 men = 24 men. Step 9: We know 3 men take 120 days to complete the job, so total work = 360 man days. Step 10: If 24 men work, days required = total man days / men = 360 / 24 = 15 days. Verification / Alternative check: We can verify by computing the daily work rate. Since 3 men complete the job in 120 days, daily work rate of 3 men together is 1/120 of the job per day, so 24 men (which is 8 times 3 men) will complete 8/120 = 1/15 of the job per day. Thus the job finishes in 15 days, consistent with our man day calculation. Why Other Options Are Wrong: 12 days and 14 days are too small, implying a higher overall efficiency that does not match the given equivalence between men and women. 18 days and 10 days also result in inconsistent implied work rates when calculated back from the total work. Only 15 days is consistent with both the equivalence between men and women and the total man day requirement for the job. Common Pitfalls: Students sometimes misinterpret the equivalence of three men and four women and simply sum the numbers of men and women without conversion. Another common mistake is to average the numbers or days instead of using total man days. Always convert all workers into the same unit, such as equivalent men, and then compute the total work and time carefully. Final Answer: The mixed team of 12 men and 16 women will complete the job in 15 days.

Question 10

A can complete a piece of work in 6 days working 8 hours per day, while B can complete the same work in 4 days working 10 hours per day. If A and B work together and the work must be finished in 5 days, how many hours per day should they work togeth…

Accepted Answer

Introduction / Context: This problem combines time and work with variable daily working hours. Workers A and B have different daily schedules and total days when working alone. We are asked to determine how many hours per day they should work together in order to complete the job in a specified number of days. This type of problem is very relevant in project planning where both number of days and daily effort can change. Given Data / Assumptions: A completes the work in 6 days when working 8 hours per day. B completes the work in 4 days when working 10 hours per day. The work must be completed in 5 days if A and B work together. We must find the required number of working hours per day when they work together. Total work is the same in all cases and is measured in effective man hours. Concept / Approach: We first compute the total amount of work in man hours using each person data. This gives a consistent measure for the entire job. Next, we compute A and B work rates per hour. When they work together, the combined rate is the sum of their hourly rates. To finish the job in 5 days, the total work must equal the product of days, hours per day, and the combined hourly rate. Solving this relation yields the required number of hours per day. Step-by-Step Solution: Step 1: Compute total work using A schedule. Step 2: A works 6 days at 8 hours per day, so total man hours = 6 * 8 = 48 man hours. Step 3: Therefore, the entire job equals 48 effective man hours. Step 4: A hourly work rate = 1 job / 48 hours = 1/48 job per hour. Step 5: Using B schedule, B works 4 days at 10 hours per day which is 40 hours to complete the same job. Step 6: B hourly work rate = 1 job / 40 hours = 1/40 job per hour. Step 7: Combined hourly rate of A and B working together = 1/48 + 1/40. Step 8: LCM of 48 and 40 is 240, so 1/48 = 5/240 and 1/40 = 6/240. Step 9: Combined rate = (5/240 + 6/240) = 11/240 job per hour. Step 10: Let the required working hours per day when they work together be H. Step 11: They need to finish the job in 5 days, so total work = 5 * H * (11/240). Step 12: Since total work is 1 job, we get 5 * H * (11/240) = 1. Step 13: Solve for H: H = 1 / (5 * 11/240) = 240 / 55 = 48/11 hours per day. Verification / Alternative check: To verify, multiply the found H by the time and combined rate. With H = 48/11 hours per day, total hours worked over 5 days = 5 * 48/11 = 240/11 hours. Total work done = (combined rate) * total hours = (11/240) * (240/11) = 1 job. This confirms that in 5 days, working 48/11 hours per day at their combined rate, A and B will complete exactly one full job. Why Other Options Are Wrong: Any value other than 48/11 hours per day will lead to total work different from 1 job when multiplied by 5 days and the combined rate of 11/240 job per hour. Values such as 59/11, 70/11, 65/11, or simply 6 hours per day either overshoot or undershoot the required total work. Therefore, these options are inconsistent with the given data and combined rate. Common Pitfalls: Learners may incorrectly assume that the number of hours per day is simply an average of 8 and 10, which ignores different completion times and rates. Another mistake is confusing rate per day with rate per hour, which leads to wrong combined rates. It is essential to convert everything to a common measure such as job per hour, then apply the formula for total work carefully. Final Answer: A and B must work together for 48/11 hours per day to complete the work in 5 days.

Question 11

A and B together complete three fifths of a piece of work and C alone completes the remaining part. If A, B and C receive a total payment of Rs 5000 for the entire job, what is the share of C?

Accepted Answer

Introduction / Context:This aptitude question checks understanding of how wages or payment are distributed when different workers complete different fractions of the same job. The key idea is that money is shared in direct proportion to the amount of work done by each person. Given Data / Assumptions: A and B together complete three fifths of the total work. C alone completes the remaining two fifths of the work. Total payment for the full work is Rs 5000. Payment is assumed to be shared in direct proportion to the fraction of work completed by each party. Concept / Approach:The basic concept is proportional division. If a worker completes a certain fraction of the total job, that worker is entitled to the same fraction of the total payment. Therefore, once we know what fraction of the work C completes, we can multiply that fraction by the total payment to obtain the amount due to C. Step-by-Step Solution:Step 1: Let the total work be 1 unit. Then the work done by A and B together is 3/5.Step 2: Since A and B together complete 3/5 of the work, the remaining work is 1 - 3/5 = 2/5 of the total.Step 3: This remaining 2/5 of the work is completed entirely by C.Step 4: Total payment for the complete work is Rs 5000.Step 5: Share of C is equal to the fraction of work completed by C multiplied by the total payment, that is (2/5) * 5000.Step 6: Compute (2/5) * 5000 = 2 * 1000 = Rs 2000. Verification / Alternative check:We can also find the combined share of A and B. Since they together do 3/5 of the work, their combined share is (3/5) * 5000 = 3000. Adding the share of C, which we found as 2000, gives 3000 + 2000 = 5000, which matches the total payment. This consistency check supports that the answer Rs 2000 is correct. Why Other Options Are Wrong: Rs 3000: This would be the combined share of A and B for three fifths of the work, not the share of C. Rs 2500: This amount corresponds to half of the total payment, but C does not complete half the work. C only completes two fifths of the work. Rs 1500: This corresponds to three tenths of the total payment and does not match the required fraction of 2/5 of the work. Common Pitfalls:Students sometimes divide the payment equally among all three workers, ignoring the fact that they do not all complete equal fractions of the work. Another common mistake is to misinterpret three fifths and two fifths, or to mix up the fraction that belongs to C. Remember that payment must follow the fraction of work done when no other conditions are mentioned. Final Answer:C receives Rs 2000 as his share of the total payment.

Question 12

Tushar can complete a piece of work in 6 hours and Amar can complete the same work in 10 hours. If both of them work together on the entire job, how much time will they take to finish it?

Accepted Answer

Introduction / Context:This question tests the concept of combined work and time, where two workers with different individual times to complete a job work together. The core idea is to add their rates of working rather than their times, and from the combined rate derive the required time for completing the job together. Given Data / Assumptions: Tushar can complete the work alone in 6 hours. Amar can complete the same work alone in 10 hours. Both Tushar and Amar start the work together and continue without any break until the work is finished. The work is uniform and their rates of working remain constant throughout. Concept / Approach:When dealing with time and work problems, it is convenient to consider the work done per hour by each worker, which is called the rate. If one worker completes a job in T hours, then the rate of that worker is 1 / T of the job per hour. For workers working together, the total rate is obtained by adding individual rates. Once the combined rate is known, the total time taken is the reciprocal of that combined rate. Step-by-Step Solution:Step 1: Let the total work be 1 unit.Step 2: Rate of Tushar = 1 / 6 of the work per hour.Step 3: Rate of Amar = 1 / 10 of the work per hour.Step 4: Combined rate when both work together = 1 / 6 + 1 / 10.Step 5: Compute 1 / 6 + 1 / 10 = (5 + 3) / 30 = 8 / 30 = 4 / 15 of the work per hour.Step 6: Time taken when working together = 1 / (4 / 15) = 15 / 4 hours.Step 7: Convert 15 / 4 hours into hours and minutes. 15 / 4 = 3.75 hours, which is 3 hours and 0.75 * 60 minutes = 45 minutes. Verification / Alternative check:We can verify by checking how much of the work they complete in 3 hours 45 minutes. In 3.75 hours, Tushar completes (1 / 6) * 3.75 = 0.625 of the job and Amar completes (1 / 10) * 3.75 = 0.375 of the job. Together they complete 0.625 + 0.375 = 1.0 of the job, which confirms that they finish the entire work in 3 hours 45 minutes. Why Other Options Are Wrong: 3 hours: At 3 hours, their combined work would be (1 / 6 + 1 / 10) * 3 = (4 / 15) * 3 = 4 / 5 of the job, which is not complete. 3 hours 15 minutes: This is 3.25 hours, which leads to total work (4 / 15) * 3.25, less than 1, so the job is still incomplete. 3 hours 30 minutes: This is 3.5 hours, which gives total work (4 / 15) * 3.5, also less than 1, so the work is not fully done. Common Pitfalls:One common mistake is to take the average of the two times, such as (6 + 10) / 2 = 8 hours, which is completely incorrect because time values are not additive in this way. Another error is forgetting to convert fractional hours into minutes properly. Always work with rates first and then convert the final time into the desired units. Final Answer:The work will be completed in 3 hours 45 minutes when Tushar and Amar work together.

Question 13

Worker A can make a cupboard in 10 days and worker B can make it in 50 days. Together with a third worker C, they finish making one cupboard in 6.25 days. In how many days can C alone make the cupboard?

Accepted Answer

Introduction / Context:This question is a typical work and time problem involving three workers with different efficiencies. Two workers have known individual times to complete the job, and the third works with them so that together they finish in a given number of days. We need to determine the time taken by the third worker when working alone. Given Data / Assumptions: Worker A alone can make the cupboard in 10 days. Worker B alone can make the cupboard in 50 days. Workers A, B and C working together complete the cupboard in 6.25 days. The work rate of each worker remains constant over time and there are no interruptions. Concept / Approach:The key approach is to convert the given times to rates of work, since rate is equal to 1 divided by the time taken for the whole job. The combined rate of A, B and C is known from the joint time. We subtract the known rates of A and B from this combined rate to obtain the rate of C. Finally, we take the reciprocal of C's rate to find the number of days C would need alone. Step-by-Step Solution:Step 1: Let the total work be 1 cupboard.Step 2: Rate of A = 1 / 10 cupboard per day.Step 3: Rate of B = 1 / 50 cupboard per day.Step 4: Time taken by A, B and C together = 6.25 days = 25 / 4 days.Step 5: Therefore combined rate of A, B and C = 1 / (25 / 4) = 4 / 25 cupboard per day.Step 6: Combined rate of A and B = 1 / 10 + 1 / 50 = (5 + 1) / 50 = 6 / 50 = 3 / 25 cupboard per day.Step 7: Rate of C = combined rate of A, B and C minus rate of A and B = 4 / 25 - 3 / 25 = 1 / 25 cupboard per day.Step 8: Time taken by C alone = 1 divided by C's rate = 1 / (1 / 25) = 25 days. Verification / Alternative check:We can verify by calculating the total work done in 6.25 days. In one day, A, B and C together do 4 / 25 of the work. In 25 / 4 days, work done = (4 / 25) * (25 / 4) = 1 cupboard. This matches exactly with the full work, confirming that the rates are consistent, and that C needing 25 days alone is correct. Why Other Options Are Wrong: 20 days: This would correspond to a rate of 1 / 20, which is faster than the rate we obtained and would make the combined work finish earlier than 6.25 days. 16 days: This implies an even faster rate, giving a total combined rate larger than 4 / 25 and so reducing the total time below 6.25 days. 15 days: This is also too fast and would not match the given joint completion time. Common Pitfalls:A frequent mistake is to try to take a simple average of the given days, which is incorrect because rates, not times, add for simultaneous work. Another error is mishandling the fractional time 6.25 days. Converting 6.25 to the fraction 25 / 4 makes algebra clearer and reduces calculation errors. Final Answer:C alone can make the cupboard in 25 days.

Question 14

Two labourers A and B together earn Rs 750 per day. If A is paid 150 percent of the daily wage paid to B, how much is B paid per day in rupees?

Accepted Answer

Introduction / Context:This problem tests basic proportional reasoning in sharing wages between two workers with a known ratio of payment. Rather than work and time, the focus is on money distribution based on a given percentage relationship between two wages and the total daily payment. Given Data / Assumptions: Total combined daily wage of A and B is Rs 750. A is paid 150 percent of what B is paid. We assume the wage relationship is fixed and applies every day. We need to find the exact daily wage of B in rupees. Concept / Approach:When one amount is a given percentage of another, we can express this relationship using a simple algebraic equation. If the wage of B is taken as a variable, then the wage of A is expressed in terms of that variable using the percentage relation. The sum of these two gives the known total wage, and solving the equation yields the wage of B. This is a direct application of ratios and percentages. Step-by-Step Solution:Step 1: Let the daily wage of B be x rupees.Step 2: A is paid 150 percent of B. One hundred fifty percent is equal to 150 / 100 = 3 / 2 times. So wage of A = (3 / 2) * x.Step 3: Total wage of A and B together is given as Rs 750.Step 4: Set up the equation: x + (3 / 2) * x = 750.Step 5: Combine like terms: x + 1.5x = 2.5x. So 2.5x = 750.Step 6: Solve for x: x = 750 / 2.5.Step 7: Compute 750 / 2.5. Multiply numerator and denominator by 10 to remove the decimal: (750 * 10) / (2.5 * 10) = 7500 / 25 = 300.Step 8: Therefore, B is paid Rs 300 per day. Verification / Alternative check:Using B = Rs 300, A is paid 150 percent of 300. That is (3 / 2) * 300 = 450. Summing A's and B's wages gives 450 + 300 = 750, which matches the total daily wage given in the question. This confirms that the computed value for B's wage is correct. Why Other Options Are Wrong: Rs 450: This would be the wage of A, not B, since A is the one paid more. Rs 250: If B were paid 250, then A would get 375, and total wage would be only 625, which is less than 750. Rs 500: If B were paid 500, then A would get 750, and the total would be 1250, which is greater than 750 and inconsistent with the given data. Common Pitfalls:Candidates sometimes misinterpret 150 percent as 1.5 percent or forget to convert the percentage into a multiplier. Another error is to mix up who is getting more money, incorrectly assuming B is paid 150 percent of A. Keeping track of which worker receives the higher wage is crucial for setting up the correct equation. Final Answer:B is paid Rs 300 per day.

Question 15

Workers A and B together can finish a job in 12 days, while A alone can finish the same job in 20 days. If B works alone on the job, how many days will B take to complete it?

Accepted Answer

Introduction / Context:This time and work question asks us to determine the time required by a single worker when we know how long it takes for that worker together with another person, and also how long it takes for one of them individually. This is a standard use of combined work rates to isolate the unknown worker's rate. Given Data / Assumptions: A and B together complete the job in 12 days. A alone completes the same job in 20 days. B alone works at a constant rate throughout. We need to find the number of days B alone takes to complete the job. Concept / Approach:The central concept is that a worker who can complete a job in T days has a work rate of 1 / T of the job per day. When two workers work together, their combined rate is the sum of their individual rates. We are given the combined rate of A and B and the individual rate of A. Subtracting A's rate from the combined rate yields B's rate. The reciprocal of this rate provides the time needed by B alone. Step-by-Step Solution:Step 1: Let the total work be 1 unit.Step 2: Rate of A alone = 1 / 20 of the work per day.Step 3: Rate of A and B together = 1 / 12 of the work per day.Step 4: Rate of B alone = rate of (A and B together) minus rate of A alone.Step 5: Compute rate of B = 1 / 12 - 1 / 20.Step 6: To subtract, take the common denominator: LCM of 12 and 20 is 60. So 1 / 12 = 5 / 60 and 1 / 20 = 3 / 60.Step 7: Rate of B = 5 / 60 - 3 / 60 = 2 / 60 = 1 / 30 of the work per day.Step 8: Time required by B alone to complete the job = 1 divided by (1 / 30) = 30 days. Verification / Alternative check:If B alone takes 30 days, his rate is 1 / 30 of the work per day. Combined with A's rate of 1 / 20, their total rate is 1 / 20 + 1 / 30. With common denominator 60, we get 3 / 60 + 2 / 60 = 5 / 60 = 1 / 12. That is exactly the given combined rate, confirming that B taking 30 days alone is consistent with the data. Why Other Options Are Wrong: 25 days: This corresponds to a rate of 1 / 25 per day, which would give a combined rate of 1 / 20 + 1 / 25 = 9 / 100, not equal to 1 / 12. 24 days: This rate of 1 / 24 per day would result in a combined rate of 1 / 20 + 1 / 24 which is different from 1 / 12. 15 days: This is much faster and leads to a combined rate that is too high, implying a total time less than 12 days, which contradicts the information given. Common Pitfalls:Sometimes students try to use averages such as (20 + 12) / 2 instead of working with rates, which is incorrect. Another error is subtracting times directly rather than converting to rates. Always convert days into rates of work, perform additions or subtractions on those rates, and then convert back to time using reciprocals. Final Answer:B alone will take 30 days to complete the job.

Question 16

Workers X, Y and Z can complete a certain piece of work in 30, 40 and 50 days respectively. All three start together, but Y leaves 4 days before the work is finished. In how many days in total is the work completed?

Accepted Answer

Introduction / Context:This time and work problem combines different individual work rates and a change in team composition shortly before completion. It requires setting up an equation involving the number of days worked at different combined rates and then solving for the total time taken to finish the work. Given Data / Assumptions: X alone can complete the work in 30 days. Y alone can complete the work in 40 days. Z alone can complete the work in 50 days. All three start working together. Y leaves 4 days before the work is completed; during the last 4 days only X and Z continue. We assume constant work rates and no idle time. Concept / Approach:The approach is to express the total work as 1 unit and write the total work done as the sum of work done in two phases: an initial phase where all three work together, and a final phase where only two workers continue. If T is the total time, then during (T minus 4) days all three work together, and during the last 4 days only X and Z work. We use the sum of the two contributions to equal 1 and solve for T. Step-by-Step Solution:Step 1: Let total work = 1 unit.Step 2: Rate of X = 1 / 30 work per day.Step 3: Rate of Y = 1 / 40 work per day.Step 4: Rate of Z = 1 / 50 work per day.Step 5: Rate of X + Y + Z together = 1 / 30 + 1 / 40 + 1 / 50.Step 6: Compute this sum with denominator 600: 1 / 30 = 20 / 600, 1 / 40 = 15 / 600, 1 / 50 = 12 / 600, so combined rate = (20 + 15 + 12) / 600 = 47 / 600.Step 7: Rate of X + Z together = 1 / 30 + 1 / 50 = 20 / 600 + 12 / 600 = 32 / 600 = 4 / 75.Step 8: Let total time taken be T days. For (T - 4) days, all three work; for last 4 days, only X and Z work.Step 9: Total work equation: (T - 4) * (47 / 600) + 4 * (4 / 75) = 1.Step 10: Note that 4 * (4 / 75) = 16 / 75.Step 11: Rearrange: (T - 4) * (47 / 600) = 1 - 16 / 75.Step 12: Compute 1 - 16 / 75 = (75 / 75 - 16 / 75) = 59 / 75.Step 13: So (T - 4) = (59 / 75) * (600 / 47) = (59 * 8) / 47 = 472 / 47.Step 14: Therefore T = 472 / 47 + 4 = (472 + 188) / 47 = 660 / 47 days. Verification / Alternative check:We can verify quickly by computing approximate value. 660 / 47 is about 14.04 days. In this period, X works the entire time, Y works for about 10.04 days, and Z also works the entire time. Approximating total work done using these durations and rates will give close to 1 unit, which is consistent and confirms that the exact algebraic solution 660 / 47 days is correct. Why Other Options Are Wrong: 640/47 days: This choice slightly underestimates the time, so total work done would be less than 1 unit. 680/47 days: This gives a slightly larger time, leading to total work more than 1 unit. 665/47 days: This is a close but incorrect fraction and does not satisfy the work equation when substituted. Common Pitfalls:Students may mistakenly assume that all three work for the same duration or forget that Y leaves exactly 4 days before completion. Another common mistake is miscalculating combined rates or mishandling fractions, especially when denominators like 30, 40 and 50 are involved. Careful use of a common denominator such as 600 helps avoid these issues. Final Answer:The work is completed in 660/47 days.

Question 17

Workers P, Q and R together can make one table in 40 minutes, while P and Q together can make the same table in 60 minutes. How many minutes will R alone take to make one table?

Accepted Answer

Introduction / Context:This question involves three workers making one table, where the combined time for all three is known and the combined time for only two of them is also known. The aim is to isolate the contribution of the third worker by working with rates of work in tables per minute. Given Data / Assumptions: P, Q and R together can make one table in 40 minutes. P and Q together can make the table in 60 minutes. Each worker works at a constant rate. We need to find the time taken by R alone to finish one table. Concept / Approach:The idea is similar to standard work rate problems. A worker who completes one table in T minutes has a rate of 1 / T table per minute. The combined rate of P, Q and R is known, as well as the combined rate of only P and Q. Subtracting these rates will give the rate of R alone. The reciprocal of R's rate provides the required time in minutes. Step-by-Step Solution:Step 1: Let the total work be 1 table.Step 2: Time taken by P, Q and R together = 40 minutes, so their combined rate = 1 / 40 table per minute.Step 3: Time taken by P and Q together = 60 minutes, so their combined rate = 1 / 60 table per minute.Step 4: Rate of R alone = rate of (P + Q + R) minus rate of (P + Q).Step 5: Compute rate of R = 1 / 40 - 1 / 60.Step 6: Use common denominator 120. Then 1 / 40 = 3 / 120 and 1 / 60 = 2 / 120.Step 7: Rate of R = 3 / 120 - 2 / 120 = 1 / 120 table per minute.Step 8: Time taken by R alone = 1 divided by (1 / 120) = 120 minutes. Verification / Alternative check:We can confirm by computing the total rate again. If R's rate is 1 / 120, then rate of P + Q + R should be 1 / 60 + 1 / 120 = 2 / 120 + 1 / 120 = 3 / 120 = 1 / 40 table per minute. This matches the given time of 40 minutes for all three together, confirming that 120 minutes for R alone is correct. Why Other Options Are Wrong: 100 minutes: This corresponds to a rate of 1 / 100, which would make the combined time of all three less than 40 minutes. 90 minutes: This rate is 1 / 90, also larger than 1 / 120, giving a faster combined completion than specified. 150 minutes: This is a slower rate (1 / 150) and would give a different combined rate that does not match the 40 minute requirement. Common Pitfalls:Some students accidentally add the times instead of the rates or take an average of the times 40 and 60 directly, which is incorrect. Others forget to use a common denominator when subtracting fractions. Always remember that in work problems, it is the rates (work per unit time) that add or subtract, not the times themselves. Final Answer:R alone will take 120 minutes to make one table.

Question 18

A can finish a piece of work in 12 days and B can finish the same work in 24 days. If A and B work together on the job for 4 days, what fraction of the total work remains unfinished?

Accepted Answer

Introduction / Context:This problem evaluates understanding of combined work rates and asks for the fraction of work left after a certain number of days of joint work. Instead of computing the total time to finish, we calculate how much work has been completed and then subtract from the whole job. Given Data / Assumptions: A alone can finish the work in 12 days. B alone can finish the work in 24 days. A and B work together for 4 days and then stop. Their work rates are constant for the entire duration. Concept / Approach:We again use the idea that a worker who finishes a job in T days has a daily rate of 1 / T of the work. Adding A's and B's rates gives the joint daily work. Multiplying this joint rate by the number of days they actually work gives the fraction of the job completed. The remaining fraction is simply 1 minus this completed fraction. Step-by-Step Solution:Step 1: Assume total work = 1 unit.Step 2: Rate of A = 1 / 12 of the work per day.Step 3: Rate of B = 1 / 24 of the work per day.Step 4: Combined rate of A and B together = 1 / 12 + 1 / 24.Step 5: Convert to a common denominator 24. Then 1 / 12 = 2 / 24, so combined rate = 2 / 24 + 1 / 24 = 3 / 24 = 1 / 8 of the work per day.Step 6: In 4 days, work completed = 4 * (1 / 8) = 4 / 8 = 1 / 2 of the total work.Step 7: Therefore fraction of work remaining = 1 - 1 / 2 = 1 / 2. Verification / Alternative check:We can verify by imagining that if they continued working together at the same rate, they would need another 4 days to complete the remaining 1 / 2 of the work, because 4 days more at rate 1 / 8 per day equals another 1 / 2. That suggests the total time to finish together is 8 days, which is consistent with the rate calculation since time = 1 divided by rate = 1 / (1 / 8) = 8 days. Thus after 4 days only half the job is completed. Why Other Options Are Wrong: 1/3: This would imply that two thirds of the work is completed, which does not match the computed total work done of 1 / 2. 1/4: This suggests three quarters of the work has been completed, which is incorrect for 4 days at rate 1 / 8 per day. 1/5: This implies completion of four fifths of the work, which contradicts the rate based calculation. Common Pitfalls:Some learners incorrectly add the days (12 and 24) or average them instead of using rates. Others forget to subtract the completed fraction from 1 and mistakenly treat the completed fraction as the remaining fraction. Always calculate work done first, then subtract from 1 to find how much work remains. Final Answer:The fraction of the work left unfinished after 4 days is 1/2.

Question 19

Akash is three times as efficient a workman as Baldev and therefore finishes a certain job in 40 days less than Baldev. If they work together on the same job, in how many days will they complete it?

Accepted Answer

Introduction / Context:This question combines relative efficiency and time difference information. One worker is said to be three times as good, or three times as efficient, as another worker, and their individual completion times differ by a fixed number of days. The objective is to deduce their individual times and then find how long they would take if they worked together. Given Data / Assumptions: Akash is three times as efficient as Baldev. Akash finishes the job 40 days earlier than Baldev. Both work at constant rates. We want the time required for Akash and Baldev working together to complete the job. Concept / Approach:If Akash is three times as efficient as Baldev, then Akash's daily work rate is three times Baldev's rate. Since time is inversely proportional to efficiency for the same job, the time taken by Akash will be one third of the time taken by Baldev. We set up an equation using the information that Akash's time is 40 days less than Baldev's time, solve for each time separately, and then compute their combined rate for the job. Step-by-Step Solution:Step 1: Let time taken by Baldev alone to finish the job be T days.Step 2: Since Akash is three times as efficient as Baldev, his time to finish the job is T / 3 days.Step 3: According to the question, Akash finishes the job 40 days earlier than Baldev, so T - T / 3 = 40.Step 4: Simplify the equation: T - T / 3 = (3T / 3 - T / 3) = 2T / 3.Step 5: So 2T / 3 = 40, which gives T = 40 * 3 / 2 = 60 days.Step 6: Therefore Baldev alone takes 60 days, and Akash alone takes T / 3 = 60 / 3 = 20 days.Step 7: Rate of Akash = 1 / 20 of the work per day, rate of Baldev = 1 / 60 of the work per day.Step 8: Combined rate of Akash and Baldev = 1 / 20 + 1 / 60.Step 9: Use common denominator 60: 1 / 20 = 3 / 60, so combined rate = 3 / 60 + 1 / 60 = 4 / 60 = 1 / 15.Step 10: Time taken together = 1 divided by (1 / 15) = 15 days. Verification / Alternative check:We can verify by imagining their joint work over 15 days. In 15 days Akash completes 15 * (1 / 20) = 3 / 4 of the job, and Baldev completes 15 * (1 / 60) = 1 / 4 of the job. Together this is 3 / 4 + 1 / 4 = 1 whole job, confirming that 15 days is correct. Why Other Options Are Wrong: 60 days: This is the time taken by Baldev alone, not the joint time. 20 days: This is the time taken by Akash alone, not the time when both work together. 10 days: This would require a combined rate of 1 / 10, which is larger than 1 / 15 and not consistent with the given separate times. Common Pitfalls:One common error is to assume that if Akash is three times as efficient, his time is three times Baldev's time, which is the opposite of the truth. Another mistake is to mis-handle the time difference and set up the equation incorrectly, such as T / 3 - T = 40, which would produce a negative value. Always remember efficiency and time are inversely related for the same job. Final Answer:Akash and Baldev together can finish the work in 15 days.

Question 20

Twenty eight women can complete a certain piece of work in 36 days. If 4 women leave after working for 18 days, how many additional days will be needed to finish the remaining work?

Accepted Answer

Introduction / Context:This time and work problem involves a change in the size of the workforce halfway through a job. After a certain period some workers leave, and we are asked to compute the extra time needed to complete the remaining work with fewer workers. It tests understanding of total work in man days or woman days and the effect of changing the number of workers. Given Data / Assumptions: 28 women can complete the work in 36 days. All 28 women initially work for 18 days. After 18 days, 4 women leave, so 24 women continue. The work rate of each woman is constant throughout. We must find how many days the remaining 24 women need to complete the remaining work. Concept / Approach:We treat the total work in terms of woman days. If W is the total work, then W = number of women multiplied by number of days, assuming one unit of work per woman per day. After computing W, we calculate how much work is done in the first phase with 28 women for 18 days, subtract that from the total work and then divide the remainder by the daily capacity of 24 women to get the additional days required. Step-by-Step Solution:Step 1: Let work done per woman per day be 1 unit. Then total work W = 28 women * 36 days = 1008 woman days.Step 2: Work completed in the first 18 days with 28 women = 28 * 18 = 504 woman days.Step 3: Remaining work = total work - work already done = 1008 - 504 = 504 woman days.Step 4: After 4 women leave, number of women remaining = 28 - 4 = 24.Step 5: Daily work capacity of the remaining group = 24 woman days per day.Step 6: Days required to finish remaining work = remaining work / daily capacity = 504 / 24.Step 7: Compute 504 / 24. Divide numerator and denominator by 8: 504 / 24 = 63 / 3 = 21 days. Verification / Alternative check:We can check that the total calendar days are 18 (before women leave) plus 21 (after some leave) = 39 days. If 24 women had worked alone from the beginning at the same rate, time would have been total work divided by 24, that is 1008 / 24 = 42 days, which is longer. This makes sense because initially there were more workers. The internal calculation remains consistent, confirming that 21 days of additional work is correct. Why Other Options Are Wrong: 24 days: This would mean total woman days after reduction is 24 * 24 = 576, which exceeds the remaining work of 504. 27 days: This would give 24 * 27 = 648 woman days, also more than needed. 25 days: This implies 600 woman days, still more than 504, making it inconsistent with the original total work. Common Pitfalls:Some students miscalculate the total work or forget that the number of workers changes midway. Another frequent mistake is to average the number of workers over the entire duration instead of correctly splitting the work into phases. Always compute total work first, then account for each phase of the work with the correct number of workers. Final Answer:The remaining work will be completed in 21 additional days.

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