Question 1

Amit can complete a piece of work in 45 days, while Bharath can complete the same work in 5 days less than Amit, that is, in 40 days. Amit and Bharath start the work together at their full efficiencies, but after working together for some days Bhara…

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Introduction / Context: This is a multi-step time and work problem that combines full and reduced efficiencies. Two workers, Amit and Bharath, start working together at full efficiency. After some time, Bharath leaves and Amit continues alone but at half his usual efficiency. You must use the given times to determine the number of days they worked together initially. This problem tests your understanding of variable work rates and how to translate them into equations. Given Data / Assumptions: Amit alone can complete the work in 45 days. Bharath alone can complete the same work in 40 days (5 days less). Both work together at full efficiency for an unknown number of days. After that, Bharath leaves and Amit continues alone for 56 days but at half of his usual efficiency. The total work is one complete job, and all efficiency changes are clearly specified and constant during the periods described. Concept / Approach: We first find the usual daily work rates of Amit and Bharath. Let Amit’s full-efficiency rate be a and Bharath’s rate be b. During the period they work together, the combined rate is a + b. During the period when Amit works alone at half efficiency, his rate is a / 2. If t is the number of days they work together, then the total work equation is t(a + b) + 56(a / 2) = 1. We substitute the known values of a and b from 45 and 40 days, solve for t, and interpret the result as the number of days of joint work. Step-by-Step Solution: Step 1: Let the total work be 1 unit. Step 2: Amit alone completes the work in 45 days, so his full-efficiency rate is a = 1 / 45 of the work per day. Step 3: Bharath alone completes the work in 40 days, so his rate is b = 1 / 40 of the work per day. Step 4: Let t be the number of days during which Amit and Bharath work together at full efficiency. Step 5: In those t days, total work done by Amit and Bharath together is t(a + b) = t(1 / 45 + 1 / 40). Step 6: After t days, Bharath leaves. Amit continues alone for 56 days but at half of his usual efficiency, so his rate in this period is a / 2 = (1 / 45) / 2 = 1 / 90 of the work per day. Step 7: Work done by Amit during these 56 days is 56 * (1 / 90) = 56 / 90 = 28 / 45 of the total work. Step 8: Total work done is the sum of the work done during the joint period and Amit’s half-efficiency period: t(a + b) + 28 / 45 = 1. Step 9: Compute a + b: 1 / 45 + 1 / 40 = (40 + 45) / (45 * 40) = 85 / 1800. This simplifies to 17 / 360. Step 10: Substitute into the work equation: t * (17 / 360) + 28 / 45 = 1. Step 11: Convert 28 / 45 to a denominator of 360: 28 / 45 = (28 * 8) / (45 * 8) = 224 / 360. Step 12: The equation becomes t * (17 / 360) + 224 / 360 = 360 / 360. Step 13: Multiply both sides by 360: 17t + 224 = 360. Step 14: Solve for t: 17t = 360 - 224 = 136, so t = 136 / 17 = 8. Step 15: Thus, Amit and Bharath worked together at full efficiency for 8 days. Verification / Alternative check: We can confirm by computing the total work done using t = 8 days. Combined rate a + b is 17 / 360 per day. In 8 days, they complete 8 * (17 / 360) = 136 / 360 = 34 / 90 of the work. Amit then works 56 days at rate 1 / 90 per day, completing 56 / 90 = 56 / 90 of the work. Total work done is 34 / 90 + 56 / 90 = 90 / 90 = 1 job. This matches the requirement that the entire work is completed, confirming that t = 8 days is correct. Why Other Options Are Wrong: Values such as 6, 9, 12, or 15 days would change the total work done during the joint period and would not balance the equation t(a + b) + 28 / 45 = 1. For example, if t were 6 days, the combined work during the joint period would be 6 * (17 / 360) = 102 / 360, and adding Amit’s 28 / 45 would not produce exactly 1. Only t = 8 days satisfies the exact equality required for the complete job. Common Pitfalls: A major pitfall is forgetting that Amit works at half efficiency during the 56 days, leading to an incorrect rate for that period. Some learners also misinterpret the statement “5 days less than Amit” and incorrectly assign Bharath’s time. Another frequent error is to average the times of 45 and 40 days rather than building the correct algebraic equation for the total work. Always convert times into rates and construct a precise equation reflecting each phase of the work. Final Answer: Amit and Bharath worked together at full efficiency for 8 days before Bharath left.

Question 2

P alone can complete a piece of work in 24 days. The time taken by P to finish one third of the work is equal to the time taken by Q to finish half of the same work. How many days will P and Q working together take to complete the entire work?

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Introduction / Context:This time and work question checks understanding of individual work rates and combined work rates. Many aptitude tests use such problems to test how well candidates convert verbal statements into algebraic relations. Here we compare how fast two workers, P and Q, can complete fractions of the same job, and then use that information to find how long they will take together to complete the entire work. The key skill is to translate the condition about one third and half of the work into daily work rates, then combine them in a clean and systematic way. Given Data / Assumptions: P alone can complete the whole work in 24 days. Time taken by P to complete one third of the work equals time taken by Q to complete half of the work. We assume both P and Q work at constant rates every day. We are asked for total days when P and Q work together on the entire job. Concept / Approach:The basic concept is rate of work. If a person completes a job in D days, then the daily work rate is 1 / D of the job per day. Work done equals rate multiplied by time. We first find the time P needs for one third of the work using P's known rate. That same time is used to compute Q's rate for half of the work. Once both daily rates are known, we add the rates to obtain the combined rate when P and Q work together. Finally we invert the combined rate to get the total time required to finish one complete job. Step-by-Step Solution: Step 1: Rate of P is 1 / 24 of the work per day. Step 2: Time taken by P to complete one third of the work is (1/3) / (1/24) = 8 days. Step 3: In the same 8 days, Q completes half of the work, so Q's rate is (1/2) / 8 = 1 / 16 per day. Step 4: Combined daily rate of P and Q together is 1/24 + 1/16. Step 5: 1/24 + 1/16 = (2 + 3) / 48 = 5 / 48 of the work per day. Step 6: Total time to complete one full job together is 1 / (5/48) = 48 / 5 = 9.6 days, which is 9 3/5 days. Verification / Alternative check:We can verify by checking total work done in 9.6 days at the combined rate. Combined rate is 5 / 48 of the work per day. Multiplying 5 / 48 by 9.6 gives 5 / 48 times 48 / 5 which simplifies exactly to 1. This confirms that the entire work is completed in 9.6 days when both P and Q work together. The fractions match perfectly, so the calculation is consistent and there is no approximation involved in this solution. Why Other Options Are Wrong: 7 days would require a much higher combined rate than 5 / 48, which is not supported by the given individual rates. 6 3/7 days is less than 7 days and is even more unrealistic for such slow individual rates. 8 days would mean they finish faster than the time P alone takes to finish one third of the work, which contradicts the given condition. Common Pitfalls:Many learners confuse time taken for a fraction of work with the daily rate directly. Another common mistake is to add the times taken by P and Q instead of adding their rates. Some students also misread the statement and assume one third and half refer to combined work instead of individual work. Carefully distinguishing between rate, time, and fraction of work is crucial for solving problems like this accurately in exams. Final Answer:The time required for P and Q working together to complete the entire work is 9 3/5 days.

Question 3

Six men and eight boys can complete a piece of work in ten days, while twenty six men and forty eight boys can complete the same work in two days. In how many days will fifteen men and twenty boys together complete the entire work?

Accepted Answer

Introduction / Context:This is a classic time and work question involving two different types of workers, men and boys, each with different efficiencies. The problem provides two different team combinations that complete the same job in different times. From these combinations we can infer the individual work rates of a man and a boy. Once those rates are known, we can combine them again for a new group size and compute the total time needed to finish the work. Such questions commonly appear in aptitude tests and competitive examinations. Given Data / Assumptions: Six men and eight boys complete the work in 10 days. Twenty six men and forty eight boys complete the same work in 2 days. All men work at the same constant rate per day. All boys work at the same constant rate per day. We need the time taken by fifteen men and twenty boys together to finish the job. Concept / Approach:The key concept is total work equals rate multiplied by time. Let one man do m units of work per day and one boy do b units per day. We write two equations from the two groups given, both equal to one complete job. Solving the system of linear equations gives values of m and b. Then we compute the daily work rate of fifteen men and twenty boys, add the contributions of men and boys, and finally take the reciprocal to find the number of days needed to complete one job. Step-by-Step Solution: Step 1: Let daily work of one man be m and that of one boy be b. Step 2: From the first group, (6m + 8b) times 10 days equals 1 job, so 6m + 8b = 1 / 10. Step 3: From the second group, (26m + 48b) times 2 days equals 1 job, so 26m + 48b = 1 / 2. Step 4: Solving these equations gives m = 0.01 and b = 0.005. Step 5: Daily work of fifteen men and twenty boys is 15m + 20b = 15(0.01) + 20(0.005) = 0.15 + 0.10 = 0.25 of the job per day. Step 6: Time taken is 1 divided by 0.25, which equals 4 days. Verification / Alternative check:We can quickly verify by checking total work units. Total work can be considered as 1 unit. Fifteen men and twenty boys complete 0.25 units per day. Over 4 days, they would complete 4 times 0.25 which equals 1 unit of work. This means the whole job is done exactly in 4 days with no remainder. The consistency between the derived rates and the original equations confirms that the system of equations has been solved correctly. Why Other Options Are Wrong: 6 days would correspond to a daily rate of only 1 / 6, which is smaller than 0.25, thus contradicting the rate obtained from the original data. 7 days makes the daily rate even smaller and therefore does not satisfy the equations derived from the initial groups. 5 days would require a daily rate of 1 / 5, which does not match the computed combined rate of 0.25 per day. Common Pitfalls:One common mistake is to try to compare the number of men and boys as if they had equal efficiency. Another error is incorrectly forming or solving the simultaneous equations. Some learners also forget to divide by the number of days when converting to rates. Being systematic about defining variables, writing equations, and solving them step by step prevents these misunderstandings and leads to the correct answer efficiently. Final Answer:Fifteen men and twenty boys working together will complete the entire work in 4 days.

Question 4

P and Q together can complete a job in 24 days. P alone can complete the same job in 32 days. If they work together for 8 days and then P leaves, how many additional days will Q alone take to finish the remaining work?

Accepted Answer

Introduction / Context:This is a standard work and time problem in which two people start a task together and later one person leaves. The aim is to determine how long the remaining person will need to finish the job. Questions like this appear frequently in banking, management, and government exams to test a candidate's ability to handle fractional work and sequential stages of a task. Correctly tracking how much work is completed at each stage is the core skill required here. Given Data / Assumptions: P and Q together complete the job in 24 days. P alone completes the job in 32 days. P and Q work together for 8 days, and then P stops working. Q works alone after that until the job is completed. Both P and Q work at constant daily rates. Concept / Approach:The key idea is to calculate the daily work rates of P and Q using the total time they take individually and together. The combined rate gives the fraction of work done per day when both work together. We first compute how much of the job is finished in the initial 8 days when both are working. Subtracting this from the full job gives the fraction of work remaining. Then we divide the remaining work by Q's individual rate to obtain the time taken by Q alone to complete the task. Step-by-Step Solution: Step 1: Combined daily rate of P and Q is 1 / 24 of the job per day. Step 2: Daily rate of P alone is 1 / 32 of the job per day. Step 3: Therefore Q's daily rate is 1/24 minus 1/32 which is (4 minus 3) / 96 = 1 / 96 of the job per day. Step 4: In 8 days working together, P and Q complete 8 times 1/24 which equals 1 / 3 of the job. Step 5: Fraction of work remaining is 1 minus 1/3 which equals 2 / 3 of the job. Step 6: Time for Q alone to finish the remaining 2 / 3 at a rate of 1 / 96 is (2/3) divided by (1/96) which is (2/3) times 96 = 64 days. Verification / Alternative check:As a check, imagine the entire work as 96 equal units. P and Q together complete 96 / 24 = 4 units per day. P alone completes 96 / 32 = 3 units per day, so Q alone must complete 1 unit per day. In 8 days together, they complete 8 times 4 which equals 32 units. Remaining work is 64 units. Q alone at 1 unit per day will indeed take 64 days. This unit method gives the same answer and confirms our fraction based calculations. Why Other Options Are Wrong: 56 days would correspond to a daily rate higher than Q's actual rate, which contradicts the derived value. 54 days and 60 days similarly do not match the required remaining work divided by Q's rate. Only 64 days matches exactly with the remaining two thirds of the job at Q's daily work rate. Common Pitfalls:Learners sometimes subtract times directly instead of subtracting rates, which is incorrect. Another pitfall is miscalculating the fraction of work completed in the initial days or forgetting to subtract it from the whole job. It is important to remember that work changes linearly with time only when the rate is constant, so we must always work with rates first and then compute fractions of work using multiplication, not by subtracting days. Final Answer:Once P leaves, Q alone will need 64 days to finish the remaining work.

Question 5

Arun works alone for 8 days and finds that only one third of the work has been completed. He then employs Akhil, who is 60 percent as efficient as Arun, and allows Akhil to complete the remaining work alone. How many more days will Akhil take to fin…

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Introduction / Context:This question involves changing workers in the middle of a job and comparing their efficiencies. Arun starts the work and later Akhil, who is less efficient, completes what is left. Many aptitude tests include such problems to check understanding of fractional work and relative efficiencies. The task is to quantify the rate at which Arun works, use it to compute Akhil's slower rate, and then determine how long Akhil alone will need to finish the remaining work. Given Data / Assumptions: Arun works alone for 8 days and completes one third of the work. Akhil is 60 percent as efficient as Arun. After the first 8 days, only Akhil works on the remaining part of the job. We assume both Arun and Akhil work at constant daily rates. We are asked to find how many more days Akhil alone will take to complete the remaining two thirds of the work. Concept / Approach:The main concept used is work rate. If a worker completes W fraction of the job in T days, the rate is W / T per day. First we compute Arun's daily work rate from the statement that one third of the work is done in 8 days. Then we use the fact that Akhil is 60 percent as efficient, meaning his rate is 0.6 times Arun's rate. Once we know Akhil's rate, we divide the remaining work, which is two thirds of the job, by this slower rate to get the additional time he needs. Step-by-Step Solution: Step 1: Arun completes one third of the work in 8 days, so Arun's rate is (1/3) / 8 = 1 / 24 of the job per day. Step 2: Akhil is 60 percent as efficient as Arun, so Akhil's rate is 0.6 times 1/24, which is 1 / 40 of the job per day. Step 3: Fraction of work remaining after Arun stops is 1 minus 1/3 which equals 2 / 3 of the job. Step 4: Time required by Akhil alone is (2/3) divided by (1/40) which equals (2/3) times 40 = 80 / 3 days. Step 5: 80 / 3 days is approximately 26.666 days, which is written as 26.6 days in the options. Verification / Alternative check:If we think of the total work as 120 units, Arun completes 40 units in 8 days, so his rate is 5 units per day. Akhil, at 60 percent of that rate, completes 3 units per day. Remaining work is 80 units. At 3 units per day, Akhil needs 80 / 3 days, which again equals 26.666 days. This confirms the fraction based calculation and shows that the approximate option 26.6 days is correct when rounded to one decimal place. Why Other Options Are Wrong: 24.5 days and 25 days underestimate the time needed since Akhil is slower than Arun and has to complete two thirds of the work. 20 days is much too small and would imply a significantly higher work rate for Akhil than given. Only 26.6 days correctly represents 80 divided by 3 when rounded to one decimal place. Common Pitfalls:Students sometimes mistakenly let Arun and Akhil work together on the remaining work or misinterpret 60 percent efficiency as a time ratio instead of a rate ratio. Another frequent error is to subtract 60 percent from the number of days rather than from the rate. Always remember that efficiency percentages apply to work rate, not directly to the number of days, and that remaining work must be calculated accurately before dividing by the new worker's rate. Final Answer:Akhil alone will take approximately 26.6 days to finish the remaining work.

Question 6

Ten men can complete a certain piece of work in 15 days. If thirty seven men are employed instead, approximately how many days will they take to complete the same work, assuming all men work at the same constant rate?

Accepted Answer

Introduction / Context:This question is about direct proportion between the number of workers and the time taken to complete a fixed amount of work. When all workers have the same efficiency and the total work is fixed, time is inversely proportional to the number of workers. Such questions appear very frequently in basic quantitative aptitude sections and are usually considered straightforward, though sometimes fractional results need to be interpreted carefully with respect to the options given. Given Data / Assumptions: Ten men complete the work in 15 days. All men have identical and constant working efficiency. We now employ thirty seven men to do the same job. We are asked how many days the thirty seven men will require to complete the same work. Concept / Approach:If a fixed job requires W man days of effort, then W equals the product of the number of men and the number of days. When the number of men changes, the time adjusts so that the product remains constant. We first compute the total amount of work in man days using the first scenario. Then we divide this total work by the new number of men to obtain the time needed in days when more men are employed. Since the exact value is slightly above an integer, we choose the closest available option. Step-by-Step Solution: Step 1: Total work in man days is 10 men times 15 days which equals 150 man days. Step 2: When thirty seven men are working, their daily total capacity is 37 man days per day. Step 3: Time required equals total work divided by daily total capacity which is 150 divided by 37 days. Step 4: 150 divided by 37 is approximately 4.054 days. Step 5: From the given options, the nearest reasonable whole number of days is 4 days. Verification / Alternative check:To confirm, check how much work is done in 4 days by thirty seven men. That is 37 times 4 which equals 148 man days, slightly less than the total 150 man days. In a real world situation, the team would need a little more than 4 days to finish exactly. Given that the options are rounded to whole days, 4 days is clearly the only reasonable choice. No other listed option is closer to the exact computed value of about 4.05 days. Why Other Options Are Wrong: 3 days would give only 111 man days, far less than the required 150 man days. 5 days would give 185 man days, which is significantly more than needed and not minimal. 6 days would produce 222 man days, much larger than required and clearly not the correct minimal time. Common Pitfalls:A common mistake is to think that if the number of men increases by around four times, the time must reduce exactly to one quarter. Here the increase is from 10 to 37, which is not an integer multiple, so the resulting time is not a neat whole number. Another pitfall is not recognizing that exam options are sometimes rounded, and thus the closest option to the computed value must be chosen. Always calculate the exact value first and then match it to the nearest option as needed. Final Answer:Thirty seven men will require approximately 4 days to complete the work.

Question 7

Six men can complete a piece of work in 12 days, eight women can complete the same work in 18 days, and eighteen children can complete it in 10 days. Four men, twelve women and twenty children work together for 2 days. If only men are to complete th…

Accepted Answer

Introduction / Context:This mixed work rate question involves three categories of workers with different efficiencies: men, women, and children. The problem first gives separate completion times for each group and then asks for the number of men required at the end to finish the remaining portion of the work in one day. Such scenarios are frequently used in aptitude exams to test comfort with fractional work, rate comparisons, and combining contributions from different groups over different time intervals. Given Data / Assumptions: Six men complete the work in 12 days. Eight women complete the same work in 18 days. Eighteen children complete the work in 10 days. Four men, twelve women, and twenty children work together for 2 days. After this, only men will work and must complete the remaining work in 1 day. All workers of the same type have equal and constant efficiency. Concept / Approach:We use the idea of daily work rates for each type of worker. From the total time required by each group we calculate the daily work done by one man, one woman, and one child. Then we find how much work is completed in the first two days by the mixed group. Subtracting that from the total leaves the remaining work. Finally, we compute how many men are needed so that, at the known rate of one man, the remaining work is finished in exactly one day. Step-by-Step Solution: Step 1: Total work can be taken as 1 job. Rate of one man is 1 divided by (6 times 12) which is 1 / 72 per day. Step 2: Rate of one woman is 1 divided by (8 times 18) which is 1 / 144 per day. Step 3: Rate of one child is 1 divided by (18 times 10) which is 1 / 180 per day. Step 4: Daily work of 4 men, 12 women and 20 children is 4(1/72) + 12(1/144) + 20(1/180) which simplifies to 1 / 4 of the job per day. Step 5: Work done in 2 days by this group is 2 times 1/4 which equals 1 / 2 of the job. Step 6: Remaining work is 1 minus 1/2 which equals 1 / 2 of the job. Step 7: Let N be the number of men required on the last day. Then N times 1/72 times 1 day equals 1/2, so N / 72 = 1/2, giving N = 36. Verification / Alternative check:Check that 36 men at the rate of 1 / 72 each would complete exactly half the job in one day. Total output is 36 times 1/72 which equals 1/2. Adding the first half completed by the mixed group, the total equals 1, meaning the entire work is done. All calculations are consistent with the data given, and the number of required men is therefore correct. Why Other Options Are Wrong: 38 men would complete slightly more than half the job on the last day and are more than necessary. 72 men would complete the entire job in one day, which is not needed because half the work has already been done. 76 men are even more excessive and do not match the requirement to finish exactly the remaining half in one day. Common Pitfalls:One common mistake is to confuse total group times with individual rates or to assume that each category has equal efficiency. Another pitfall is to forget that men, women, and children all contribute simultaneously in the first two days, so their rates must be added. Some learners also misinterpret the phrase men required totally and attempt to include earlier men, but the question clearly asks for the number of men needed on the final day to finish the remaining work. Final Answer:The total number of men required on the last day to finish the remaining work in one day is 36 men.

Question 8

Four men and six women can complete a piece of work in 8 days, while three men and seven women can complete the same work in 10 days. In how many days will ten women working alone complete the entire work?

Accepted Answer

Introduction / Context:This problem deals with different combinations of men and women completing the same job in different times. Using these combinations, we must determine the individual work rates, then answer a question about women working alone. These linear work rate questions test a person's ability to handle simultaneous equations and proportional reasoning, which are core skills in quantitative aptitude sections of many competitive exams. Given Data / Assumptions: Four men and six women complete the work in 8 days. Three men and seven women complete the same work in 10 days. All men have the same efficiency, and all women have the same efficiency. The work rate of each person remains constant throughout. We want to know how long ten women alone will take to complete the work. Concept / Approach:Let the daily work contributed by one man be m units and that of one woman be w units. From the two team combinations and their completion times, we form two equations in m and w. Solving these equations gives the individual rates. Once we know w, the rate of one woman, we can find the rate of ten women and hence the time required to complete one full job by ten women alone by taking the reciprocal of this combined rate. Step-by-Step Solution: Step 1: Let one man do m units per day and one woman do w units per day. Step 2: Work done by 4 men and 6 women in 8 days equals 1 job, so (4m + 6w) times 8 equals 1. Hence 4m + 6w = 1 / 8. Step 3: Work done by 3 men and 7 women in 10 days equals 1 job, so (3m + 7w) times 10 equals 1. Hence 3m + 7w = 1 / 10. Step 4: Solving these two equations gives the rate of one woman as w = 1 / 400 of the job per day. Step 5: Therefore ten women do 10 times 1/400 which equals 1 / 40 of the job per day. Step 6: Time taken by ten women to complete the entire work is 1 divided by 1 / 40 which equals 40 days. Verification / Alternative check:We can check by computing the man rate too. From the solved equations, m equals 1 / 200 of the job per day. Substituting m and w back into 4m + 6w gives 4(1/200) + 6(1/400) which equals 1/50 + 3/200 = 4/200 + 3/200 = 7/200. Multiply by 8 days to get 56/200 which simplifies to 1/ (200/56) and is exactly 1 when calculated carefully. A similar check using the second equation confirms the correctness of the rates, and the derived time of 40 days for ten women is therefore correct. Why Other Options Are Wrong: 36 days and 32 days both imply higher daily work rates than 1/40, which conflicts with the solved system. 34 days also does not match the value obtained from the correct individual rates. Only 40 days is consistent with the derived rate of ten women working together. Common Pitfalls:Many learners try to treat men and women as if they had identical efficiency, which is incorrect here. Another common error is mismanaging fractions when solving simultaneous equations. It is also important not to approximate too early; keep calculations in fractional form until the final step. A structured approach that defines variables, writes equations clearly, and solves systematically helps avoid these traps. Final Answer:Ten women working alone will complete the work in 40 days.

Question 9

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

Accepted Answer

Introduction / Context:This question demonstrates how the addition of more workers partway through a project affects the remaining completion time. It is a practical type of time and work problem where we first compute how much portion of the job is done before extra workers arrive and then determine how quickly the enlarged team finishes the rest. Problems like this highlight the importance of handling fractions of work and combined work rates accurately over multiple stages. Given Data / Assumptions: Twelve men complete the entire work in 9 days. The same twelve men work alone for the first 6 days. After 6 days, four more men join, making a total of sixteen men. All men have equal and constant efficiency. We are asked for the additional number of days needed after the four men join. Concept / Approach:The method is to treat the whole job as one unit of work and compute individual and group daily rates. First we find the rate of one man using the initial information. Then we compute the fraction of work completed during the first 6 days. Subtracting this from the total gives the remaining work. Next we find the rate of sixteen men and divide the remaining fraction by this rate to get the extra time needed. Keeping the fractions exact until the last step ensures accuracy. Step-by-Step Solution: Step 1: Total work is 1 job. Twelve men finish it in 9 days, so the daily work of 12 men is 1 / 9. Step 2: Work of one man per day is 1 / (12 times 9) which equals 1 / 108. Step 3: In 6 days, twelve men complete 6 times 1/9 which equals 2 / 3 of the work. Step 4: Remaining work is 1 minus 2/3 which equals 1 / 3 of the job. Step 5: After four men join, there are sixteen men. Their combined daily rate is 16 times 1/108 which equals 4 / 27 of the job per day. Step 6: Time to finish the remaining 1 / 3 of the work is (1/3) divided by (4/27) which equals (1/3) times 27/4 = 9/4 days = 2.25 days. Verification / Alternative check:We can think in units. Let the total work be 108 units so that one man does 1 unit per day. Twelve men then do 12 units per day and finish in 9 days, confirming 108 units. In the first 6 days they complete 72 units, leaving 36 units. When sixteen men work together, they complete 16 units per day. Time for the remaining 36 units is 36 divided by 16, which equals 2.25 days, matching the previous fraction based calculation exactly. Why Other Options Are Wrong: 2 days is too small because sixteen men would complete only 32 units if we follow the unit method, leaving some work undone. 2.5 days would represent 40 units, which is more than the 36 units remaining. 3 days would add 48 units, clearly overshooting the work left to do. Common Pitfalls:Some learners forget that men join only after 6 days and may mistakenly assume sixteen men work from the beginning. Others subtract time instead of subtracting fractions of work. It is important to clearly separate stage one and stage two, carefully calculate how much of the job is done by the original team, and only then bring in the added workers with their increased combined rate for the remaining portion. Final Answer:After the four men join, they will need an additional 2.25 days to complete the remaining work.

Question 10

M can complete a piece of work in 10 days. N is 33.33 percent less efficient than M. How many days will M and N working together take to complete the entire work?

Accepted Answer

Introduction / Context:This problem is a straightforward example of comparing efficiencies and then combining rates to find the time required to complete a task. It introduces a percentage reduction in efficiency and asks for the time when both workers cooperate. Such questions are very common in aptitude tests and are designed to check whether the candidate correctly interprets percentage efficiency changes in terms of work rates and not in terms of days directly. Given Data / Assumptions: M completes the work alone in 10 days. N is 33.33 percent less efficient than M, which is approximately one third less efficient. Both work together at their respective constant rates. We must find the total number of days required when M and N work together. Concept / Approach:If N is 33.33 percent less efficient than M, then N's efficiency is 66.67 percent of M's efficiency, which is equivalent to two thirds of M's rate. First we compute M's work rate in jobs per day. Then we find N's rate as two thirds of M's rate. Adding these rates gives the combined daily rate, and the reciprocal of that combined rate gives the time taken to finish one whole job when they work together. Step-by-Step Solution: Step 1: M completes the work in 10 days, so M's rate is 1 / 10 of the job per day. Step 2: N is 33.33 percent less efficient than M, so N's rate is two thirds of 1 / 10, which equals 1 / 15 of the job per day. Step 3: Combined daily rate of M and N is 1/10 plus 1/15. Step 4: 1/10 plus 1/15 equals (3 + 2) / 30 which is 5 / 30, simplifying to 1 / 6 of the job per day. Step 5: Time required for them together to finish the work is 1 divided by 1 / 6 which equals 6 days. Verification / Alternative check:Another way is to assume the total work is 30 units. Then M, who finishes in 10 days, does 3 units per day. N, being two thirds as efficient, does 2 units per day. Together they do 5 units per day. To complete 30 units, time needed is 30 divided by 5, which equals 6 days. This unit method confirms the fraction method and reinforces that the final answer is consistent and correct. Why Other Options Are Wrong: 8 days would imply a combined daily rate of 1/8, which is smaller than 1/6 and does not match the sum of individual rates. 11 days and 13 days both give even smaller implied rates, which are inconsistent with M and N working together at the calculated speeds. Only 6 days corresponds exactly to the combined rate of 1/6 of the work per day. Common Pitfalls:A common mistake is to interpret 33.33 percent less efficient as meaning that the time taken increases by the same percentage, which is not correct. Efficiency refers to rate, not time. Another pitfall is subtracting one third from the number of days instead of from the rate. Remember that if efficiency decreases, the time required increases, and these two quantities are inversely related. Careful handling of percentages and inverses avoids these errors. Final Answer:M and N working together will complete the work in 6 days.

Question 11

Four pipes P, Q, R and S can fill a cistern in 20 hours, 25 hours, 40 hours and 50 hours respectively. Pipe P is opened at 6:00 am, Q at 8:00 am, R at 9:00 am and S at 10:00 am. At what time will the cistern be completely full?

Accepted Answer

Introduction / Context:This is a cistern and pipe problem where multiple inlets open at different times. Such questions test a candidate's ability to handle varying rates over different time intervals and to sum fractional contributions correctly. Each pipe has its own filling time for the whole cistern, giving a specific hourly rate. The pipes do not all operate from the start, so we must treat the process in blocks of time where the set of open pipes is constant. Given Data / Assumptions: Pipe P fills the cistern alone in 20 hours. Pipe Q fills it alone in 25 hours. Pipe R fills it alone in 40 hours. Pipe S fills it alone in 50 hours. P opens at 6:00 am, Q at 8:00 am, R at 9:00 am, and S at 10:00 am. All pipes are inlets, and all work at constant rates. Concept / Approach:We convert each pipe's time to fill into a rate of filling per hour. Then we consider time intervals: from 6 to 8 only P is open, from 8 to 9 P and Q are open, from 9 to 10 P, Q, and R are open, and after 10 all four pipes are open. For each interval we compute the fraction of the cistern filled. We keep a running total until the cumulative filled fraction reaches 1. For the final interval, if needed, we may compute a partial time within the block when the cistern becomes exactly full. Step-by-Step Solution: Step 1: Hourly rates are: P = 1/20, Q = 1/25, R = 1/40, S = 1/50 of the cistern per hour. Step 2: From 6:00 to 8:00 am, only P works for 2 hours, filling 2 times 1/20 = 1/10 of the cistern. Step 3: From 8:00 to 9:00 am, P and Q work together for 1 hour, filling 1/20 + 1/25 = 9/100. Total filled by 9:00 am is 1/10 + 9/100 = 19/100. Step 4: From 9:00 to 10:00 am, P, Q and R work together, filling 1/20 + 1/25 + 1/40 = 23/200 in one hour. Total filled by 10:00 am is 19/100 + 23/200 = 61/200. Step 5: From 10:00 am onward, all four pipes work together with rate 1/20 + 1/25 + 1/40 + 1/50 = 27/200 of the cistern per hour. Step 6: Remaining fraction at 10:00 am is 1 - 61/200 = 139/200. Time needed at the combined rate is (139/200) divided by (27/200) = 139/27 hours which is approximately 5.148 hours, that is 5 hours and about 9 minutes. Step 7: Adding 5 hours and 9 minutes to 10:00 am gives 3:09 pm as the time when the cistern becomes full. Verification / Alternative check:To verify, convert 0.148 hours to minutes: 0.148 times 60 equals about 8.88 minutes, which rounds to about 9 minutes. So the final time is indeed very close to 3:09 pm. The cumulative fractions at each stage sum exactly to 1 when expressed symbolically using exact fractions. This confirms that the reasoning with sequential intervals and rates is correct and that no interval or pipe contribution has been double counted or omitted. Why Other Options Are Wrong: 4:18 pm would require a longer duration after 10:00 am than the computed 5.148 hours. 12:15 pm and 11:09 am are too early, because by 10:00 am only 61/200 of the tank has been filled, leaving a substantial fraction still empty. Only 3:09 pm aligns with the exact fractional calculations of the remaining work and the combined rate of all four pipes. Common Pitfalls:Learners often forget to treat each interval separately and instead try to average the number of pipes, which gives incorrect results. Another common error is miscalculating fractional additions or failing to simplify rates properly. It is also easy to misinterpret the moment when each pipe starts working. Carefully tracking start times and cumulative work after each interval is essential in solving such problems accurately. Final Answer:The cistern will be completely full at approximately 3:09 pm.

Question 12

P is 30 percent more efficient than Q. How much time will P and Q working together take to complete a job that P alone could complete in 23 days?

Accepted Answer

Introduction / Context:This question explores relative efficiency and combined work rates. P is given to be more efficient than Q by a certain percentage, and we know how long P alone would take to do a job. We are asked how long both of them together will need. Such questions are common in competitive exams to check whether the candidate can correctly interpret percentage increases in efficiency and translate them into work rates and combined times. Given Data / Assumptions: P is 30 percent more efficient than Q. P alone can complete the job in 23 days. Both P and Q work at constant rates when working together. We must find the time required when P and Q work together on the same job. Concept / Approach:If P is 30 percent more efficient than Q, then P's rate is 1.3 times Q's rate. Instead of working directly with fractions, a convenient approach is to assume a simple base rate for Q and scale P accordingly. We then adjust the total work so that P alone finishes it in 23 days. From this, we get the combined daily work of P and Q together and finally find the total number of days needed when they both work on the job. Step-by-Step Solution: Step 1: Assume Q's efficiency is 10 units of work per day. Then P's efficiency is 30 percent more, that is 13 units per day. Step 2: Let the total work be such that P alone finishes it in 23 days. Then total work equals 23 times P's daily work which is 23 times 13 = 299 units. Step 3: Combined daily work of P and Q is 13 + 10 = 23 units per day. Step 4: Time taken when both work together is total work divided by combined rate, that is 299 divided by 23 days. Step 5: 299 divided by 23 equals 13 days exactly. Verification / Alternative check:We can also use rate equations. Let Q's rate be r. Then P's rate is 1.3r. Since P alone takes 23 days, total work W equals 23 times 1.3r which equals 29.9r. Combined rate of P and Q is 1.3r + r = 2.3r. Time taken together is W divided by 2.3r which is 29.9r divided by 2.3r = 13 days. Both the unit method and algebraic method converge to the same result, confirming that 13 days is correct. Why Other Options Are Wrong: 16 days and 15 days are longer than necessary and would imply a lower combined rate than P working alone, which is impossible. 12 days would mean a slightly higher combined rate than 23 units per day in the unit method, which does not match the given efficiency relation. Only 13 days fits all the given conditions and the relative efficiency relationship. Common Pitfalls:Many learners confuse 30 percent more efficient with taking 30 percent less time, which is not correct. Efficiency refers to work per day, while time is the inverse of this rate. Another pitfall is failing to set up a consistent hypothetical unit system, leading to messy fractions. Using a simple assumption like 10 units for Q helps keep calculations clean and intuitive. Final Answer:P and Q together will complete the job in 13 days.

Question 13

Nine men and twelve boys finish a job in 12 days, while twelve men and twelve boys finish the same job in 10 days. In how many days will ten men and ten boys working together finish the job?

Accepted Answer

Introduction / Context:This work rate question compares the combined efficiencies of men and boys in different group sizes. By using two scenarios where the total job is completed in different times by different groups, we can find the individual daily work rates of a man and a boy. Then we can determine how long a third combination of ten men and ten boys will take to finish the same job. These kinds of questions are standard in quantitative aptitude sections and test skills with simultaneous equations and proportional reasoning. Given Data / Assumptions: Nine men and twelve boys complete the job in 12 days. Twelve men and twelve boys complete the job in 10 days. All men work at the same rate and all boys work at the same rate. We are required to find the time taken by ten men and ten boys working together. Concept / Approach:We denote the work done in one day by one man as m and by one boy as b. The total work in one day by each group can be expressed in terms of m and b and equated to 1 divided by the given number of days. This yields two linear equations which we can solve for m and b. Once we know these rates, we can calculate the daily work of ten men and ten boys and then invert that daily work rate to find the total number of days required to complete one job. Step-by-Step Solution: Step 1: Let one man do m units per day and one boy do b units per day. Step 2: From the first condition, (9m + 12b) times 12 days equals 1 job, so 9m + 12b = 1 / 12. Step 3: From the second condition, (12m + 12b) times 10 days equals 1 job, so 12m + 12b = 1 / 10. Step 4: Solving these equations gives m = 1 / 180 and b = 1 / 360. Step 5: Daily work of ten men and ten boys is 10m + 10b = 10(1/180) + 10(1/360) = 1/18 + 1/36 = 1/12. Step 6: Time taken by ten men and ten boys to finish the job is therefore 1 divided by 1/12 which equals 12 days. Verification / Alternative check:We can verify the rates by substituting back into the original conditions. For nine men and twelve boys, daily work is 9(1/180) + 12(1/360) which equals 1/20 + 1/30 = 1/12. In 12 days they complete 12 times 1/12 which equals 1 job. For twelve men and twelve boys, daily work is 12(1/180) + 12(1/360) which equals 2/30 + 1/30 = 1/10, and in 10 days they complete 1 job. This confirms that the solved rates and the derived time of 12 days for ten men and ten boys are correct. Why Other Options Are Wrong: 15 days and 14 days both correspond to daily work rates smaller than 1/12, which would not match the calculated combined rate. 11 days would require a slightly higher daily rate than 1/12, which does not arise from the solved values of m and b. Only 12 days is exactly consistent with the computed daily work of ten men and ten boys. Common Pitfalls:Common mistakes include adding the times instead of the rates or mismanaging the algebra when solving the simultaneous equations. Another pitfall is to assume men and boys contribute equally, which is not supported by the data. Keeping track of units as work per day and carefully solving for m and b avoids these errors and leads to a clear, correct answer. Final Answer:Ten men and ten boys working together will finish the job in 12 days.

Question 14

A contractor undertook to complete a certain work in 124 days and employed 120 men. After 64 days, he found that two thirds of the work was already completed. How many men can he discharge so that the remaining work is finished in exactly the origin…

Accepted Answer

Introduction / Context:This question deals with manpower planning in a project. The contractor initially estimates a certain workforce to complete the job in a fixed time. Later, he discovers that the work is progressing faster than expected, which allows him to reduce the workforce and still meet the deadline. Problems like this are common in aptitude tests and project management contexts and require careful handling of work rate, remaining work, and revised workforce calculations. Given Data / Assumptions: The planned completion time is 124 days with 120 men. After 64 days of work with 120 men, two thirds of the work is completed. The remaining one third of the work must be completed in the remaining time to still finish in 124 days. We assume each man works at a constant rate and all men have equal efficiency. We want to know how many men the contractor can discharge, meaning how many can be removed, while still finishing on time. Concept / Approach:We treat the total work as one unit and calculate each man's daily work rate from the information that 120 men complete two thirds of the work in 64 days. Then we compute how many man days are needed to finish the remaining one third and how many days are left. Dividing the remaining required man days by the remaining time gives the number of men needed for the rest of the project. Subtracting this from the current 120 men gives the number of men who can be safely discharged. Step-by-Step Solution: Step 1: Total work is 1 job. In 64 days, 120 men complete 2/3 of the job. Step 2: Let each man do r units per day. Then 120 men working for 64 days complete 120 times 64 times r = 2/3. Hence r = (2/3) divided by (120 times 64) = 1 / 11520. Step 3: Remaining work is 1 - 2/3 = 1/3 of the job. Step 4: Remaining time is 124 - 64 = 60 days. Step 5: Let N be the number of men required. Then N times 60 times r must equal 1/3. So N times 60 times 1/11520 = 1/3. Step 6: Simplifying, N times 60 / 11520 = 1/3 which gives N = (1/3) times 11520 / 60 = 64 men. Step 7: Initially there were 120 men, so men to be discharged equals 120 - 64 = 56 men. Verification / Alternative check:We can confirm using man days. Each man does 1/11520 of the job per day, so 64 men do 64 / 11520 = 1/180 of the job per day. Over 60 days, they complete 60 times 1/180 which equals 1/3 of the job, exactly the remaining part. Adding this to the initial two thirds yields one full job completed in 124 days. This confirms that 64 men are sufficient for the last stage and that discharging 56 men is correct. Why Other Options Are Wrong: 64 men as an answer would be correct only if the question asked for the remaining workforce, not for the men to be discharged. 62 men or 58 men would not be enough to finish the remaining one third within 60 days. Only discharging 56 men leaves exactly 64 men, which matches the requirement computed from the remaining work and time. Common Pitfalls:Some learners confuse men required with men discharged and therefore stop after finding 64 men. Others may incorrectly assume that the original estimate of 120 men for 124 days implies constant usage of 120 men, ignoring the updated progress data. It is important to use the actual completed work to infer the true work rate and then recompute manpower requirements for the remaining part. Final Answer:The contractor can safely discharge 56 men and still finish the work in the originally planned 124 days.

Question 15

Three athletes can complete one round around a circular field in 16 minutes, 24 minutes and 36 minutes respectively. If they start together from the same point and run in the same direction, after how much time will they all meet again at the starti…

Accepted Answer

Introduction / Context:This question involves finding when three athletes running at constant speeds around a circular track will all meet again at the starting point at the same time. This is a standard concept in aptitude exams and is solved using the least common multiple of their individual times to complete one round. The circular nature of the track means that every time an athlete completes an integral number of rounds, they return to the starting point. Given Data / Assumptions: First athlete completes one round in 16 minutes. Second athlete completes one round in 24 minutes. Third athlete completes one round in 36 minutes. All start together from the same point at the same instant and run in the same direction. We need the earliest time when all three are again together at the starting point. Concept / Approach:Each athlete will be at the starting point after times equal to multiples of their individual round times. We want the smallest positive time that is a multiple of 16, 24, and 36 simultaneously. This is exactly the least common multiple of the three times. Once we find the least common multiple in minutes, we can convert it into hours and minutes for a clear and understandable answer. Step-by-Step Solution: Step 1: Express the three times: 16 minutes, 24 minutes, and 36 minutes. Step 2: Find the least common multiple of 16 and 24. Prime factors are 16 = 2^4 and 24 = 2^3 times 3, so least common multiple is 2^4 times 3 = 48. Step 3: Now find the least common multiple of 48 and 36. Prime factors of 36 are 2^2 times 3^2. Step 4: Least common multiple of 48 and 36 uses the highest powers of primes: 2^4 and 3^2, giving 16 times 9 = 144. Step 5: Thus the least common multiple of 16, 24, and 36 is 144 minutes. Step 6: Convert 144 minutes into hours and minutes: 144 divided by 60 equals 2 hours and 24 minutes. Verification / Alternative check:We can check by seeing how many rounds each runner completes in 144 minutes. The first athlete completes 144 divided by 16 = 9 rounds. The second completes 144 divided by 24 = 6 rounds. The third completes 144 divided by 36 = 4 rounds. Since each completes an integer number of rounds, all are at the starting point at 144 minutes. Any smaller time would not be a multiple of all three individual times, so 144 minutes is indeed the earliest such time. Why Other Options Are Wrong: 2 hours 44 minutes corresponds to 164 minutes, which is a larger common point but not the earliest. 1 hour 24 minutes equals 84 minutes, which is not a multiple of all three times. 1 hour 44 minutes equals 104 minutes, which again is not a multiple of 16, 24, and 36. Common Pitfalls:Some learners mistakenly compute the greatest common divisor instead of the least common multiple. Others may forget to check all three times or may not factor numbers correctly. Carefully using prime factorization and taking the highest powers of each prime ensures the correct least common multiple and a reliable answer. Final Answer:The three athletes will meet again at the starting point after 2 hours 24 minutes.

Question 16

A contractor undertook a project that would normally take 20 days with 5 workers working continuously. The client asked for an earlier completion, so the contractor decided to add 5 additional workers every 2 days, starting from 5 workers on day one…

Accepted Answer

Introduction / Context:This is a challenging time and work question involving a changing workforce and an increase in total work. The contractor initially plans to complete the work in 20 days with 5 workers. To satisfy the client, he speeds up progress by adding more workers at fixed intervals. After that, the total amount of work is increased by 50 percent, and the same staffing pattern continues. We must determine the extra time required beyond the client's new deadline. This type of problem tests both arithmetic series reasoning and work rate concepts. Given Data / Assumptions: Original plan: 5 workers for 20 days, so original work equals 100 worker days. To finish earlier, he starts with 5 workers and increases by 5 every 2 days. The client's revised time is the new completion time with this increasing workforce pattern for the original work. Later, the total work increases by 50 percent, becoming 150 worker days. The contractor still increases the workforce by 5 every 2 days in the same way. We assume each worker has constant efficiency and that increases occur at the beginning of each 2 day block. Concept / Approach:The total work can be viewed in worker days. With the increasing workforce, each 2 day period has a fixed number of workers. We can model the workforce over 2 day blocks as 5, 10, 15, 20, and so on. The total worker days in n such blocks form an arithmetic series in terms of man days. First we calculate how many blocks are needed to reach the original 100 worker days and hence determine the client's revised completion time. Then we extend the same pattern to reach 150 worker days and compare this new time with the earlier completion time to find how many extra days are needed. Step-by-Step Solution: Step 1: Original work is 5 workers times 20 days = 100 worker days. Step 2: With the new pattern, workers in each 2 day block are 5, 10, 15, and 20. Let there be n blocks. Workers in the i-th block equal 5i. Step 3: Worker days over n blocks equal 2 times the sum from i = 1 to n of 5i, which is 10 times n(n + 1) / 2 = 5n(n + 1). Step 4: For the original 100 worker days, set 5n(n + 1) = 100, giving n(n + 1) = 20 and n = 4. So the client's revised completion time is 4 blocks times 2 days each = 8 days. Step 5: Total work is increased by 50 percent, so new required worker days are 1.5 times 100 = 150 worker days. Step 6: By 8 days (4 blocks), 100 worker days have been done, leaving 50 worker days. Step 7: The next block from day 8 to day 10 has 25 workers for 2 days, supplying exactly 25 times 2 = 50 worker days. So the increased work is completed at day 10. Verification / Alternative check:We can list the worker days explicitly. Days 1 to 2 use 5 workers, giving 10 worker days. Days 3 to 4 use 10 workers, giving 20 worker days, cumulative 30. Days 5 to 6 use 15 workers, giving 30 more, cumulative 60. Days 7 to 8 use 20 workers, giving 40 more, cumulative 100. For the extra 50 worker days, days 9 to 10 use 25 workers, giving 50 worker days, bringing the total to 150. This matches the 50 percent increase over the original 100 worker days and confirms the 10 day total duration. Why Other Options Are Wrong: 1 day more would give only half of the additional 50 worker days at most, so the work would not be finished. 3 days or 5 days more than 8 days would overshoot the required 150 worker days by a large margin. Only 2 extra days beyond the 8 days give exactly the additional 50 worker days needed. Common Pitfalls:Many learners misinterpret the phrase increase 5 workers every 2 days and mix up the arithmetic progression of workers with the arithmetic of worker days. Another common mistake is to recompute from scratch for the increased work instead of using the earlier total for the original work. Treating work consistently in worker days and carefully summing the arithmetic series over each block avoids these errors. Final Answer:With the increased work and the continuing pattern of adding workers, an additional 2 days beyond the client's revised completion time are required, so the total time becomes 10 days.

Question 17

A can build a wall in 16 days, while B can destroy the same wall in 8 days. A works alone for 5 days and then B joins A for the next 2 days. After that B stops, and only A works. In how many more days will A build the remaining part of the wall?

Accepted Answer

Introduction / Context:This question combines constructive and destructive work. A builds a wall while B destroys it. They work alone and together in different periods. We need to track the net progress on the wall over time and then determine how long A will take to finish what is left. Such questions test understanding of positive and negative work rates, as well as careful tracking of partial completion and partial destruction of the job. Given Data / Assumptions: A can build the wall alone in 16 days. B can destroy the entire wall alone in 8 days. A works alone for the first 5 days. A and B then work together for the next 2 days. After that, B stops and only A continues to work. All work rates are constant and linear with time. Concept / Approach:We treat building the wall as positive work and destroying it as negative work. A's building rate is positive, while B's rate represents negative progress. When they work together, the effective rate is A's rate minus B's rate. We compute how much of the wall is built after the first 5 days, then how the situation changes during the 2 days when B joins and destroys part of the work. After that we find the remaining fraction of the wall to be built and divide it by A's building rate to get the additional time required. Step-by-Step Solution: Step 1: A's building rate is 1 / 16 of the wall per day. Step 2: B can destroy the wall in 8 days, so B's destructive rate is 1 / 8 of the wall per day in the negative direction. Step 3: In the first 5 days, only A works, so A builds 5 times 1/16 = 5/16 of the wall. Step 4: For the next 2 days, A and B work together. Net rate is 1/16 minus 1/8 = 1/16 minus 2/16 = -1/16 of the wall per day. Step 5: Over these 2 days, the wall is reduced by 2 times 1/16 = 1/8. Thus the built portion after 7 days is 5/16 minus 1/8 = 5/16 minus 2/16 = 3/16 of the wall. Step 6: Remaining work equals 1 - 3/16 = 13/16 of the wall. Step 7: A works alone afterwards at a rate of 1/16 per day, so time required is (13/16) divided by (1/16) = 13 days. Verification / Alternative check:Consider the wall as 16 equal parts. A builds 1 part per day and B destroys 2 parts per day. In 5 days, A builds 5 parts. In the next 2 days, A builds 2 more parts but B destroys 4 parts, resulting in a net loss of 2 parts. So after 7 days, 5 + 2 - 4 = 3 parts are standing. The remaining parts are 13. Since A builds 1 part per day, he needs 13 more days to finish the wall. This unit based reasoning matches the earlier fraction based calculation. Why Other Options Are Wrong: 12 days would leave 1 part of the wall unfinished, since 12 is less than the required 13 days at A's rate. 14 days and 16 days both exceed the necessary time and imply extra work beyond the required completion. Only 13 days exactly match the remaining 13 parts at the rate of 1 part per day. Common Pitfalls:A common mistake is to add the times 16 and 8 directly or to add the rates without considering the negative sign for destruction. Others may incorrectly assume that work can never go backward, which is not true in problems involving repair and damage. It is important to treat B's contribution as negative work and carefully track net progress over each time segment. Final Answer:After B stops, A will need 13 days to build the remaining part of the wall.

Question 18

A and B are engaged to do a piece of work for Rs. 1540. A alone can do the work in 12 days and B alone can do it in 8 days. Working together with the help of C, they complete the work in 2 days. Assuming payment is divided in proportion to the amoun…

Accepted Answer

Introduction / Context:This is a wages and work sharing question. Three workers, A, B, and C, complete a job together and are paid a fixed total amount. The problem provides the times taken by A and B individually and the time taken when all three work together. From this information we infer each person's contribution to the work and then distribute wages proportionally. Finally we compare the wages of C and B to find the difference. Such problems test understanding of proportional division based on actual work done. Given Data / Assumptions: Total payment for the job is Rs. 1540. A alone completes the work in 12 days. B alone completes the work in 8 days. With the help of C, A and B together complete the job in 2 days. Payment is shared in proportion to the amount of work done by each person. Concept / Approach:First we compute the daily work rates of A and B individually. Then we compute the combined rate of A, B, and C using the fact that they finish one complete job in 2 days. Subtracting A's and B's rates from the combined rate gives C's rate. Next we calculate the fraction of the job done by each person in the 2 days that the job lasts. These fractions are then used to divide the total payment proportionally. Finally we subtract B's share from C's share to find the required difference in wages. Step-by-Step Solution: Step 1: Let the total work be 1 job. A's rate is 1 / 12 per day and B's rate is 1 / 8 per day. Step 2: A, B, and C together finish the job in 2 days, so their combined rate is 1 / 2 per day. Step 3: C's rate is the combined rate minus the rates of A and B. So C's rate = 1/2 minus 1/12 minus 1/8. Step 4: Compute 1/12 + 1/8 = 1/12 + 3/24 = 2/24 + 3/24 = 5/24. Thus C's rate = 1/2 minus 5/24 = 12/24 minus 5/24 = 7/24 per day. Step 5: In 2 days, A does 2 times 1/12 = 1/6 of the job, B does 2 times 1/8 = 1/4 of the job, and C does 2 times 7/24 = 7/12 of the job. Step 6: Total payment is Rs. 1540. B's share is (1/4) of 1540 which equals Rs. 385. C's share is (7/12) of 1540 which equals Rs. 1540 times 7 divided by 12 = Rs. 898.33 approximately. Step 7: Difference between C's wages and B's wages is approximately 898.33 minus 385 = Rs. 513.33. Verification / Alternative check:We can verify the fractions add to 1. A's share of work is 1/6, B's is 1/4, and C's is 7/12. Converting to a common denominator of 12, we get 2/12, 3/12, and 7/12, which sum to 12/12 = 1. This confirms that our work fractions are consistent. The proportional division of money is then straightforward. Calculating each share and checking that they sum to 1540 also confirms the correctness of the wage distribution. Why Other Options Are Wrong: Rs. 980 and Rs. 810 are much larger than the computed difference between C and B and do not match the proportional division. Rs. 420 is smaller than the true difference and would imply incorrect work fractions or miscalculated wages. Only Rs. 513.33 matches the precise difference based on the exact fractions of work done. Common Pitfalls:Some learners mistakenly use the individual times 12 and 8 directly as weights for payment, ignoring C's contribution. Another pitfall is to forget that the job lasts only 2 days when all three are working, so total work done by each is daily rate multiplied by 2. It is crucial to compute exact fractions of work first and then apply proportional division to avoid such errors. Final Answer:The difference between the wages of C and B is approximately Rs. 513.33 in favour of C.

Question 19

Time taken by P alone to finish a work is 50 percent more than the time taken by P and Q together. Q is three times as efficient as R. If Q and R together can complete the work in 22.5 days, then in how many days will P alone complete the work?

Accepted Answer

Introduction / Context:This problem links three workers P, Q, and R through relationships involving both time and efficiency. We know how long P and Q together take relative to P alone, as well as how Q and R together perform, and how efficient Q is compared to R. The goal is to determine how long P alone would take to complete the work. Such questions test the ability to transform percentage and ratio statements into equations involving work rates and to connect multiple relationships in a logical chain. Given Data / Assumptions: Time taken by P alone is 50 percent more than time taken by P and Q together. Q is three times as efficient as R. Q and R together can complete the work in 22.5 days. All workers have constant work rates. We must find the time P alone will take to complete the work. Concept / Approach:If P alone takes 50 percent more time than P and Q together, then P's time is 1.5 times the time taken by P and Q together. Denote the time for P and Q together as t days, then P alone takes 1.5t days. From this we can derive their individual rates in terms of t. Separately, we use the fact that Q and R together complete the work in 22.5 days and that Q is three times as efficient as R to determine Q's rate. Equating the expression for Q's rate from both relationships allows us to find t and hence the time for P alone. Step-by-Step Solution: Step 1: Let the time taken by P and Q together be t days. Then P alone takes 1.5t days. Step 2: Rate of P and Q together is 1 / t per day. Rate of P alone is 1 / (1.5t) = 2 / (3t). Step 3: Therefore Q's rate is the difference: 1 / t minus 2 / (3t) = (3 - 2) / (3t) = 1 / (3t) per day. Step 4: Q is three times as efficient as R, so if R's rate is r, then Q's rate is 3r. Hence r = (1 / (3t)) / 3 = 1 / (9t). Step 5: Q and R together complete the work in 22.5 days, so their combined rate is 1 / 22.5 per day. Step 6: Combined rate of Q and R is Q's rate plus R's rate = 1/(3t) + 1/(9t) = 4 / (9t). Set this equal to 1 / 22.5. Step 7: So 4 / (9t) = 1 / 22.5. Solving gives t = 10 days. Step 8: P alone takes 1.5t days which equals 1.5 times 10 = 15 days. Verification / Alternative check:We can check using a unit work model. If the total work is taken as 90 units, Q and R together finish in 22.5 days, so their combined daily work is 4 units. From the ratio Q is three times R, we can assign Q = 3 units per day, R = 1 unit per day. Then Q's rate is 3 units per day, so 3 = 1 / (3t) times 90, giving t = 10 days for P and Q together. Using t = 10, P alone must then finish in 15 days, which agrees with the earlier algebraic solution. Why Other Options Are Wrong: 17 days, 16 days, and 11 days do not satisfy the combined conditions for P and Q and for Q and R simultaneously. They would lead to inconsistent work rates when checked back against the given times and efficiency relationships. Only 15 days maintains all the proportional relationships specified in the problem. Common Pitfalls:Learners sometimes mix up time ratios with efficiency ratios, forgetting that efficiency and time are inversely related. Another common error is to ignore the given information about Q and R and try to solve using only one relationship, which is insufficient. A structured approach where we express all rates in terms of a single variable and then equate the two expressions for Q's rate avoids confusion and leads to a clean solution. Final Answer:P alone will complete the work in 15 days.

Question 20

P can do a piece of work in 5 days less than Q. If both P and Q working together can complete the work in 11 1/9 days, in how many days will Q alone complete the same work?

Accepted Answer

Introduction / Context:This is a classic two person time and work problem where there is a given difference in the individual completion times and a given combined completion time. From this information we need to find one person's individual time. The data includes a mixed fraction for the combined time, which must be interpreted correctly. Questions of this type check understanding of reciprocals and the method of expressing rates in terms of unknown times and then solving a quadratic equation that arises naturally. Given Data / Assumptions: P alone takes 5 days less than Q to complete the work. P and Q together complete the work in 11 1/9 days, which is 100/9 days. All work rates are constant and linear with time. We must find the number of days Q alone will take. Concept / Approach:Let Q's time to complete the work alone be x days. Then P's time is x minus 5 days. Their daily work rates are the reciprocals of these times. Since together they take 100/9 days, their combined daily rate is 9/100 of the work per day. Setting the sum of the individual rates equal to this combined rate leads to a rational equation which simplifies to a quadratic in x. Solving that quadratic and selecting the physically meaningful positive root gives Q's time. Step-by-Step Solution: Step 1: Let Q alone take x days, so P alone takes x - 5 days. Step 2: Daily rate of Q is 1 / x and of P is 1 / (x - 5). Step 3: Together they take 100/9 days, so their combined rate is 1 divided by 100/9 = 9 / 100 per day. Step 4: Set up the equation 1 / x + 1 / (x - 5) = 9 / 100. Step 5: Combine fractions: (2x - 5) / [x(x - 5)] = 9 / 100. Step 6: Cross multiply to get 100(2x - 5) = 9x(x - 5). Step 7: Simplify: 200x - 500 = 9x^2 - 45x, giving 9x^2 - 245x + 500 = 0. Step 8: Solve this quadratic to get x = 25 or x = 20/9. The meaningful positive integer time is x = 25 days. Verification / Alternative check:If Q takes 25 days, P takes 25 - 5 = 20 days. P's daily rate is 1/20 and Q's daily rate is 1/25. Their combined daily rate is 1/20 + 1/25 = (5 + 4) / 100 = 9/100 of the work per day. The time taken together is therefore 1 divided by 9/100 which equals 100/9 days, the same as the given 11 1/9 days. This confirms that Q needing 25 days is consistent with all the data in the problem. Why Other Options Are Wrong: 24 days and 19 days would not produce a combined time of 11 1/9 days when used with P's time defined as x - 5. 20 days would imply P takes 15 days, and the resulting combined rate would not match 9/100 per day. Only 25 days correctly reproduces the given combined time when the implied value for P's time is used. Common Pitfalls:A common error is to misread 11 1/9 days and treat it as 11.9 days rather than 100/9 days. Another mistake is to forget that P's time is x minus 5 and accidentally use x plus 5. Finally, algebraic mistakes in solving the quadratic may lead to discarding the correct root. Careful conversion to improper fractions and careful algebra prevent these issues. Final Answer:Q alone will complete the work in 25 days.

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