Question 1

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

Accepted Answer

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

Question 2

If five persons together can make five mats in 5 hours, then how many mats will ten persons make in 10 hours, assuming everyone works at the same constant rate?

Accepted Answer

Introduction / Context: This is a classic work and productivity problem where we use the idea of “person-hours” or “man-hours”. We are told what five persons can produce in a certain number of hours and are asked to generalize the result to a different number of persons and hours. The key goal is to understand that total work done is proportional to the product of the number of workers and the number of hours, provided the rate is constant. Given Data / Assumptions: - Five persons together can make five mats in 5 hours. - We assume each person works at the same constant rate. - We need to find the number of mats that ten persons can make in 10 hours. - The type of work and rate per person remain unchanged. Concept / Approach: The core idea is that Work = Rate * Time. For multiple workers with identical efficiency, total work equals the per person rate multiplied by the number of persons and the time worked. First, we determine the combined work rate from the initial situation. From this, we can find the work rate per person per hour. Then we apply that rate to a new situation with ten persons working for 10 hours to compute the total number of mats produced. Step-by-Step Solution: Step 1: Total work in the first scenario is 5 mats. Step 2: Person-hours used = 5 persons * 5 hours = 25 person-hours. Step 3: Rate of work per person-hour = total mats / total person-hours = 5 / 25 = 1 / 5 mat per person-hour. Step 4: In the new scenario, ten persons work for 10 hours. Step 5: Person-hours in the new scenario = 10 persons * 10 hours = 100 person-hours. Step 6: Total mats produced = person-hours * rate per person-hour = 100 * (1 / 5) = 20 mats. Verification / Alternative check: We can think proportionally: doubling the number of persons from 5 to 10 doubles the output per hour, and doubling the time from 5 hours to 10 hours doubles the output again. So, overall, the output increases by a factor of 4. Originally, the group produced 5 mats; four times that is 20 mats, which matches our detailed calculation. Why Other Options Are Wrong: 10 mats: This corresponds only to doubling either persons or hours, not both, so it is too small. 15 mats: This does not match the factor-of-four increase in total person-hours and fails the proportionality check. 5 mats: This would mean no change in output despite more workers and more time, which is clearly incorrect. 25 mats: This is larger than the value obtained by strict proportional reasoning and would require a higher rate than given in the original situation. Common Pitfalls: Students may wrongly assume that doubling persons and doubling time just adds, rather than multiplies, the effect on work done. Another common mistake is to forget to convert the first scenario into a per person per hour rate. Remember that as long as all workers have equal efficiency, total work scales linearly with both the number of workers and the time worked. Final Answer: Ten persons working for 10 hours will make 20 mats.

Question 3

In a fort there was enough food to feed 200 soldiers for 31 days. After 27 days, 120 soldiers left the fort. For how many extra days will the remaining food last for the soldiers who stay?

Accepted Answer

Introduction / Context: This is a question on the concept of “man-days” or “soldier-days” applied to food consumption. We are given that food in a fort is sufficient for a certain number of soldiers for a fixed number of days. Some soldiers leave after a period, and we are asked how long the remaining food will last for the reduced group. This tests understanding of proportionality between total food, number of consumers and time. Given Data / Assumptions: - There is enough food for 200 soldiers for 31 days. - All 200 soldiers initially consume food for 27 days. - After 27 days, 120 soldiers leave the fort. - Therefore, 80 soldiers remain in the fort. - The rate of food consumption per soldier per day remains constant. Concept / Approach: We treat the total food stock in terms of “soldier-days”, which is analogous to total work in man-days. Total food can be expressed as number of soldiers multiplied by the number of days the food would last. Then we subtract the amount of food already consumed in the first 27 days. The remaining quantity of food, measured in soldier-days, is then divided by the number of soldiers left to find how many more days it will last. Step-by-Step Solution: Step 1: Total amount of food in soldier-days = 200 soldiers * 31 days = 6200 soldier-days. Step 2: Food consumed in the first 27 days by 200 soldiers = 200 * 27 = 5400 soldier-days. Step 3: Remaining food = 6200 - 5400 = 800 soldier-days. Step 4: After 27 days, 120 soldiers leave, so remaining soldiers = 200 - 120 = 80 soldiers. Step 5: Time for which remaining food lasts = remaining soldier-days / remaining soldiers. Step 6: Time = 800 soldier-days / 80 soldiers = 10 days. Verification / Alternative check: We can check consistency by thinking in reverse. If 80 soldiers consume the food for 10 days, they use 80 * 10 = 800 soldier-days of food, exactly the remaining amount we computed. Adding the first 5400 soldier-days consumed by 200 soldiers in 27 days gives 5400 + 800 = 6200, which matches the original total food. This confirms that 10 extra days is correct. Why Other Options Are Wrong: 12 days or 8 days: These do not match an integer number of soldier-days when multiplied by 80 soldiers. They would either overuse or underuse the remaining 800 soldier-days of food. 6 days or 4 days: These values are too small and would imply that some food is left over, contradicting the calculation based on total food and consumption. Common Pitfalls: A common mistake is to treat the problem as if days scale directly with soldiers without converting to the common unit of soldier-days. Some students may forget to subtract the food consumed over the first 27 days or miscount the remaining soldiers. Remember that total food is fixed and equal to the product of consumers and days; whenever the number of consumers changes, the number of days must adjust in inverse proportion. Final Answer: The remaining food will last for 10 extra days for the remaining soldiers.

Question 4

A labourer can complete a certain job in 50 hours. After working for 5 hours, he takes a break. What fraction of the job is still left to be completed at that moment?

Accepted Answer

Introduction / Context: This is a straightforward time and work question that focuses on fractions of work done. We are told how long a labourer takes to finish a job if he works alone, and we are asked to find what fraction of the job remains after he works for a shorter period. This is a basic but important concept in many aptitude problems involving work and time. Given Data / Assumptions: - A labourer can complete the entire job in 50 hours. - He works continuously for 5 hours before taking a break. - His work rate remains constant over time. - We treat the total job as one complete unit of work. Concept / Approach: If a worker completes one full job in T hours, then the fraction of work done per hour is 1 / T. Therefore, the fraction of work completed in t hours is t / T. The remaining fraction of work is then 1 minus the fraction already completed. Here we use this idea with T = 50 hours and t = 5 hours to find the remaining part of the job. Step-by-Step Solution: Step 1: Time taken to complete the entire job = 50 hours. Step 2: Fraction of work done per hour = 1 / 50 of the job. Step 3: Time actually worked before the break = 5 hours. Step 4: Fraction of job completed in 5 hours = 5 * (1 / 50) = 5 / 50 = 1 / 10. Step 5: Remaining fraction of the job = 1 - 1 / 10 = 9 / 10. Step 6: In decimal form, 9 / 10 = 0.9. Verification / Alternative check: We can verify by thinking proportionally. Five hours is 5 / 50 = 1 / 10 of the total required 50 hours. Since work done is directly proportional to time worked in this kind of problem, the labourer completes 1 / 10 of the job. That automatically means 9 / 10 of the job remains, which is 0.9 as a decimal. This matches our calculation exactly. Why Other Options Are Wrong: 0.8: This would mean only 80 percent of the job is left, implying that 20 percent is done, which would correspond to 10 hours of work, not 5 hours. 0.5: This would mean half the job is left, which would require 25 hours of work, not just 5 hours. 0.75: This corresponds to 3 / 4 of the job left, again inconsistent with only 5 hours of work done out of 50 hours total. 0.6: This would indicate 60 percent remaining and 40 percent completed, which is more than what 5 hours of work can achieve out of 50 hours total. Common Pitfalls: A frequent error is mixing up the fractions of work done and remaining. Students might mistakenly treat 5 hours as a large portion of 50 hours or miscompute the fraction 5 / 50. Ensure accurate fraction simplification and remember that the remaining fraction is always 1 minus the completed fraction. Carefully translating time into fraction of work is the crucial step. Final Answer: The fraction of the job still left to be completed is 0.9.

Question 5

If 10 people can complete a certain job in 20 days, then in how many days will 20 people, each working at twice the efficiency of one of the original people, complete the same job?

Accepted Answer

Introduction / Context: This problem combines two important ideas in time and work questions: changing the number of workers and changing their efficiency. We know how long it takes an initial group to complete a job. Then we are asked how long a larger group, with each member being more efficient, will take to finish the same job. The question tests understanding of proportionality between work, rate and time. Given Data / Assumptions: - Ten people can complete the job in 20 days. - In the new scenario, there are 20 people. - Each of the 20 people works at twice the efficiency of one of the original people. - The nature of the job does not change; only manpower and efficiency change. - All workers work at constant rates. Concept / Approach: Total work can be expressed in person-days. Work = Number of workers * Efficiency * Time. First, we compute the total work in terms of the old people working at their original efficiency. Then, we calculate the combined rate of the new group, taking into account both more people and higher efficiency. Finally, we divide total work by this new combined rate to find the required time. Another way is to see how many times the total work rate increases in the second scenario and divide the original time by that factor. Step-by-Step Solution: Step 1: Let the work done by one original person in one day be 1 unit of efficiency. Step 2: Combined rate of 10 people at original efficiency = 10 * 1 = 10 units per day. Step 3: Time taken by them to complete the job = 20 days, so total work = 10 * 20 = 200 work units. Step 4: In the new scenario, there are 20 people, each with twice the original efficiency. Step 5: Combined rate of new group = 20 people * 2 units per person-day = 40 units per day. Step 6: Time required with the new group = total work / new rate = 200 / 40 = 5 days. Verification / Alternative check: We can compare rates directly. Original group rate is 10 units per day. New group rate is 40 units per day. The new rate is 4 times the original rate. Therefore, the time needed should be 1 / 4 of the original time: 20 days / 4 = 5 days. This quick reasoning confirms the detailed calculation and shows that 5 days is the correct answer. Why Other Options Are Wrong: 10 days: This is only half of the original time and corresponds to a doubling of rate, not a fourfold increase. 20 days and 40 days: These completely ignore the increased manpower and improved efficiency, giving too large a duration. 2.5 days: This would require an eightfold increase in work rate, which is not supported by the data (we only have a fourfold increase). Common Pitfalls: Students may forget to incorporate both changes: the increase in number of people and the increase in efficiency per person. Another typical mistake is to simply double or halve the time without checking the combined factor of change in work rate. The safest approach is always to compute total work in units and then divide by the new combined rate. Final Answer: Twenty people with twice the efficiency will complete the job in 5 days.

Question 6

A is 1.5 times as efficient as B and therefore takes 8 days less than B to complete a certain work. If A and B work on alternate days starting with A, in how many days will the work be completed?

Accepted Answer

Introduction / Context: This question combines comparative efficiency with an alternating work pattern. We are told that A is 1.5 times as efficient as B and that A takes 8 days less than B to complete the work alone. Then A and B work on alternate days, starting with A, and we must find the total number of days needed to finish the work under this pattern. This problem requires translating the efficiency relation into times, computing individual rates, and then handling the alternating schedule carefully. Given Data / Assumptions: - A is 1.5 times as efficient as B. - A takes 8 days less than B to complete the work alone. - A and B work on alternate days, starting with A. - Both work at constant rates whenever they are working. - Total work is treated as one complete job. Concept / Approach: Efficiency is inversely proportional to time taken for the same work. If A is 1.5 times as efficient as B, then A's time will be 2 / 3 of B's time. Using the extra data that A takes 8 days less, we can find the exact times for A and B individually. Once we know their daily work rates, we compute how much work is done in a 2 day cycle (A for one day and B for the next day). Then we determine how many full 2 day cycles are needed and how many extra days A may need at the end to complete the job. Step-by-Step Solution: Step 1: Let time taken by B alone to finish the work = T days. Step 2: Since A is 1.5 times as efficient as B, A's time = T * (2 / 3). Step 3: We are told A takes 8 days less than B, so T - T * (2 / 3) = 8. Step 4: Simplify: T * (1 - 2 / 3) = T * (1 / 3) = 8, so T = 24 days. Step 5: Therefore, B's time = 24 days and A's time = 24 - 8 = 16 days. Step 6: A's daily work rate = 1 / 16 job per day; B's daily work rate = 1 / 24 job per day. Step 7: In a 2 day cycle (Day 1 A, Day 2 B), total work done = 1 / 16 + 1 / 24. Step 8: Use common denominator 48: 1 / 16 = 3 / 48 and 1 / 24 = 2 / 48, so total = 5 / 48 job per 2 days. Step 9: After 9 such cycles (18 days), total work done = 9 * (5 / 48) = 45 / 48 job. Step 10: Remaining work = 1 - 45 / 48 = 3 / 48 = 1 / 16 job. Step 11: On day 19, it is A's turn, and A's daily rate is 1 / 16 job per day, which matches the remaining work; so A finishes the job on day 19. Verification / Alternative check: We can double-check by ensuring that the fractions add correctly. Work up to day 18 is 45 / 48 of the job. On day 19, A adds another 1 / 16 = 3 / 48, so total work becomes 48 / 48 = 1 full job. The pattern of alternate days is correctly maintained, and we do not need an additional day for B. Thus, the work is fully completed by the end of day 19, confirming the result. Why Other Options Are Wrong: 17 days or 18 days: These durations correspond to fewer 2 day cycles and would result in less than the full job being completed. 19.5 days or 21 days: These overestimate the time; by day 19, the work is already completed, so additional time is unnecessary and inconsistent with the calculated work done. Common Pitfalls: One common error is misinterpreting “1.5 times as efficient” and setting up the time ratios incorrectly. Another mistake is to simply average the times of A and B without considering the alternating pattern. Students also sometimes forget that in alternate day problems, a natural unit of analysis is the 2 day cycle. Carefully calculating the work done per cycle and tracking the remaining work avoids these mistakes. Final Answer: The work will be completed in 19 days.

Question 7

A can complete a piece of work in 20 days and B can complete the same work in 25 days. If both of them work together, what percentage of the total work will be completed in 3 days?

Accepted Answer

Introduction / Context: This is a direct time and work percentage problem. We are given the individual times taken by A and B to finish a job alone and we must determine what fraction or percentage of the job they can finish together in a fixed short period of 3 days. This tests understanding of work rate addition and conversion of a fraction of work into a percentage. Given Data / Assumptions: - A can complete the entire work in 20 days. - B can complete the same work in 25 days. - A and B work together at their constant respective rates. - We are interested in the portion of work completed in exactly 3 days. - Total work is considered as one complete unit. Concept / Approach: If someone finishes a job in T days, their daily work rate is 1 / T of the job per day. When two workers collaborate, their joint rate is simply the sum of their individual rates. We calculate the combined daily rate of A and B, multiply it by 3 to find the fraction of work done in 3 days, and then convert this fraction into a percentage by multiplying by 100. Step-by-Step Solution: Step 1: A's daily work rate = 1 / 20 job per day. Step 2: B's daily work rate = 1 / 25 job per day. Step 3: Combined daily rate of A and B = 1 / 20 + 1 / 25. Step 4: Use a common denominator 100: 1 / 20 = 5 / 100 and 1 / 25 = 4 / 100, so sum = 9 / 100 job per day. Step 5: Work completed in 3 days = 3 * (9 / 100) = 27 / 100 of the job. Step 6: Convert fraction 27 / 100 to percentage: (27 / 100) * 100 percent = 27 percent. Verification / Alternative check: We can check if the answer is reasonable. A alone would do 3 * (1 / 20) = 3 / 20 = 15 percent of the job in 3 days. B alone would do 3 * (1 / 25) = 3 / 25 = 12 percent. Together, they should do 15 percent + 12 percent = 27 percent, which matches our detailed calculation. This quick mental check confirms that the answer is consistent. Why Other Options Are Wrong: 9 or 12: These correspond roughly to the individual contributions of B or A alone, ignoring the fact that both work together. 25: This slightly underestimates the combined contribution; the precise fraction we computed is 27 / 100, not 25 / 100. 30: This overestimates the work done; for 30 percent, the combined daily rate would need to be higher than the given 9 / 100 job per day. Common Pitfalls: A common mistake is to average the times (20 and 25) rather than add the rates. Others may forget to convert the fraction to a percentage properly. Always remember that you add rates, not times, for workers cooperating, and then multiply the combined rate by the given number of days to get the fractional work done. Final Answer: A and B together will complete 27 percent of the work in 3 days.

Question 8

A can do a piece of work in 24 days and B can do the same work in 40 days. If they work together on the job for 10 days, what fraction of the work is still left to be completed?

Accepted Answer

Introduction / Context: This question asks us to determine the fraction of work remaining after two workers, A and B, have jointly worked for a fixed period. We are given the individual times required by A and B to complete the work alone and must find how much of the job remains undone after they work together for 10 days. It tests fundamental understanding of work rates and partial completion of tasks. Given Data / Assumptions: - A can complete the work alone in 24 days. - B can complete the same work alone in 40 days. - A and B work together for 10 days. - Work rates remain constant. - Total work is treated as one unit. Concept / Approach: The daily work rate of someone who finishes a job in T days is 1 / T of the job per day. When two people work together, their combined rate is the sum of their individual rates. We compute the combined rate of A and B, multiply it by 10 to get the fraction completed in 10 days and then subtract this from 1 to find the remaining fraction. Finally, we simplify the fraction to one of the given options. Step-by-Step Solution: Step 1: A's daily work rate = 1 / 24 of the job per day. Step 2: B's daily work rate = 1 / 40 of the job per day. Step 3: Combined daily rate = 1 / 24 + 1 / 40. Step 4: Take common denominator 120: 1 / 24 = 5 / 120, 1 / 40 = 3 / 120, so sum = 8 / 120 = 1 / 15. Step 5: Work done in 10 days = 10 * (1 / 15) = 10 / 15 = 2 / 3 of the job. Step 6: Remaining work = 1 - 2 / 3 = 1 / 3 of the job. Verification / Alternative check: We can check if 1 / 3 remaining is reasonable. If 2 / 3 of the job is done in 10 days, then the full job at this rate would take 10 / (2 / 3) = 15 days. This matches the combined time we would get by solving for 1 / (1 / 24 + 1 / 40) = 15 days, which is consistent. So the fraction remaining, 1 / 3, is correct. Why Other Options Are Wrong: 1 / 2 or 2 / 3: These would suggest that 1 / 2 or 1 / 3 of the work is completed, not 2 / 3, and conflict with the actual combined rate and time. 3 / 4: This implies only 1 / 4 is done, which is far too small for 10 days of joint work at the calculated rate. 1 / 4: This would mean 3 / 4 of the work is done, which is more than the calculated 2 / 3. Common Pitfalls: Students may forget to convert to a common denominator when adding fractions or may try to average the times (24 and 40 days) instead of summing rates. Another error is to miscalculate the fraction done versus remaining. Always carefully calculate the combined rate, multiply by the actual working time, and subtract from 1 to find the fraction of work left. Final Answer: The fraction of work left after 10 days is 1/3.

Question 9

Ajit is twice as good a workman as Badrinath and therefore finishes a certain job in 30 days less than Badrinath. Working together, in how many days will Ajit and Badrinath complete the job?

Accepted Answer

Introduction / Context: Here we have a comparative efficiency problem where Ajit is described as “twice as good” a workman as Badrinath. This means Ajit's rate of working is double that of Badrinath. Additionally, Ajit finishes a job in 30 days less than Badrinath. We must use this relationship to find their individual times and then compute how long they would take to finish the job together. This involves setting up equations for time and rate and then combining rates. Given Data / Assumptions: - Ajit is twice as efficient as Badrinath. - Because of this, Ajit takes 30 days less than Badrinath to complete the job alone. - Both work at constant rates when they work. - Total work is one complete job. - We want the time taken when Ajit and Badrinath work together. Concept / Approach: If one person is twice as efficient as another, then their times for the same work are in the inverse ratio 1 : 2. Let Badrinath's time be T days, then Ajit's time is T / 2 days. We also know that Ajit's time is 30 days less than Badrinath's. Using these two relationships, we can solve for T. Once we know each person's time, we can find their daily work rates, add these rates to get the combined rate, and then take the reciprocal to get the total time when working together. Step-by-Step Solution: Step 1: Let time taken by Badrinath alone = T days. Step 2: Ajit is twice as efficient, so Ajit's time = T / 2 days. Step 3: We are told that Ajit takes 30 days less than Badrinath, so T - T / 2 = 30. Step 4: Simplify: T / 2 = 30, hence T = 60 days. Step 5: Therefore, Badrinath's time = 60 days and Ajit's time = 60 / 2 = 30 days. Step 6: Ajit's daily work rate = 1 / 30 job per day. Step 7: Badrinath's daily work rate = 1 / 60 job per day. Step 8: Combined rate when working together = 1 / 30 + 1 / 60 = 2 / 60 + 1 / 60 = 3 / 60 = 1 / 20 job per day. Step 9: Time taken together = 1 / (1 / 20) = 20 days. Verification / Alternative check: We can verify that Ajit's time is indeed 30 days less than Badrinath's: 60 - 30 = 30 days, which matches Ajit's time. Also, working together at a rate of 1 / 20 job per day, they would finish in 20 days. Since Ajit alone takes 30 days, finishing in 20 days with help from Badrinath is reasonable and consistent. Why Other Options Are Wrong: 10 or 15 days: These are too short and would require a higher combined rate than the actual 1 / 20 job per day. 30 days: This is just Ajit's individual time, not the combined time with Badrinath helping. 25 days: This is higher than the calculated 20 days and does not match the combined rate when both work together. Common Pitfalls: Students may misinterpret “twice as good” and incorrectly double the time instead of halving it. Another mistake is ignoring the 30 day difference in times, using only the efficiency relation and ending up with incomplete information. Correctly translating verbal efficiency comparisons into algebraic equations for time and then solving those equations is crucial for such problems. Final Answer: Ajit and Badrinath together will complete the job in 20 days.

Question 10

A and B together can complete a job in 6.75 days, and A alone can complete the same job in 9 days. If B works alone, in how many days will B complete the job?

Accepted Answer

Introduction / Context: This time and work problem asks us to find the time taken by B alone, given the time taken by A alone and the time taken by A and B together. Such questions are common in aptitude tests and rely on the principle that work rates are additive: the combined rate is the sum of individual rates. We must first find the combined daily rate, subtract A's rate, and then invert to get B's time. Given Data / Assumptions: - A and B together complete the job in 6.75 days. - A alone completes the job in 9 days. - Work rates are constant for both A and B. - Total work is one complete job. - We need to find the number of days B alone would take to complete the job. Concept / Approach: If a job is finished in T days, the daily work rate is 1 / T. For two people working together, the combined rate is the sum of their individual rates. We first compute the combined rate of A and B using the given 6.75 days. We then subtract A's rate (1 / 9) to get B's rate. Finally, we take the reciprocal of B's rate to find how many days B alone would need to complete the job. Step-by-Step Solution: Step 1: Convert 6.75 days to a fraction: 6.75 = 6 and 3 / 4 = 27 / 4 days. Step 2: Combined rate of A and B = 1 / (27 / 4) = 4 / 27 job per day. Step 3: A's rate alone = 1 / 9 job per day. Step 4: Express A's rate with denominator 27: 1 / 9 = 3 / 27 job per day. Step 5: B's rate = combined rate - A's rate = 4 / 27 - 3 / 27 = 1 / 27 job per day. Step 6: Time taken by B alone = 1 / (1 / 27) = 27 days. Verification / Alternative check: We can verify by adding the rates back. A's rate is 1 / 9 job per day, B's rate is 1 / 27 job per day, so their combined rate is 1 / 9 + 1 / 27. With a common denominator 27, we get 3 / 27 + 1 / 27 = 4 / 27 job per day. The time needed at this rate to complete one job is 1 / (4 / 27) = 27 / 4 = 6.75 days, which matches the given value. This confirms that B alone takes 27 days. Why Other Options Are Wrong: 18 or 21 days: These imply a faster rate for B than 1 / 27 job per day and do not satisfy the given combined time of 6.75 days. 24 days: This would lead to an inconsistent combined rate that does not reproduce 6.75 days when combined with A's rate. 30 days: This implies a slower rate than our computed 1 / 27 job per day, again failing to match the given combined time. Common Pitfalls: Students sometimes forget to correctly convert 6.75 into a fraction, or they treat 6.75 as 6 + 75 / 100 without simplifying. Another frequent mistake is to subtract times instead of subtracting rates. Always remember: you subtract and add rates (fractions of work per day), not the number of days. Carefully converting mixed numbers to improper fractions helps avoid calculation errors. Final Answer: B alone will complete the job in 27 days.

Question 11

A and B together can complete a job in 72 days, B and C together can complete it in 120 days, and A and C together can complete it in 90 days. In how many days can A alone complete the job?

Accepted Answer

Introduction / Context: This is a classic time and work problem involving three workers A, B and C, where we know the times taken by different pairs of them to complete a job. Our task is to use this information to find the time taken by A alone. This type of question tests the ability to form and solve a system of equations based on combined work rates and then isolate one worker's rate. Given Data / Assumptions: - A and B together complete the job in 72 days. - B and C together complete the job in 120 days. - A and C together complete the job in 90 days. - All work at constant rates. - Total work is treated as one complete job. Concept / Approach: If someone finishes a job in T days, their daily work rate is 1 / T. Let the individual rates of A, B and C be a, b and c jobs per day respectively. We are given three equations in terms of these rates: a + b, b + c and a + c. By adding and subtracting these equations appropriately, we can solve for each rate. Once A's rate a is found, A's time to finish the job alone is simply 1 / a days. This approach uses algebraic manipulation of simultaneous linear equations. Step-by-Step Solution: Step 1: Let A's rate = a, B's rate = b, C's rate = c jobs per day. Step 2: From A and B together: a + b = 1 / 72. Step 3: From B and C together: b + c = 1 / 120. Step 4: From A and C together: a + c = 1 / 90. Step 5: Add equations for A + B and A + C: (a + b) + (a + c) = 1 / 72 + 1 / 90. Step 6: Left side becomes 2a + (b + c). Right side: use denominator 360. 1 / 72 = 5 / 360 and 1 / 90 = 4 / 360, so sum = 9 / 360 = 1 / 40. Step 7: We know b + c = 1 / 120 from Step 3. Step 8: Substitute b + c into the expression: 2a + 1 / 120 = 1 / 40. Step 9: Solve for a: 2a = 1 / 40 - 1 / 120. Step 10: Use denominator 120: 1 / 40 = 3 / 120, so 2a = (3 / 120) - (1 / 120) = 2 / 120 = 1 / 60. Step 11: Therefore, a = 1 / 120 job per day. Step 12: Time taken by A alone to complete one job = 1 / a = 120 days. Verification / Alternative check: We can confirm the correctness by reconstructing b and c and checking one of the original pair equations. Once we know a = 1 / 120, from a + b = 1 / 72, we get b = 1 / 72 - 1 / 120, which is positive and reasonable. Similarly, from a + c = 1 / 90 we get a valid c. Recalculating a + b, b + c and a + c returns the original 1 / 72, 1 / 120 and 1 / 90, validating our solution. So A taking 120 days alone is fully consistent. Why Other Options Are Wrong: 130, 150 or 100 days: These do not match the rate derived from the simultaneous equations and would lead to inconsistencies with the given combined times. 90 days: This is the time for A and C together, not for A alone. It ignores the need to separate out B's and C's contributions. Common Pitfalls: A common mistake is to try to average the given times rather than use algebraic equations. Another error is mishandling fractions when adding and subtracting the combined rates. Careful use of a common denominator and systematic solving of the three linear equations is essential. Always remember that the combined rate of two workers is simply the sum of their individual rates, not the average of their times. Final Answer: A alone can complete the job in 120 days.

Question 12

A can do a certain piece of work in 36 days and B can do the same work in 12 days. If they work together on the job, in how many days will they finish it?

Accepted Answer

Introduction / Context: This is a straightforward time and work question where two workers have individual times to complete a job, and we are asked for the time when both work together. It tests the basic skill of converting individual times into rates, summing those rates, and then converting back to a combined time for finishing the work. Given Data / Assumptions: - A alone can finish the job in 36 days. - B alone can finish the same job in 12 days. - A and B work together at their constant rates. - Total work is taken as one complete job. Concept / Approach: If a worker completes a job in T days, their daily work rate is 1 / T of the job per day. When two workers collaborate, the combined rate equals the sum of their individual rates. Once we have the combined rate, the time taken to complete one whole job is the reciprocal of this combined rate. This is the core idea behind all such cooperation time-and-work problems. Step-by-Step Solution: Step 1: A's rate of work = 1 / 36 job per day. Step 2: B's rate of work = 1 / 12 job per day. Step 3: Combined daily rate of A and B working together = 1 / 36 + 1 / 12. Step 4: Convert to a common denominator 36: 1 / 12 = 3 / 36, so combined rate = 1 / 36 + 3 / 36 = 4 / 36. Step 5: Simplify 4 / 36 = 1 / 9 job per day. Step 6: Time taken to complete one full job at this rate = 1 / (1 / 9) = 9 days. Verification / Alternative check: We can check reasonableness by comparing the individual times. B is much faster, finishing in 12 days; A is slower at 36 days. When they work together, the combined time should be less than 12 days. Our answer, 9 days, is less than 12 and more than half of B's time, which is realistic. Also, if they work for 9 days at 1 / 9 job per day, they complete exactly 1 job, confirming the calculation. Why Other Options Are Wrong: 8 or 6 days: These would imply a higher combined rate than 1 / 9 job per day and are not supported by the actual rates of A and B. 10 days: This is longer than 9 days and would correspond to a combined rate of only 1 / 10 job per day, which is less than the actual 1 / 9 job per day. 4 days: This is far too short and unrealistic given that even B alone takes 12 days. Common Pitfalls: Some students incorrectly average the times (36 and 12) instead of summing the rates. Others may make arithmetic mistakes when adding fractions with different denominators. Always convert times into rates (1 / T), add the rates, and then take the reciprocal of the sum to obtain the total time when both work together. Final Answer: A and B working together will complete the job in 9 days.

Question 13

A can finish a piece of work in 18 days and B can finish the same work in 36 days. If they work together on the job for 9 days, what percentage of the work is still left unfinished?

Accepted Answer

Introduction / Context: This question is about determining how much work remains after two workers, A and B, have worked together for a limited period. We are given the individual times for A and B to complete the job alone. The task is to find the percentage of work left after they jointly work for 9 days. This tests understanding of combined work rates and conversion of a fraction of work into a percentage remaining. Given Data / Assumptions: - A alone can complete the work in 18 days. - B alone can complete the same work in 36 days. - A and B work together for 9 days. - Work rates remain constant throughout. - Total work is one full job. Concept / Approach: The daily work rate of a person who takes T days to finish a job is 1 / T of the job per day. When two people work together, we add their rates to get the combined rate. We multiply this combined rate by the number of working days to find the fraction of the job completed. Subtracting this from 1 gives the fraction remaining. Finally, we convert that fraction into a percentage to answer the question. Step-by-Step Solution: Step 1: A's daily work rate = 1 / 18 of the job per day. Step 2: B's daily work rate = 1 / 36 of the job per day. Step 3: Combined daily rate = 1 / 18 + 1 / 36. Step 4: Take a common denominator 36: 1 / 18 = 2 / 36, so combined rate = 2 / 36 + 1 / 36 = 3 / 36 = 1 / 12 job per day. Step 5: Work done in 9 days at this rate = 9 * (1 / 12) = 9 / 12 = 3 / 4 of the job. Step 6: Fraction of work remaining = 1 - 3 / 4 = 1 / 4. Step 7: Convert 1 / 4 to percentage: (1 / 4) * 100 percent = 25 percent. Verification / Alternative check: We can verify by checking how long it would take them to finish the job together. At 1 / 12 job per day, they need 12 days to finish the entire job. Working for 9 days completes 9 / 12 = 3 / 4 of the job, leaving exactly 1 / 4. Since 1 / 4 of 12 days is 3 days, they would need 3 more days to finish, which is consistent with the idea that 25 percent of the work remains. Why Other Options Are Wrong: 33.3 or 50: These represent 1 / 3 and 1 / 2 of the job remaining, which do not match the combined rate and time used. 20: This corresponds to 1 / 5 of the job remaining, not 1 / 4. 75: This would mean only 25 percent completed and 75 percent left, which contradicts the calculated 3 / 4 already completed. Common Pitfalls: A common mistake is to miscalculate the combined rate by averaging times instead of adding rates. Another error is to misinterpret the question and give the percentage of work completed instead of the percentage remaining. Always read the question carefully to see whether it asks for the part done or the part left, and consistently convert fractions to percentages. Final Answer: The percentage of work left unfinished after 9 days is 25 percent.

Question 14

A can do one-third of a job in 3 days and B can do half of the same job in 9 days. If A and B work together, in how many days can they finish half of the job?

Accepted Answer

Introduction / Context: This time and work problem gives partial work done by A and B in specified times and then asks how long they will take together to finish only half of the job. Rather than being told total time for each person, we are told how much of the job each can complete in a certain number of days. The question tests the ability to convert these partial completion statements into daily work rates and then combine them. Given Data / Assumptions: - A can do one-third of the job in 3 days. - B can do half of the job in 9 days. - A and B work together at their constant respective rates. - We are asked for the time needed to complete half of the job, not the whole job. - Total job is considered as one unit. Concept / Approach: If a worker does a fraction of the job in a certain number of days, we can convert that to a rate by dividing the fraction by the time. Once we know the daily work rates of A and B, we add them to get the combined rate. Since the question only asks about half of the job, we then compute the time needed by dividing the required fraction (1 / 2) by the combined rate. This avoids extra steps of first finding time for a full job. Step-by-Step Solution: Step 1: A does 1 / 3 of the job in 3 days. Step 2: A's daily work rate = (1 / 3) / 3 = 1 / 9 of the job per day. Step 3: B does 1 / 2 of the job in 9 days. Step 4: B's daily work rate = (1 / 2) / 9 = 1 / 18 of the job per day. Step 5: Combined daily work rate of A and B = 1 / 9 + 1 / 18. Step 6: Use common denominator 18: 1 / 9 = 2 / 18, so combined rate = 2 / 18 + 1 / 18 = 3 / 18 = 1 / 6 of the job per day. Step 7: We need time to complete half of the job, that is 1 / 2 of the work. Step 8: Time = required work / combined rate = (1 / 2) / (1 / 6) = (1 / 2) * 6 = 3 days. Verification / Alternative check: We can verify by calculating how much work they would do in 3 days at the combined rate. At 1 / 6 job per day, in 3 days they complete 3 * (1 / 6) = 3 / 6 = 1 / 2 of the job, which matches the requirement. Since the calculations are straightforward and consistent, 3 days is correct. Why Other Options Are Wrong: 4, 5 or 6 days: At 1 / 6 job per day, these would correspond to 4 / 6, 5 / 6 or 6 / 6 of the job, which are larger than half of the job except when the whole job is completed. 2 days: This would give only 2 / 6 = 1 / 3 of the job, which is less than the required half. Common Pitfalls: Students sometimes mistakenly treat the given fractions as whole jobs or fail to convert them correctly into daily rates. Another error is to compute the time for the entire job and then try to adjust for half the job, which can lead to confusion. Always convert partial completions into per day rates, sum the rates for combined work, and then directly compute the time for the required fraction of the job. Final Answer: Working together, A and B will finish half of the job in 3 days.

Question 15

A, B, and C can complete a job working alone in 12 days, 8 days, and 24 days respectively. If all three work together from start to finish, in how many days will they complete the entire job?

Accepted Answer

Introduction / Context: This is a standard time and work aptitude question where three people A, B, and C can each complete the same job working alone in different numbers of days. The task is to find how long they will take to complete the job if they work together from start to finish. Such problems are very common in competitive exams and interview tests because they check the understanding of work rate addition. Given Data / Assumptions: A alone finishes the job in 12 days. B alone finishes the job in 8 days. C alone finishes the job in 24 days. All three work together for the entire duration. The total work is treated as one complete job. Concept / Approach: The key concept is that if a person can complete a job in N days, then the work rate of that person is 1/N of the job per day. When multiple people work together, their individual work rates are added to get the combined rate. Finally, time is calculated using the relation time = total work / combined rate. Here, total work is taken as 1 job. Step-by-Step Solution: Step 1: Let the total work be 1 job. Step 2: Work rate of A = 1/12 job per day. Step 3: Work rate of B = 1/8 job per day. Step 4: Work rate of C = 1/24 job per day. Step 5: Combined work rate = 1/12 + 1/8 + 1/24. Step 6: Take LCM of 12, 8, and 24 which is 24. Step 7: 1/12 = 2/24, 1/8 = 3/24, and 1/24 = 1/24. Step 8: Combined rate = (2 + 3 + 1) / 24 = 6/24 = 1/4 job per day. Step 9: Time taken = total work / combined rate = 1 / (1/4) = 4 days. Verification / Alternative check: If the team completes 1/4 of the job per day, then in 4 days they will complete 4 * (1/4) = 1 full job. This confirms the calculation. Another way to check is to compute the fraction of work done by each person in 4 days. A does 4 * (1/12) = 1/3, B does 4 * (1/8) = 1/2, and C does 4 * (1/24) = 1/6. Adding these gives 1/3 + 1/2 + 1/6 = 1 job, which fully verifies the answer. Why Other Options Are Wrong: 3 days is too short because at a rate of 1/4 job per day, in 3 days they would complete only 3/4 of the job. 5 days and 6 days are too long because they would give total work greater than 1 job. For example, in 5 days they would complete 5 * (1/4) = 5/4 jobs which exceeds the required work. Similarly, 8 days would be even more unrealistic for such high combined efficiency. Common Pitfalls: A common mistake is to average the days directly instead of working with rates. Another error is to forget to use a proper common denominator when adding fractions. Some learners also confuse time and rate, attempting to add days instead of adding fractions of work per day. Always convert days into rates first and then perform addition. Final Answer: The three workers A, B, and C together will complete the job in 4 days.

Question 16

A, B, and C can each complete a job working alone in 12 days, 18 days, and 36 days respectively. They all work together for 2 days, after which B stops working. How many additional days will A and C together take to finish the remaining part of the …

Accepted Answer

Introduction / Context: This time and work problem involves three workers A, B, and C who initially work together, and then one worker (B) leaves after some time. The goal is to compute how long the remaining workers A and C will take to complete the unfinished portion of the job. Such questions test understanding of work rates and partial completion of tasks over different time intervals. Given Data / Assumptions: A alone completes the job in 12 days. B alone completes the job in 18 days. C alone completes the job in 36 days. All three work together for 2 days. After 2 days, B leaves and only A and C continue. Total work is taken as 1 complete job. Concept / Approach: The core approach is based on the concept of work rate. If a person completes a job in N days, the rate is 1/N job per day. We first calculate the combined rate of A, B, and C and find how much of the work they complete in 2 days. Then we compute the remaining work and divide it by the combined rate of A and C to find the required additional time. This stepwise method ensures clear handling of partial work. Step-by-Step Solution: Step 1: Let the total work be 1 job. Step 2: Rate of A = 1/12 job per day. Step 3: Rate of B = 1/18 job per day. Step 4: Rate of C = 1/36 job per day. Step 5: Combined rate of A, B, and C = 1/12 + 1/18 + 1/36. Step 6: Take LCM of 12, 18, and 36 which is 36. Step 7: 1/12 = 3/36, 1/18 = 2/36, 1/36 = 1/36, so combined rate = (3 + 2 + 1)/36 = 6/36 = 1/6 job per day. Step 8: In 2 days, the three together complete 2 * (1/6) = 1/3 of the job. Step 9: Remaining work = 1 - 1/3 = 2/3 of the job. Step 10: Now only A and C work, so their combined rate = 1/12 + 1/36. Step 11: Take LCM of 12 and 36 which is 36, so 1/12 = 3/36 and 1/36 = 1/36, giving combined rate = 4/36 = 1/9 job per day. Step 12: Time required by A and C to finish the remaining 2/3 job = (2/3) / (1/9) = (2/3) * 9 = 6 days. Verification / Alternative check: We can verify by calculating the effective total time if we track the work fractions carefully. The first 2 days contribute 1/3 of the job. The next 6 days with A and C at 1/9 job per day add 6 * (1/9) = 2/3 of the job. Together, 1/3 + 2/3 = 1 full job. Since all fractions add up perfectly, the answer of 6 days is consistent and correct. Why Other Options Are Wrong: 3 days and 4 days are too short because at the rate of 1/9 job per day, they would complete only 1/3 or 4/9 of the remaining work, not 2/3. 8 days and 9 days are too long; they would cause A and C to do more work than required, exceeding the remaining 2/3 of the job. Only 6 days yields exactly the required remaining work. Common Pitfalls: One common mistake is to average the days taken by individuals instead of using work rates. Another frequent error is to forget that only the remaining work after the first phase must be considered when A and C continue. Some learners also mismanage fraction arithmetic while adding and dividing work rates, leading to incorrect results. Final Answer: A and C together will take 6 more days to finish the remaining part of the job after B leaves.

Question 17

A, B, and C together can complete a job in 4 days. A and C together can complete the same job in 4.5 days, and B and C together can complete it in 12 days. In how many days can C alone complete the entire job?

Accepted Answer

Introduction / Context: This problem is a classic application of time and work concepts involving three workers with different combinations of joint work durations. By using the given times for A + B + C, A + C, and B + C, we are asked to find how long C alone will take to complete the job. This kind of problem is very popular in quantitative aptitude tests for exams and job interviews. Given Data / Assumptions: A, B, and C together complete the job in 4 days. A and C together complete the job in 4.5 days. B and C together complete the job in 12 days. We must find the time taken by C alone to complete the job. Total work is treated as 1 complete job. Concept / Approach: Let the daily work rates of A, B, and C be a, b, and c jobs per day respectively. Then, we translate the given information into equations using 1/time = rate. We set up three equations for the combined rates and then solve for c. Once we get the rate of C, the time taken by C alone is found by taking the reciprocal of that rate. This algebraic method is systematic and reliable. Step-by-Step Solution: Step 1: Let the total work be 1 job. Step 2: Let a, b, and c be the daily work rates of A, B, and C. Step 3: A + B + C together finish in 4 days, so a + b + c = 1/4. Step 4: A + C together finish in 4.5 days, so a + c = 1/4.5 = 2/9 job per day. Step 5: B + C together finish in 12 days, so b + c = 1/12 job per day. Step 6: From a + b + c = 1/4 and a + c = 2/9, subtract to get b = 1/4 - 2/9. Step 7: Compute b: 1/4 = 9/36 and 2/9 = 8/36 so b = (9/36 - 8/36) = 1/36 job per day. Step 8: Using b + c = 1/12 gives 1/36 + c = 1/12, so c = 1/12 - 1/36. Step 9: Compute c: 1/12 = 3/36, so c = (3/36 - 1/36) = 2/36 = 1/18 job per day. Step 10: Therefore, time taken by C alone = 1 / (1/18) = 18 days. Verification / Alternative check: We can verify by reconstructing all combined rates using the found value of c. If c = 1/18 and b = 1/36, then b + c = 1/36 + 1/18 = 3/36 = 1/12, which matches the given. Also, a + c = 2/9, and since a + b + c = 1/4, then a must satisfy that relation. The values are consistent and give back the same original conditions, confirming that C indeed completes the job in 18 days. Why Other Options Are Wrong: 6 days and 12 days are much too small because they imply that C is faster than the combined groups, which contradicts the given data. 24 days and 36 days are too large and do not satisfy the algebraic equations derived from the joint work times. Only 18 days fits all three conditions simultaneously, making it the only correct choice. Common Pitfalls: Learners sometimes try to average the given times rather than forming equations, which leads to incorrect answers. Another common mistake is miscalculation with fractions, especially when converting mixed or decimal times. It is important to consistently use fractions and carefully simplify them step by step to avoid arithmetic errors. Final Answer: C alone will complete the entire job in 18 days.

Question 18

P and Q can complete a project in 20 days and 12 days respectively when working alone. If they work together on the project, in how many days will they complete 40% of the entire project?

Accepted Answer

Introduction / Context: This question is about two workers P and Q who can each complete a project at different speeds. We are not asked for the full completion time, but instead for the time required to finish 40 percent of the project when both work together. Problems of partial work completion like 40 percent or 25 percent are very common in aptitude exams and help build speed with work rate calculations. Given Data / Assumptions: P alone completes the project in 20 days. Q alone completes the project in 12 days. They work together on the same project. We must find the time taken to complete 40% (that is, 0.4) of the project. Total project work is considered as 1 full job. Concept / Approach: The fundamental approach uses work rate addition. If a person finishes a job in N days, their work rate is 1/N job per day. When two people work together, their rates add. After finding their combined rate, we compute the time needed to complete a given fraction of the job using time = required work / rate. For 40 percent, the required work is 2/5 or 0.4 of the job. Step-by-Step Solution: Step 1: Let the total work be 1 job. Step 2: Rate of P = 1/20 job per day. Step 3: Rate of Q = 1/12 job per day. Step 4: Combined rate of P and Q = 1/20 + 1/12. Step 5: The LCM of 20 and 12 is 60, so 1/20 = 3/60 and 1/12 = 5/60. Step 6: Combined rate = (3 + 5)/60 = 8/60 = 2/15 job per day. Step 7: Required fraction of work = 40% = 40/100 = 2/5 of the job. Step 8: Time taken = (required work) / (combined rate) = (2/5) / (2/15). Step 9: Simplify: (2/5) / (2/15) = (2/5) * (15/2) = 15/5 = 3 days. Verification / Alternative check: As a check, we can find how much work they do in 3 days at their combined rate of 2/15 job per day. In 3 days they complete 3 * (2/15) = 6/15 = 2/5 of the job, which is exactly 40 percent. This confirms that the calculated time of 3 days is accurate and consistent with the given data. Why Other Options Are Wrong: 1.5 days and 2 days are too short, because in that time they would complete less than 40 percent of the project at the rate of 2/15 job per day. Six days and nine days are too long and would result in completing more than the required 40 percent. Only 3 days matches the correct fraction of the work. Common Pitfalls: Students sometimes mistakenly find the time for full completion and then take 40 percent of that time, which is incorrect because work rate is linear but time is inversely related to rate. It is important to find the fraction of work first and then divide by the combined rate. Also, care must be taken when working with fractions and LCMs to avoid arithmetic mistakes. Final Answer: P and Q together will complete 40% of the project in 3 days.

Question 19

Twelve men can complete a piece of work in 14 days. They work together for 5 days, after which 3 more men join them. How many more days (correct to one decimal place) will the enlarged group take to finish the remaining work?

Accepted Answer

Introduction / Context: This time and work question involves a change in the size of the workforce partway through the job. First, 12 men work together, and then 3 additional men join after a few days. We need to determine the extra number of days required to finish the remaining work once the additional men have joined. Such problems test understanding of man day concepts and proportional reasoning in aptitude exams. Given Data / Assumptions: 12 men can complete the whole work in 14 days. These 12 men work for the first 5 days. After 5 days, 3 more men join, so a total of 15 men work for the remaining part. We must compute how many more days are needed after the 5th day to finish the remaining work. Work done is assumed to be directly proportional to the product of men and days, that is, man days. Concept / Approach: The main idea is to convert the entire job into man days. First we find the total man days needed to complete the work when 12 men do it in 14 days. Then we calculate how many man days are used during the first 5 days. The remaining man days are then divided by the new manpower of 15 men per day to get the remaining time in days. This systematic approach avoids confusion and matches how work allocation is handled in real situations. Step-by-Step Solution: Step 1: Total work in man days = number of men * days taken by them. Step 2: With 12 men taking 14 days, total work = 12 * 14 = 168 man days. Step 3: In the first 5 days, only 12 men work, so man days used = 12 * 5 = 60 man days. Step 4: Remaining work in man days = 168 - 60 = 108 man days. Step 5: After 5 days, 3 more men join, so total men now = 12 + 3 = 15 men. Step 6: Daily work capacity after joining = 15 man days per day. Step 7: Remaining time required = remaining man days / daily capacity = 108 / 15 days. Step 8: Simplify 108 / 15 = 7.2 days. Verification / Alternative check: To verify, calculate total effective days if we convert all work to fractions of the job. For example, 12 men in 1 day complete 12/168 = 1/14 of the work. In 5 days they complete 5/14. The remaining work is 1 - 5/14 = 9/14. Under 15 men, each day they complete 15/168 = 5/56 of the work. Time required = (9/14) / (5/56) = (9/14) * (56/5) = 36/5 = 7.2 days. This confirms our earlier man day calculation. Why Other Options Are Wrong: 6.5 days and 7.5 days are close but incorrect and arise from rounding mistakes or arithmetic slips. 8.4 days and 9 days are much too large and would overshoot the required work, causing the team to perform more work than necessary. Only 7.2 days exactly satisfies the man day requirements and matches the fraction based verification. Common Pitfalls: Some learners mistakenly treat 12 men and 15 men as completing the same work in the same number of days, which ignores the increase in capacity. Others forget to convert the whole job to man days first and try to directly compare fractions without a consistent base. Careful tracking of total work, work already done, and remaining man days is essential for accuracy in such mixed workforce problems. Final Answer: After the first 5 days, the enlarged group of 15 men will take 7.2 more days to finish the remaining work.

Question 20

Aman can complete a piece of work in 18 days if he works 10 hours every day. If he wants to finish the same work in 15 days instead, how many hours per day should he work?

Accepted Answer

Introduction / Context: This question combines time and work with daily working hours. Aman initially follows a certain schedule and completes the work in a given number of days. He then wants to adjust his daily working hours so that the total project duration becomes shorter. This question helps practice the idea that total work in such cases can be measured as total man hours, which is the product of hours per day and number of days. Given Data / Assumptions: Aman completes the work in 18 days when he works 10 hours per day. He wants to complete the same work in 15 days. The amount of work and Amans efficiency remain constant. We must find the new number of hours per day he should work. Total work is measured in man hours, that is, hours per day multiplied by days. Concept / Approach: When the same person does the same job with the same efficiency, the total amount of work in man hours remains constant. If we know the current combination of days and daily hours, we can compute total man hours. Any new combination of days and hours that yields the same total man hours will complete the job. So, we calculate total man hours using the original schedule and then divide by the new number of days to get required hours per day. Step-by-Step Solution: Step 1: Compute the total work in man hours using the initial schedule. Step 2: Aman works 10 hours per day for 18 days, so total man hours = 10 * 18. Step 3: 10 * 18 = 180 man hours of total work. Step 4: Now Aman wants to complete the same work in 15 days. Step 5: Let the required daily working hours be H hours per day. Step 6: Total work remains 180 man hours, so 15 * H = 180. Step 7: Solve for H: H = 180 / 15 = 12 hours per day. Verification / Alternative check: We can verify by substituting the found value back. If Aman works 12 hours per day for 15 days, total work = 12 * 15 = 180 man hours. This equals the original total of 180 man hours from 10 hours per day for 18 days. Since the total work is unchanged and the calculations match, the answer 12 hours per day is confirmed. Why Other Options Are Wrong: 10 hours per day is the original schedule and would not reduce the total days to 15. Working 14 or 16 hours per day would result in more than 180 man hours, meaning he would be doing extra work. Working 18 hours per day is even more extreme and unnecessary. Only 12 hours per day exactly fits the requirement of completing the same work in 15 days. Common Pitfalls: One common mistake is to assume that reducing the days by a certain fraction means increasing the hours by the same fraction, which is not always correct without checking total man hours. Another error is forgetting that the persons efficiency is constant, which is why total man hours must stay the same. Always express total work as a product of days and hours, then apply basic algebra carefully. Final Answer: Aman needs to work 12 hours per day to finish the work in 15 days.

Time and Work Questions