Question 1

A total of 35 persons are engaged to complete a certain piece of work in 60 days. After 32 days it is found that only two-fifths of the work has been completed. How many persons in total must be working so that the remaining work is finished within …

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Introduction / Context: This question is about workforce adjustment when the actual progress of a project is slower than planned. The work was originally scheduled based on 35 persons working for 60 days, but after 32 days, only two-fifths of the job is complete. We must determine the total number of persons that should now be employed so that the remaining work is finished within the original deadline. This is a typical manpower and time trade-off question in aptitude tests. Given Data / Assumptions: Initial workforce: 35 persons. Planned total duration: 60 days. Actual progress after 32 days: 2/5 of the work completed. Remaining duration: 60 − 32 = 28 days. All persons are assumed to work at the same constant rate. We are asked for the total number of persons required for the remaining period. Concept / Approach: We use the idea of "man-days" as the unit of total effort required for the job. First we compute the amount of work (in man-days) already done and relate it to the fraction of work completed to infer the total work in man-days. Then we find how many man-days are still needed for the remaining fraction. Finally, dividing the required man-days by the remaining days gives the workforce needed. This total includes the original persons plus any additional persons to be engaged. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: In 32 days, 35 persons have completed 2/5 of the work. Step 3: Man-days consumed so far = 35 × 32 = 1120 man-days, which correspond to 2/5 of the work. Step 4: Total man-days required for the whole work = 1120 × (5/2) = 2800 man-days. Step 5: Remaining fraction of work = 1 − 2/5 = 3/5. Step 6: Man-days required for remaining work = 2800 × 3/5 = 1680 man-days. Step 7: Remaining time available = 60 − 32 = 28 days. Step 8: Required workforce = 1680 man-days / 28 days = 60 persons. Verification / Alternative check: If 60 persons now work for the remaining 28 days, the additional man-days contributed are 60 × 28 = 1680 man-days. Adding the original 1120 man-days gives a total of 2800 man-days, which is exactly the total effort we calculated earlier for completing the job. Also, 2800 man-days divided by 60 planned days is about 46.67 persons on average, which is consistent with needing more than the original 35 persons because the work is behind schedule. Why Other Options Are Wrong: 45, 50, and 55 persons would supply fewer than 1680 man-days in 28 days, so the remaining work would not be finished in time. For example, 50 persons would give only 50 × 28 = 1400 man-days, short of the 1680 required. 40 persons would be even less. Only 60 persons provide exactly the required 1680 man-days in the remaining 28 days. Common Pitfalls: Some students mistakenly compute how many extra persons to add instead of the total, which would be 25 additional persons beyond the original 35. Others forget to translate the fraction of work completed into man-days and directly scale persons in proportion to days, which can be misleading. Always think in terms of total man-days for the whole job, then work backward to the required workforce. Final Answer: The total number of persons required to finish the work on time is 60.

Question 2

Three labourers A, B and C together can finish a certain piece of work in 8 days. A and C together can do the same work in 12 days, while A and B together can do it in 13 1/3 days. If the total payment for the work is Rs. 750, in what ratio should i…

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Introduction / Context: This question asks for the ratio of wages to be paid to three labourers A, B and C based on their individual contributions to a job. They have worked together and in pairs for different durations, and we are given how long various combinations take to complete the full work. Payment must be divided proportional to the amount of work each has done, so we must find their individual work rates. Such problems appear often in aptitude tests involving partnerships and work-sharing. Given Data / Assumptions: A + B + C together can complete the work in 8 days. A + C together can complete the work in 12 days. A + B together can complete the work in 13 1/3 days (which is 40/3 days). Total payment for the job is Rs. 750, to be divided based on individual contributions. All work rates are constant over time. Concept / Approach: We treat the total work as 1 unit. Let the daily work rates of A, B and C be a, b and c respectively. Then we can write equations for A + B + C, A + C, and A + B based on their respective completion times. Solving these equations gives the individual rates a, b and c. The ratio of these rates equals the ratio of work done by each in any common time interval, which matches the ratio of wages they should receive. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: From A + B + C finishing in 8 days, we have a + b + c = 1/8. Step 3: From A + C finishing in 12 days, we have a + c = 1/12. Step 4: From A + B finishing in 13 1/3 days = 40/3 days, we have a + b = 1 / (40/3) = 3/40. Step 5: To find a, add the last two equations and subtract the first: (a + b) + (a + c) − (a + b + c) = 3/40 + 1/12 − 1/8. Step 6: This simplifies to a = 3/40 + 1/12 − 1/8 = 1/30. Step 7: Then b = (a + b) − a = 3/40 − 1/30 = 1/24, and c = (a + c) − a = 1/12 − 1/30 = 1/20. Step 8: Thus daily work rates of A, B, C are 1/30, 1/24, and 1/20 respectively. Step 9: Multiply all rates by the common denominator 120 to get the ratio A : B : C = 4 : 5 : 6. Verification / Alternative check: We can double-check by recomputing each pair and all three. For example, A + C has rate 1/30 + 1/20 = 1/12, matching the given 12 days. A + B has rate 1/30 + 1/24 = 3/40, matching 40/3 days. A + B + C has rate 1/30 + 1/24 + 1/20 = 1/8, matching 8 days. Since all conditions are satisfied, the ratio 4 : 5 : 6 is correct and consistent. Why Other Options Are Wrong: Any other ratio, such as 4 : 7 : 5, 5 : 7 : 4, or 5 : 6 : 8, would correspond to different relative rates for A, B and C. Reconstructing completion times using those ratios would not match the given 8 days, 12 days and 40/3 days for the various combinations. Only 4 : 5 : 6 fits all three completion-time conditions simultaneously. Common Pitfalls: A frequent mistake is to assume that time is directly proportional to payment without considering individual rates. Another issue is mismanaging the fractional days, particularly 13 1/3 days; it is important to convert this to 40/3 days before taking the reciprocal. Carefully forming and solving the system of equations for the rates a, b and c avoids these issues. Final Answer: The wages should be divided among A, B and C in the ratio 4 : 5 : 6.

Question 3

A, B and C together can finish a task in 12 days. A is twice as productive as B, and C alone can finish the task in 36 days. If C goes on leave and only A and B work, in how many days will A and B together complete the task?

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Introduction / Context: This problem mixes information about a group working together with relative efficiency information and a single individual time. We know how long A, B and C take together, we know that A is twice as productive as B, and we know how long C alone takes. We are asked how long A and B together would take if C is not available. This kind of question checks your skill in setting up and solving equations for work rates. Given Data / Assumptions: A, B and C together can finish the task in 12 days. A is twice as productive as B. C alone can finish the task in 36 days. All three work at constant rates. We are asked for the time taken by A and B together when C is absent. Concept / Approach: We treat the task as 1 unit of work. Let B's rate be a base variable and express A's rate in terms of B's rate using the efficiency ratio. C's rate comes directly from the single-worker time. We then use the total time for A + B + C to form an equation for the sum of their rates. Solving this gives B's rate, and from there A's rate. Adding A and B's rates yields the combined rate when they work without C, and its reciprocal gives the required time. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Let B's daily rate = b units per day. Step 3: A is twice as productive as B, so A's rate = 2b units per day. Step 4: C alone finishes the work in 36 days, so C's rate = 1/36 per day. Step 5: A + B + C together finish in 12 days, so 2b + b + 1/36 = 1/12. Step 6: This becomes 3b + 1/36 = 1/12. Step 7: Subtract 1/36 from both sides: 3b = 1/12 − 1/36 = 2/36 = 1/18. Step 8: Therefore, b = 1/54 and A's rate = 2b = 1/27. Step 9: A + B together have rate = 1/27 + 1/54 = 1/18 of the work per day. Step 10: Time taken by A and B together = 1 / (1/18) = 18 days. Verification / Alternative check: We can verify by recomputing the joint rate of A, B and C. Using A's rate 1/27, B's rate 1/54, and C's rate 1/36, we have 1/27 + 1/54 + 1/36 = 2/54 + 1/54 + 1/36. Converting to denominator 108 gives 4/108 + 2/108 + 3/108 = 9/108 = 1/12, matching the given completion time of 12 days. This confirms that the computed rates are correct and that 18 days for A and B together is consistent. Why Other Options Are Wrong: 10 or 15 days are too small and would require a higher combined rate than 1/18, which contradicts the derived individual rates. 20 and 24 days are too large and would imply that A and B together are slower than they actually are. Only 18 days matches the reciprocal of the combined rate obtained from the given information. Common Pitfalls: Students often misinterpret "twice as productive" and mistakenly think A takes twice the time of B instead of half. Another mistake is to forget C's contribution when using the 12-day completion data. Using symbols for the rates and carefully setting up the equation ensures that each piece of information is used correctly and avoids sign or fraction errors. Final Answer: A and B together will complete the task in 18 days when C is absent.

Question 4

If A alone can do a job in 40 days and A and B together can complete the same job in 8 days, then in how many days can B alone complete the job working at a constant rate?

Accepted Answer

Introduction / Context: This time and work problem combines the individual time of one worker with the combined time of two workers to find the time needed by the second worker alone. It checks your ability to handle work rates and to subtract one worker's contribution from the combined rate. Such problems are very common in competitive exams and provide a clear illustration of rate subtraction. Given Data / Assumptions: A alone can do the job in 40 days. A and B together can do the job in 8 days. Work rates of A and B are constant throughout. We are asked to find how many days B alone would take to complete the job. Concept / Approach: We treat the job as 1 unit of work. If a worker finishes the job in N days, the daily work rate is 1/N units per day. We compute A's rate and the combined rate of A and B. Subtracting A's rate from the combined rate gives B's rate. Finally, we take the reciprocal of B's rate to find the time taken by B alone. This straightforward add-and-subtract approach is the key to these types of questions. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: A alone completes the work in 40 days, so A's rate = 1/40 of the work per day. Step 3: A and B together complete the work in 8 days, so their combined rate = 1/8 of the work per day. Step 4: Let B's daily rate be r units of work per day. Step 5: Then 1/40 + r = 1/8. Step 6: Subtract 1/40 from 1/8 to find r: r = 1/8 − 1/40. Step 7: Using a common denominator of 40, r = 5/40 − 1/40 = 4/40 = 1/10. Step 8: Therefore, B's rate = 1/10 of the work per day, so B alone needs 10 days to complete the work. Verification / Alternative check: If B alone takes 10 days, then B's daily rate is 1/10. Together, A and B would have a rate of 1/40 + 1/10 = 1/40 + 4/40 = 5/40 = 1/8, meaning they complete the job in 8 days, which matches the given information. This confirms that B's time of 10 days is consistent with both the individual and combined data. Why Other Options Are Wrong: 15, 20, 25, and 30 days correspond to rates of 1/15, 1/20, 1/25 and 1/30 respectively. Adding any of these to A's rate 1/40 would not give 1/8. For instance, 1/40 + 1/15 is clearly more than 1/8, and 1/40 + 1/30 is clearly less than 1/8. Only the rate 1/10, corresponding to 10 days, produces the correct combined rate of 1/8 per day. Common Pitfalls: A frequent mistake is to try to subtract days instead of rates, for example doing 40 − 8 = 32 and trying to interpret this as B's time, which is incorrect. Another pitfall is to mismanage the fractions when subtracting 1/40 from 1/8. Always remember that rates add and subtract, while times do not. Using a common denominator helps keep the arithmetic accurate. Final Answer: B alone can complete the job in 10 days.

Question 5

P and Q can complete a task in 50 days and 20 days respectively when working alone. If they work together from the beginning, how many days will they take to complete 70% of the task?

Accepted Answer

Introduction / Context: This question involves two workers P and Q with different individual completion times. Instead of asking for the time to finish the entire task, it asks how long they will take working together to complete 70 percent of the task. This tests your ability to apply the concept of combined work rates to a fraction of the total job, not just the full job. Given Data / Assumptions: P alone can complete the full task in 50 days. Q alone can complete the full task in 20 days. P and Q work together for the entire period considered. We need the time taken to complete 70% (or 0.7) of the full task. All rates are constant over time. Concept / Approach: We treat the full task as 1 unit of work. P and Q's individual rates are 1 divided by their times. The combined rate is the sum of these rates. To find the time to complete only 70% of the work, we multiply the full completion time by 0.7, or more directly divide 0.7 by the combined rate. This method generalizes to any fraction of work, not just 70% or 100%. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: P's rate = 1/50 of the work per day. Step 3: Q's rate = 1/20 of the work per day. Step 4: Combined rate of P and Q = 1/50 + 1/20. Step 5: Using common denominator 100, 1/50 = 2/100 and 1/20 = 5/100, so combined rate = 7/100 of the work per day. Step 6: We need time to finish 70% of the work = 0.7 units. Step 7: Time = required work / rate = 0.7 / (7/100) = 0.7 × (100/7) = 70/7 = 10 days. Verification / Alternative check: If P and Q work for 10 days, total work done = 10 × (7/100) = 70/100 = 0.7 of the full task, which is exactly 70%. Also, the full task would take 1 / (7/100) = 100/7 ≈ 14.29 days when they work together, so finishing 70% in 10 days is consistent with this overall rate. Nothing contradicts the given individual times. Why Other Options Are Wrong: 12, 14, 18 and 20 days all correspond to more than 70% of the work done at the rate of 7/100 per day. For example, in 14 days they would complete 14 × 7/100 = 0.98 of the work, which is 98%, not 70%. Therefore, none of those options match the required fraction. Only 10 days produces exactly 70% of the job. Common Pitfalls: One common mistake is to compute the time for the full task and then forget to scale it down for 70%. Another is to multiply 50 and 20 or average them instead of adding their rates. Some learners also directly take 70% of each individual time, which is not correct because work combines through rates, not times. Always work with rates and required fractions of work for accurate results. Final Answer: P and Q together will complete 70% of the task in 10 days.

Question 6

A can complete a work in 36 days and B can complete the same work in 12 days. If they work together on the job for 3 days, what fraction of the work will still remain unfinished?

Accepted Answer

Introduction / Context: This problem looks at partial completion when two workers A and B work together for only a few days. You are given how long each would take individually to complete the whole job. The question is to determine what fraction of the job remains after they have worked together for 3 days. This type of question is fundamental for understanding how rates combine and how to track progress over time. Given Data / Assumptions: A alone can complete the work in 36 days. B alone can complete the work in 12 days. A and B work together for 3 days only. We assume constant work rates for both workers. We must find the fraction of the total work that remains unfinished after these 3 days. Concept / Approach: We treat the total work as 1 unit. A's rate and B's rate are each 1 divided by their respective total times. The combined rate is the sum of these two rates. Multiplying the combined rate by 3 (the number of days worked) gives the fraction of the job completed. Subtracting this from 1 yields the remaining fraction of the job. This is a direct application of work rate arithmetic. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: A's daily rate = 1/36 of the work per day. Step 3: B's daily rate = 1/12 of the work per day. Step 4: Combined rate of A and B = 1/36 + 1/12. Step 5: Use a common denominator of 36: 1/12 = 3/36, so combined rate = 1/36 + 3/36 = 4/36 = 1/9 of the work per day. Step 6: Work completed in 3 days together = 3 × (1/9) = 3/9 = 1/3 of the total work. Step 7: Remaining work = total work − work completed = 1 − 1/3 = 2/3 of the work. Verification / Alternative check: We can check by reasoning. If A and B keep working together at 1/9 of the job per day, they will need 9 days to finish the full job. In 3 days, they have done 3/9 of the job, which is 1/3. Therefore, 2/3 remains. This matches the result, and the full-time check also aligns, since 9 days is less than both 36 and 12 days, as expected for combined work. Why Other Options Are Wrong: 1/3 would be the fraction completed, not the fraction remaining. 1/4, 1/5 and 3/4 do not match the arithmetic of 1 − 1/3. 3/4 actually exceeds the completed fraction of 1/3 and would correspond to having done much more work than the pair actually did in just 3 days. Therefore, the only correct fraction for the remaining work is 2/3. Common Pitfalls: One frequent mistake is to compute the fraction completed correctly and then incorrectly identify it as the remaining fraction. Another is to misadd the rates 1/36 and 1/12 without using a proper common denominator. Some also forget to multiply the combined rate by the number of days. Keeping the sequence "find rate, multiply by time, subtract from total" helps avoid these issues. Final Answer: After 3 days of working together, two-thirds (2/3) of the work still remains unfinished.

Question 7

Pankaj has completed half of a job in 12 days. Sainath then completes the remaining half of the same job in 6 days. If Pankaj and Sainath work together on the entire job from the beginning, in how many days will they finish it?

Accepted Answer

Introduction / Context: This question involves two workers, Pankaj and Sainath, each completing half of a job in different times. We are asked to find how long it would take if they started together and worked jointly on the entire job. This is a straightforward application of converting partial-completion times into work rates and then combining those rates. Given Data / Assumptions: Pankaj completes half (1/2) of the job in 12 days. Sainath completes the remaining half (1/2) in 6 days. Each works at a constant rate whenever working. We are asked for the time for the whole job when both work together from the beginning. Concept / Approach: We consider the full job as 1 unit. From the statement, we derive Pankaj's daily work rate as half the job divided by 12 days. Similarly, Sainath's rate is half the job divided by 6 days. Adding the two rates gives the combined rate when both work simultaneously. The total time to complete the job is then 1 divided by this combined rate. This is a direct and clean application of basic work-rate arithmetic. Step-by-Step Solution: Step 1: Let total job = 1 unit. Step 2: Pankaj completes 1/2 of the job in 12 days, so his rate = (1/2) / 12 = 1/24 of the job per day. Step 3: Sainath completes 1/2 of the job in 6 days, so his rate = (1/2) / 6 = 1/12 of the job per day. Step 4: Combined rate when both work together = 1/24 + 1/12. Step 5: Convert 1/12 to 2/24, so combined rate = 1/24 + 2/24 = 3/24 = 1/8 of the job per day. Step 6: Time to complete the entire job together = 1 / (1/8) = 8 days. Verification / Alternative check: To verify, consider their work in 8 days at the combined rate of 1/8 per day. Total work done = 8 × (1/8) = 1 unit, which is the full job. Check the relative speeds: Sainath is twice as fast as Pankaj since 1/12 is double 1/24, so together they should finish in less time than either alone. Pankaj alone would need 24 days for the full job, Sainath alone 12 days, so 8 days together is reasonable and consistent. Why Other Options Are Wrong: 4 days is too small; at 1/8 per day, only half the job would be done. 12 and 16 days are too large, implying a slower combined rate than either individual, which is impossible. 6 days is also too small because at 1/8 per day only 6/8 = 3/4 of the job would be completed. Therefore, 8 days is the only option that matches the combined rate correctly. Common Pitfalls: A frequent error is to average the given times 12 and 6 to get 9 days, which would be incorrect as work rates do not average linearly in this way. Another mistake is to treat half-jobs as full jobs, leading to wrong rates. Carefully converting "half of the job in N days" into a rate of 1/(2N) job per day is the key to solving this correctly. Final Answer: Pankaj and Sainath together will complete the job in 8 days.

Question 8

Eight men can complete a certain work in 16 days, and sixteen women can complete the same work in 24 days. If a team of 4 men and 8 women work together at their respective constant rates, in how many days will they complete the work?

Accepted Answer

Introduction / Context: This problem compares the work rates of men and women and then combines a mixed team of both. We know how long it takes for only men and only women to complete the same work. Then we form a new group consisting of 4 men and 8 women and must determine the time they need together. This type of question tests your ability to convert group times into individual rates and then recombine them correctly. Given Data / Assumptions: Eight men can complete the work in 16 days. Sixteen women can complete the same work in 24 days. We consider a mixed group of 4 men and 8 women. All men are assumed to have the same constant rate, and all women likewise. We must find the number of days for 4 men and 8 women together to complete the work. Concept / Approach: The total work is taken as 1 unit. From the men-only and women-only completion times, we compute the daily work rate of one man and one woman by dividing the total work by the corresponding total man-days or woman-days. Then we calculate the combined rate of 4 men and 8 women by summing their individual contributions. Finally, we find the required time as 1 divided by this combined daily rate. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Eight men finish the work in 16 days, so total man-days = 8 × 16 = 128 man-days. Step 3: Work done by 1 man in 1 day = 1 / 128 of the work. Step 4: Sixteen women finish the work in 24 days, so total woman-days = 16 × 24 = 384 woman-days. Step 5: Work done by 1 woman in 1 day = 1 / 384 of the work. Step 6: Combined daily rate of 4 men = 4 × (1/128) = 4/128 = 1/32 of the work per day. Step 7: Combined daily rate of 8 women = 8 × (1/384) = 8/384 = 1/48 of the work per day. Step 8: Total daily rate of 4 men and 8 women together = 1/32 + 1/48. Step 9: Using common denominator 96, 1/32 = 3/96 and 1/48 = 2/96, so total rate = 5/96 of the work per day. Step 10: Time to complete the work = 1 / (5/96) = 96/5 = 19.2 days. Verification / Alternative check: We can approximate: since 8 men alone need 16 days, 4 men alone would need 32 days. Similarly, since 16 women need 24 days, 8 women alone would need 48 days. A team that combines both should take less time than either 32 or 48 days. Our result 19.2 days is indeed smaller than both and matches the exact rate calculation. This makes the answer reasonable and consistent. Why Other Options Are Wrong: 16 and 18 days are smaller than 19.2 days and would require a higher combined daily rate than 5/96, which is not supported by the given individual rates. 24 and 20 days are larger than 19.2 days; 24 days is even equal to the time for 16 women alone and contradicts the fact that adding 4 men should speed up the work. Only 19.2 days exactly corresponds to the combined rate derived from the given information. Common Pitfalls: Some students forget to divide total man-days by the number of men to get the rate for a single man. Others incorrectly average the times 16 and 24 without considering the group sizes. A careful breakdown into per-person rates, then recombining them for the new group, is the correct and robust approach for such questions. Final Answer: Four men and eight women together will complete the work in 19.2 days.

Question 9

A and B can do a piece of work separately in 24 days and 30 days respectively. They work together for 6 days, then B leaves and C joins A. A and C together complete the remaining work in 11 days. In how many days can C alone complete the entire work?

Accepted Answer

Introduction / Context: This is a multi-stage time and work problem. First A and B work together for a few days, after which B leaves and C joins A to finish the remaining work. We are asked to find C's individual time to complete the entire work alone. The problem tests your ability to track fractions of work across stages and to use the resulting information to determine an unknown worker's rate. Given Data / Assumptions: A alone can complete the work in 24 days. B alone can complete the work in 30 days. A and B work together for 6 days. Then B leaves and C works with A for 11 more days to finish the work. All workers have constant rates during the time they work. We must find the number of days C alone would need to complete the entire work. Concept / Approach: We take the total work as 1 unit. First we compute the combined rate of A and B, then calculate how much work they complete in 6 days. Subtracting this from 1 gives the remaining work. Next, using the time taken by A and C together to finish this remaining work, we find their combined rate and subtract A's individual rate to derive C's rate. Finally, we take the reciprocal of C's rate to get C's individual completion time. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: A's rate = 1/24 of the work per day. Step 3: B's rate = 1/30 of the work per day. Step 4: Combined rate of A and B = 1/24 + 1/30. Step 5: Using common denominator 120, 1/24 = 5/120 and 1/30 = 4/120, so combined rate = 9/120 = 3/40. Step 6: Work done by A and B together in 6 days = 6 × (3/40) = 18/40 = 9/20 of the total work. Step 7: Remaining work after 6 days = 1 − 9/20 = 11/20 of the job. Step 8: A and C together finish this remaining 11/20 of the job in 11 days, so their combined rate = (11/20) / 11 = 1/20 of the work per day. Step 9: A's rate is known as 1/24 per day, so C's rate = (A + C) rate − A's rate = 1/20 − 1/24. Step 10: Using common denominator 120, 1/20 = 6/120 and 1/24 = 5/120, so C's rate = 1/120 of the work per day. Step 11: Time taken by C alone to complete the whole job = 1 / (1/120) = 120 days. Verification / Alternative check: We can verify by recomputing the work distribution. In 6 days, A and B do 9/20 of the job. The remaining 11/20 is done in 11 days by A and C at 1/20 per day, giving exactly 11/20 more work. Also, checking rates: A + C = 1/24 + 1/120 = 5/120 + 1/120 = 6/120 = 1/20 as required. All stages match the information given, confirming that C's time of 120 days is correct. Why Other Options Are Wrong: 80, 90, 100, and 130 days correspond to different daily work rates for C that would disrupt the precise balance of rates and times. They would not reproduce the 11 days required for A and C to complete the remaining 11/20 of the work. Only 120 days gives C a rate that fits all parts of the problem exactly. Common Pitfalls: Common mistakes include miscalculating the fraction of work completed in the first 6 days, or forgetting to subtract this correctly from 1. Another frequent error is mishandling the fractions when computing the rate of A and C together. Always proceed step by step: find A + B work, find remaining work, use A + C time, then isolate C's rate. This structured method prevents most errors. Final Answer: C alone can complete the entire work in 120 days.

Question 10

Ehsaan is twice as good a worker as Kamal, meaning he is twice as efficient. Together they finish a piece of work in 29 days. If both work at constant rates, in how many days will Kamal alone finish the same work?

Accepted Answer

Introduction / Context: This is another efficiency comparison problem similar in style to other "twice as good" questions. We are told that one worker, Ehsaan, is twice as efficient as Kamal and that together they complete a job in 29 days. We must find how long Kamal alone would take to finish the same job. Problems like this reinforce the relationship between efficiency, work rate, and time. Given Data / Assumptions: Ehsaan is twice as efficient as Kamal. Ehsaan and Kamal together finish the work in 29 days. Both work at constant rates throughout. We are asked for Kamal's individual completion time. Concept / Approach: Efficiency is directly tied to work rate. If Ehsaan is twice as efficient as Kamal, his daily rate is twice Kamal's rate. Let Kamal's rate be a base variable and set Ehsaan's rate as twice that. The combined rate is the sum of the two and is known from the 29-day completion time. Solving this equation gives Kamal's rate, and taking the reciprocal of that rate gives the time required for Kamal alone to complete the work. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Let Kamal's daily rate = k units per day. Step 3: Ehsaan is twice as efficient, so Ehsaan's rate = 2k units per day. Step 4: Combined rate of Ehsaan and Kamal together = k + 2k = 3k units per day. Step 5: They complete the job in 29 days, so 3k = 1/29. Step 6: Solve for k: k = 1 / (29 × 3) = 1/87. Step 7: Kamal's rate is 1/87 of the work per day, so Kamal alone will need 87 days to complete the work. Verification / Alternative check: If Kamal takes 87 days, then Ehsaan, being twice as efficient, would take half that time: 87/2 = 43.5 days. Their combined rate is 1/87 + 1/43.5 = 1/87 + 2/87 = 3/87 = 1/29, which means together they finish in 29 days as stated. This confirms that the computed time of 87 days for Kamal is consistent with the given information. Why Other Options Are Wrong: 58 days would imply that Kamal is faster than the slower of the pair implied by the efficiency ratio, and it would not produce the correct joint completion time. Values like 70 or 116 days also lead to combined rates that do not match the 29-day completion time for the pair. Only 87 days yields an exact combined rate of 1/29 per day when combined with Ehsaan's rate of 2/87 per day. Common Pitfalls: Many learners misread "twice as good" and mistakenly give Ehsaan twice the time instead of twice the speed. Another common mistake is to treat 29 as directly related to Kamal's time without first using the combined rate. The correct method is to work systematically with rates, starting with the base variable for Kamal's rate and using the given combined duration. Final Answer: Kamal alone will finish the work in 87 days.

Question 11

A can do half of a certain piece of work in 8 days, while B can do one-third of the same work in 8 days. If A and B work together from the start at their constant rates, in how many days will they complete the entire work?

Accepted Answer

Introduction / Context: This problem provides partial completion times for two workers, A and B, and asks for the total time required when they work together from the beginning. A completes half of the work in a given time, and B completes one-third of the work in the same time. This type of question checks your understanding of how to convert partial work into rates and then combine those rates to find the joint completion time. Given Data / Assumptions: A completes 1/2 of the work in 8 days. B completes 1/3 of the work in 8 days. Both work together from the start. Each worker's rate is constant over time. We must find the number of days required to complete the full work together. Concept / Approach: We represent the total work as 1 unit. From the information given, we derive A's daily rate as half the job divided by 8 days, and B's daily rate as one-third of the job divided by 8 days. Adding these two rates gives the combined daily rate of A and B. The total time to complete the full job is then 1 divided by this combined rate. This method is direct and avoids unnecessary complications. Step-by-Step Solution: Step 1: Let the total work = 1 unit. Step 2: A completes 1/2 of the work in 8 days, so A's rate = (1/2) / 8 = 1/16 of the work per day. Step 3: B completes 1/3 of the work in 8 days, so B's rate = (1/3) / 8 = 1/24 of the work per day. Step 4: Combined daily rate of A and B = 1/16 + 1/24. Step 5: Using a common denominator of 48, 1/16 = 3/48 and 1/24 = 2/48. Step 6: Combined rate = 3/48 + 2/48 = 5/48 of the work per day. Step 7: Time to complete the full job = 1 / (5/48) = 48/5 days. Step 8: 48/5 days = 9.6 days. Verification / Alternative check: Approximate reasoning supports this result. A's full-time if working alone would be 16 days for the whole work, and B's alone would be 24 days. Working together must therefore take less than 16 days. Our answer 9.6 days is indeed less than 16 and closer to the value you expect when rates of 1/16 and 1/24 are added. Also, if they work for 9.6 days at 5/48 per day, total work = 9.6 × 5/48 = 48/5 × 5/48 = 1 unit, confirming correctness. Why Other Options Are Wrong: 10.5, 11.2, 16 and 8 days all correspond to effective rates that do not match 5/48 per day. For instance, 16 days would indicate a rate of only 1/16 per day, which is A's rate alone, ignoring B's contribution. 8 days would require a higher rate than even both working together can provide. Therefore, 9.6 days is the only duration consistent with the combined rate. Common Pitfalls: Some students mistakenly average the partial times or assume that 1/2 and 1/3 can be simply added as if they corresponded to the same time period without adjusting for the 8 days factor. Another mistake is miscalculating the fractions when finding 1/16 and 1/24. Keeping careful track of fractions and always working in terms of daily rates prevents these errors. Final Answer: A and B together will complete the entire work in 9.6 days.

Question 12

Painter A can paint a house alone in 40 days and painter B can paint the same house alone in 60 days. With the additional help of painter C, they are able to finish painting the house in only 20 days when all three work together. In how many days ca…

Accepted Answer

Introduction / Context: In this time and work aptitude question, three painters A, B, and C are working together to paint a single house. You are given the individual times of A and B, and the combined time of A, B, and C. The concept tested is work rate or efficiency and how to compute the time taken by one worker when the combined time of several workers is known. Given Data / Assumptions: Painter A can complete the work alone in 40 days. Painter B can complete the same work alone in 60 days. Painters A, B, and C together complete the work in 20 days. The total work is assumed to be 1 unit (one complete house painted). All painters work at constant rates and their efficiencies do not change over time. Concept / Approach: The key idea is to convert time taken into daily work rates. If a worker finishes 1 unit of work in T days, then the worker's rate is 1/T units per day. When multiple workers operate together, their rates add algebraically. Once the combined rate of all three is known, we subtract the rates of A and B from the total to obtain the rate of C alone and then invert that rate to get C's time. Step-by-Step Solution: Step 1: Let the total work be 1 unit. Step 2: Rate of painter A = 1/40 units per day. Step 3: Rate of painter B = 1/60 units per day. Step 4: A, B, and C together finish the work in 20 days, so their combined rate = 1/20 units per day. Step 5: Let rate of painter C be Rc. Then 1/40 + 1/60 + Rc = 1/20. Step 6: Compute 1/40 + 1/60. Using the LCM of 120, we get 1/40 = 3/120 and 1/60 = 2/120, so 1/40 + 1/60 = 5/120 = 1/24. Step 7: Substitute into the equation: 1/24 + Rc = 1/20. Step 8: Therefore Rc = 1/20 - 1/24. Step 9: Using the LCM of 120 again, 1/20 = 6/120 and 1/24 = 5/120, so Rc = 1/120 units per day. Step 10: Time taken by C alone = 1 / Rc = 1 / (1/120) = 120 days. Verification / Alternative check: We can verify by checking combined work in 20 days. In 20 days, A does 20 * (1/40) = 1/2 of the work and B does 20 * (1/60) = 1/3 of the work. Together A and B complete 1/2 + 1/3 = 5/6 of the work. The remaining 1 - 5/6 = 1/6 of the work must be done by C in 20 days. C's daily work then is (1/6) / 20 = 1/120, which matches our calculated rate and confirms that C alone needs 120 days. Why Other Options Are Wrong: 100 days: This would correspond to a higher rate than 1/120, which does not match the given combined time of 20 days. 90 days: Implies an even higher work rate for C, causing the combined team to finish in less than 20 days. 80 days: Also makes the combined rate too large and contradicts the given total time of 20 days. Common Pitfalls: Students may mistakenly add times directly instead of adding rates, or they may confuse the idea of “more efficient” with fewer days. Another frequent error is miscalculating fractions during subtraction of the rates. Always convert to a common denominator and carefully simplify the fractions to avoid arithmetic mistakes. Final Answer: Painter C alone will complete the entire painting job in 120 days.

Question 13

Rohan and Mohit together can build a wall in 8 days. Mohit and Vikas can build the same wall in 10 days, and Vikas and Rohan together can build it in 12 days. In how many days will Rohan, Mohit, and Vikas working all together be able to complete the…

Accepted Answer

Introduction / Context: This problem is a classic example of a work and time question where we know how long different pairs of workers take to complete a job. The goal is to find how long all three workers, Rohan, Mohit, and Vikas, take when they work together. It tests understanding of combined work rates and solving simple linear equations with multiple unknowns. Given Data / Assumptions: Rohan and Mohit together complete the wall in 8 days. Mohit and Vikas together complete the wall in 10 days. Vikas and Rohan together complete the wall in 12 days. The total work is considered as 1 unit (one complete wall). All work rates are constant and additive. Concept / Approach: Assign individual daily work rates to Rohan, Mohit, and Vikas. Use the information about their pairwise times to build equations in terms of these rates. Then solve for the sum of the three individual rates, which equals the combined rate when all three work together. Finally, take the reciprocal of that rate to find the required number of days for the combined team. Step-by-Step Solution: Step 1: Let R, M, and V be the daily work rates of Rohan, Mohit, and Vikas respectively. Step 2: Rohan and Mohit together finish in 8 days, so R + M = 1/8. Step 3: Mohit and Vikas together finish in 10 days, so M + V = 1/10. Step 4: Vikas and Rohan together finish in 12 days, so V + R = 1/12. Step 5: Add all three equations: (R + M) + (M + V) + (V + R) = 1/8 + 1/10 + 1/12. Step 6: The left side becomes 2(R + M + V). The right side is 1/8 + 1/10 + 1/12. Step 7: Compute 1/8 + 1/10 + 1/12. Using common denominator 120, we get 15/120 + 12/120 + 10/120 = 37/120. Step 8: Therefore 2(R + M + V) = 37/120, so R + M + V = 37/240. Step 9: When all three work together, their combined rate is 37/240 units per day. Hence time taken = 1 / (37/240) = 240/37 days. Verification / Alternative check: We can approximate the value 240/37. Since 37 * 6 = 222 and 37 * 7 = 259, the time is between 6 and 7 days, approximately 6.49 days. That is reasonable because each pair alone takes between 8 and 12 days, so three people together should take less time than any pair, which matches our result. Why Other Options Are Wrong: 120/37 days: This is roughly 3.24 days, which is unrealistically fast compared with the pairwise times. 150/37 days: This is around 4.05 days, still too fast compared with the individual pairwise times of 8, 10, and 12 days. 180/37 days: This is close to 4.86 days, which does not match the derived combined rate and is an incorrect result of manipulating the equations. Common Pitfalls: One common mistake is to average the three given times directly, which does not work for rate problems. Another error is forgetting to divide the sum of the three pairwise equations by 2 to get R + M + V. Students may also mishandle fraction addition, so it is important to use a common denominator carefully and simplify step by step. Final Answer: All three, Rohan, Mohit, and Vikas, working together will complete the wall in 240/37 days.

Question 14

Person X can complete a task alone in 8 days, whereas person Y can complete the same task alone in 10 days. If X and Y work together on the task from start to finish, in how many days will they complete the entire task?

Accepted Answer

Introduction / Context: This is a basic time and work problem involving two people, X and Y, each capable of completing a task alone in different amounts of time. The question checks whether you understand how to combine individual work rates to find the time required when both persons work together on the same job. Given Data / Assumptions: X alone can complete the work in 8 days. Y alone can complete the work in 10 days. Both X and Y work together for the entire duration. Total work is considered as 1 unit. Work rates are constant and additive. Concept / Approach: We convert the time taken by each person into a daily work rate expressed as a fraction of the total work completed per day. Then we add their rates to obtain the combined rate. Finally, we take the reciprocal of that combined rate to find the total time needed for them to complete the task when working together. Step-by-Step Solution: Step 1: Assume the total work is 1 unit. Step 2: Rate of X = 1/8 units per day. Step 3: Rate of Y = 1/10 units per day. Step 4: Combined rate of X and Y working together = 1/8 + 1/10. Step 5: Find a common denominator for 1/8 and 1/10. The least common multiple of 8 and 10 is 40. Step 6: Convert the fractions: 1/8 = 5/40 and 1/10 = 4/40. Step 7: Add the two rates: 5/40 + 4/40 = 9/40 units per day. Step 8: Time taken when working together = total work / combined rate = 1 / (9/40) = 40/9 days. Verification / Alternative check: We can check the reasonableness of the answer by approximation. X alone takes 8 days and Y alone takes 10 days. When they work together, the time must be less than 8 days. 40/9 is approximately 4.44 days, which is indeed less than 8 days and greater than 0, so the answer is logical and consistent with expectations. Why Other Options Are Wrong: 9/40 days: This is actually the combined rate, not the time. It is the reciprocal of the correct answer. 44/9 days: This is greater than 8 days, which cannot be correct because working together must be faster than either person alone. 9/44 days: This is a very small fraction of a day and does not match the combined rate calculation. Common Pitfalls: A frequent mistake is to average the two times as (8 + 10) / 2 = 9 days, which is incorrect. Time cannot be averaged directly in such problems; rates must be added instead. Another pitfall is forgetting to invert the combined rate to obtain the time. Care must also be taken when adding fractions to ensure the proper least common multiple is used. Final Answer: The task will be completed in 40/9 days when X and Y work together.

Question 15

Two flood gates A and B can be used to fill a reservoir. If both gates A and B are opened together, the reservoir becomes full in 6 hours. Gate A is faster and can fill the reservoir 5 hours sooner than gate B can. How many hours does the faster flo…

Accepted Answer

Introduction / Context: This question deals with the classic pipes and cisterns or inflow problems, which are mathematically similar to time and work problems. Here, two flood gates are filling a reservoir, and we know the combined time and the relationship between the times of the two individual gates. The task is to find the time taken by the faster gate to fill the reservoir alone. Given Data / Assumptions: Gates A and B together can fill the reservoir in 6 hours. Gate A is faster and fills the reservoir 5 hours sooner than gate B. Let the time taken by gate A be Ta hours. Then the time taken by gate B is Ta + 5 hours. Total work (reservoir filled once) is considered as 1 unit. Concept / Approach: The method is to express the work rates of the gates in terms of Ta. If a gate takes T hours to fill the reservoir, its rate is 1/T units per hour. Since both gates together fill the reservoir in 6 hours, the sum of their rates must equal 1/6. This leads to an equation in Ta, which is then solved to find the time taken by the faster gate. Step-by-Step Solution: Step 1: Let Ta be the time taken by gate A alone. Then time taken by gate B alone is Ta + 5. Step 2: Rate of gate A = 1/Ta units per hour. Step 3: Rate of gate B = 1/(Ta + 5) units per hour. Step 4: Together, they fill the reservoir in 6 hours, so combined rate = 1/6. Step 5: Set up the equation: 1/Ta + 1/(Ta + 5) = 1/6. Step 6: Multiply through by 6 * Ta * (Ta + 5) to clear denominators: 6(Ta + 5) + 6Ta = Ta(Ta + 5). Step 7: Simplify the left side: 6Ta + 30 + 6Ta = 12Ta + 30. Step 8: The equation becomes Ta(Ta + 5) = 12Ta + 30. Step 9: Expand and rearrange: Ta^2 + 5Ta - 12Ta - 30 = 0, which simplifies to Ta^2 - 7Ta - 30 = 0. Step 10: Solve the quadratic equation Ta^2 - 7Ta - 30 = 0. The roots are Ta = 10 and Ta = -3. Since time cannot be negative, Ta = 10 hours. Verification / Alternative check: If gate A takes 10 hours, then gate B takes 15 hours. Rate of A = 1/10, rate of B = 1/15. Combined rate = 1/10 + 1/15 = (3/30 + 2/30) = 5/30 = 1/6, meaning they fill the reservoir in 6 hours together, which matches the problem statement and confirms the solution. Why Other Options Are Wrong: 7 hours: This would not yield a combined time of 6 hours when paired with a gate that is 5 hours slower. 13 hours: Leads to inconsistent combined rate and does not satisfy the equation 1/Ta + 1/(Ta + 5) = 1/6. 5 hours: Implies gate B takes 10 hours, and together they would fill much faster than 6 hours. Common Pitfalls: Learners often misinterpret the phrase “5 hours faster” and sometimes incorrectly set the time of the slower gate as Ta - 5 instead of Ta + 5. Another frequent error is algebraic manipulation when solving the quadratic equation, such as sign mistakes or missing terms during expansion. Carefully forming and solving the equation is crucial to arriving at the correct answer. Final Answer: The faster flood gate alone will fill the reservoir in 10 hours.

Question 16

A alone can complete a piece of work in 20 days. Worker B is 25% more efficient than A, that is, B's efficiency is 125% of A's efficiency. A and B work together for 4 days, after which the remaining work is completed by C alone in 22 days. How many …

Accepted Answer

Introduction / Context: This is a multi-step time and work problem that involves comparing efficiencies of workers A and B and then adding a third worker C who finishes the remaining work. The question checks your understanding of percentage-based efficiency, conversion of efficiency into time, combined work, and then determining the individual time of a third worker based on partial work information. Given Data / Assumptions: A alone finishes the work in 20 days. B is 25% more efficient than A (that is, B's efficiency is 125% of A's efficiency). A and B work together for 4 days. The remaining work is completed by C alone in 22 days. The work rate of each worker is constant throughout. Total work is assumed to be 1 unit. Concept / Approach: We first convert efficiency information into work rates. Since B is 25% more efficient than A, B's rate is 1.25 times A's rate. Using A's time, we find A's work rate and then B's work rate. We compute how much work A and B complete in 4 days, subtract that from the total work, and then use the remaining work and time taken by C to find C's rate. Finally, we invert C's rate to find C's total time for the entire job. Step-by-Step Solution: Step 1: Let the total work be 1 unit. Step 2: A's time = 20 days, so A's rate = 1/20 units per day. Step 3: B is 25% more efficient than A, so B's rate = 1.25 * (1/20) = 1/16 units per day (which corresponds to B taking 16 days for the whole work). Step 4: Combined rate of A and B = 1/20 + 1/16 = (4/80 + 5/80) = 9/80 units per day. Step 5: In 4 days, work done by A and B together = 4 * 9/80 = 36/80 = 9/20 of the work. Step 6: Remaining work after 4 days = 1 - 9/20 = 11/20. Step 7: C alone completes this remaining 11/20 of the work in 22 days. Step 8: Therefore, C's daily work rate = (11/20) / 22 = 11 / 440 = 1/40 units per day. Step 9: Time taken by C to do the entire work alone = 1 / (1/40) = 40 days. Verification / Alternative check: To verify, calculate actual work done. A and B together do 9/20 of the work. The remaining 11/20 is done by C. At rate 1/40, in 22 days C does 22 * (1/40) = 22/40 = 11/20, which matches the remaining work. This confirms the correctness of C's rate and the final time of 40 days for C alone. Why Other Options Are Wrong: 30 days: This would make C's rate 1/30, in which case 22 days of work would complete 22/30 of the job, which is not equal to 11/20. 35 days: This gives C a rate of 1/35, which does not match the requirement that 11/20 of the work is done in 22 days. 42 days: This again leads to a mismatched rate and incorrect remaining work calculation. Common Pitfalls: One common mistake is to misinterpret "125% more efficient" as 225% of A's efficiency instead of 125% of A's efficiency. Another error is incorrectly adding the work done or forgetting to subtract the work already completed by A and B. It is crucial to keep track of fractions carefully and ensure that each step builds accurately on the previous calculations. Final Answer: C alone would take 40 days to complete the entire work from the beginning.

Question 17

Worker A is 50% more efficient than worker B. They start working together on a job, and after 8 days, 7/12 of the work is still left unfinished. In how many days would B alone be able to complete the entire work from the beginning?

Accepted Answer

Introduction / Context: This question examines how to use relative efficiency information and partial work done to determine the time required by a single worker. Knowing that A is 50% more efficient than B lets us express one worker's rate in terms of the other. The question also provides how much work remains after a specific time period when both work together, which allows us to solve for B's individual time to complete the full job alone. Given Data / Assumptions: Worker A is 50% more efficient than worker B. A and B work together for 8 days. After 8 days, 7/12 of the work is still left. That means 5/12 of the work has been completed in 8 days. We assume total work is 1 unit and all work rates are constant. Concept / Approach: First, we express A's work rate in terms of B's rate because efficiency is proportional to rate. If B's rate is b units per day, then A's rate is 1.5b units per day. Their combined rate is then 2.5b. We use the information that 5/12 of the job is completed in 8 days to calculate b. Once we know B's rate, we can invert it to find the number of days B alone would take to do the whole job. Step-by-Step Solution: Step 1: Let B's rate be b units per day. Step 2: A is 50% more efficient, so A's rate = 1.5b units per day. Step 3: Combined rate of A and B = b + 1.5b = 2.5b units per day. Step 4: In 8 days, the work done by them together = 8 * 2.5b = 20b units. Step 5: We know that this equals 5/12 of the total work (since 7/12 is left). So, 20b = 5/12. Step 6: Solve for b: b = (5/12) / 20 = 5 / 240 = 1/48 units per day. Step 7: B's time to complete the full job alone = 1 / b = 1 / (1/48) = 48 days. Verification / Alternative check: Check the combined work done in 8 days using the found rates. A's rate = 1.5b = 1.5 * (1/48) = 1/32 units per day. B's rate = 1/48 units per day. Combined rate = 1/32 + 1/48. Using a common denominator of 96, we get 3/96 + 2/96 = 5/96 units per day. In 8 days, total work = 8 * (5/96) = 40/96 = 5/12, which matches the given information that 5/12 of the work is completed. Hence the calculation is correct. Why Other Options Are Wrong: 36 days: This would give B a rate of 1/36, which does not produce the correct fraction of work completed in 8 days. 40 days: Implies a rate of 1/40, again inconsistent with 5/12 of work being done in 8 days by both workers. 60 days: This is slower than the correct value and does not satisfy the relationship derived from the partial work information. Common Pitfalls: Some students confuse "50% more efficient" with "works in 50% of the time." Efficiency relates to rate, not directly to days, so we must be careful to scale rates, not times. Others forget that only 5/12 of the work is completed, not 7/12, which leads to incorrect equations. Being consistent about which fraction of work has been done and carefully solving the algebraic equation are crucial steps. Final Answer: B alone would take 48 days to complete the entire work from the beginning.

Question 18

Type 1 workers are three times as productive as Type 2 workers. Twelve Type 1 workers can complete a certain task in 10 days. If instead a group of 4 Type 1 workers and 8 Type 2 workers work together, in how many days will they complete the same tas…

Accepted Answer

Introduction / Context: This time and work question compares two types of workers with different productivities. Type 1 workers are more efficient than Type 2 workers. We know how long a group of Type 1 workers takes to complete a task and must find the time required by a mixed group containing both worker types. This tests proportional reasoning and the concept of equivalent work rates. Given Data / Assumptions: Type 1 workers can each do three times as much work as a Type 2 worker in the same time. Twelve Type 1 workers complete the task in 10 days. We need the time taken when 4 Type 1 workers and 8 Type 2 workers work together. All workers of the same type are equally efficient, and work rates are constant. Total work is assumed to be 1 unit. Concept / Approach: First, we determine the work rate of a single Type 1 worker by using the total work completed by 12 such workers in 10 days. Knowing that Type 1 workers are three times as efficient as Type 2 workers allows us to find the rate of a Type 2 worker. We then compute the combined daily work rate of the new group (4 Type 1 and 8 Type 2 workers) and invert that rate to find the total time required for the job. Step-by-Step Solution: Step 1: Let the total work be 1 unit. Step 2: Twelve Type 1 workers complete the work in 10 days, so total work = 12 * (rate of Type 1) * 10 = 1. Step 3: Let the rate of one Type 1 worker be r units per day. Then 12 * r * 10 = 1, so 120r = 1, giving r = 1/120 units per day. Step 4: A Type 2 worker is one third as efficient as a Type 1 worker, so rate of one Type 2 worker = r / 3 = (1/120) / 3 = 1/360 units per day. Step 5: In the new group, we have 4 Type 1 workers and 8 Type 2 workers. Step 6: Combined rate of 4 Type 1 workers = 4 * (1/120) = 4/120 = 1/30 units per day. Step 7: Combined rate of 8 Type 2 workers = 8 * (1/360) = 8/360 = 1/45 units per day. Step 8: Total rate of the mixed group = 1/30 + 1/45. Use LCM of 90: 1/30 = 3/90 and 1/45 = 2/90, so total rate = 5/90 = 1/18 units per day. Step 9: Time taken to complete the work = 1 / (1/18) = 18 days. Verification / Alternative check: We can check the reasonableness by comparing rates. Twelve Type 1 workers alone take 10 days, so their rate is 1/10 of the job per day. The new group has fewer Type 1 workers and some slower Type 2 workers. Hence, its rate must be smaller than 1/10, meaning the time must be more than 10 days. Our answer 18 days is indeed greater than 10, which makes logical sense. Why Other Options Are Wrong: 16 days: This suggests a higher rate than 1/18 and does not match the calculated combined rate of the group. 17 days: Not supported by any correct combination of rates and is slightly faster than the correct answer. 20 days: This would imply a slower rate of 1/20 units per day, which does not agree with the derived total rate of 1/18 units per day. Common Pitfalls: Students may misinterpret the statement "Type 1 workers can do three times the work of Type 2 workers" and invert the ratio. Another error is to average the days directly instead of using rates. Properly calculating individual and combined work rates and keeping track of fractions is essential to solving such problems accurately. Final Answer: The group of 4 Type 1 and 8 Type 2 workers will complete the task in 18 days.

Question 19

Worker A can complete a certain piece of work in 24 days. Worker B is 20% more efficient than A, so B can finish the same work faster. Worker C can complete the work in 10 more days than B takes. How many days will A and C, working together from the…

Accepted Answer

Introduction / Context: This time and work problem involves three workers A, B, and C with varying efficiencies. The efficiency of B is related to A by a percentage, and C is slower than B by a fixed number of days. The question asks for the time taken by A and C together to complete the task, which requires converting efficiency relationships into exact work rates and then combining those rates appropriately. Given Data / Assumptions: A alone can complete the work in 24 days. B is 20% more efficient than A. C takes 10 more days than B to complete the same work alone. A and C work together from the beginning. Total work is considered as 1 unit, and all workers have constant rates. Concept / Approach: We first determine A's work rate, then use the fact that B is 20% more efficient to find B's work rate and corresponding time. Using B's time, we find C's time, which is 10 days more than B's. We then convert C's time into a rate. The combined rate of A and C is the sum of their individual rates, and taking the reciprocal of this combined rate gives the total time required when A and C work together. Step-by-Step Solution: Step 1: Assume total work is 1 unit. Step 2: A can do the work in 24 days, so A's rate = 1/24 units per day. Step 3: B is 20% more efficient than A, so B's rate = 1.2 * (1/24) = 1/20 units per day. Hence B's time = 20 days. Step 4: C takes 10 more days than B, so C's time = 20 + 10 = 30 days. Therefore C's rate = 1/30 units per day. Step 5: A and C work together, so combined rate = 1/24 + 1/30. Step 6: Use the least common multiple of 24 and 30, which is 120. Then 1/24 = 5/120 and 1/30 = 4/120. Step 7: Combined rate = 5/120 + 4/120 = 9/120 = 3/40 units per day. Step 8: Time taken by A and C together = total work / combined rate = 1 / (3/40) = 40/3 days. Verification / Alternative check: We can estimate numerically. A alone would take 24 days, and C alone would take 30 days, so their average is about 27 days. When they work together, the time must be less than either 24 or 30 days. The answer 40/3 is about 13.33 days, which is comfortably smaller and reasonable since two workers are sharing the workload. Rechecking the rates confirms that the arithmetic is consistent. Why Other Options Are Wrong: 43/3 days: This is more than 14 days and does not match the correctly calculated combined rate. 50/3 days: This is around 16.67 days, corresponding to a slower combined rate than 3/40 and does not align with the data. 41/2 days: This represents 20.5 days, which is even closer to the time of a single worker and clearly inconsistent with two workers combining their efforts. Common Pitfalls: A common error is to treat "20% more efficient" as adding 20% of the time instead of 20% of the rate. Another mistake is to average days instead of combining rates. Students may also mis-handle fractions when adding 1/24 and 1/30, so using the correct least common multiple and carefully adding the converted fractions is very important. Final Answer: Workers A and C together will finish the work in 40/3 days.

Question 20

Prabhat completes one-half of a job in 12 days. Santosh completes the remaining half of the same job in 6 days. If both Prabhat and Santosh work together from the beginning on the entire job, in how many days will they complete it?

Accepted Answer

Introduction / Context: This problem involves two workers, Prabhat and Santosh, each completing half of a job in different times when working alone. The question asks for the total time needed if they work together from the start. It tests your ability to interpret partial work information and transform it into full-work rates for each person before combining them. Given Data / Assumptions: Prabhat completes 1/2 of the job in 12 days. Santosh completes the remaining 1/2 of the job in 6 days. We assume total work is 1 unit. Both workers have constant work rates. We need the time when both work together on the full job from the beginning. Concept / Approach: From the time taken to complete half the work, we can infer the time each worker would need for the full job. This leads to daily work rates for both Prabhat and Santosh. Once we know their individual rates, we add them to find the combined rate and then invert that rate to obtain the total time required when they work together on the complete job. This approach uses the idea that work rates add for simultaneous workers. Step-by-Step Solution: Step 1: Let total work be 1 unit. Step 2: Prabhat completes 1/2 of the work in 12 days. So his rate = (1/2) / 12 = 1/24 units per day. Hence, he would finish the full job in 24 days. Step 3: Santosh completes 1/2 of the work in 6 days. So his rate = (1/2) / 6 = 1/12 units per day. Hence, he would finish the full job in 12 days. Step 4: When they work together, combined rate = 1/24 + 1/12. Step 5: Express 1/12 with denominator 24: 1/12 = 2/24. Step 6: Combined rate = 1/24 + 2/24 = 3/24 = 1/8 units per day. Step 7: Time taken to complete the full job together = 1 / (1/8) = 8 days. Verification / Alternative check: Check the work done in 8 days. At a combined rate of 1/8 units per day, in 8 days they do 8 * (1/8) = 1 unit of work, which exactly equals the full job. The time is less than both 24 days (Prabhat alone) and 12 days (Santosh alone), which is consistent because combining workers should always reduce the total time. Why Other Options Are Wrong: 12 days: This is the time Santosh would need alone, and it ignores the contribution of Prabhat. 4 days: This would require a combined rate of 1/4 units per day, which is much larger than the actual rate of 1/8 units per day derived from the given data. 16 days: This is longer than Santosh's individual time and therefore cannot be correct when both are working together. Common Pitfalls: Students may incorrectly assume that if half is done in 12 days and half in 6 days, then the total time together is some direct average of 12 and 6. However, time cannot be added in such a way here; instead, we must work with rates. Another pitfall is forgetting that Prabhat and Santosh are working on the entire job together, not only on their original halves. Final Answer: Prabhat and Santosh together will complete the entire job in 8 days.

Time and Work Questions