Question 1

Worker A can complete a piece of work alone in 24 days. The time taken by A to complete one-third of the work is equal to the time taken by worker B to complete one-half of the same work. If both A and B start working together from the beginning, in…

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Introduction / Context: This question connects the partial work done by two workers, A and B, and uses that relation to deduce B's efficiency. A's time to complete one-third of the work equals B's time to complete one-half of the work. Once we know the individual times of A and B for the full job, we can combine their rates to find how long they take to complete the job together. Given Data / Assumptions: A can complete the full work in 24 days. Time taken by A to complete one-third of the work equals the time taken by B to complete one-half of the work. Both A and B work together on the entire job. Total work is 1 unit. Work rates remain constant throughout. Concept / Approach: We use the relationship between the partial portions (one-third and one-half) to find B's full-time for the job. Time equals work divided by rate. Therefore, we equate the time expressions and solve for B's rate. After obtaining individual rates for both A and B, we add them to get the combined rate. Finally, we take the reciprocal of that rate to find the total time for completing the work together. Step-by-Step Solution: Step 1: Let total work be 1 unit. Step 2: A completes the work in 24 days, so A's rate = 1/24 units per day. Step 3: Time taken by A to finish one-third of the work = (1/3) / (1/24) = 8 days. Step 4: This same amount of time, 8 days, is the time B takes to finish one-half of the work. Step 5: So, for B, (1/2) / (rate of B) = 8 days. Let B's rate be r units per day. Then (1/2) / r = 8, so r = (1/2) / 8 = 1/16 units per day. Step 6: Therefore, B alone would take 16 days to complete the full job. Step 7: Combined rate of A and B = 1/24 + 1/16. Step 8: Use the least common multiple of 24 and 16, which is 48. Then 1/24 = 2/48 and 1/16 = 3/48. Step 9: Combined rate = 2/48 + 3/48 = 5/48 units per day. Step 10: Time taken together = 1 / (5/48) = 48/5 days. Verification / Alternative check: We can check by computing the work completed in 48/5 days (which is 9.6 days). At rate 5/48 units per day, the work done in 48/5 days is (5/48) * (48/5) = 1 unit, confirming that the job is completed. The time is less than both 24 days and 16 days, which is expected when two workers cooperate. Why Other Options Are Wrong: 21/3 days: This simplifies to 7 days, which is unrealistically small given that A alone needs 24 days and B needs 16 days. 48 days: This is the time it would take a far slower worker, not the combined time of A and B. 40 days: Also too large, and it does not correspond to the combined rate derived from the data. Common Pitfalls: A common mistake is to mix up the fractions and equate the wrong parts of the work, for example confusing one-third and one-half. Some students also incorrectly handle the step of computing B's rate. It is important to clearly set up the equation for time as work divided by rate and solve carefully. Additionally, avoid simply averaging 24 and 16 days, which does not give the correct combined time. Final Answer: A and B together will complete the work in 48/5 days.

Question 2

A team of 48 people can complete a certain task in 17 days. After they have worked together for 6 days, 4 workers leave the team. Starting from that moment, in how many additional days will the remaining workers take to complete the remaining work?

Accepted Answer

Introduction / Context: This is a straightforward work and manpower distribution problem. We are told how long a team of 48 people takes to complete a task, and then some workers leave before the work is finished. The objective is to calculate how much additional time the reduced team needs to complete the remaining part of the work. This question tests your understanding of total work, work already done, and the work rate of the remaining team. Given Data / Assumptions: 48 people complete the task in 17 days. They work together for 6 days initially. After 6 days, 4 people leave, leaving 44 people. We assume constant efficiency for each worker and that work is distributed uniformly. Total work is taken as 1 unit. Concept / Approach: We first compute the total work in terms of man-days, that is, number of workers multiplied by days. We then calculate how much of this work is completed in the first 6 days by the initial team. The remaining work can then be calculated. With fewer workers, we compute the new daily work rate and finally determine the number of additional days needed to finish the remaining work. Step-by-Step Solution: Step 1: Compute total work in man-days. Total work = 48 workers * 17 days = 816 man-days. Step 2: Work done in the first 6 days by 48 workers = 48 * 6 = 288 man-days. Step 3: Remaining work in man-days = total man-days - work already done = 816 - 288 = 528 man-days. Step 4: After 6 days, 4 workers leave. Remaining workers = 48 - 4 = 44. Step 5: Let the number of additional days required be D. Then work done by 44 workers in D days = 44 * D man-days. Step 6: This must equal the remaining work of 528 man-days, so 44 * D = 528. Step 7: Solve for D: D = 528 / 44 = 12 days. Verification / Alternative check: We can quickly verify by checking total time. First 6 days with 48 workers plus 12 more days with 44 workers completes the task. It is consistent with the total of 816 man-days, so the answer is coherent. Additionally, as the number of workers has only slightly decreased, the extra time compared with the original 17 days is not very large, which aligns with our result. Why Other Options Are Wrong: 13 days: This would give 44 * 13 = 572 man-days, which exceeds the remaining work requirement. 15 days: 44 * 15 = 660 man-days, which is far more than needed and does not fit the total work calculation. 16 days: 44 * 16 = 704 man-days, which again is too large and inconsistent with the total work of 816 man-days. Common Pitfalls: Learners may forget to think in terms of total man-days and may attempt to directly manipulate the days without accounting for the changing team size. Another common error is subtracting days instead of calculating the remaining work explicitly. Also, mixing up the order and incorrectly using 17 - 6 instead of working with man-days can lead to incorrect results. Final Answer: From the moment 4 workers leave, the remaining team will need 12 days to finish the task.

Question 3

Two typists of different speeds can together complete a typing job in 6 minutes. If the first typist types alone for 4 minutes and then the second typist types alone for 6 minutes, 1/5 of the job is still left unfinished. How many minutes would it t…

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Introduction / Context: This problem describes two typists with different typing speeds. Together they can finish a job in a short time, and we also know how much of the job is completed when each works alone for a specific number of minutes. The scenario tests your understanding of work rates, partial work contribution by different persons, and identifying the slower worker based on calculated rates. Given Data / Assumptions: The two typists together can finish the job in 6 minutes. If the first typist works alone for 4 minutes and then the second typist works alone for 6 minutes, 1/5 of the job remains incomplete. Therefore, 4 minutes of the first typist plus 6 minutes of the second typist completes 4/5 of the job. We need the time taken by the slower typist to do the whole job alone. Total work is taken as 1 unit. Concept / Approach: Assign work rates to the two typists and translate the verbal description into equations. The combined work time of 6 minutes gives a relation between their rates. The second condition that 4 minutes of the first plus 6 minutes of the second typist complete 4/5 of the job gives another equation. Solving these simultaneously provides individual rates. The typist with the smaller rate is the slower one, and the reciprocal of that rate gives the time to complete the job alone. Step-by-Step Solution: Step 1: Let the rate of the first typist be f jobs per minute and the rate of the second typist be s jobs per minute. Step 2: Together they complete the job in 6 minutes, so f + s = 1/6. Step 3: When the first works for 4 minutes and the second for 6 minutes, they complete 4/5 of the job. So, 4f + 6s = 4/5. Step 4: From f + s = 1/6, we can write f = 1/6 - s. Step 5: Substitute f into the second equation: 4(1/6 - s) + 6s = 4/5. Step 6: Simplify: 4/6 - 4s + 6s = 4/5, so 2/3 + 2s = 4/5. Step 7: Subtract 2/3 from both sides. 4/5 - 2/3 = (12/15 - 10/15) = 2/15, so 2s = 2/15. Step 8: Therefore s = 1/15 jobs per minute. This is the rate of the second typist. Step 9: Since f + s = 1/6, we get f = 1/6 - 1/15. Find the LCM of 6 and 15 which is 30, so 1/6 = 5/30 and 1/15 = 2/30. Step 10: Thus f = 5/30 - 2/30 = 3/30 = 1/10 jobs per minute. Step 11: The slower typist has the smaller rate, which is s = 1/15 jobs per minute. Hence, time taken by the slower typist alone = 1 / (1/15) = 15 minutes. Verification / Alternative check: We can verify the combined rate: f + s = 1/10 + 1/15 = (3/30 + 2/30) = 5/30 = 1/6, so they indeed finish the job in 6 minutes together. For the second condition, 4f + 6s = 4 * (1/10) + 6 * (1/15) = 4/10 + 6/15 = 2/5 + 2/5 = 4/5, which matches the statement that only 1/5 of the job remains. This confirms the correctness of the calculations. Why Other Options Are Wrong: 10 minutes: This corresponds to the faster typist's time, not the slower one. 12 minutes: There is no rate consistent with all conditions that would yield 12 minutes as the slower typist's time. 20 minutes: This implies an even slower typist and would not satisfy the equations based on the given data. Common Pitfalls: Common errors include misinterpreting "1/5 of the work is left" as "1/5 completed," which reverses the fraction in the equation. Another pitfall is incorrectly setting up or solving the system of simultaneous equations, especially when working with fractions. Careful algebraic manipulation and clear identification of which typist is slower based on their rate are crucial for correct solutions. Final Answer: The slower typist would take 15 minutes to complete the typing job alone.

Question 4

If person X can complete a piece of work in p days and person Y can complete the same piece of work in q days (with p and q positive), then in how many days will X and Y working together be able to complete that work?

Accepted Answer

Introduction / Context: This is a conceptual time and work question that asks for the general formula for the combined time when two people, X and Y, work together on a task. Instead of numerical values, the question uses variables p and q to represent the individual times. The goal is to derive a formula that expresses the number of days required when both work simultaneously. Given Data / Assumptions: X alone finishes the work in p days. Y alone finishes the work in q days. Both X and Y work together from the beginning. Total work is considered as 1 unit. p and q are positive real numbers, and p + q is not zero. Concept / Approach: The key idea is that time and work questions are easier when we work with rates rather than times directly. If X finishes the work in p days, then X's work rate is 1/p units per day. Similarly, Y's rate is 1/q units per day. When they work together, their combined rate is the sum of these rates, that is, 1/p + 1/q. The time taken together equals total work divided by combined rate. Simplifying this expression gives the required formula in terms of p and q. Step-by-Step Solution: Step 1: Assume total work is 1 unit. Step 2: Since X completes 1 unit of work in p days, X's rate = 1/p units per day. Step 3: Since Y completes 1 unit of work in q days, Y's rate = 1/q units per day. Step 4: When X and Y work together, their combined rate = 1/p + 1/q. Step 5: Combine the fractions: 1/p + 1/q = (q + p) / (p*q). Step 6: Time taken when working together = total work / combined rate = 1 / ( (p + q) / (p*q) ). Step 7: This simplifies to (p*q) / (p + q) days. Verification / Alternative check: We can test the formula with simple numeric values. Suppose p = 8 days and q = 12 days. Then the formula gives (8*12)/(8+12) = 96/20 = 4.8 days. Checking with rates: 1/8 + 1/12 = (3/24 + 2/24) = 5/24. Time together should be 1 / (5/24) = 24/5 = 4.8 days, which matches the formula. This confirms the correctness of the expression (p*q)/(p+q). Why Other Options Are Wrong: (p+q)/2 days: This is a simple arithmetic average of the times, which does not correctly model combined work rates. p+q days: This is even larger and represents the sum of individual times, which is meaningless in the context of simultaneous work. (p*q)/(p-q) days: This form appears similar to the correct one but uses a difference instead of a sum in the denominator. It generally gives incorrect and sometimes negative values when p is less than q. Common Pitfalls: Many students incorrectly average or add times instead of using rates. It is crucial to remember that work rates add for workers operating together, not their times. Another common mistake is failing to simplify 1 / (1/p + 1/q) correctly, leading to algebraic errors. Working through the algebra step by step helps avoid such mistakes. Final Answer: The time taken by X and Y working together to complete the work is (p*q)/(p+q) days.

Question 5

A piece of work is completed by three workers A, B, and C together. A and B together complete 60% of the work, while B and C together complete 70% of the work. Based on this information, who among A, B, and C is the most efficient worker (that is, w…

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Introduction / Context: This conceptual question focuses on comparing the efficiencies of three workers A, B, and C based on how much work various pairs of them complete together. Instead of asking for the time to complete the work, it asks which worker is the most efficient, meaning who contributes the largest share to the total work. The problem tests reasoning with percentages and systems of equations for work contributions. Given Data / Assumptions: A, B, and C together complete 100% (the whole work). A and B together complete 60% of the work. B and C together complete 70% of the work. Work rates of A, B, and C are constant. Percentages refer to the same total job. Concept / Approach: We interpret the percentages as fractions of the total work. Let the total work be 1 unit and let a, b, and c represent the shares of work done by A, B, and C respectively when all three work together. The total contribution satisfies a + b + c = 1. The given percentages indicate that a + b = 0.6 and b + c = 0.7. Solving this system of equations allows us to find relative contributions of A, B, and C and then identify the largest contribution and thus the most efficient worker. Step-by-Step Solution: Step 1: Let the total work be 1 unit. Step 2: Let a, b, and c be the parts of the work that A, B, and C effectively contribute when the whole job is done. Step 3: According to the problem, A and B together complete 60% of the work, so a + b = 0.6. Step 4: Similarly, B and C together complete 70% of the work, so b + c = 0.7. Step 5: Since the entire work is finished by A, B, and C together, we have a + b + c = 1. Step 6: From a + b + c = 1 and a + b = 0.6, subtract the second equation from the first to get c = 1 - 0.6 = 0.4. Step 7: From a + b + c = 1 and b + c = 0.7, subtract the second equation to get a = 1 - 0.7 = 0.3. Step 8: Now use a + b = 0.6 to find b = 0.6 - a = 0.6 - 0.3 = 0.3. Step 9: Therefore, the contributions are a = 0.3, b = 0.3, and c = 0.4. Step 10: Since c = 0.4 is the largest share, C is the most efficient worker. Verification / Alternative check: We can verify consistency. A and B together contribute 0.3 + 0.3 = 0.6, which matches the 60% given. B and C together contribute 0.3 + 0.4 = 0.7, consistent with 70%. The total contribution a + b + c = 0.3 + 0.3 + 0.4 = 1, confirming that the entire work is accounted for. Therefore, our solution is correct and internally consistent. Why Other Options Are Wrong: A: A contributes 0.3 of the work, which is less than C's 0.4. B: B also contributes 0.3, again less than C's contribution. None: This would be correct only if there were a tie or if the data were inconsistent, but here C clearly does the greatest share of work. Common Pitfalls: Some students may attempt to divide the percentages directly among workers without setting up equations, leading to confusion. Others may incorrectly assume that if two pairs involving B show high percentages, then B must be the most efficient, ignoring the actual algebraic relationships. Carefully forming and solving the equations a + b = 0.6, b + c = 0.7, and a + b + c = 1 is the best way to avoid mistakes. Final Answer: Among A, B, and C, the most efficient worker is C.

Question 6

One tap can fill an empty water tank completely in 50 minutes, while another tap can empty a completely filled tank in 75 minutes. If both taps are opened together when the tank is already half full, in how many minutes will the tank become complete…

Accepted Answer

Introduction / Context: This is a pipes and cisterns problem, which is conceptually identical to a time and work problem. One tap fills the tank and another empties it. We are given the individual times for filling and emptying, and the tank is initially half full. The task is to determine how long it will take for the tank to become completely full when both taps are open simultaneously. Given Data / Assumptions: Inlet tap (filling tap) alone fills the tank in 50 minutes. Outlet tap (emptying tap) alone empties a full tank in 75 minutes. Initially, the tank is half full. Both taps are opened together. The rates of filling and emptying are constant. Total capacity of the tank is considered as 1 unit. Concept / Approach: We first compute the filling and emptying rates in units per minute, then find the net rate when both taps are open simultaneously. Since the tank is already half full, only half of the tank's capacity remains to be filled. Dividing the remaining volume by the net rate gives the time to fill the tank completely. Step-by-Step Solution: Step 1: Let the total capacity of the tank be 1 unit. Step 2: Inlet tap fills 1 unit in 50 minutes, so filling rate = 1/50 units per minute. Step 3: Outlet tap empties 1 unit in 75 minutes, so emptying rate = 1/75 units per minute. Step 4: When both taps are open, the net rate = filling rate - emptying rate = 1/50 - 1/75. Step 5: Find a common denominator for 1/50 and 1/75. The least common multiple of 50 and 75 is 150. Step 6: Convert rates: 1/50 = 3/150 and 1/75 = 2/150. Step 7: Net rate = 3/150 - 2/150 = 1/150 units per minute. Step 8: Initially, the tank is half full, so remaining volume to be filled = 1/2 unit. Step 9: Time required = remaining volume / net rate = (1/2) / (1/150) = 1/2 * 150 = 75 minutes. Verification / Alternative check: We can reason about the result qualitatively. If there were no outlet, the remaining half would be filled in 25 minutes (since full tank is 50 minutes). But because water is also draining, it should take more than 25 minutes. Our result is 75 minutes, which seems plausible because the outlet is slowing down the filling significantly. The exact rate calculation confirms the answer. Why Other Options Are Wrong: 60 minutes: Implies a higher net filling rate than 1/150 units per minute, which does not match the given individual rates. 125 minutes: This is too long, and at that rate the outlet would almost cancel the inlet more than our calculation allows. 150 minutes: This is the time required to fill the entire tank at the net rate starting from empty, not from half full. Common Pitfalls: Common mistakes include adding the rates instead of subtracting them, or forgetting that the tank is initially half full and treating it as empty. Another pitfall is mishandling the fractions when computing 1/50 - 1/75. Always find the correct common denominator and subtract accurately to avoid arithmetic errors. Final Answer: The half full tank will become completely full in 75 minutes when both taps are opened together.

Question 7

Workers A, B, and C can complete a piece of work alone in 8 days, 10 days, and 12 days respectively. If all three work together on the job and they receive a total payment of Rs 7400 for completing it, what is the share (in rupees) of worker B, assu…

Accepted Answer

Introduction / Context: This problem combines time and work concepts with proportional division of money. Workers A, B, and C each have different times to complete the same task individually. When they work together and receive a total payment, it is fair to distribute the money in proportion to the work contributed by each worker. The question asks specifically for B's share of the total payment. Given Data / Assumptions: A alone completes the work in 8 days. B alone completes the work in 10 days. C alone completes the work in 12 days. Together, they complete one whole job and are paid Rs 7400. The payment is distributed in proportion to the amount of work each worker performs. Total work is considered as 1 unit. Concept / Approach: Since payment is proportional to work contribution, we first determine the daily work rate of each worker. If a worker completes a job in T days, then his rate is 1/T units per day. When they work together for the time necessary to complete one job, each person's contribution to the total work will be proportional to their rate. Therefore, the ratio of their rates equals the ratio of their shares of the total money. Step-by-Step Solution: Step 1: Let total work be 1 unit. Step 2: A's rate = 1/8 units per day. Step 3: B's rate = 1/10 units per day. Step 4: C's rate = 1/12 units per day. Step 5: Combined rate of A, B, and C = 1/8 + 1/10 + 1/12. Step 6: Find common denominator. The least common multiple of 8, 10, and 12 is 120. Step 7: Convert each rate: 1/8 = 15/120, 1/10 = 12/120, 1/12 = 10/120. Step 8: Combined rate = 15/120 + 12/120 + 10/120 = 37/120 units per day. Step 9: Time to complete the work together = 1 / (37/120) = 120/37 days (this time is not directly needed for share calculation but confirms they finish the job). Step 10: The share ratio of A : B : C equals their rates 1/8 : 1/10 : 1/12, that is 15/120 : 12/120 : 10/120, which simplifies to 15 : 12 : 10. Step 11: Sum of ratio parts = 15 + 12 + 10 = 37. Step 12: B's share of the total payment = (12 / 37) * 7400. Step 13: Compute B's share: 7400 / 37 = 200, then 200 * 12 = 2400. Verification / Alternative check: We can check by finding the other shares. A's share = (15/37) * 7400 = 15 * 200 = 3000. C's share = (10/37) * 7400 = 10 * 200 = 2000. Adding them yields 3000 + 2400 + 2000 = 7400, which matches the total payment. This confirms that the allocation is consistent and correct. Why Other Options Are Wrong: 2600: This value does not correspond to the correct proportional ratio of 15:12:10. 3000: This is actually A's share, not B's. 2000: This is C's share, which is lower than B's because C is slower than B. Common Pitfalls: A common mistake is to treat the days directly as the ratio for sharing money instead of using work rates. Remember that a smaller number of days means a higher rate, so the share must be proportional to 1/T, not T itself. Another error is to ignore simplification of fractions, which can lead to complicated numbers; using a common denominator helps to see simple integer ratios. Final Answer: Worker B's share of the payment is Rs 2400.

Question 8

Amit can complete a piece of work alone in 25 days, and Punit can complete the same work alone in 20 days. Punit works alone on the job for 10 days and then leaves. In how many days will Amit alone complete the remaining work?

Accepted Answer

Introduction / Context: This is a two-person time and work problem where one worker (Punit) starts the work and the other worker (Amit) finishes it. We are given how long each worker would take individually to complete the full job, and we know the amount of time Punit works before leaving. The goal is to determine the additional time Amit needs to finish the remaining part of the job on his own. Given Data / Assumptions: Amit alone can complete the work in 25 days. Punit alone can complete the same work in 20 days. Punit works alone for 10 days and then leaves. Amit finishes the remaining work alone. Total work is considered as 1 unit. All work rates are constant. Concept / Approach: We first calculate how much work Punit completes in 10 days using his individual work rate. Subtract this from the total work to find the remaining amount. Then we use Amit's work rate to compute how long it will take him to finish that remaining portion. This approach is direct and uses the basic formula: work done = rate * time. Step-by-Step Solution: Step 1: Let total work be 1 unit. Step 2: Punit can complete the work in 20 days, so his rate = 1/20 units per day. Step 3: In 10 days, the amount of work done by Punit = 10 * (1/20) = 10/20 = 1/2 of the job. Step 4: Remaining work after Punit leaves = 1 - 1/2 = 1/2 of the job. Step 5: Amit can complete the full work in 25 days, so his rate = 1/25 units per day. Step 6: Time taken by Amit to complete the remaining 1/2 of the work = (1/2) / (1/25). Step 7: Simplify the expression: (1/2) / (1/25) = 1/2 * 25 = 25/2 = 12.5 days. Verification / Alternative check: We can check by computing the work done by Amit in 12.5 days. At rate 1/25 units per day, Amit does 12.5 * (1/25) = 12.5 / 25 = 1/2 of the job, which matches the remaining amount. Thus, Punit does the first half of the job, and Amit does the second half, so together they complete the full job as required. Why Other Options Are Wrong: 10.5 days, 11.5 days, and 13.5 days all correspond to incorrect calculations of Amit's time or misunderstanding of how much work Punit completed. Any value other than 12.5 days would imply that Amit either over-completes or under-completes the remaining half of the work. Common Pitfalls: Some students mistakenly think that since Punit works for half of his required time (10 out of 20 days), he must have done half the work, which is actually correct here but should always be checked using rate * time. Others may average the days of Amit and Punit or add them directly, which does not reflect the actual work distribution. Always compute the exact work done by each person using rates and time intervals. Final Answer: Amit will take 12.5 days to complete the remaining work alone.

Question 9

Worker A can complete a piece of work alone in 10 days, and worker B can complete the same work alone in 12 days. With the additional help of worker C, all three together finish the work in 4 days. In how many days can C alone complete the same work?

Accepted Answer

Introduction / Context: This question concerns three workers A, B, and C who together complete a single job faster than any of them could alone. We know the individual times for A and B but only the combined time when all three work together. Using this information, we are asked to determine how long C alone would take to complete the entire job, which requires subtracting known rates from the combined rate. Given Data / Assumptions: A alone can complete the work in 10 days. B alone can complete the work in 12 days. A, B, and C together complete the work in 4 days. Total work is taken as 1 unit. Work rates of A, B, and C are constant. Concept / Approach: We convert A's and B's times into daily work rates. The combined time of 4 days for all three gives us the joint rate of A, B, and C together. Subtracting A's and B's individual rates from the total combined rate yields C's rate. Finally, we take the reciprocal of C's rate to find the number of days C would need to complete the work alone. Step-by-Step Solution: Step 1: Assume total work is 1 unit. Step 2: A's rate = 1/10 units per day. Step 3: B's rate = 1/12 units per day. Step 4: A, B, and C together complete the work in 4 days, so combined rate of A + B + C = 1/4 units per day. Step 5: Let C's rate be c units per day. Then 1/10 + 1/12 + c = 1/4. Step 6: Compute 1/10 + 1/12. The least common multiple of 10 and 12 is 60. So, 1/10 = 6/60 and 1/12 = 5/60. Step 7: Therefore, 1/10 + 1/12 = 6/60 + 5/60 = 11/60. Step 8: Substitute into the equation: 11/60 + c = 1/4. Step 9: Express 1/4 with denominator 60: 1/4 = 15/60. Step 10: So, 11/60 + c = 15/60, which gives c = (15/60) - (11/60) = 4/60 = 1/15 units per day. Step 11: Time taken by C alone to complete the work = 1 / (1/15) = 15 days. Verification / Alternative check: We can verify by checking the combined rate. A works at 1/10 units per day, B at 1/12 units per day, and C at 1/15 units per day. Their total rate is 1/10 + 1/12 + 1/15. Using a common denominator of 60, we get 6/60 + 5/60 + 4/60 = 15/60 = 1/4, meaning they finish the work in 4 days, which matches the given condition. Why Other Options Are Wrong: 12 days: This would correspond to a faster rate of 1/12 and would not satisfy the combined rate equation. 18 days: This gives a slower rate of 1/18 and leads to a combined rate that is less than 1/4, so they would take more than 4 days. 24 days: Even slower, with a rate of 1/24, clearly contradicting the given 4-day completion time. Common Pitfalls: Some students incorrectly average the times of A and B instead of using work rates, or they forget to subtract both A and B's rates from the total combined rate, leading to an incorrect rate for C. Errors in fraction arithmetic, particularly when finding common denominators, can also lead to wrong answers. Systematic step-by-step calculation helps avoid these pitfalls. Final Answer: Worker C alone can complete the work in 15 days.

Question 10

Workers A, B, and C can complete a piece of work alone in 10 days, 12 days, and 15 days respectively. All three start working together at the same time. A leaves after working for 2 days, and B leaves 3 days before the work is completed, so C finish…

Accepted Answer

Introduction / Context: This is a more involved time and work problem where three workers A, B, and C join and leave the job at different times. We know their individual times to complete the work and the sequence of their participation on the job. The objective is to find the total number of days required to complete the work, taking into account changing combinations of workers over different intervals. Given Data / Assumptions: A alone can complete the work in 10 days. B alone can complete the work in 12 days. C alone can complete the work in 15 days. All three start working together from day 0. A leaves after 2 days of work. B continues but leaves 3 days before the work is completed. C then completes the remaining work alone. Total work is considered as 1 unit. Concept / Approach: We break the total time into three intervals, each with a different set of workers: first interval with A, B, and C; second interval with B and C; and third interval with C alone. We express the work done in each interval in terms of the total time T and the known rates of the workers. Setting the sum of the work over all intervals equal to 1 and solving for T yields the total time required to finish the job. Step-by-Step Solution: Step 1: Let total work be 1 unit and let the total time of completion be T days. Step 2: A's rate = 1/10 units per day, B's rate = 1/12 units per day, C's rate = 1/15 units per day. Step 3: For the first 2 days, all three (A, B, and C) work together. Work done in this period = 2 * (1/10 + 1/12 + 1/15). Step 4: After 2 days, A leaves. From day 2 to day (T - 3), only B and C work together. The length of this interval is (T - 3) - 2 = T - 5 days. Work done in this period = (T - 5) * (1/12 + 1/15). Step 5: For the last 3 days, only C works. Work done in this period = 3 * (1/15). Step 6: Now compute the rates. First, 1/10 + 1/12 + 1/15. With a common denominator of 60, we get 6/60 + 5/60 + 4/60 = 15/60 = 1/4 units per day. Step 7: Next, 1/12 + 1/15. The common denominator is 60 again, giving 5/60 + 4/60 = 9/60 = 3/20 units per day. Step 8: And 1/15 = 1/15 units per day, so C alone does 3 * (1/15) = 3/15 = 1/5 of the work in the last 3 days. Step 9: Work in first 2 days = 2 * (1/4) = 1/2. Step 10: Work in middle period = (T - 5) * (3/20). Step 11: Work in last 3 days = 1/5. Step 12: Total work done must be 1, so 1/2 + (T - 5) * (3/20) + 1/5 = 1. Step 13: Combine constant terms. 1/2 + 1/5 = (5/10 + 2/10) = 7/10. Step 14: The equation becomes 7/10 + (3/20)(T - 5) = 1. Step 15: Subtract 7/10 from both sides giving (3/20)(T - 5) = 1 - 7/10 = 3/10. Step 16: Multiply both sides by 20/3 to solve for T - 5: T - 5 = (3/10) * (20/3) = 2. Step 17: Therefore T = 2 + 5 = 7 days. Verification / Alternative check: Check the work distribution with T = 7 days. First 2 days with A, B, and C give 2 * (1/4) = 1/2 of the work. Next T - 5 = 2 days with B and C give 2 * (3/20) = 6/20 = 3/10 of the work. Last 3 days with C give 1/5 of the work. Summing all: 1/2 + 3/10 + 1/5 = 5/10 + 3/10 + 2/10 = 10/10 = 1, confirming the correctness of T = 7 days. Why Other Options Are Wrong: 5 days and 6 days: These are too short and would not allow the workers to complete the full work at the calculated rates. 8 days: Longer than necessary and would result in more than 1 unit of work being completed. Common Pitfalls: A frequent mistake is mishandling the time intervals, such as forgetting that B leaves 3 days before completion and miscomputing the length of the middle interval. Another issue is not properly adding the work contributions and mixing up the workers' rates. Setting up the equation with clear intervals and checking the arithmetic step by step helps avoid these problems. Final Answer: The entire work is completed in a total of 7 days.

Question 11

A and B can together complete a piece of work in 10 days. If A works at twice his usual efficiency and B works at an efficiency that is one-third less than his usual efficiency, then the same work is finished in 6 days. In how many days can A alone …

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Introduction / Context: This is a classic time and work question that tests your understanding of individual work rates and how changes in efficiency affect the overall time required to complete a job. Two workers, A and B, can finish a task together in a known number of days, and then their efficiencies are altered. From the new completion time, we have to reverse engineer their original individual times to complete the same work alone. Given Data / Assumptions: The whole work is considered as 1 unit of job. Original time when A and B work together is 10 days. When A works at twice his usual efficiency and B works at one-third less (that is two-thirds) of his usual efficiency, the work finishes in 6 days. Work rates are assumed constant over time, and there are no breaks or interruptions. Concept / Approach: We use the concept that Work = Rate * Time and that individual rates add up when people work together. Let the original daily work rates of A and B be a and b respectively. Then their combined rate is a + b. From the given conditions, we will form two linear equations in terms of a and b and solve those equations to find each worker's rate and hence their individual times to complete the work alone. Step-by-Step Solution: Let total work = 1 unit. Since A and B together finish the work in 10 days, (a + b) * 10 = 1, so a + b = 1/10. When A works at twice his efficiency, his new rate is 2a. When B works at one-third less efficiency, his new rate is (2/3)b. Together in the modified scenario they complete the same 1 unit of work in 6 days, so (2a + (2/3)b) * 6 = 1, hence 2a + (2/3)b = 1/6. We now solve the system: a + b = 1/10 and 2a + (2/3)b = 1/6. From a + b = 1/10, we get a = 1/10 - b. Substitute in the second equation: 2(1/10 - b) + (2/3)b = 1/6. This gives 1/5 - 2b + (2/3)b = 1/6, so 1/5 - (4/3)b = 1/6. Rearranging, (4/3)b = 1/5 - 1/6 = (6 - 5)/30 = 1/30, so b = (1/30) * (3/4) = 1/40. Then a = 1/10 - 1/40 = 4/40 - 1/40 = 3/40. Time taken by A alone = 1/a = 1 / (3/40) = 40/3 days. Time taken by B alone = 1/b = 1 / (1/40) = 40 days. Verification / Alternative check: Check the original combined time: a + b = 3/40 + 1/40 = 4/40 = 1/10, so together they take 10 days, which matches the given data. Now check the modified rates: 2a + (2/3)b = 2*(3/40) + (2/3)*(1/40) = 6/40 + 2/120 = 6/40 + 1/60 = 9/60 + 1/60 = 10/60 = 1/6, which means 6 days for the whole job. Both conditions are satisfied, so the solution is consistent. Why Other Options Are Wrong: Option 20/3 days, 20 days gives a combined rate that does not equal 1/10 and fails the second condition as well. Option 30 days, 20/3 days swaps and distorts the time values and does not satisfy either equation for the modified efficiencies. Option 50/3 days, 25 days leads to slower workers and predicts a combined time greater than 10 days, so it contradicts the problem statement. Option 30 days, 30 days assumes equal speeds, which again does not match the linear system formed from the given conditions. Common Pitfalls: A common mistake is to directly scale times instead of rates when efficiency changes. Another mistake is to misinterpret "one-third less" as one-third of the original instead of reducing the rate to two-thirds of the original. Sign errors while solving simultaneous equations are also frequent. Always define rates carefully and form equations systematically. Final Answer: Therefore, A alone completes the work in 40/3 days and B alone completes the work in 40 days, so the correct option is 40/3 days, 40 days.

Question 12

Amar can complete a certain piece of work in 30 days, while Raman can finish the same work in 15 days when working alone. If both Amar and Raman work together continuously for 4 days, what percentage of the total work will be completed by them in th…

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Introduction / Context: This time and work problem checks your understanding of how to add individual work rates and then convert the completed fraction of work into a percentage. Two workers, Amar and Raman, have different capacities, and you are asked to determine how much of the job they complete together in a fixed period of time. Given Data / Assumptions: Amar alone completes the work in 30 days. Raman alone completes the same work in 15 days. They work together for exactly 4 days at constant rates. The total work is considered as one complete unit of job. Concept / Approach: We use the basic relationship Work = Rate * Time. First, convert the given times into daily work rates for Amar and Raman. Since they are working together, their combined rate is the sum of their individual rates. Multiply that combined rate by the time they work together (4 days) to find the fraction of the work completed. Finally, convert that fraction into a percentage to match the requirement of the question. Step-by-Step Solution: Let the total work be 1 unit. Daily work rate of Amar = 1/30 of the work per day. Daily work rate of Raman = 1/15 of the work per day. Combined daily work rate when they work together = 1/30 + 1/15. Compute this: 1/30 + 1/15 = 1/30 + 2/30 = 3/30 = 1/10 of the work per day. In 4 days, the fraction of work completed = combined rate * time = (1/10) * 4 = 4/10. Simplify 4/10 to 2/5 of the work. Convert this fraction to a percentage: (2/5) * 100% = 40% of the total work. Verification / Alternative check: You can double check by imagining the total work as 30 units (the least common multiple of 30 and 15). Amar does 1 unit per day and Raman does 2 units per day, so together they do 3 units per day. In 4 days they will complete 3 * 4 = 12 units. Since the total is 30 units, the portion completed is 12/30 = 2/5, which again is 40%. This confirms that our earlier calculation is accurate. Why Other Options Are Wrong: Option 15% is too low because even Amar alone would complete more than that in 4 days. Option 37% does not correspond to a simple fraction of the total work and does not match any correct calculation. Option 45% is greater than the actual 40% and results from overestimating their combined rate. Option 50% would mean half of the work is done, which would require 5 days at a rate of 1/10 per day, not 4 days. Common Pitfalls: A common error is to average the days (30 and 15) instead of adding their rates. Another frequent mistake is converting the fraction into a percentage incorrectly. Always remember to sum rates, not times, when people work together. Also be careful not to confuse 4/10 with 40/100; in both cases the simplified form is 2/5, which is exactly 40%. Final Answer: Thus, Amar and Raman together complete 40% of the total work in 4 days, so the correct option is 40%.

Question 13

Forty-five men or sixty boys can complete a piece of work in 20 days when working separately. If a group consists of 15 men and 20 boys working together on the same job, in how many days will they finish the entire work?

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Introduction / Context: This is a mixed time and work problem that involves two different types of workers, men and boys, with different individual efficiencies. You are given how long a group of only men or only boys would take to complete a piece of work. Using this information, you must determine how long a mixture of both men and boys takes to finish the same work. Given Data / Assumptions: Forty-five men alone can complete the work in 20 days. Sixty boys alone can complete the same work in 20 days. We assume all men have identical efficiency and all boys have identical efficiency among themselves. We have a new group of 15 men and 20 boys working together.All workers maintain constant rates, and no one joins or leaves during the job. Concept / Approach: We treat the total work as 1 unit and compute the work done per day by one man and one boy separately using the given information. Since 45 men finish in 20 days, we get the per-man rate. Similarly, we obtain the per-boy rate. Then we combine these rates for a group of 15 men and 20 boys and finally compute how many days this combined group needs to finish the entire job. Step-by-Step Solution: Let total work = 1 unit. Since 45 men finish in 20 days, total man-days = 45 * 20 = 900 man-days. So, work rate of one man per day = 1/900. Similarly, 60 boys finish in 20 days, so boy-days = 60 * 20 = 1200 boy-days. Work rate of one boy per day = 1/1200. Now for 15 men and 20 boys together: daily work rate = 15*(1/900) + 20*(1/1200). Compute 15/900 = 1/60 and 20/1200 = 1/60. So combined rate = 1/60 + 1/60 = 2/60 = 1/30 of the work per day. Time taken to complete the whole work = 1 / (1/30) = 30 days. Verification / Alternative check: To verify, in 30 days at a rate of 1/30 per day, the group completes exactly 1 unit of work, which matches the definition of total work. Also, notice that 15 men and 20 boys are less than half of the original 45 men and 60 boys combined, so it is reasonable that they take more than 20 days. The derived time of 30 days is therefore logically consistent with the relative group sizes and efficiencies. Why Other Options Are Wrong: Option 23 days is too small because the new group is significantly smaller than either original full group, so it cannot complete the work faster than 20 days. Option 45 days overestimates the time; at the computed rate of 1/30 per day, 45 days would correspond to 1.5 units of work, which is impossible for a single job. Option 25 days is also too low and does not match the calculated combined rate of 1/30 per day. Option 20 days assumes that the efficiency of the new mixed group is equal to one of the original larger groups, which is not true because the group is much smaller. Common Pitfalls: Students sometimes treat 45 men and 60 boys as if they had equal efficiency, which leads to incorrect averaging. Another error is to average the times instead of computing individual rates. Always convert to per-person work rates first, then sum up the contributions of all members of the new group. Keeping track of units like man-days and boy-days helps to avoid confusion. Final Answer: Thus, 15 men and 20 boys working together will complete the work in 30 days, so the correct option is 30 days.

Question 14

A field can be completely reaped by 12 men in 14 days or by 18 women in the same 14 days. If 8 men and 16 women work together at their respective constant rates, in how many days will they harvest the entire field?

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Introduction / Context: This time and work problem involves two different categories of workers, men and women, each with different efficiencies. We know how long groups of only men or only women take to reap the field. Using this, we must compute how long a mixed group of men and women will take to complete the same field work when they work together. Given Data / Assumptions: Twelve men can reap the field in 14 days. Eighteen women can reap the same field in 14 days. We assume that all men have equal efficiency among themselves and all women have equal efficiency among themselves. We consider a new group of 8 men and 16 women working simultaneously at constant rates. The total work (the whole field) is taken as one complete unit. Concept / Approach: We first find the daily work rate of one man and one woman using the total work concept and man-days or woman-days. Once the individual rates are obtained, the combined rate of 8 men and 16 women is found by adding their contributions. Finally, time is computed by dividing total work by the combined daily work rate. This approach is standard for work and manpower distribution problems. Step-by-Step Solution: Let total work = 1 unit (one entire field). 12 men finish the field in 14 days, so total man-days = 12 * 14 = 168 man-days. Hence, one man's daily rate = 1/168 of the field per day. 18 women finish the field in 14 days, so total woman-days = 18 * 14 = 252 woman-days. Thus, one woman's daily rate = 1/252 of the field per day. Now consider 8 men and 16 women together. Daily work done by 8 men = 8 * (1/168) = 8/168 = 1/21. Daily work done by 16 women = 16 * (1/252) = 16/252 = 2/31.5, which simplifies to 16/252 = 2/31.5, but it is easier to compute as 16/252 = 8/126 = 4/63. So combined daily work rate = 1/21 + 4/63. Convert 1/21 to denominator 63: 1/21 = 3/63. Therefore, combined rate = 3/63 + 4/63 = 7/63 = 1/9 of the field per day. Time required to reap the field = 1 / (1/9) = 9 days. Verification / Alternative check: To check, in 9 days at the rate of 1/9 of the field per day, the team completes exactly 1 field, which matches the defined total work. Also, observe that 8 men and 16 women are fewer than 12 men plus 18 women, so it is reasonable that they require more than 7 or 8 days but fewer than 14 days. The calculated answer of 9 days fits well between these extremes and is consistent with the relative workforce size and efficiencies. Why Other Options Are Wrong: Option 26 days and 24 days are far too large; such values would imply very low combined efficiency, which contradicts the given data that either 12 men or 18 women alone can finish in 14 days. Option 8 days is slightly smaller than the correct value. It corresponds to a higher combined rate than what calculations yield. Option 12 days is also incorrect; at the rate of 1/9 per day, 12 days would mean 4/3 of the work, which is impossible for a single field. Common Pitfalls: Students sometimes average the days directly between men and women, which is incorrect. Another common error is mishandling fractions when adding the rates of men and women. Always convert to a common denominator before summing fractions to avoid arithmetic mistakes. Keeping the total work as 1 unit helps to maintain clarity throughout the calculation. Final Answer: Therefore, 8 men and 16 women working together will reap the field in 9 days, so the correct option is 9 days.

Question 15

A, B and C can each complete a certain job working alone in 12 days, 16 days and 24 days respectively. If all three work together continuously at their usual constant rates, in how many days will they complete the entire job?

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Introduction / Context: This is a straightforward time and work question where three workers, A, B and C, have known individual times to complete a job. You must determine the time required when all three work together simultaneously. Such questions test your ability to convert individual times into work rates and then combine those rates correctly. Given Data / Assumptions: A alone can complete the job in 12 days. B alone can complete the job in 16 days. C alone can complete the job in 24 days. All of them work at constant rates and there are no breaks or interruptions. The total work is treated as 1 complete unit. Concept / Approach: The key idea is that if a person can complete a job in T days, the daily work rate of that person is 1/T of the job per day. When multiple people work together, their rates simply add. After finding the combined rate of A, B and C, we use the formula Time = Total Work / Combined Rate to find how many days they need to finish the job together. Step-by-Step Solution: Let total work = 1 unit. Daily rate of A = 1/12 of the job per day. Daily rate of B = 1/16 of the job per day. Daily rate of C = 1/24 of the job per day. Combined daily rate of A, B and C = 1/12 + 1/16 + 1/24. Take the LCM of 12, 16 and 24 which is 48. Convert each fraction: 1/12 = 4/48, 1/16 = 3/48, 1/24 = 2/48. Add them: 4/48 + 3/48 + 2/48 = 9/48. So combined rate = 9/48 = 3/16 of the job per day. Time taken together = total work / combined rate = 1 / (3/16) = 16/3 days. Verification / Alternative check: To verify, think of the job as 48 equal parts (the LCM). A completes 4 parts per day, B completes 3 parts per day and C completes 2 parts per day. Together they complete 4 + 3 + 2 = 9 parts per day. Total parts are 48, so days needed = 48 / 9 = 16/3 days, which matches our earlier calculation. This double check confirms the accuracy of the result. Why Other Options Are Wrong: Option 15/4 days corresponds to 3.75 days, which would require a combined rate greater than 3/16, not supported by the data. Option 17/3 days is slightly more than the correct value and comes from incorrect fraction addition or mistaken LCM usage. Option 19/4 days is 4.75 days, again incompatible with the calculated combined rate of 3/16 per day. Option 4 days would require an extremely high combined rate of 1/4 per day, which exceeds the sum of the three given rates. Common Pitfalls: A common mistake is to average the days instead of adding the rates. Another error is to mishandle the fractions when converting them to a common denominator. Always remember that time values cannot be added directly; only work rates (reciprocals of time) combine linearly. Using the LCM of denominators makes addition of fractions easier and less error prone. Final Answer: Hence, A, B and C together can complete the work in 16/3 days, making 16/3 days the correct option.

Question 16

A can complete a piece of work alone in 15 days and B can complete the same work alone in 9 days. If A and B work together on the job for 5 consecutive days, what fraction of the total work will still remain unfinished at the end?

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Introduction / Context: This question is about combining work rates of two people and then finding how much work remains after they have worked together for a certain period. It reinforces ideas about fractions of work completed and the relationship between time, rate and work in time and work problems. Given Data / Assumptions: A alone can finish the work in 15 days. B alone can finish the same work in 9 days. A and B work together for 5 days continuously at constant efficiencies. The total work is considered as one complete job (1 unit). Concept / Approach: If someone completes a job in T days, the person's daily rate is 1/T. When two people work together, their combined rate is the sum of their individual rates. After calculating how much fraction of the work A and B perform in the 5 days, we subtract that from 1 to find the remaining fraction of the job that is still incomplete. Step-by-Step Solution: Let total work = 1 unit. Daily work rate of A = 1/15 of the work per day. Daily work rate of B = 1/9 of the work per day. Combined daily rate of A and B together = 1/15 + 1/9. Take LCM of 15 and 9, which is 45. Convert fractions: 1/15 = 3/45 and 1/9 = 5/45. Combined rate = 3/45 + 5/45 = 8/45 of the work per day. In 5 days, work completed = 5 * (8/45) = 40/45. Simplify 40/45 by dividing numerator and denominator by 5, giving 8/9 of the work completed. Remaining work = 1 - 8/9 = 1/9 of the total work. Verification / Alternative check: As a check, if 8/9 of the work is done in 5 days, then the average rate of the team is (8/9)/5 = 8/45 per day, which matches our calculated combined rate. Also, since A alone needs 15 days and B alone needs 9 days, it is reasonable that together they almost finish the work in 5 days but leave a small fraction. The remaining 1/9 is consistent with these intuitive expectations. Why Other Options Are Wrong: Option 1/6 suggests less work has been done, meaning more is left, but our calculation shows a higher combined rate and more work completed. Option 3/7 does not come from any correct manipulation of the fractions and is not a natural result of the combined rate 8/45 over 5 days. Option 1/3 would mean only two-thirds has been done, which is too little given the strong combined efficiency of A and B. Option 2/9 is also incorrect because it corresponds to a completed fraction of 7/9, not 8/9, and does not match the exact multiplication of rate and time. Common Pitfalls: A common error is to average the times (15 and 9) instead of summing the work rates. Another frequent mistake is to miscalculate the LCM or the fractions, leading to an incorrect combined rate. Always convert to a common denominator and work carefully with fractions. Also, do not forget to subtract the completed fraction from 1 to find what is left. Final Answer: Therefore, the fraction of the work still left unfinished after 5 days is 1/9, so the correct option is 1/9.

Question 17

A can complete a work alone in 20 days and B can complete the same work alone in 50 days. If they work together on this job for 5 days, what fraction of the total work will still remain incomplete at the end of the fifth day?

Accepted Answer

Introduction / Context: This problem is another application of the time and work concept, where two workers A and B have known individual completion times. You are asked to calculate how much of the job is still incomplete after they work together for a certain number of days. This type of question is common in quantitative aptitude exams. Given Data / Assumptions: A alone can finish the work in 20 days. B alone can finish the work in 50 days. A and B work together for exactly 5 days at constant speed. The entire job is treated as one unit of work. Concept / Approach: We use the fact that a person who finishes a work in T days does 1/T of the work per day. The combined daily rate of two workers is the sum of their individual rates. After computing how much fraction of the work they complete in 5 days, we subtract that from 1 to obtain the fraction of work that remains unfinished. Step-by-Step Solution: Let total work = 1 unit. Daily rate of A = 1/20 of the work per day. Daily rate of B = 1/50 of the work per day. Combined daily rate of A and B = 1/20 + 1/50. Find the LCM of 20 and 50, which is 100. Convert fractions: 1/20 = 5/100 and 1/50 = 2/100. Combined rate = 5/100 + 2/100 = 7/100 of the work per day. In 5 days, work completed = 5 * (7/100) = 35/100. Simplify 35/100 to 7/20 of the work completed. Remaining work = 1 - 7/20 = 13/20 of the work. Verification / Alternative check: To verify, imagine the job consists of 100 equal units. A does 5 units per day (since 20 days for 100 units) and B does 2 units per day (since 50 days for 100 units). Together they do 7 units per day. In 5 days they complete 7 * 5 = 35 units. That leaves 100 - 35 = 65 units, which is 65/100 = 13/20 of the work, matching our earlier calculation. This confirms that 13/20 of the job is still incomplete. Why Other Options Are Wrong: Option 1/3 is too small; it would mean that two-thirds of the work is already finished, which contradicts the calculated 7/20 completed portion. Option 1/6 is even smaller and suggests that most of the job has been done, which is not supported by the rates. Option 2/9 does not align with any natural fraction from these numbers and does not result from correct calculations. Option 7/10 is closer but still incorrect; this would require that only 3/10 of the work had been done, not 7/20. Common Pitfalls: Learners often confuse the fraction of work completed with the fraction remaining. Some students also incorrectly average 20 and 50 instead of adding the rates 1/20 and 1/50. Always remember to work with rates (reciprocals of time), carefully find a common denominator and then sum the fractions. Subtract the completed fraction from 1 to get what is left. Final Answer: Thus, after working together for 5 days, the fraction of the work still left incomplete is 13/20, so the correct option is 13/20.

Question 18

Raja and Kundan together can build a wall in 20 days, Kundan and Mahesh together can build the same wall in 30 days, and Mahesh and Raja together can build it in 24 days. Working at their usual constant rates, in how many days will Raja, Kundan and …

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Introduction / Context: This question involves three workers, Raja, Kundan and Mahesh, who can build a wall in pairs at different combined speeds. The task is to determine how long they would take to build the same wall if all three worked together. This type of problem tests your understanding of simultaneous equations using work rates. Given Data / Assumptions: Raja and Kundan together can build the wall in 20 days. Kundan and Mahesh together can build the wall in 30 days. Mahesh and Raja together can build the wall in 24 days. All three work at constant individual rates throughout, and the total wall is treated as one unit of work. Concept / Approach: Let the daily work rates of Raja, Kundan and Mahesh be r, k and m respectively. The combined work rates for each pair give us three equations. By adding these equations, we can find the sum r + k + m, which is exactly the rate when all three work together. Once that combined rate is known, we invert it to obtain the time required for the three of them to complete the wall together. Step-by-Step Solution: Let total work = 1 wall. From Raja and Kundan together: r + k = 1/20. From Kundan and Mahesh together: k + m = 1/30. From Mahesh and Raja together: m + r = 1/24. Add all three equations: (r + k) + (k + m) + (m + r) = 1/20 + 1/30 + 1/24. Left side simplifies to 2(r + k + m). Compute the right side using LCM of 20, 30 and 24 which is 120. 1/20 = 6/120, 1/30 = 4/120, 1/24 = 5/120. Sum = 6/120 + 4/120 + 5/120 = 15/120 = 1/8. So 2(r + k + m) = 1/8, which gives r + k + m = 1/16. Thus, when all three work together, their combined rate is 1/16 of the wall per day. Time taken by Raja, Kundan and Mahesh together = 1 / (1/16) = 16 days. Verification / Alternative check: A quick check is to note that if all three together work at a rate of 1/16 per day, they finish the wall in 16 days. Now compare this to the pairwise times of 20, 30 and 24 days; it is reasonable that adding a third worker increases the overall speed enough to reduce the time to a value slightly less than the fastest pair time, which is 20 days. Therefore, 16 days is a consistent and realistic answer. Why Other Options Are Wrong: Option 18 days corresponds to a combined rate of 1/18 per day, which does not satisfy the pair equations when decomposed. Option 20 days is actually the time for Raja and Kundan only, ignoring Mahesh's contribution. Option 14 days implies an even higher combined rate, inconsistent with the given pairwise speeds. Option 15 days also does not match the algebraic result of r + k + m = 1/16 and leads to contradictions if checked with the pair equations. Common Pitfalls: Students often try to average the pairwise times instead of working with rates and equations. Another common error is forgetting that when you add the three pairwise equations, you get twice the sum of the three individual rates. Carefully handling fractions and using the LCM for denominators reduces mistakes and leads to the correct combined rate. Final Answer: Hence, Raja, Kundan and Mahesh together can build the wall in 16 days, so the correct option is 16 days.

Question 19

A can complete a piece of work alone in 36 days, whereas B can complete the same work alone in 12 days. If A and B work together at their usual constant rates, in how many days will they finish the entire work?

Accepted Answer

Introduction / Context: This is a basic time and work question where you know how long each person, A and B, takes to complete a job individually. You are asked how long they will take if they work together. It is a direct application of adding work rates and then computing the total time. Given Data / Assumptions: A alone can complete the work in 36 days. B alone can complete the same work in 12 days. Both A and B work together without interruption at constant efficiencies. The total work is considered as 1 unit of job. Concept / Approach: If a worker finishes a job in T days, the rate of work is 1/T of the job per day. When multiple workers cooperate, their rates add up. After finding the combined rate of A and B, we simply take the reciprocal of the combined rate to find how many days they need to complete one full unit of work. Step-by-Step Solution: Let total work = 1 unit. Daily rate of A = 1/36 of the work per day. Daily rate of B = 1/12 of the work per day. Combined daily rate of A and B = 1/36 + 1/12. Take the LCM of 36 and 12, which is 36. Convert fractions: 1/36 stays 1/36, and 1/12 = 3/36. Combined rate = 1/36 + 3/36 = 4/36. Simplify 4/36 to 1/9 of the work per day. Time taken to finish the whole work together = 1 / (1/9) = 9 days. Verification / Alternative check: Think of the job as 36 equal parts. A completes 1 part per day and B completes 3 parts per day. Together they complete 4 parts per day. Total parts = 36, so days needed = 36 / 4 = 9 days, which matches the earlier calculation. This consistency confirms that the answer is correct. Why Other Options Are Wrong: Option 8 days would require a combined rate of 1/8 per day which is higher than 1/9 and not supported by the given individual rates. Option 6 days is even faster and would imply a combined rate of 1/6 per day, which does not equal 1/36 + 1/12. Option 10 days would correspond to a combined rate of 1/10 per day; substituting back shows it does not match the sum of individual rates. Option 12 days is simply B's individual time and ignores the help provided by A, so it is clearly incorrect. Common Pitfalls: Some students mistakenly average the times (36 and 12) to get 24 days, which is wrong because you must add rates, not times. Others mis-handle the fraction addition or forget to simplify. Always convert times to rates, add those rates and then invert to get the total time when everyone works together. Final Answer: Therefore, A and B working together can complete the work in 9 days, so the correct option is 9 days.

Question 20

P can complete a work in 15 days and Q can complete the same work in 24 days. They begin the work together, but Q leaves the job 2 days before it is finished. In how many days in total was the work completed?

Accepted Answer

Introduction / Context: This problem involves two workers, P and Q, who start a job together but one of them stops working before the work is completed. You need to determine the total number of days taken to finish the work. This combines ideas of work rates and careful time accounting when one worker leaves early. Given Data / Assumptions: P alone can complete the work in 15 days. Q alone can complete the work in 24 days. They start working together from the beginning. Q leaves the work 2 days before completion, while P continues until the end. Work rates are constant and the total work is one complete job. Concept / Approach: We let the total time taken to finish the work be t days. P works for all t days, but Q works only for t - 2 days. We express their contributions in terms of their daily rates and set the sum equal to 1 (the whole work). This gives a linear equation in t, which we solve to get the total time required. Step-by-Step Solution: Let total work = 1 unit. P's daily rate = 1/15 of the work per day. Q's daily rate = 1/24 of the work per day. Let total time to complete the work be t days. P works for t days, so P's contribution = t * (1/15). Q leaves 2 days before completion, so Q works for (t - 2) days, giving contribution = (t - 2) * (1/24). Total work done by both = t/15 + (t - 2)/24 = 1. Find a common denominator; LCM of 15 and 24 is 120. Rewrite as (8t)/120 + (5(t - 2))/120 = 1. Simplify numerator: 8t + 5t - 10 = 13t - 10. So (13t - 10)/120 = 1, which implies 13t - 10 = 120. Solve 13t = 130, so t = 10 days. Verification / Alternative check: If t = 10 days, then P works for all 10 days, doing 10/15 = 2/3 of the work. Q works for 8 days (since t - 2 = 8), doing 8/24 = 1/3 of the work. Together they complete 2/3 + 1/3 = 1 whole unit of work, which matches the requirement. This confirms that the total time to finish the job is indeed 10 days. Why Other Options Are Wrong: Option 12 days leads to P doing 12/15 = 4/5 and Q doing 10/24 = 5/12, which does not sum to 1. Option 9 days gives P's work as 9/15 and Q's as 7/24, which also does not equal 1 when added. Option 11 days yields P's contribution of 11/15 and Q's of 9/24, again not equal to the full job. Option 8 days is clearly too short given the individual times of 15 and 24 days and results in less than one complete job when contributions are summed. Common Pitfalls: A typical mistake is to assume that Q works for exactly half the time or to forget that Q leaves 2 days before the end, not halfway. Another pitfall is mismanaging fractions when combining the work expressions. Carefully set up the equation using t, and always check the final contributions of each worker to ensure they add up to 1. Final Answer: Thus, the total work is completed in 10 days, so the correct option is 10 days.

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