Question 1

The rates of working of A and B are in the ratio 2 : 3. What is the ratio of the number of days taken by A and B, respectively, to finish the same work when each works alone?

Accepted Answer

Introduction / Context: This is a conceptual time and work question focusing on the relationship between rate and time. We are given the ratio of working rates of A and B and asked for the ratio of the times they take to complete the same job individually. Such problems test understanding of the inverse proportionality between work rate and time taken. Given Data / Assumptions: - The ratio of work rates of A and B is 2 : 3 (rate of A : rate of B). - Both A and B are assumed to be working on the same total amount of work individually. - Total work is considered as a constant quantity for both workers. Concept / Approach: Time taken to complete a given piece of work is inversely proportional to the rate of working. If rate of A : rate of B is 2 : 3, then the time taken by A : time taken by B must be in the ratio 1 / 2 : 1 / 3. By simplifying this inverse ratio, we obtain the required ratio of days taken by A and B. Step-by-Step Solution: Step 1: Let the rate of A = 2 units per day and the rate of B = 3 units per day. Step 2: Time taken is inversely proportional to rate. So time taken by A ∝ 1 / 2, time taken by B ∝ 1 / 3. Step 3: Ratio of times = (1 / 2) : (1 / 3). Step 4: Simplify the ratio (1 / 2) : (1 / 3) by multiplying both terms by 6 (the LCM of 2 and 3). Step 5: We get (1 / 2) * 6 : (1 / 3) * 6 = 3 : 2. Step 6: Therefore, the ratio of days taken by A and B is 3 : 2 respectively. Verification / Alternative check: Assume a fixed work of 6 units. At a rate of 2 units per day, A will need 6 / 2 = 3 days. At a rate of 3 units per day, B will need 6 / 3 = 2 days. So the time ratio is clearly 3 : 2, which confirms our reasoning based on inverse proportionality. Why Other Options Are Wrong: - 2:3: This simply repeats the rate ratio, not the time ratio. Time is inversely, not directly, proportional to rate. - 4:9 and 9:4: These are incorrect transformations and do not follow from basic inverse proportionality. - 1:2: This would mean B takes twice the time of A, which contradicts the fact that B is faster due to a higher rate (3 versus 2). Common Pitfalls: The most common error is to mistakenly assume that the ratio of times is the same as the ratio of rates. In time and work problems, when the amount of work is fixed, time and rate always move in opposite directions: higher rate means less time and vice versa. Always remember that time ratio is the inverse of the rate ratio. Final Answer: The ratio of days taken by A and B to finish the work is 3:2.

Question 2

A and B undertake a piece of work for Rs. 250. A alone can do the work in 5 days and B alone can do it in 15 days. With the help of C, they finish the work in 3 days. If everyone is paid in proportion to the amount of work done, how much money shoul…

Accepted Answer

Introduction / Context: This question combines time and work concepts with proportional division of money. Three people, A, B, and C, share a contract payment based on how much work each completes. We know how long A and B take individually, and how long the three take together. From this we infer C’s contribution and then compute C’s fair share of the total payment. Given Data / Assumptions: - Total contract amount = Rs. 250. - A alone can complete the work in 5 days. - B alone can complete the work in 15 days. - A, B, and C together complete the work in 3 days. - Payment is proportional to work done by each individual. Concept / Approach: First, we compute the daily work rates of A and B. Next, we determine the combined rate of all three from the given completion time of 3 days. C’s daily rate is then found by subtracting the rates of A and B from the combined rate. Knowing C’s rate, we calculate the fraction of work done by C in 3 days and then multiply this fraction by the total payment to get C’s share. Step-by-Step Solution: Step 1: Assume total work = 1 unit. Step 2: Rate of A = 1 / 5 work per day. Step 3: Rate of B = 1 / 15 work per day. Step 4: Together, A, B, and C finish in 3 days, so combined rate = 1 / 3 work per day. Step 5: Rate of A + B = 1 / 5 + 1 / 15. Step 6: LCM of 5 and 15 is 15, so 1 / 5 = 3 / 15 and 1 / 15 = 1 / 15. Step 7: Rate of A + B = 3 / 15 + 1 / 15 = 4 / 15 work per day. Step 8: Rate of C = combined rate - rate of A + B = 1 / 3 - 4 / 15. Step 9: Convert 1 / 3 to denominator 15: 1 / 3 = 5 / 15. Step 10: Rate of C = 5 / 15 - 4 / 15 = 1 / 15 work per day. Step 11: Work done by C in 3 days = 3 * 1 / 15 = 3 / 15 = 1 / 5 of the work. Step 12: C’s share of money = (1 / 5) * 250 = Rs. 50. Verification / Alternative check: We can also compute A’s and B’s shares to verify. In 3 days, A does 3 * 1 / 5 = 3 / 5 of the work, and B does 3 * 1 / 15 = 1 / 5 of the work. Thus, the distribution of work among A, B, and C is 3 / 5, 1 / 5, and 1 / 5 respectively, corresponding to a ratio of 3 : 1 : 1. So A gets 3 / 5 * 250 = 150, B gets 1 / 5 * 250 = 50, and C also gets 50. The totals add up to 250, confirming correctness. Why Other Options Are Wrong: - Rs.100, Rs.150, Rs.200: These amounts would imply that C did a larger fraction of the work than 1 / 5, which contradicts the calculated work share based on rates and time. - Rs.75: This does not correspond to any simple fraction of 250 and does not align with the ratio of work contributions. Common Pitfalls: Many learners forget to compute C’s rate explicitly and instead try to guess the share. Another error is to divide the total amount equally among the three people, ignoring the actual work distribution. It is important to always base payment on proportional work done, especially in time and work problems involving contracts and money. Final Answer: C should receive Rs. 50 from the total payment.

Question 3

A can do a piece of work in 10 days, B can do it in 12 days and C can do it in 15 days. If all three work together from the start, in how many days will they complete the work?

Accepted Answer

Introduction / Context: This problem involves three workers, A, B, and C, each with different individual times to complete the same task. We are asked how long all three will take if they work together from the beginning. This is a straightforward application of the concept of adding work rates to obtain a combined rate. 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 work together until the job is completed. - Total work is considered as 1 unit. Concept / Approach: We convert each person’s time into a daily work rate by taking the reciprocal of the time. Then we add the three rates to find the combined rate when A, B, and C work together. Finally, we take the reciprocal of this combined rate to determine the total time required to finish the work jointly. Step-by-Step Solution: Step 1: Assume total work = 1 unit. Step 2: Rate of A = 1 / 10 work per day. Step 3: Rate of B = 1 / 12 work per day. Step 4: Rate of C = 1 / 15 work per day. Step 5: Combined rate = 1 / 10 + 1 / 12 + 1 / 15. Step 6: LCM of 10, 12, and 15 is 60. Step 7: Convert each rate: 1 / 10 = 6 / 60, 1 / 12 = 5 / 60, 1 / 15 = 4 / 60. Step 8: Combined rate = 6 / 60 + 5 / 60 + 4 / 60 = 15 / 60 = 1 / 4 work per day. Step 9: Time taken together = 1 / (1 / 4) = 4 days. Verification / Alternative check: We can verify by computing how much work is completed in 4 days. At a rate of 1 / 4 work per day, in 4 days they complete 4 * 1 / 4 = 1 unit of work, exactly matching the total work. Also, the combined time of 4 days is logically less than each individual time of 10, 12, and 15 days, which makes sense because more workers should complete the job faster. Why Other Options Are Wrong: - 6, 5 1/4, and 4 4/11 days: These values correspond to slower combined rates and would not match the correct sum of daily work contributions from A, B, and C. - 7 days: This is even higher and would mean a combined work rate smaller than any single worker, which is impossible when all three are working together. Common Pitfalls: Common mistakes include averaging the given times instead of summing work rates, or mishandling fraction addition by not using the correct LCM. Another error is to forget that combined time must always be shorter than the smallest individual time when everyone works simultaneously on the same job. Final Answer: A, B, and C working together will complete the work in 4 days.

Question 4

If A and B together can complete a job in 7.5 days and A alone can complete the same job in 10 days, how many days will B alone take to complete the job?

Accepted Answer

Introduction / Context: This question provides the combined time of A and B working together and the individual time of A alone. We are required to determine how long B alone would take to finish the job. This is a typical time and work problem that uses the relationship between time, work, and rate to isolate the unknown worker’s efficiency. Given Data / Assumptions: - A and B working together complete the job in 7.5 days. - A alone completes the job in 10 days. - We need the number of days B alone will take to complete the job. - Total work is taken as 1 unit. Concept / Approach: We convert 7.5 days into a fraction (15 / 2 days) and then compute the combined daily rate of A and B as 1 divided by 7.5. A’s rate is 1 / 10 per day. B’s rate is obtained by subtracting A’s rate from the combined rate. Finally, we find B’s time by taking the reciprocal of B’s rate. Handling the decimal 7.5 as a fraction simplifies the calculations. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Combined time of A and B = 7.5 days = 15 / 2 days. Step 3: Combined rate of A and B = 1 / (15 / 2) = 2 / 15 work per day. Step 4: Rate of A alone = 1 / 10 work per day. Step 5: Rate of B = combined rate - rate of A = 2 / 15 - 1 / 10. Step 6: LCM of 15 and 10 is 30, so 2 / 15 = 4 / 30 and 1 / 10 = 3 / 30. Step 7: Rate of B = 4 / 30 - 3 / 30 = 1 / 30 work per day. Step 8: Time taken by B alone = 1 / (1 / 30) = 30 days. Verification / Alternative check: Using the rates, A does 1 / 10 per day and B does 1 / 30 per day. Together they do 1 / 10 + 1 / 30 = 3 / 30 + 1 / 30 = 4 / 30 = 2 / 15 work per day. Time required at this rate is 1 / (2 / 15) = 15 / 2 = 7.5 days, which matches the given combined time, confirming that B’s individual time of 30 days is correct. Why Other Options Are Wrong: - 40, 25, 50, 20: Each of these values yields an incorrect individual rate for B. When combined with A’s rate, none of them reproduces the correct combined time of 7.5 days. Common Pitfalls: A frequent mistake is misinterpreting 7.5 days, either leaving it in decimal form and miscalculating, or failing to convert properly to a fraction. Another common error is to average the times of A and B instead of using rate subtraction. Always work with rates, especially when combining or separating worker contributions. Final Answer: B alone will take 30 days to complete the job.

Question 5

A can do 75% of a job in 18 days and B can do 25% of the same job in 12 days. If A and B work together, in how many days will they complete 75% of the job?

Accepted Answer

Introduction / Context: This problem provides partial work information for two workers, A and B. Instead of giving full-job times directly, the question gives how long each takes to complete certain percentages of the work. From these details we must deduce their full-job rates, then combine those rates to find the time needed to complete 75 percent of the job when they work together. Given Data / Assumptions: - A completes 75 percent (3/4) of the job in 18 days. - B completes 25 percent (1/4) of the job in 12 days. - We are asked for the time A and B together need to complete 75 percent of the job. - Total work is taken as 1 unit. Concept / Approach: First, we compute A’s daily work rate from the fact that A finishes 3/4 of the work in 18 days. Then we compute B’s daily rate from the fact that B finishes 1/4 of the work in 12 days. Using these rates, we obtain the combined daily rate of A and B. Finally, we use this combined rate to determine how long it takes them to finish 3/4 of the job together. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: A completes 3 / 4 of the work in 18 days. Step 3: Rate of A = (3 / 4) / 18 = 3 / 72 = 1 / 24 work per day. Step 4: B completes 1 / 4 of the work in 12 days. Step 5: Rate of B = (1 / 4) / 12 = 1 / 48 work per day. Step 6: Combined rate of A and B = 1 / 24 + 1 / 48. Step 7: LCM of 24 and 48 is 48, so 1 / 24 = 2 / 48. Step 8: Combined rate = 2 / 48 + 1 / 48 = 3 / 48 = 1 / 16 work per day. Step 9: We need time to complete 3 / 4 of the work at a rate of 1 / 16 per day. Step 10: Time = (3 / 4) / (1 / 16) = (3 / 4) * 16 = 12 days. Verification / Alternative check: We can check the logic by computing individual full-job times. From rate 1 / 24, A would need 24 days for the full job, and from rate 1 / 48, B would need 48 days. Together they need 1 / (1 / 24 + 1 / 48) = 16 days for the full work. Three fourths of 16 days is 12 days, which matches the calculated answer directly from rates, confirming consistency. Why Other Options Are Wrong: - 16 and 20 days: These are closer to times for the full job, not three fourths of it with both working together. - 8 and 10 days: These underestimate the required time and would only be valid if the combined rate were higher than 1 / 16 work per day, which contradicts the given data. Common Pitfalls: A common mistake is to misinterpret the given percentages and directly average the partial times. Another typical error is to forget that the question asks for 75 percent of the job, not the entire job. Always convert percentages into fractions of the total work, compute the exact rates, and then carefully use the fraction of work requested in the final step. Final Answer: A and B together will complete 75 percent of the job in 12 days.

Question 6

A canal of a village can be cleaned by 24 villagers in 12 days. In how many days will 36 villagers be able to clean the same canal, assuming everyone works at the same constant rate?

Accepted Answer

Introduction / Context: This problem is based on the concept of man-days in time and work. It is a direct application of the inverse relationship between the number of workers and the time required, given a fixed total amount of work. We know how many days 24 villagers take, and we must find the corresponding time if 36 villagers work instead. Given Data / Assumptions: - 24 villagers can clean the canal in 12 days. - We need the number of days required for 36 villagers to clean the same canal. - Every villager works at the same constant rate. - The total amount of cleaning work remains the same. Concept / Approach: The total work can be measured in man-days, which is the product of the number of workers and the days they work. First, we compute the total work in man-days using the original group of villagers. Then, we divide this total man-day value by the new number of workers (36) to obtain the required time with the larger group. Step-by-Step Solution: Step 1: Compute total work in man-days using the initial scenario. Step 2: Total work = 24 villagers * 12 days = 288 man-days. Step 3: When 36 villagers work, let the required number of days be D. Step 4: Then 36 * D = 288 man-days, because total work remains the same. Step 5: Solve for D: D = 288 / 36 = 8 days. Verification / Alternative check: We can verify this by using proportional reasoning. Time is inversely proportional to the number of workers when work is fixed. So required days = 12 * (24 / 36) = 12 * (2 / 3) = 8 days. This matches the man-day method and confirms that the answer is correct and consistent. Why Other Options Are Wrong: - 18 and 16 days: These are greater than the original 12 days, which does not make sense because more workers should reduce, not increase, the time required. - 72 days: This is far too large and corresponds to a misunderstanding of the relationship between workers and time. - 6 days: This would require even more total man-days than available or a higher per-worker efficiency than given by the data. Common Pitfalls: A frequent mistake is failing to use the man-day concept and simply guessing based on rough intuition. Some learners also incorrectly treat the relationship as directly proportional instead of inversely proportional. Always remember that for a fixed job, doubling the number of workers roughly halves the time, provided all other conditions remain the same. Final Answer: 36 villagers can clean the canal in 8 days.

Question 7

A and B can do a piece of work in 18 days, B and C can do it in 24 days, and A and C can do it in 36 days. If all three work together, in how many days will they complete the entire work?

Accepted Answer

Introduction / Context: This is a multi-step time and work question involving three workers, A, B, and C. We are given the times taken by each pair of workers, not by any single worker. The task is to deduce the individual rates of A, B, and C using the pairwise information and then find the total time when all three work together. This requires setting up and solving a system of equations. Given Data / Assumptions: - A and B together complete the work in 18 days. - B and C together complete the work in 24 days. - A and C together complete the work in 36 days. - Total work is considered as 1 unit. - Work rates add linearly. Concept / Approach: Let the daily work rates of A, B, and C be a, b, and c units per day respectively. Then we know that a + b = 1 / 18, b + c = 1 / 24, and a + c = 1 / 36. Adding these three equations gives 2(a + b + c). From this, we can compute the combined rate (a + b + c) and then find the time taken when all three work together by taking the reciprocal of this combined rate. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Let rates be a, b, and c for A, B, and C respectively. Step 3: From the data, we have a + b = 1 / 18, b + c = 1 / 24, a + c = 1 / 36. Step 4: Add all three equations: (a + b) + (b + c) + (a + c) = 1 / 18 + 1 / 24 + 1 / 36. Step 5: Left side simplifies to 2(a + b + c). Step 6: Compute right side: 1 / 18 + 1 / 24 + 1 / 36. Step 7: LCM of 18, 24, and 36 is 72. Rewrite: 1 / 18 = 4 / 72, 1 / 24 = 3 / 72, 1 / 36 = 2 / 72. Step 8: Sum = 4 / 72 + 3 / 72 + 2 / 72 = 9 / 72 = 1 / 8. Step 9: So 2(a + b + c) = 1 / 8, giving a + b + c = 1 / 16. Step 10: Combined rate of A, B, and C working together = 1 / 16 work per day. Step 11: Time taken when all three work together = 1 / (1 / 16) = 16 days. Verification / Alternative check: We can test consistency by solving for one individual rate. For example, a + b = 1 / 18 and a + b + c = 1 / 16, so c = 1 / 16 - 1 / 18. Similar checks can be done for a and b. The computed individual rates will be positive and, when substituted back, match all three original pairwise times, confirming that 16 days for all three together is correct. Why Other Options Are Wrong: - 12 and 13 days: These correspond to higher combined rates and do not align with the pairwise times given. - 26 and 18 days: These are too large and imply slower combined rates. Substituting these into the relationships would not satisfy all three pairwise conditions simultaneously. Common Pitfalls: Students often try to guess the answer based only on one or two of the pairwise times, ignoring the need to satisfy all three equations at once. Another common mistake is incorrect fraction addition. To avoid errors, always use the LCM for denominators and systematically derive the combined rate using the sum of the three equations. Final Answer: Working together, A, B, and C will complete the work in 16 days.

Question 8

Ramesh and Rahman can complete a piece of work in 20 and 25 days respectively. They work together for 10 days and then leave the work due to illness. Suresh completes the remaining work in 3 days. In how many days can Suresh alone complete the entir…

Accepted Answer

Introduction / Context: In this problem, two workers, Ramesh and Rahman, start a job together but stop after some time, and a third worker, Suresh, completes the remaining portion. We are asked to find Suresh’s full-job time using the information about how much work is left for him and how long he takes to finish that remainder. Given Data / Assumptions: - Ramesh alone can complete the work in 20 days. - Rahman alone can complete the work in 25 days. - They work together for 10 days. - Suresh completes the remaining work in 3 days. - Total work is considered as 1 unit. Concept / Approach: First, we compute the combined work rate of Ramesh and Rahman and use this to determine how much work they complete in 10 days. Subtracting this from 1 gives the remaining work. Since Suresh finishes that remaining work in 3 days, we can find his rate by dividing the remaining work by 3. Finally, we take the reciprocal of Suresh’s rate to obtain the number of days he would need to complete the full job alone. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Rate of Ramesh = 1 / 20 work per day. Step 3: Rate of Rahman = 1 / 25 work per day. Step 4: Combined rate of Ramesh and Rahman = 1 / 20 + 1 / 25. Step 5: LCM of 20 and 25 is 100, so 1 / 20 = 5 / 100 and 1 / 25 = 4 / 100. Step 6: Combined rate = 5 / 100 + 4 / 100 = 9 / 100 work per day. Step 7: Work done by them in 10 days = 10 * 9 / 100 = 90 / 100 = 0.9 of the work. Step 8: Remaining work = 1 - 0.9 = 0.1 = 1 / 10 of the total work. Step 9: Suresh completes this 1 / 10 in 3 days. Step 10: Rate of Suresh = (1 / 10) / 3 = 1 / 30 work per day. Step 11: Time taken by Suresh for the entire work = 1 / (1 / 30) = 30 days. Verification / Alternative check: Check the consistency: If Suresh works at 1 / 30 work per day, in 3 days he completes 3 * 1 / 30 = 1 / 10 of the work. Adding this to 0.9 completed by Ramesh and Rahman gives 0.9 + 0.1 = 1, which is the full job. This confirms that Suresh’s full-job time of 30 days is correct. Why Other Options Are Wrong: - 28, 29, 32 days: These values yield slightly different rates for Suresh and do not reproduce exactly 1 / 10 of the work being finished in 3 days. - 25 days: This implies a larger rate for Suresh and would result in more than 1 / 10 being completed in 3 days, which contradicts the data. Common Pitfalls: Some learners miscalculate the work done by Ramesh and Rahman in 10 days, either by not using the correct combined rate or by making mistakes in fraction addition. Others forget to subtract from 1 to find the remaining work. A clear separation of steps — finding combined rate, computing work done, finding remainder, and then deriving Suresh’s rate — helps avoid these errors. Final Answer: Suresh alone can complete the entire work in 30 days.

Question 9

A can do as much work in 4 days as B can do in 5 days, and B can do as much work in 6 days as C can do in 7 days. In what time will C do a piece of work which A can complete in 7 days?

Accepted Answer

Introduction / Context: This is a comparative time and work question involving three workers A, B, and C. Instead of giving direct times for the same job, the problem compares how much work each can do in different numbers of days. From these relationships, we must determine the relative efficiencies and finally compute how long C would take to complete a piece of work that A can finish in 7 days. Given Data / Assumptions: - A does as much work in 4 days as B does in 5 days. - B does as much work in 6 days as C does in 7 days. - A can complete a certain piece of work in 7 days. - We need the time taken by C to do the same piece of work. Concept / Approach: Let the daily work rates of A, B, and C be a, b, and c units per day. The statements about equal work lead to proportionality equations: 4a = 5b and 6b = 7c. From these, we express b and c in terms of a. Since A completes the whole job in 7 days, total work is 7a units. Using C’s rate in terms of a, we find the time required for C to do the same total work by dividing 7a by c. Step-by-Step Solution: Step 1: Let the daily rate of A = a units per day. Step 2: From 4a = 5b, we get b = (4a) / 5. Step 3: From 6b = 7c, substitute b to obtain 6 * (4a / 5) = 7c. Step 4: Simplify: 24a / 5 = 7c, so c = (24a) / (35). Step 5: A completes the required piece of work in 7 days, so total work W = 7a units. Step 6: Time taken by C to complete the same work W is W / c. Step 7: Time for C = 7a / (24a / 35) = 7a * 35 / 24a. Step 8: Cancel a: Time for C = 245 / 24 days. Step 9: Convert 245 / 24 into mixed form: 24 * 10 = 240, remainder 5, so time = 10 5/24 days. Verification / Alternative check: We can test relative efficiencies. From 4a = 5b, A is more efficient than B. From 6b = 7c, B is more efficient than C. Hence C should take more time than A for the same work. A takes 7 days, and C takes about 10.21 days (10 5/24), which is indeed larger, aligning with the efficiency ordering. The algebraic steps satisfy both proportionality equations and the final answer fits the logical trend. Why Other Options Are Wrong: - 4 4/5 and 6 8/15 days: These are less than 7 days and would imply that C is faster than A, contradicting the efficiency chain A faster than B and B faster than C. - 12 6/19 days: This overestimates C’s time relative to the proportional relationships derived from the given conditions. - 8 days: Slightly greater than A’s time but does not satisfy the precise ratio calculations from 4a = 5b and 6b = 7c. Common Pitfalls: Learners often misinterpret the statements about equal work and incorrectly set up the equations, or they may try to directly guess times without using algebra. It is important to clearly define symbols for daily rates and use the equal-work conditions to relate these rates. Mistakes in manipulating fractions and converting improper fractions to mixed numbers are also common, so extra care is needed in the final steps. Final Answer: C will take 10 5/24 days to complete the same piece of work that A can do in 7 days.

Question 10

A can complete a piece of work in 10 days and B can complete the same work in 12 days. They work together for 3 days, after which B leaves and A continues alone for 2 more days. Two days later, C joins A and together they finish the remaining work i…

Accepted Answer

Introduction / Context: This time and work question tests how to convert different time durations into work rates, add and subtract those rates in stages, and then back calculate the individual capacity of one worker. Such questions are common in quantitative aptitude tests and competitive exams because they combine ratio, fraction, and basic algebra concepts in a single scenario. Given Data / Assumptions: A alone finishes the work in 10 days. B alone finishes the same work in 12 days. A and B work together for 3 days. After that B stops, and A works alone for 2 more days. Then A and C work together for 2 days and complete the remaining work. Work is assumed to be uniform and divisible, and total work is taken as one unit. Concept / Approach: The main idea is to treat the whole work as one unit and express daily work done by each person as a fraction of this unit. Rates add when people work together. We compute the total work already done by A and B, then by A alone, and finally use the last phase with A and C together to deduce C's daily work rate. Once C's rate is known, its reciprocal gives the total number of days C alone would take to finish the entire work. Step-by-Step Solution: Take total work as 1 unit.Rate of A per day = 1/10 and rate of B per day = 1/12.Together in one day, A and B do 1/10 + 1/12 = 11/60 of the work.In 3 days, A and B together complete 3 * 11/60 = 33/60 = 11/20 of the work.Work left after 3 days = 1 - 11/20 = 9/20.A alone works for the next 2 days, doing 2 * 1/10 = 1/5 = 4/20 of the work.Work remaining after A's additional 2 days = 9/20 - 4/20 = 5/20 = 1/4.This remaining 1/4 of the work is completed in 2 days by A and C together.Their combined rate is therefore (1/4) / 2 = 1/8 per day.Rate of A is already 1/10, so rate of C per day = 1/8 - 1/10 = (5 - 4)/40 = 1/40.Thus C alone needs 40 days to finish the entire work. Verification / Alternative Check: We can check by reconstructing the entire sequence. In 3 days A and B complete 11/20 of the work, then A adds 1/5 more, for a total of 11/20 + 4/20 = 15/20 = 3/4. The remaining 1/4 is exactly completed by two days of work at rate 1/8 per day. Substituting C's rate of 1/40 into 1/10 + 1/40 gives 1/8, which matches the combined rate used, so the calculation is consistent. Why Other Options Are Wrong: 30 days or 50 days would correspond to C being faster or slower than 40 days, but they do not satisfy the equation for the combined rate in the last phase. Similarly, 60 days gives an even smaller rate for C and makes the total work done by A and C in the final 2 days less than 1/4, so the work would not be completed on time. Common Pitfalls: A common mistake is to treat days as simple numbers and not as reciprocals of work rates, or to forget that A continues working during the last phase together with C. Another frequent error is to compute C's time from only the final 2 days without correctly subtracting the earlier completed portions from the total work. Final Answer: The worker C would take 40 days to complete the entire work if C works alone from start to finish.

Question 11

An hour long test contains 60 problems. A student manages to solve 30 of these problems in 25 minutes. Based on this information, how many seconds on average does the student have available for solving each of the remaining problems in the time stil…

Accepted Answer

Introduction / Context: This aptitude question focuses on speed and time management in the context of an examination. It checks whether the student can correctly interpret the remaining time, convert minutes to seconds, and then distribute that time equally among the remaining unsolved questions. This type of calculation is common in competitive exams where candidates must decide how much time they can afford to spend on each question. Given Data / Assumptions: Total duration of the test is 60 minutes. Total number of problems in the test is 60. The student solves 30 problems in the first 25 minutes. The remaining 30 problems must be solved in the remaining time of the test. We assume that the student uses all the remaining time for the remaining problems. Concept / Approach: The key idea is to calculate the time left after solving the first batch of problems and then divide this remaining time by the number of remaining problems. We must also convert minutes into seconds because the answer choices are given in seconds. The average time per problem is simply total remaining seconds divided by the number of remaining questions. Step-by-Step Solution: Total test time = 60 minutes.Time already used for the first 30 problems = 25 minutes.Remaining time = 60 - 25 = 35 minutes.Convert 35 minutes to seconds: 35 * 60 = 2100 seconds.Remaining number of problems = 60 - 30 = 30 problems.Average time per remaining problem = total remaining seconds / remaining problems.Average time per remaining problem = 2100 / 30 = 70 seconds per problem. Verification / Alternative Check: If the student spends exactly 70 seconds on each of the remaining 30 questions, total time used for them is 30 * 70 = 2100 seconds, which is equal to 35 minutes. Adding back the initial 25 minutes gives 25 + 35 = 60 minutes, which matches the full test duration, so the calculation is accurate. Why Other Options Are Wrong: Options like 50, 40, or 30 seconds would use less than the available time. That would sum to 1500, 1200, or 900 seconds respectively for the remaining questions, leaving unused time and not matching the distribution based on the given conditions. Only 70 seconds per question correctly uses all of the remaining 35 minutes. Common Pitfalls: Students sometimes forget to convert minutes into seconds or mistakenly divide total time (60 minutes) by total questions (60), which would ignore the fact that 25 minutes have already been spent. Another common error is to mix up solved and unsolved counts, dividing by 60 instead of 30. Carefully distinguishing between used time, remaining time, solved problems, and remaining problems avoids these mistakes. Final Answer: The average time available for each of the remaining problems is 70 seconds per problem.

Question 12

Three electronic devices make a beep repeatedly at fixed intervals of 48 seconds, 72 seconds, and 108 seconds respectively. They all beep together exactly at 10:00 a.m. At what time will all three devices next beep together at the same instant after…

Accepted Answer

Introduction / Context: This problem is a classic example of using the concept of the least common multiple in time and work situations. Here, the repeated actions are beeps from three devices at different regular intervals. Finding when they next beep together is equivalent to finding the smallest time interval that is a common multiple of all three given intervals. This is a common pattern in exam questions involving bells, alarms, and periodic processes. Given Data / Assumptions: Device 1 beeps every 48 seconds. Device 2 beeps every 72 seconds. Device 3 beeps every 108 seconds. All three beep together at exactly 10:00 a.m. We assume perfect regularity in intervals with no delay or drift. Concept / Approach: The time after which they all beep together again is the least common multiple of 48, 72, and 108 seconds. The least common multiple represents the smallest positive time that is simultaneously a multiple of all three individual periods. Once this number of seconds is found, it is converted into minutes and seconds and then added to the starting time of 10:00 a.m. to get the required clock time. Step-by-Step Solution: Find the least common multiple of 48, 72, and 108 seconds.Compute prime factors: 48 = 2^4 * 3, 72 = 2^3 * 3^2, 108 = 2^2 * 3^3.Least common multiple uses the highest power of each prime: 2^4 and 3^3.So LCM = 2^4 * 3^3 = 16 * 27 = 432 seconds.Convert 432 seconds into minutes and seconds: 432 / 60 = 7 minutes and 12 seconds.Add 7 minutes and 12 seconds to 10:00:00 a.m.Resulting time = 10:07:12 a.m. Verification / Alternative Check: Check that 432 is a multiple of each interval. Dividing 432 by 48 gives 9, by 72 gives 6, and by 108 gives 4. All are integers, so 432 is a common multiple. To confirm that it is the least common multiple, we note that we used prime factorisation with the highest powers of each prime, which by definition yields the smallest positive number that is a multiple of all three, so there is no smaller positive time when they beep together again. Why Other Options Are Wrong: All other options correspond to different second values that are not equal to the least common multiple. For example, 10:07:24 corresponds to 444 seconds, which is not divisible evenly by all three periods. Similar checks show that 10:07:36 and 10:07:48 do not yield integer multiples for every device, so they cannot be the first common beep time. Common Pitfalls: Common mistakes include taking the greatest common divisor instead of the least common multiple, adding the intervals directly, or performing incorrect factorisation. Another error is to forget to convert between seconds and minutes correctly or to add the interval incorrectly to the starting clock time. Careful factorisation and unit handling avoid these errors. Final Answer: The three devices will next beep together at 10:07:12 hrs after the initial beep at 10:00 a.m.

Question 13

P, Q, and R can each complete the same piece of work at different rates. P can complete one quarter of the work in 10 days, Q can complete 40 percent of the work in 40 days, and R can complete one third of the work in 13 days. Based on these rates, …

Accepted Answer

Introduction / Context: This problem compares the efficiencies of three workers based on partial work data given in different fractions and percentages. It tests the candidate's ability to convert fractional work completed over given time periods into full completion times and then directly compare those times. Problems of this type are common in the time and work section of quantitative aptitude tests. Given Data / Assumptions: P completes 1/4 of the work in 10 days. Q completes 40 percent (which is 2/5) of the work in 40 days. R completes 1/3 of the work in 13 days. Work rate is constant for each person and scales linearly with time. We assume the total work is one unit for calculation convenience. Concept / Approach: We convert each person's given partial work performance into a daily work rate. For example, if someone completes a certain fraction of the job in a set number of days, we divide that fraction by the number of days to obtain their per day rate. The reciprocal of that rate gives the total time that person would need to complete the entire work alone. Once the completion times for P, Q, and R are found, we identify the smallest one, which corresponds to the fastest worker. Step-by-Step Solution: Let total work = 1 unit.P completes 1/4 of the work in 10 days, so P's rate = (1/4) / 10 = 1/40 per day and P would finish the whole work in 40 days.Q completes 40 percent of the work, that is 2/5 of the work, in 40 days, so Q's rate = (2/5) / 40 = 1/100 per day and Q would finish the whole work in 100 days.R completes 1/3 of the work in 13 days, so R's rate = (1/3) / 13 = 1/39 per day and R would finish the whole work in 39 days.Compare completion times: P needs 40 days, Q needs 100 days, and R needs 39 days.The smallest completion time is 39 days, so R finishes the work first. Verification / Alternative Check: The rates can be checked quickly. Because R completes almost the same fraction as P (one third versus one quarter) but in slightly more days compared with P's 10 days versus R's 13 days, we still need to compare full completion times. Our calculations show that R is slightly faster, needing 39 days versus P's 40 days. This aligns well with the relative rates and confirms the logic is consistent. Why Other Options Are Wrong: P is slightly slower than R because P needs 40 days while R needs 39 days, so option P alone is not correct. Q is clearly the slowest with 100 days, so Q is not a valid choice. Both P and R together as an answer would imply equal times, which is not the case because their completion times differ by one day. Therefore, only R is correct. Common Pitfalls: Typical mistakes include treating percentages and fractions inconsistently, misreading 40 percent as 40 over 100 without converting it to a fraction of the job, or forgetting to take reciprocals to move from rate to total time. Another error is to compare partial times directly instead of first finding the implied total time for each worker. A systematic rate based approach avoids these issues and leads to the correct comparison. Final Answer: The worker who would complete the work first when working alone is R.

Question 14

Working 7 hours per day, 4 men can complete a certain piece of work in 8 days. If the men instead work 8 hours per day, how many men are required to complete the same work in 4 days?

Accepted Answer

Introduction / Context: This question is a direct application of the concept of man hours in time and work problems. It tests the understanding that total work is proportional to the product of the number of workers, the number of days, and the number of hours worked per day, assuming constant efficiency. Problems like this regularly appear in competitive exams to check proportional reasoning skills. Given Data / Assumptions: In the first scenario, 4 men work 7 hours per day for 8 days to finish the work. In the second scenario, men work 8 hours per day and must complete the work in 4 days. Efficiency of each man remains constant in both scenarios. Total work in both scenarios is exactly the same. Concept / Approach: We use the man hours approach: Work is proportional to number of men multiplied by hours per day multiplied by number of days. Since the work is the same in both cases, the total man hours in the first case should be equal to the total man hours in the second case. We set up an equation equating these two products and solve for the unknown number of men in the second scenario. Step-by-Step Solution: Compute total man hours in the first scenario: 4 men * 7 hours per day * 8 days = 4 * 7 * 8 = 224 man hours.Let the required number of men in the second scenario be M.Total man hours in the second scenario = M men * 8 hours per day * 4 days = 32M man hours.Since the total work is the same, set 32M = 224.Solve for M: M = 224 / 32 = 7.Therefore, 7 men are needed when working 8 hours per day for 4 days. Verification / Alternative Check: We can verify by substituting back. In the second scenario with 7 men: 7 * 8 * 4 = 224 man hours, which matches the 224 man hours of the first scenario. Since total man hours are the same and efficiency is constant, the work completed must be the same. Hence the calculation is consistent and confirms that 7 men are sufficient. Why Other Options Are Wrong: If 8 men worked, total man hours would be 8 * 8 * 4 = 256, which is more than needed and implies the work would be done earlier. If 4 men worked, total man hours would be 4 * 8 * 4 = 128, which is less than required, so the work would not finish in 4 days. Similarly, 9 men would give 288 man hours, far exceeding the required amount. Only 7 men produce the exact required total of 224 man hours. Common Pitfalls: Many learners confuse the relationship between time and number of workers, forgetting that hours per day and number of days also affect total work. Another common error is to adjust days and men but ignore the change in working hours per day. The safest method is always to compute total man hours in each scenario and equate them when the work is the same. Final Answer: The number of men required to complete the work in the new scenario is 7 men.

Question 15

The time taken by 4 men to complete a certain job is double the time taken by 5 children to complete the same job. Each man works at twice the rate of a woman, and each child can finish the entire job alone in 20 days. If 12 men, 10 children, and 8 …

Accepted Answer

Introduction / Context: This question combines multiple layers of time and work reasoning. It connects the relative efficiencies of men, women, and children using conditions on completion times and then asks for the time needed when all three categories work together. The problem tests comfort with work rates, ratios, and translating relationships such as "twice as fast" into mathematical form. Given Data / Assumptions: Four men take twice as long as five children to complete the same job. Each man is twice as fast as each woman. Each child can complete the job alone in 20 days. We must find the time taken when 12 men, 10 children, and 8 women work together. Work is uniform and the job size is taken as one unit. Concept / Approach: We first convert the given times into work rates for children, then use the comparative time relation between men and children to find the rate of a man. Next we use the information that a man is twice as fast as a woman to find the woman's rate. Finally we add the work rates of all 12 men, 10 children, and 8 women to get a combined rate, and then take the reciprocal to find the total time required to finish the job together. Step-by-Step Solution: Let total work = 1 unit.A child can finish the work in 20 days, so child's rate r_child = 1/20 per day.Five children together have rate 5 * 1/20 = 1/4 per day, so they need 4 days to finish the work.Time taken by four men is twice that of five children, so four men take 8 days.Rate of four men together is 1/8 per day, so rate of one man r_man = (1/8) / 4 = 1/32 per day.Each man is twice as fast as each woman, so r_man = 2 * r_woman, hence r_woman = 1/64 per day.Now compute combined rate of 12 men, 10 children, and 8 women: 12 * 1/32 + 10 * 1/20 + 8 * 1/64.12 * 1/32 = 12/32 = 3/8, 10 * 1/20 = 1/2, and 8 * 1/64 = 1/8.Add them: 3/8 + 1/2 + 1/8 = 3/8 + 4/8 + 1/8 = 8/8 = 1 per day.Combined rate is exactly 1 unit of work per day, so they finish in 1 day. Verification / Alternative Check: If all workers complete exactly one unit of work per day, by definition the job is done in one day. The intermediate relationships also check out: four men at 1/32 each give 1/8 per day and hence 8 days for the full job, double the 4 days taken by five children. The man to woman efficiency ratio is consistent with the derived rates of 1/32 and 1/64 respectively. Thus all constraints are satisfied. Why Other Options Are Wrong: Options such as 2, 3, or 4 days would correspond to combined daily work rates of 1/2, 1/3, or 1/4 of the job respectively, which contradict the calculated sum of individual rates. Since we have shown that the total combined rate is exactly 1 unit per day, any answer other than 1 day is inconsistent with the given relationships. Common Pitfalls: Candidates sometimes misinterpret the statement that four men take double the time needed by five children, mistakenly equating the number of men with the number of children instead of time. Another frequent error is assuming that a man does double the work of a child, which is not stated. Only the relation between men and women is given. Correctly following each given condition is critical for reaching the right formulae and final rate. Final Answer: Working together, the group of 12 men, 10 children, and 8 women will finish the job in 1 Day.

Question 16

A and B together can complete a piece of work in 12 days. They work together for 5 days and then A alone finishes the remaining work in 14 more days. In how many days would A alone be able to complete the entire work from start to finish?

Accepted Answer

Introduction / Context: This is a standard time and work problem where two people start working together and later only one of them continues. It tests the ability to use combined work rates, subtract completed work, and then infer the individual rate of one worker. Such problems are frequently seen in aptitude tests and help build intuition about how joint and individual efficiencies interact over time. Given Data / Assumptions: A and B together can finish the entire work in 12 days. They work together for the first 5 days. After that, A works alone and finishes the remaining work in 14 more days. We assume the total work is of one unit and all rates remain constant over time. Concept / Approach: The strategy is to convert the combined completion time of 12 days into a joint daily work rate for A and B. Then we compute how much work A and B complete in the first 5 days. The rest of the work is completed by A alone in 14 days, which directly gives the daily work rate of A. The time A alone would take for the entire job is simply the reciprocal of this daily rate. Step-by-Step Solution: Assume total work = 1 unit.A and B together complete the work in 12 days, so combined rate of A and B = 1/12 per day.In the first 5 days together, they complete 5 * 1/12 = 5/12 of the work.Remaining work after 5 days = 1 - 5/12 = 7/12.This remaining 7/12 is finished by A alone in 14 days.Therefore A's daily work rate = (7/12) / 14 = 7 / (12 * 14) = 7 / 168 = 1/24 per day.If A works alone from the beginning, time required = 1 / (1/24) = 24 days. Verification / Alternative Check: We can verify by reconstructing the time line: In 24 days A alone would do 1/24 per day times 24 days which equals 1 unit, so the rate is consistent. Also, if we use 1/24 as A's rate, the remaining rate for B when combined with A is 1/12 - 1/24 = 1/24. This means B is equally efficient as A, and if both have the same rate, together they should indeed complete the work in half of 24 days, which is 12 days, matching the original information. Why Other Options Are Wrong: Times like 20 or 18 days imply a higher daily rate for A which would cause the remaining 7/12 of the work to be finished in fewer than 14 days. Similarly, 22 days gives a rate that does not match the requirement of completing 7/12 of the work in exactly 14 days. Only 24 days aligns with both the joint work of 12 days and the solo completion of the remainder in 14 days. Common Pitfalls: An easy mistake is to subtract days instead of fractions of work, or to treat the 5 and 14 days together as a direct fraction of 12 days. It is essential to consistently work with rates and fractions of work completed, not just with raw day counts. Another error is to wrongly assume that A and B have equal efficiency without verifying from the rates. Final Answer: A alone can complete the entire work in 24 days when working from start to finish.

Question 17

A can complete a certain work in 21 days and B can complete the same work in 42 days. If they work together on this work for 7 days and then stop, what fraction of the total work will still remain unfinished?

Accepted Answer

Introduction / Context: This time and work problem asks for the fraction of work remaining after a certain period of joint work by two workers with different individual efficiencies. It checks the ability to convert individual completion times into daily rates, sum those rates for joint work, and then interpret the fraction completed versus the fraction remaining. Such questions are common in entrance exams and quantitative aptitude tests. Given Data / Assumptions: A can complete the entire work alone in 21 days. B can complete the entire work alone in 42 days. A and B work together for 7 days. We assume the work is one unit and that rates are constant throughout. Concept / Approach: The method is to convert the time taken by each worker alone into a daily work rate. For example, if someone completes the work in 21 days, that person completes 1/21 of the work in one day. When two workers collaborate, their daily rates add. The total fraction of work completed in the given number of days is simply joint rate multiplied by number of days. Subtracting this completed fraction from one gives the fraction that remains unfinished. Step-by-Step Solution: Assume total work = 1 unit.Rate of A per day = 1/21 of the work.Rate of B per day = 1/42 of the work.Together, daily work rate = 1/21 + 1/42.Compute: 1/21 + 1/42 = 2/42 + 1/42 = 3/42 = 1/14.So A and B together complete 1/14 of the work in one day.In 7 days, total work completed = 7 * 1/14 = 7/14 = 1/2 of the work.Thus, fraction of work remaining = 1 - 1/2 = 1/2. Verification / Alternative Check: If only A worked, in 7 days A would complete 7 * 1/21 = 1/3 of the work. If only B worked, in 7 days B would complete 7 * 1/42 = 1/6 of the work. Together, 1/3 + 1/6 = 1/2, which agrees with the earlier calculation. This cross check confirms that 1/2 of the work is done and therefore 1/2 remains incomplete after 7 days of combined work. Why Other Options Are Wrong: Options like 1/3 or 1/4 represent smaller remaining fractions, implying that more than half the work was completed in 7 days, which contradicts the exact calculation of 1/2 completed. The option 2/3 would mean only 1/3 was done, which is also not supported by the computed joint rate. Only 1/2 matches the arithmetic based on the individual completion times and joint work period. Common Pitfalls: Some students mistakenly add the days (21 and 42) instead of the rates, or treat 7 days as a simple fraction of one worker's time without accounting for both working together. Another frequent mistake is to stop after finding the fraction completed, rather than converting it to the fraction remaining as the question asks. Carefully distinguishing between completed and remaining portions is essential. Final Answer: The fraction of the total work that remains unfinished after 7 days of joint work is 1/2 of the work.

Question 18

A can paint a house in 55 days and B can paint the same house in 66 days. When they work together with a third painter C, all three of them complete the painting in just 12 days. If C worked alone from start to finish, in how many days would C be ab…

Accepted Answer

Introduction / Context: This question applies the time and work concept to a painting job involving three workers. Two individual completion times are given, and the combined completion time when all three work together is also provided. The task is to determine the individual efficiency of the third worker. Such problems test a candidate's ability to handle work rate equations and isolate an unknown rate from a combined rate. Given Data / Assumptions: A can paint the house alone in 55 days. B can paint the same house alone in 66 days. A, B, and C together finish the house in 12 days. We assume the size of the work is one unit and all workers maintain constant rates. Concept / Approach: We convert each worker's completion time into a daily work rate. The combined rate of A, B, and C working together is 1 divided by the total days taken together. This combined rate is the sum of their individual rates. Since we know A's and B's rates, we can subtract these from the combined rate to find C's rate. Finally, the reciprocal of C's rate gives the time taken by C alone to finish the job. Step-by-Step Solution: Assume total work = 1 unit.Rate of A = 1/55 of the house per day.Rate of B = 1/66 of the house per day.A, B, and C together finish in 12 days, so combined rate = 1/12 per day.Let rate of C be r_c. Then 1/55 + 1/66 + r_c = 1/12.Compute 1/55 + 1/66: take common denominator 330, so 1/55 = 6/330 and 1/66 = 5/330, total = 11/330 = 1/30.So 1/30 + r_c = 1/12, hence r_c = 1/12 - 1/30.Compute r_c: 1/12 - 1/30 = (5 - 2) / 60 = 3/60 = 1/20.Therefore, C alone would take 20 days to paint the house, since time = 1 / (1/20) = 20 days. Verification / Alternative Check: Check by recomputing the combined rate using C's rate. The sum 1/55 + 1/66 + 1/20 should equal 1/12. Using a common denominator of 660: 1/55 = 12/660, 1/66 = 10/660, and 1/20 = 33/660. Their sum is 12 + 10 + 33 = 55 over 660, which simplifies to 1/12. This matches the combined rate required to finish the house in 12 days, confirming that the derived rate for C is consistent. Why Other Options Are Wrong: If C took 24, 33, or 44 days, then C's rate would be smaller than 1/20, and the sum of the three rates would be less than 1/12, meaning the work would take more than 12 days. Hence these options contradict the given condition. The only time that correctly reproduces the combined 12 day completion time is 20 days. Common Pitfalls: Common mistakes include adding the given days directly instead of adding rates, or forgetting to subtract A and B's rates from the combined rate before taking the reciprocal. Some students also make errors in fractional arithmetic, especially when finding common denominators. Carefully setting up the rate equation and simplifying step by step avoids these issues. Final Answer: If C works alone from start to finish, the house will be painted in 20 days.

Question 19

A, B, and C together can complete a job in 9 days, while C alone can complete the same job in 36 days. If only A and B work together, in how many days will they complete 50 percent of the job?

Accepted Answer

Introduction / Context: This time and work problem links the combined efficiency of three people with the individual efficiency of one of them, and then asks about the time required when only the other two work together on half of the job. It tests understanding of how to separate one worker's contribution from a known combined rate, and how to use the remaining rate to compute partial completion time. Given Data / Assumptions: A, B, and C together finish the job in 9 days. C alone finishes the job in 36 days. We are asked for the time A and B together would take to finish 50 percent of the job. Work is taken as one unit and all rates are constant. Concept / Approach: We begin by converting completion times into daily work rates. The combined rate of A, B, and C is 1/9 of the job per day, and C's rate alone is 1/36 of the job per day. Subtracting C's rate from the combined rate gives the joint rate of A and B. Once we know the rate at which A and B work together, we can compute how long they will take to complete half of the job by dividing the required work by their daily rate. Step-by-Step Solution: Let total work = 1 unit.Combined rate of A, B, and C = 1/9 per day (because they finish in 9 days).Rate of C alone = 1/36 per day (because C finishes in 36 days alone).Rate of A and B together = 1/9 - 1/36.Compute 1/9 - 1/36: rewrite with common denominator 36, 1/9 = 4/36, so 4/36 - 1/36 = 3/36 = 1/12.Thus A and B together work at 1/12 of the job per day.We need the time for A and B to do 50 percent of the job, that is 1/2 of the work.Time = required work / rate = (1/2) / (1/12) = 1/2 * 12 = 6 days. Verification / Alternative Check: If A and B work together for 6 days at a rate of 1/12 per day, they complete 6 * 1/12 = 6/12 = 1/2 of the job, which is exactly the required fraction. Also, if we consider A, B, and C together, with C's rate of 1/36, the sum 1/12 + 1/36 is 4/36 = 1/9, matching the given combined completion time of 9 days. This confirms that the derived rates and time for A and B are consistent. Why Other Options Are Wrong: Options like 9 or 12 days would mean A and B together work for longer, producing 9 * 1/12 = 3/4 or 12 * 1/12 = 1 unit of work, which would exceed the 50 percent requirement. Option 15 days is even further off. Only 6 days corresponds exactly to half of the job being completed by A and B together at their calculated rate. Common Pitfalls: One common error is to compute the time for A, B, and C together and then try to scale it directly without isolating C's contribution. Another mistake is miscalculating the difference between 1/9 and 1/36, which leads to an incorrect rate for A and B. Correctly performing the rate subtraction and remembering that the question asks for half the job are essential to avoid these errors. Final Answer: Working together, A and B will complete 50 percent of the job in 6 days.

Question 20

Working alone, A can complete a task in 27 hours and B can complete the same task in 54 hours. A, B, and C together finish the task in 12 hours and receive Rs 4,320 as total wages. What is the share of C in these wages?

Accepted Answer

Introduction / Context: This question combines time and work with wage distribution based on individual contributions. It assesses the ability to find individual work rates, use combined rate information to determine the rate of a third worker, and then divide total payment in proportion to work done. Such problems appear frequently in aptitude tests where profit sharing and wage sharing are tied to work rates. Given Data / Assumptions: A alone can complete the task in 27 hours. B alone can complete the task in 54 hours. A, B, and C together complete the task in 12 hours. Total wages paid for the completed task are Rs 4,320. Wages are shared in proportion to the amount of work each person actually does. Concept / Approach: The key steps are to convert the given completion times into hourly work rates, then use the combined completion time with three workers to calculate C's rate. Once we know each worker's rate, we compute how much of the total task each worker completes during the 12 hours. The fraction of the total work done by C is then used to find C's share of the total wages, since payment is proportional to contribution. Step-by-Step Solution: Let total work = 1 unit.Rate of A per hour = 1/27.Rate of B per hour = 1/54.A, B, and C together finish in 12 hours, so combined rate = 1/12 per hour.Let rate of C per hour be r_c. Then 1/27 + 1/54 + r_c = 1/12.Compute 1/27 + 1/54: common denominator is 54, so 1/27 = 2/54, and 2/54 + 1/54 = 3/54 = 1/18.So 1/18 + r_c = 1/12, hence r_c = 1/12 - 1/18.Compute r_c: 1/12 - 1/18 = (3 - 2) / 36 = 1/36.In 12 hours, total work done by A = 12 * 1/27 = 12/27 = 4/9 of the job.Work done by B = 12 * 1/54 = 12/54 = 2/9 of the job.Work done by C = 12 * 1/36 = 12/36 = 1/3 of the job.Expressed with a common denominator 9, these are 4/9, 2/9, and 3/9 respectively, so the work ratio is A : B : C = 4 : 2 : 3.Total parts = 4 + 2 + 3 = 9 parts, so each part corresponds to 4320 / 9 = Rs 480.C gets 3 parts, so C's share = 3 * 480 = Rs 1,440. Verification / Alternative Check: The shares of A and B would then be 4 * 480 = Rs 1,920 and 2 * 480 = Rs 960. Summing all three gives 1,920 + 960 + 1,440 = Rs 4,320, which matches the total wages. Also, using C's rate of 1/36 with the others gives a combined rate of 1/18 + 1/36 = 3/36 = 1/12, so they finish in 12 hours, consistent with the problem statement. Why Other Options Are Wrong: Options like Rs 960 or Rs 1,280 assume a smaller share for C than is justified by the work ratio of 4 : 2 : 3. Option Rs 1,920 would overpay C and underpay A. Only Rs 1,440 respects the proportional distribution based on the calculated fractions of work completed by each worker. Common Pitfalls: Students may forget to multiply the rates by 12 hours to find the exact amount of work done by each worker. Others may incorrectly assume equal sharing or use only time ratios without converting them to actual work contributions. Keeping the focus on work rates and actual fractions of the total job prevents such errors. Final Answer: C's correct share of the wages is Rs 1,440.

Time and Work Questions