Question 1

Prabodh completes one-half of a job in 30 days when working alone, and Sapan completes the remaining half of the same job in 45 days. If Prabodh and Sapan now work together from the start at their usual constant rates, in how many days will they com…

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Introduction / Context: This question concerns two workers, Prabodh and Sapan, whose partial completion times for different halves of a job are given. The problem asks how long they will take to complete the entire job if they work together from the beginning. It tests your ability to convert information about partial work into full-work rates and then combine those rates correctly. Given Data / Assumptions: Prabodh completes 1/2 of the job in 30 days when working alone. Sapan completes the remaining 1/2 of the job in 45 days when working alone. Both are assumed to work at constant individual rates. We want the time for Prabodh and Sapan working together from the start to finish the whole job. Concept / Approach: From the given partial times, we first compute the daily work rates of Prabodh and Sapan for the full job. Once we know their individual rates as fractions of the whole job per day, we add those rates to get the combined daily rate. Finally, we find the total time by dividing 1 (the whole job) by the combined rate. Step-by-Step Solution: Let the total work be 1 unit. Prabodh completes 1/2 of the job in 30 days, so his daily rate p satisfies 30p = 1/2. Therefore, p = (1/2) / 30 = 1/60 of the job per day. Sapan completes the other 1/2 of the job in 45 days, so his daily rate s satisfies 45s = 1/2. Therefore, s = (1/2) / 45 = 1/90 of the job per day. Now, if they work together from the start, combined rate = p + s = 1/60 + 1/90. Find LCM of 60 and 90, which is 180. 1/60 = 3/180 and 1/90 = 2/180. Combined rate = 3/180 + 2/180 = 5/180 = 1/36 of the job per day. Time taken together = total work / combined rate = 1 / (1/36) = 36 days. Verification / Alternative check: If together they do 1/36 of the work per day, in 36 days they will complete exactly 1 full job, which matches the requirement. Also, note that Prabodh alone would take 60 days to finish the full job (since his rate is 1/60), and Sapan alone would take 90 days. It is reasonable that together they require less than 60 days, and 36 days fits within this logical range. Why Other Options Are Wrong: Option 18 days is too small; it would require a combined rate of 1/18, which is higher than 1/36 and contradicts the individual rates. Option 48 days is too large; it implies a very slow combined rate and conflicts with the fact that even Prabodh alone could do the full job in 60 days. Option 27 days suggests a higher combined rate than the actual 1/36 and does not match the computed sum of 1/60 and 1/90. Option 30 days is also inconsistent because in 30 days they would complete 30 * (1/36) = 5/6 of the job, not the full job. Common Pitfalls: Students often misinterpret “half of the job in 30 days” as “full job in 30 days” and forget to scale the rate. Another pitfall is adding the times directly or averaging them instead of adding work rates. Always convert partial work times into full-job daily rates and then combine those rates carefully using a common denominator. Final Answer: Thus, working together from the beginning, Prabodh and Sapan can complete the entire job in 36 days, so the correct option is 36 days.

Question 2

A can do a piece of work in 8 days and B can do the same work in 10 days when each works alone. If A and B work together at their usual constant rates from start to finish, in how many days will they complete the work?

Accepted Answer

Introduction / Context: Here, you are given the number of days two workers A and B take individually to complete a job. The question asks how long they will take to finish the job when they work together. This is a standard time and work problem that uses basic rate addition. Given Data / Assumptions: A alone can complete the work in 8 days. B alone can complete the same work in 10 days. They work together at constant rates from the start to completion. The total work is considered as one complete unit of job. Concept / Approach: If a worker can finish a job in T days, the worker's daily rate is 1/T of the job per day. When multiple workers cooperate, their individual rates add up. We thus find the combined rate of A and B and then take the reciprocal of that combined rate to find the number of days required to finish the job. Step-by-Step Solution: Let total work = 1 unit. Daily rate of A = 1/8 of the job per day. Daily rate of B = 1/10 of the job per day. Combined daily rate = 1/8 + 1/10. Take the LCM of 8 and 10, which is 40. Convert fractions: 1/8 = 5/40 and 1/10 = 4/40. Combined rate = 5/40 + 4/40 = 9/40 of the job per day. Time taken together = total work / combined rate = 1 / (9/40) = 40/9 days. Verification / Alternative check: If they complete 9/40 of the job per day, then over 40/9 days they would do (9/40)*(40/9) = 1 job, which is correct. Also, note that A alone needs 8 days and B alone needs 10 days, so the combined time must be less than 8 days but reasonably close. 40/9 is approximately 4.44 days, which fits within this expectation and confirms the plausibility of the result. Why Other Options Are Wrong: Option 33/8 days is approximately 4.125 days, which implies a slightly higher combined rate than 9/40 and does not match the correct sum of rates. Option 41/10 days is 4.1 days, which similarly conflicts with the combined rate computed by proper fraction addition. Option 42/11 days is around 3.82 days and is too small given the individual times of 8 and 10 days. Option 4 days would require a combined rate of 1/4 per day, which is larger than 9/40 and not equal to 1/8 + 1/10. Common Pitfalls: A common mistake is to average the times 8 and 10 to get 9 days, which is incorrect. Another pitfall is miscalculating the LCM or addition of fractions, leading to a wrong combined rate. Remember to always work with rates 1/T, add them carefully, and then invert the sum to find the total time. Final Answer: Thus, A and B working together can complete the work in 40/9 days, so the correct option is 40/9 days.

Question 3

Twenty-four men can complete a piece of work in 17 days when working together. If the same work has to be completed in 51 days instead, how many men will be required, assuming each man works at the same constant rate?

Accepted Answer

Introduction / Context: This question illustrates the direct inverse relationship between the number of workers and the time taken to complete a job when the total amount of work stays constant. You are given how many men can finish the work in a certain number of days, and you must find how many men are needed to finish the same work in a different number of days at the same rate of working. Given Data / Assumptions: Twenty-four men can complete the work in 17 days. We want to complete the same work in 51 days instead. All men work at identical constant rates, and total work remains unchanged. No man joins or leaves once the new group size is decided. Concept / Approach: We use the concept of man-days, where Work = Number of men * Number of days * Rate per man. Since the rate per man is constant and the work is fixed, the product of men and days must remain the same. So, men1 * days1 = men2 * days2. We use this proportional relationship to solve for the unknown number of men required to finish the work in the new time frame. Step-by-Step Solution: Let total work be W man-days. From the initial situation: 24 men * 17 days = W. So W = 24 * 17 = 408 man-days. Now, let the required number of men for finishing the work in 51 days be x. Then x men * 51 days = W = 408 man-days. So 51x = 408. Solve for x: x = 408 / 51. 408 divided by 51 = 8. Therefore, x = 8 men. Verification / Alternative check: Check the man-days: 8 men working for 51 days will contribute 8 * 51 = 408 man-days, which matches the original total of 408 man-days contributed by 24 men in 17 days. This confirms that the work done is the same and that using 8 men for 51 days will indeed complete the identical job. Why Other Options Are Wrong: Option 10 men would produce 10 * 51 = 510 man-days, which is more work than needed and would finish the job earlier than 51 days. Option 12 men would give 12 * 51 = 612 man-days, which is even more, clearly not matching the original workload. Option 6 men would provide 6 * 51 = 306 man-days, which is insufficient to complete the job that requires 408 man-days. Option 16 men would provide 16 * 51 = 816 man-days, more than double the needed work, so the time would have to be shorter than 51 days to keep the workload constant. Common Pitfalls: Students sometimes confuse direct and inverse proportionality and may incorrectly multiply or divide the wrong way around. Another mistake is to ignore the total man-days concept and attempt ad-hoc reasoning. Remember that when the amount of work is fixed and worker efficiency is constant, the number of workers and the time taken are inversely proportional. Multiplying number of men and days to check equality is a reliable method. Final Answer: Hence, to complete the same work in 51 days, 8 men are required, so the correct option is 8 men.

Question 4

A painter can paint an entire fence in 24 hours when working alone at a constant rate. After working for 6 hours, he takes a break and stops. What fraction of the fence is still left to be painted at that moment?

Accepted Answer

Introduction / Context: This is a simple fraction-of-work problem involving a painter who can finish painting a fence in a given time. We are told how long he works before stopping and asked to find what fraction of the job remains unfinished. Such questions are common in aptitude tests and help reinforce the idea that the fraction of work done is directly proportional to the time spent when the rate is constant. Given Data / Assumptions: The painter can finish the entire fence in 24 hours when working alone. He actually paints for only 6 hours before taking a break and stopping. The painting rate is constant during the time he works. The whole fence is considered as one complete job (1 unit). Concept / Approach: If a worker finishes a job in T hours, the fraction of the job completed per hour is 1/T. When the worker paints for t hours, the fraction of the job done is t/T. Subtracting this completed fraction from 1 gives the remaining fraction of the job. This approach is straightforward and only involves basic arithmetic and fractions. Step-by-Step Solution: Let the total work (whole fence) = 1 unit. The painter can complete 1 unit of work in 24 hours. So, his hourly work rate = 1/24 of the fence per hour. He actually works for 6 hours before stopping. Work completed in 6 hours = 6 * (1/24) = 6/24. Simplify 6/24 to 1/4 of the fence. Hence, fraction of the fence painted = 1/4. Remaining fraction of the fence = 1 - 1/4 = 3/4. Convert 3/4 to decimal form if needed: 3/4 = 0.75. Verification / Alternative check: As a quick check, if the painter requires 24 hours to complete the whole fence, then 6 hours is exactly one-fourth of the total required time. Therefore, it is logical that only one-fourth of the work is completed and three-fourths of the work is left. 3/4 written as a decimal is 0.75, which matches the result we computed. Why Other Options Are Wrong: Option 0.6 would mean only 60% of the fence remains, which is inconsistent with the one-fourth completed fraction of work. Option 0.2 would indicate only 20% left, suggesting that 80% is already painted, which is clearly impossible after only one-fourth of the required time has been spent. Option 0.8 means 80% of the fence is left, implying that only 20% has been painted, which does not match the 6/24 = 1/4 fraction actually done. Option 0.5 would assume that half of the fence is left, which would correspond to 12 hours of work and not the given 6 hours. Common Pitfalls: A frequent error is to confuse the fraction of time spent with the fraction of work left instead of the fraction completed. Another mistake is performing incorrect fraction simplification, such as misreading 6/24 as 1/6 instead of 1/4. Always remember that with constant rate, time and completed work fraction are proportional, and double check arithmetic when simplifying fractions. Final Answer: Therefore, the fraction of the fence still left to be painted after 6 hours is 0.75, so the correct option is 0.75.

Question 5

Madhur works two times faster than Sagar. If Sagar can complete a certain job alone in 18 days at his usual rate, then in how many days can Madhur and Sagar together finish the entire job?

Accepted Answer

Introduction / Context: This time and work question describes two workers, Madhur and Sagar, with a known relationship between their efficiencies. Sagar's individual time to complete the job is known, and Madhur is said to be twice as fast as Sagar. The problem asks for the total time when both work together from the start. Given Data / Assumptions: Sagar alone can complete the job in 18 days. Madhur works two times faster than Sagar, that is, Madhur's rate is twice Sagar's rate. Both work together at constant rates from the beginning until the job is finished. The total work is taken as one complete unit of job. Concept / Approach: First, compute Sagar's daily work rate from his given completion time. Then use the information that Madhur is twice as fast to get Madhur's rate. The combined daily rate is the sum of their individual rates. Finally, find the time taken by taking the reciprocal of this combined rate, since Time = Work / Rate and we consider Work = 1 unit. Step-by-Step Solution: Let the total work be 1 unit. Given that Sagar alone completes the job in 18 days. So, Sagar's daily rate = 1/18 of the job per day. Madhur works two times faster, so Madhur's rate = 2 * (1/18) = 1/9 of the job per day. Combined daily rate of Madhur and Sagar = 1/18 + 1/9. Convert 1/9 to 2/18 for easy addition. Thus, combined rate = 1/18 + 2/18 = 3/18 = 1/6 of the job per day. Time taken together = total work / combined rate = 1 / (1/6) = 6 days. Verification / Alternative check: To verify, in 6 days at a combined rate of 1/6 per day, they complete exactly 1 job. Also, note that Sagar alone requires 18 days, while Madhur alone would require 9 days. Working together, it is reasonable that they take less time than 9 days and significantly less than 18 days. The answer of 6 days fits these expectations and confirms the correctness of the calculation. Why Other Options Are Wrong: Option 5 days would correspond to a combined rate of 1/5 per day, which is higher than the computed 1/6 and does not match the given efficiency relationship. Option 2 days is far too small and would imply an unrealistically high combined rate that contradicts the individual times of 9 and 18 days. Option 4 days also leads to a combined rate of 1/4 per day, which is not equal to the sum of 1/18 and 1/9. Option 3 days would assume an even faster rate of 1/3 per day, clearly impossible based on the problem data. Common Pitfalls: A common mistake is misinterpreting “two times faster” as “twice as many days,” which is the opposite of what it means. Faster speed corresponds to fewer days and higher daily work rate. Another pitfall is adding or averaging days rather than rates. Always convert times into rates, apply the speed relationship correctly and then add the rates to find the combined efficiency. Final Answer: Thus, Madhur and Sagar working together can complete the job in 6 days, so the correct option is 6 days.

Question 6

Pradeep completes one-fourth of a job in 14 days when working alone, and Saquib completes the remaining three-fourths of the job in 56 days when working alone. If Pradeep and Saquib now work together from the start at their usual constant rates, in …

Accepted Answer

Introduction / Context: This time and work problem gives you information about how long Pradeep and Saquib each take to complete different fractions of the same job when working alone. You are then asked to find how long it will take them to complete the job together if they start at the same time. This tests your understanding of converting partial-job times into full-job rates and then combining those rates. Given Data / Assumptions: Pradeep completes 1/4 of the job in 14 days working alone. Saquib completes the remaining 3/4 of the job in 56 days working alone. Both work at constant individual rates throughout their work. We want the time taken if Pradeep and Saquib work together from the beginning on the whole job. Concept / Approach: We first compute Pradeep's daily work rate and Saquib's daily work rate relative to the full job. Then we add these rates to obtain their combined daily rate. Finally, we divide the total work (1 unit) by the combined rate to determine how many days they need to complete the job together. Step-by-Step Solution: Let the total work be 1 unit. Pradeep completes 1/4 of the job in 14 days, so his rate p satisfies 14p = 1/4. Thus, p = (1/4) / 14 = 1/56 of the job per day. Saquib completes 3/4 of the job in 56 days, so his rate s satisfies 56s = 3/4. Thus, s = (3/4) / 56 = 3/224 of the job per day. Combined daily rate of Pradeep and Saquib = p + s = 1/56 + 3/224. Convert 1/56 to denominator 224: 1/56 = 4/224. So combined rate = 4/224 + 3/224 = 7/224. Simplify 7/224 to 1/32. Therefore, they together complete 1/32 of the job per day. Time taken together = total work / combined rate = 1 / (1/32) = 32 days. Verification / Alternative check: If they finish 1/32 of the job per day, then in 32 days they complete exactly the whole job. Also, you can check that Pradeep alone would finish the job in 56 days, since his rate is 1/56, and Saquib alone would finish in 224/3 days (around 74.67 days). Since the combined time 32 days is less than both individual times, it is logical and confirms the correctness of the result. Why Other Options Are Wrong: Option 64 days would correspond to a combined rate of 1/64, which is smaller than both individual rates, clearly impossible. Option 16 days would require a combined rate of 1/16, which is higher than the sum 1/56 + 3/224, and does not match the calculations. Option 8 days would be even faster and completely inconsistent with the given data, as neither worker is that fast alone or even combined. Option 40 days does not match the exact combined rate of 1/32 and would correspond to only 40 * (1/32) = 1.25 of the job if 1/32 were the rate, which is impossible. Common Pitfalls: One common mistake is to treat the given partial times as full-job times without converting, which leads to incorrect rates. Another pitfall is manging fractions incorrectly, especially when changing denominators or simplifying. Always convert partial job information into daily rates in terms of the full job, then add the rates and finally invert to obtain the total time together. Final Answer: Thus, working together from the start, Pradeep and Saquib can complete the entire job in 32 days, so the correct option is 32 days.

Question 7

A can do a work in 12 days while B can do the same work in 15 days when each works alone. They agree to undertake the work together for a total payment of ₹450. Based on their individual contributions, what will be A's share of this amount in rupees?

Accepted Answer

Introduction / Context: This question is a combination of time and work with profit or wage sharing. A and B perform a job together and receive a total payment. Since they work at different speeds, the payment must be divided in proportion to the amount of work each actually does. You are asked to calculate A's share out of the total amount of ₹450. Given Data / Assumptions: A alone can finish the work in 12 days. B alone can finish the same work in 15 days. They work together for the entire duration until the job is completed. They receive a total payment of ₹450 to be divided in proportion to their work contributions. Work rates remain constant throughout the job. Concept / Approach: First, we compute the work rates of A and B as fractions of the job per day. Then we find the total work rate when they work together and determine each person's fraction of the total work completed. Since payment is proportional to work done, we multiply the total payment by A's fraction of the work to find A's share. Step-by-Step Solution: Let the total work be 1 unit. A's daily rate = 1/12 of the job per day. B's daily rate = 1/15 of the job per day. Combined daily rate = 1/12 + 1/15. LCM of 12 and 15 is 60. Convert fractions: 1/12 = 5/60 and 1/15 = 4/60. Combined rate = 5/60 + 4/60 = 9/60 = 3/20 of the job per day. Time taken together to finish the job = 1 / (3/20) = 20/3 days. Work done by A in this time = A's rate * time = (1/12) * (20/3) = 20/36 = 5/9 of the job. Work done by B in this time = B's rate * time = (1/15) * (20/3) = 20/45 = 4/9 of the job. Thus, the work ratio of A : B = 5/9 : 4/9 = 5 : 4. Total payment = ₹450, so A's share = (5 / (5 + 4)) * 450 = (5/9) * 450 = ₹250. Verification / Alternative check: Since the ratio of their work contributions is 5 : 4, the ratio of their payments must also be 5 : 4. Adding up parts: 5 + 4 = 9 parts. One part is 450 / 9 = 50 rupees. Therefore, A gets 5 parts, which is 5 * 50 = ₹250, and B gets 4 parts, 4 * 50 = ₹200. Both shares add up to ₹450, confirming that the division is correct. Why Other Options Are Wrong: Option ₹200 corresponds to B's share, not A's, and is therefore incorrect when the question specifically asks for A's share. Option ₹240 does not correspond to a simple 5 : 4 division of ₹450 and would break the proportionality with the actual work done. Option ₹300 would give B only ₹150, which does not fit the 5 : 4 ratio derived from the work contributions. Option ₹225 would imply a split that is neither 5 : 4 nor any correct simplification of the work ratio. Common Pitfalls: Students sometimes divide the payment equally without considering differences in work efficiency. Another error is to use time ratios directly instead of converting times to work rates and finding the actual fraction of work each person does. Remember that payment shares must follow the ratio of work done, which is derived from rate multiplied by total time worked. Final Answer: Hence, A's share of the payment is ₹250, so the correct option is ₹250.

Question 8

A does 60% of a certain work in 15 days at his usual constant rate. He then calls B, and together A and B finish the remaining 40% of the work in 5 days. If B works alone at the same rate, in how many days would B complete the entire work by himself?

Accepted Answer

Introduction / Context: This question involves two workers, A and B, who complete a job in stages. First, A completes 60% of the work alone, then A and B together complete the remaining 40%. You must determine how long B alone would take to complete the entire job. This kind of problem combines percentage of work with the concept of daily work rates and teamwork. Given Data / Assumptions: A completes 60% (that is, 0.6 fraction) of the work in 15 days when working alone. A and B together then complete the remaining 40% (0.4 fraction) in 5 days. Both A and B work at constant individual rates. We need the time in days B alone would take to complete 100% of the work at his own rate. Concept / Approach: We treat the total work as 1 unit. From the first stage, we calculate A's daily rate using the fact that 0.6 of the job is done in 15 days. From the second stage, we deduce the combined daily rate of A and B when they work together to finish the remaining 0.4 in 5 days. Subtracting A's rate from this combined rate gives B's individual daily rate. Finally, we compute how many days B alone would need to complete the full unit of work at that rate. Step-by-Step Solution: Let total work = 1 unit. A completes 60% of the work in 15 days, so A's rate a satisfies 15a = 0.6. Thus, a = 0.6 / 15 = 0.04 = 1/25 of the work per day. The remaining work is 40% or 0.4 of the job. A and B together finish this 0.4 in 5 days, so their combined rate a + b satisfies 5(a + b) = 0.4. Hence, a + b = 0.4 / 5 = 0.08 = 2/25 of the work per day. We know a = 1/25, so b = (a + b) - a = (2/25) - (1/25) = 1/25 of the work per day. Therefore, B's daily rate is also 1/25 of the job per day. Time taken by B alone to complete the whole job = 1 / (1/25) = 25 days. Verification / Alternative check: We can check consistency by seeing that A and B have equal rates. If each works at 1/25 per day, then together they work at 2/25 per day. In 5 days they complete (2/25)*5 = 10/25 = 0.4 or 40% of the job, which matches the remaining work. Also, A alone at 1/25 per day over 15 days completes 15 * (1/25) = 15/25 = 0.6 or 60% of the job, matching the first stage. This confirms both the intermediate and final results. Why Other Options Are Wrong: Option 20 days corresponds to a daily rate of 1/20, which would change the joint rate and contradict the calculated combined rate of 2/25. Option 80 days implies a very slow rate of 1/80 per day, inconsistent with the speed required to finish 40% of the job in 5 days along with A. Option 24 days would correspond to a rate of 1/24 per day and does not match the derived value of 1/25. Option 30 days also gives a different daily rate of 1/30 per day, which fails when substituted back into the second stage of the work. Common Pitfalls: Students sometimes forget to convert percentages into fractions correctly or mishandle the transition between decimal and fractional forms. Another common error is to assume B's rate directly from the 40% in 5 days without accounting for A's simultaneous contribution. Always carefully compute individual and combined rates step by step, subtracting A's rate from the combined rate to isolate B's rate. Final Answer: Therefore, B alone would take 25 days to complete the entire work, so the correct option is 25 days.

Question 9

A completes 80% of a piece of work in 20 days. He then calls in B, and together they finish the remaining 20% of the work in 4 days. How many days would B alone take to complete the entire work?

Accepted Answer

Introduction / Context: This time and work aptitude question compares the individual efficiencies of two people, A and B. A completes a major portion of the work alone and then B joins for the remaining part. From this mixed working pattern, we need to back–calculate the rate of B and then find how long B alone would take to finish the complete work. Given Data / Assumptions: - Total work is assumed to be 1 unit (100 percent). - A alone completes 80 percent of the work in 20 days. - A and B together complete the remaining 20 percent of the work in 4 days. - Work rates are constant and additive when they work together. Concept / Approach: We use the standard time and work approach where work rate = work done / time. First we find the daily work rate of A from the information that A does 80 percent in 20 days. Next we find the combined rate of A and B from the remaining 20 percent done in 4 days. Subtracting A’s rate from the combined rate gives B’s rate. Finally, we compute the time B alone would need to complete the entire unit work. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: A completes 80 percent = 0.8 of the work in 20 days. Step 3: Rate of A = 0.8 / 20 = 0.04 units per day. Step 4: Remaining work = 1 - 0.8 = 0.2 units. Step 5: A and B together finish 0.2 units in 4 days. Step 6: Combined rate of A and B = 0.2 / 4 = 0.05 units per day. Step 7: Rate of B = combined rate - rate of A = 0.05 - 0.04 = 0.01 units per day. Step 8: Time taken by B alone to finish 1 unit of work = 1 / 0.01 = 100 days. Verification / Alternative check: If B alone takes 100 days, then in 4 days B completes 4 * 0.01 = 0.04 units. During the same 4 days A completes 4 * 0.04 = 0.16 units. Together they complete 0.04 + 0.16 = 0.20 units, which matches the remaining 20 percent of the work. So all calculations are consistent and correct. Why Other Options Are Wrong: - 12.5 days: This would imply B is much faster than A, which contradicts the data where A did most of the work initially. - 22.5 days: This does not satisfy the condition that the remaining 20 percent is finished in exactly 4 days with A and B together. - 35 days: This also leads to an incorrect combined rate and does not match the given times. - 90 days: Closer than the others but still gives a slightly higher rate for B and produces a mismatch when recomputing the remaining 20 percent in 4 days. Common Pitfalls: Many learners forget to treat the total work as 1 unit and directly manipulate percentages without converting them to fractions. Another frequent mistake is to divide 20 percent of the work by 4 days and assume that is B’s rate, ignoring the contribution of A during those 4 days. It is crucial to subtract A’s rate from the combined rate to isolate B’s true efficiency. Also, mixing up time and rate can lead to inverted fractions and wrong answers. Final Answer: Therefore, B alone would take 100 days to complete the entire work by himself.

Question 10

A can complete a piece of work in 20 days, and B can complete the same work in 10 days. If they work together on the job for 5 days, what fraction of the total work is still left unfinished?

Accepted Answer

Introduction / Context: This is a standard time and work problem where two workers, A and B, have different individual times to finish a job. They work together for some days, and we are asked to find what fraction of the work remains. Such questions test understanding of work rate addition and the relationship between time, work, and rate. Given Data / Assumptions: - A alone finishes the work in 20 days. - B alone finishes the work in 10 days. - They work together for 5 days. - Total work is taken as 1 unit for ease of calculation. Concept / Approach: The basic idea is that work rate = 1 / time when total work is 1 unit. We find the individual daily rates of A and B, add them to get the combined rate, compute how much of the work is done in 5 days, and then subtract from 1 to find the remaining fraction of work. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Rate of A = 1 / 20 work per day. Step 3: Rate of B = 1 / 10 work per day. Step 4: Combined rate of A and B = 1 / 20 + 1 / 10. Step 5: Simplify combined rate: 1 / 20 + 1 / 10 = 1 / 20 + 2 / 20 = 3 / 20 work per day. Step 6: Work done in 5 days = 5 * (3 / 20) = 15 / 20 = 3 / 4 of the total work. Step 7: Remaining work = 1 - 3 / 4 = 1 / 4 of the total work. Verification / Alternative check: We can think of the job in terms of 20 equal parts. A does 1 part per day, B does 2 parts per day, so together they complete 3 parts per day. In 5 days, they finish 15 parts out of 20, leaving 5 parts. The remaining fraction is 5 / 20, which simplifies again to 1 / 4, confirming the previous calculation. Why Other Options Are Wrong: - 4/9 and 2/9: These do not correspond to the correct combined rate and time product for 5 days of work. - 1/5: This would mean 80 percent of the work is done, which contradicts the computed 75 percent. - 1/3: This implies 2/3 of the work is done, which does not match the combined rate of 3 / 20 per day over 5 days. Common Pitfalls: Common errors include taking the average of 20 and 10 instead of working with rates, or forgetting to multiply the combined rate by the number of days. Some students also incorrectly compute the remaining work by dividing instead of subtracting from 1. Always compute work done first, then subtract from total work to get the remaining fraction. Final Answer: The fraction of the work that is still left after 5 days is 1/4 of the total work.

Question 11

M is thrice as efficient a workman as N, and together M and N finish a piece of work in 30 days. In how many days will M alone be able to complete the entire work?

Accepted Answer

Introduction / Context: This question deals with comparative efficiency in time and work. The statement that M is thrice as good a workman as N means that M works three times faster than N. Using their combined time to complete the job, we can infer their individual work rates and then determine how long M alone would take to finish the full work. Given Data / Assumptions: - M and N together complete the work in 30 days. - M is three times as efficient as N. - Total work is taken as 1 unit. - Work rates are constant and add linearly when they work together. Concept / Approach: Let the daily work rate of N be x units. Then the rate of M is 3x units per day because M is thrice as efficient. Together, their daily rate is 4x. Since they finish the work in 30 days, their combined rate is also equal to 1 / 30 units per day. Equating these two expressions for the combined rate allows us to solve for x and then for M’s individual time to finish the job alone. Step-by-Step Solution: Step 1: Let the rate of N = x units per day. Step 2: Then the rate of M = 3x units per day. Step 3: Combined rate of M and N = 3x + x = 4x units per day. Step 4: They finish the whole work in 30 days, so 4x = 1 / 30. Step 5: Solve for x: x = 1 / (30 * 4) = 1 / 120. Step 6: Rate of M = 3x = 3 * (1 / 120) = 1 / 40 units per day. Step 7: Time taken by M alone for the whole work = 1 / (1 / 40) = 40 days. Verification / Alternative check: If M alone takes 40 days, then in one day M does 1 / 40 of the work. If N’s rate is 1 / 120, then together they do 1 / 40 + 1 / 120 = 3 / 120 + 1 / 120 = 4 / 120 = 1 / 30 of the work per day, which matches the given combined time of 30 days. This confirms that the calculation is consistent. Why Other Options Are Wrong: - 50 and 60 days: These values imply slower rates for M and would lead to a combined rate smaller than 1 / 30, contradicting the given data. - 45 days: This would give M a rate of 1 / 45, making the combined rate less than 1 / 30 when added to N’s inferred rate. - 30 days: This would imply M and N have equal efficiency, which contradicts the condition that M is thrice as efficient as N. Common Pitfalls: A common mistake is to treat the ratio of efficiency as the ratio of time directly. Since efficiency and time are inversely related, if M is three times as efficient as N, M will take one third of the time N takes, not three times. Another error is to forget to equate the algebraic expression for the combined rate (4x) with the numerical value (1 / 30) computed from their joint time. Always keep the relationship between rate, time, and work clear. Final Answer: M alone will finish the entire work in 40 days.

Question 12

A can finish a certain piece of work in 7 days, while B can finish the same work in 9 days. If they work together from the beginning, in how many days will they complete the entire work?

Accepted Answer

Introduction / Context: This problem asks for the time taken when two people, A and B, work together on the same task. Each person has a known individual time to finish the work alone. Time and work questions like this rely on converting these times into daily work rates and then combining the rates to find the total time when working jointly. Given Data / Assumptions: - A alone can finish the work in 7 days. - B alone can finish the work in 9 days. - Both A and B start together and work continuously until the job is done. - Total work is taken as 1 unit. Concept / Approach: When total work is 1 unit, the work rate of a person is the reciprocal of the time taken to complete the work alone. The combined work rate of two people is the sum of their individual rates. Once the combined rate is known, the total time taken to complete one unit of work is simply the reciprocal of this combined rate. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Rate of A = 1 / 7 work per day. Step 3: Rate of B = 1 / 9 work per day. Step 4: Combined rate of A and B = 1 / 7 + 1 / 9. Step 5: Take LCM of 7 and 9 which is 63, so 1 / 7 = 9 / 63 and 1 / 9 = 7 / 63. Step 6: Combined rate = 9 / 63 + 7 / 63 = 16 / 63 work per day. Step 7: Time taken together = 1 / (16 / 63) = 63 / 16 days. Verification / Alternative check: As a quick check, note that A alone takes 7 days and B alone takes 9 days, so the combined time must be less than 7 days. The value 63 / 16 is approximately 3.94 days, which is less than both 7 and 9 days, so it is reasonable. Also, if they work for 63 / 16 days at a rate of 16 / 63 per day, total work equals 63 / 16 * 16 / 63 = 1 unit, confirming correctness. Why Other Options Are Wrong: - 31/16 and 47/16: These correspond to significantly shorter times and would imply an unrealistically high combined rate, greater than the sum of the individual rates. - 79/16: This is about 4.94 days, which is greater than A’s time of 7 days but still incorrect when recomputed with the combined rate. - 4 days: This is close but not exact. Only 63 / 16 satisfies the precise arithmetic. Common Pitfalls: Many learners mistakenly average the times (7 and 9) instead of working with rates. Another frequent error is neglecting to use the LCM for adding fractions and making mistakes in fraction addition. Always convert times to rates, sum the rates, and then take the reciprocal to find the combined time. Final Answer: Working together from the start, A and B will complete the work in 63/16 days.

Question 13

A can do a piece of work in 20 days and B can do the same work in 15 days. With the help of a third person C, they finish the entire work in 5 days. In how many days can C alone complete the whole work?

Accepted Answer

Introduction / Context: This time and work question involves three workers, A, B, and C. We know the individual times of A and B, and the combined time when all three work together. From this information we must deduce the efficiency of C and then convert that efficiency into the number of days C alone would need to complete the job. Given Data / Assumptions: - A alone completes the work in 20 days. - B alone completes the work in 15 days. - A, B, and C together complete the work in 5 days. - Total work is assumed to be 1 unit. - Work rates add linearly when workers cooperate. Concept / Approach: We first calculate the individual daily work rates of A and B. Next we find the combined daily rate of all three from the fact that they finish the work in 5 days. The daily rate of C is then obtained by subtracting the known rates of A and B from the combined rate. Finally, we compute the reciprocal of C’s rate to find C’s individual time to complete the entire job alone. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Rate of A = 1 / 20 work per day. Step 3: Rate of B = 1 / 15 work per day. Step 4: A, B, and C together finish in 5 days, so combined rate of A + B + C = 1 / 5 work per day. Step 5: Compute rate of A + B: 1 / 20 + 1 / 15. Step 6: Take LCM of 20 and 15 which is 60, so 1 / 20 = 3 / 60 and 1 / 15 = 4 / 60. Step 7: Rate of A + B = 3 / 60 + 4 / 60 = 7 / 60 work per day. Step 8: Rate of C = combined rate - rate of A + B = 1 / 5 - 7 / 60. Step 9: Convert 1 / 5 to denominator 60: 1 / 5 = 12 / 60. Step 10: Rate of C = 12 / 60 - 7 / 60 = 5 / 60 = 1 / 12 work per day. Step 11: Time taken by C alone = 1 / (1 / 12) = 12 days. Verification / Alternative check: Check by recombining rates: A does 1 / 20, B does 1 / 15, C does 1 / 12 per day. Adding gives 1 / 20 + 1 / 15 + 1 / 12. The LCM of 20, 15, and 12 is 60. So the combined rate is 3 / 60 + 4 / 60 + 5 / 60 = 12 / 60 = 1 / 5, which matches the given time of 5 days. Hence C’s computed rate and time are consistent. Why Other Options Are Wrong: - 5 and 6 days: These would imply C is extremely fast and make the combined rate larger than 1 / 5 per day. - 10 days: Close but leads to a slight mismatch in the combined rate when all three are considered. - 15 days: This is slower than indicated by the difference between the combined rate and the sum of A and B’s rates. Common Pitfalls: Many learners mistakenly average the given times instead of working with rates. Another common error is to forget to subtract both A and B’s rates from the joint rate to isolate C’s rate. Always convert each time into a rate, use linear addition of rates, and then revert rates back into times by taking reciprocals. Final Answer: C alone can complete the whole work in 12 days.

Question 14

A and B together can complete a job in 15 days, and A alone can complete the same job in 20 days. If B worked alone, how many days would he take to complete half of the job?

Accepted Answer

Introduction / Context: This time and work problem focuses on finding the individual efficiency of worker B, given the combined time of A and B as well as A’s individual time. After determining B’s time for the whole job, we then compute the time required by B to complete only half of the job when working alone. Given Data / Assumptions: - A and B together can finish the job in 15 days. - A alone can finish the job in 20 days. - We need the time B alone would take to complete half the work. - Total work is taken as 1 unit. Concept / Approach: First, we convert the given times into daily work rates. The combined rate of A and B is 1 / 15, and A’s rate is 1 / 20. The rate of B is the difference between these two. Once B’s rate is known, we can find his time for the full job by taking the reciprocal of his rate. Since the question asks for the time to do half the work, we take half of that full-job time. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Combined rate of A and B = 1 / 15 work per day. Step 3: Rate of A alone = 1 / 20 work per day. Step 4: Rate of B = combined rate - rate of A. Step 5: Rate of B = 1 / 15 - 1 / 20. Step 6: Take LCM of 15 and 20 which is 60, so 1 / 15 = 4 / 60 and 1 / 20 = 3 / 60. Step 7: Rate of B = 4 / 60 - 3 / 60 = 1 / 60 work per day. Step 8: Time taken by B alone for the entire job = 1 / (1 / 60) = 60 days. Step 9: Time taken by B to complete half the work = 60 / 2 = 30 days. Verification / Alternative check: Check the combined rate: 1 / 20 + 1 / 60 = 3 / 60 + 1 / 60 = 4 / 60 = 1 / 15. This matches the given combined time of 15 days, so the rates are consistent. Half of 60 days is indeed 30 days, so the final answer is correct. Why Other Options Are Wrong: - 60: This is the time B needs for the entire job, not for half of it. - 45 and 40: These do not correspond to any direct fraction of B’s full-job time and do not fit the derived rate of 1 / 60. - 20: This would mean B is faster than he actually is, contradicting the given combined time of 15 days. Common Pitfalls: Typical mistakes include averaging times instead of subtracting rates to find the individual rate of B. Another common error is forgetting that half of the job requires half of the full time when working at a constant rate. Always distinguish between full-job time and partial-job time, especially when only a fraction of the work is asked for. Final Answer: B would take 30 days to complete half of the job when working alone.

Question 15

X, Y and Z can complete a piece of work in 30, 40 and 50 days respectively when working alone. All three begin the work together, but Y leaves 4 days before the completion of the work. In how many days in total is the work completed?

Accepted Answer

Introduction / Context: This is a slightly advanced time and work question involving three workers X, Y, and Z with different individual times. They start together, but Y leaves before the work ends. We need to handle two time intervals: a period when all three work together, and a final period when only X and Z work. This requires careful algebra and use of total work as 1 unit. Given Data / Assumptions: - X alone can finish the work in 30 days. - Y alone can finish the work in 40 days. - Z alone can finish the work in 50 days. - All three start together. - Y leaves 4 days before completion, so during the last 4 days only X and Z work. - Total work is taken as 1 unit. Concept / Approach: Let T be the total number of days the work takes. Then for the first T - 4 days, all three work together. For the last 4 days, only X and Z work. We convert individual times into rates, write an equation for the total work done, and solve for T. Fractional arithmetic is essential because the answer is a rational number rather than a neat integer. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Rate of X = 1 / 30 work per day. Step 3: Rate of Y = 1 / 40 work per day. Step 4: Rate of Z = 1 / 50 work per day. Step 5: Let T be the total time in days for completion. Step 6: For T - 4 days, X, Y, and Z work together. For the last 4 days, only X and Z work. Step 7: Total work done by X = T / 30 units. Step 8: Total work done by Y = (T - 4) / 40 units. Step 9: Total work done by Z = T / 50 units. Step 10: Since full work is 1 unit, we have T / 30 + (T - 4) / 40 + T / 50 = 1. Step 11: Solving this equation gives T = 660 / 47 days (approximately 14.04 days). Verification / Alternative check: We can verify numerically: T is about 14.04 days, so Y works for about 10.04 days and X and Z work for the full period. Using precise fractions, substitute T = 660 / 47 into T / 30, (T - 4) / 40, and T / 50, add the three values, and they sum to 1 unit of work. This confirms that T = 660 / 47 is correct. Why Other Options Are Wrong: - 640/47, 680/47, 665/47: These nearby fractions produce totals slightly less or more than the full unit of work when substituted into the equation, so they do not satisfy the work condition exactly. - 14: This is an approximation but not the exact solution, and leads to a small imbalance when total work is recomputed. Common Pitfalls: Students often assume that all three work for the same duration or forget to adjust Y’s time by subtracting 4 days. Another common mistake is not writing a proper equation for total work done, or handling fractions incorrectly when solving. It is important to separate the time intervals clearly and carefully use algebraic methods to find T. Final Answer: The total time taken to complete the work is 660/47 days.

Question 16

A, B and C contract to complete a piece of work for Rs. 440. Together, A and B are supposed to do 9/11 of the work. What should be the share of C in rupees from the total payment?

Accepted Answer

Introduction / Context: This question tests the concept of sharing money in proportion to work done. A, B, and C together take a contract for a fixed amount. The share of work done by A and B together is known as a fraction of the total, so the remaining fraction must be done by C. Payment is then divided according to the fraction of work contributed by each party. Given Data / Assumptions: - Total contract amount = Rs. 440. - A, B, and C together complete the entire work. - A and B together are to do 9/11 of the total work. - C therefore does the remaining part of the work. - Payment is proportional to the amount of work done by each party. Concept / Approach: If A and B together complete 9/11 of the work, then the remaining 2/11 of the work must be done by C. Since money is distributed in proportion to the work done, C’s share of the total payment will be the same fraction, 2/11, of the total amount. We simply multiply the total amount by 2/11 to get the rupee share of C. Step-by-Step Solution: Step 1: Total work is considered as 1 unit. Step 2: A and B together are to do 9/11 of the work. Step 3: Remaining work for C = 1 - 9/11 = 2/11 of the total work. Step 4: Total payment for the work = Rs. 440. Step 5: Since payment is proportional to work done, C’s share = (2 / 11) * 440. Step 6: Compute (2 / 11) * 440 = 2 * 40 = Rs. 80. Verification / Alternative check: We can also compute the share for A and B together: (9 / 11) * 440 = 9 * 40 = Rs. 360. The total of C’s share and the share of A and B together is 360 + 80 = 440, which matches the full contract amount. Hence the distribution is correct and consistent. Why Other Options Are Wrong: - 75, 90, 100, 60: None of these correspond to the fraction 2/11 of Rs. 440. They also do not leave a correct remaining amount for A and B when subtracted from 440, so they do not maintain proportionality with the given work fraction of 9/11 for A and B. Common Pitfalls: Some learners confuse the fraction of work done by C (2/11) with the fraction done by A and B (9/11), or attempt to split the payment equally among A, B, and C. It is essential to remember that payment should be made directly proportional to the work contribution, not to the number of workers. Another common mistake is to miscalculate (2 / 11) * 440 by not simplifying carefully. Final Answer: C’s correct share from the total payment is Rs. 80.

Question 17

A can do a piece of work in 18 days. He works alone for 12 days, and then B finishes the remaining work in 8 days. In how many days can B alone complete the entire work?

Accepted Answer

Introduction / Context: This question focuses on determining the efficiency of worker B based on partial work completed by A and the remaining work done by B. A works alone initially, and B completes whatever is left. From this, we infer how fast B works and then compute the total time B alone would need to finish the whole job. Given Data / Assumptions: - A alone can complete the work in 18 days. - A works alone for 12 days. - B completes the remaining work in 8 days. - Total work is treated as 1 unit. Concept / Approach: We first compute how much work A finishes in 12 days. Subtracting that from the total gives the remaining work. Since B finishes that remaining portion in 8 days, we can deduce B’s daily work rate. Finally, we calculate the time B alone would take to complete the entire 1 unit of work by taking the reciprocal of B’s rate. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Rate of A = 1 / 18 work per day. Step 3: Work done by A in 12 days = 12 * (1 / 18) = 12 / 18 = 2 / 3 of the work. Step 4: Remaining work after A finishes = 1 - 2 / 3 = 1 / 3 of the work. Step 5: B completes this remaining 1 / 3 of the work in 8 days. Step 6: Rate of B = (1 / 3) / 8 = 1 / 24 work per day. Step 7: Time taken by B alone to complete the full work = 1 / (1 / 24) = 24 days. Verification / Alternative check: We can verify by recomputing the work distribution. In 12 days, A completes 12 / 18 = 2 / 3 of the job. B’s rate is 1 / 24, so in 8 days B finishes 8 * 1 / 24 = 8 / 24 = 1 / 3, which exactly matches the remaining work. This confirms that both the interim calculations and B’s total time of 24 days are consistent. Why Other Options Are Wrong: - 16, 20 days: These imply that B works faster, which would result in more work being done in 8 days than the remaining 1 / 3, contradicting the data. - 28, 29 days: These imply B is slower than 24 days, which would mean that in 8 days B would complete less than the required 1 / 3 of the work. Common Pitfalls: Learners sometimes forget to convert days into work fractions and instead try to average times. Another common mistake is to assume that A and B share the remaining work together, while in this problem A stops after 12 days. Correct interpretation of who works when is crucial for setting up accurate calculations. Final Answer: B alone can complete the entire work in 24 days.

Question 18

A and B together can complete a piece of work in 6 days, and A alone can complete it in 9 days. How many days will B alone take to complete the same work?

Accepted Answer

Introduction / Context: This is a classic time and work problem where we are given the combined time of A and B working together and the individual time of A. From this, we must determine how long B would take to finish the entire work working alone. Such questions are common in aptitude exams and require understanding of how to manipulate work rates. Given Data / Assumptions: - A and B together complete the work in 6 days. - A alone completes the work in 9 days. - We need the time required for B alone to complete the work. - Total work is assumed to be 1 unit. Concept / Approach: We first convert the given times into work rates. The combined rate of A and B is 1 / 6 work per day, and A’s individual rate is 1 / 9 work per day. The rate of B is obtained by subtracting A’s rate from the combined rate. Once we have B’s rate, the time taken by B alone is simply the reciprocal of that rate. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Combined rate of A and B = 1 / 6 work per day. Step 3: Rate of A alone = 1 / 9 work per day. Step 4: Rate of B = combined rate - rate of A = 1 / 6 - 1 / 9. Step 5: Take LCM of 6 and 9 which is 18, so 1 / 6 = 3 / 18 and 1 / 9 = 2 / 18. Step 6: Rate of B = 3 / 18 - 2 / 18 = 1 / 18 work per day. Step 7: Time taken by B alone = 1 / (1 / 18) = 18 days. Verification / Alternative check: Check that A and B together indeed complete the work in 6 days: A does 1 / 9 per day, B does 1 / 18 per day, so together they do 1 / 9 + 1 / 18 = 2 / 18 + 1 / 18 = 3 / 18 = 1 / 6 per day. Hence, in 6 days, they finish 6 * 1 / 6 = 1 unit of work. This matches the given information, confirming that B’s individual time of 18 days is correct. Why Other Options Are Wrong: - 24, 12, 15 days: These values lead to inconsistent combined rates when used with A’s time of 9 days, and would not produce a combined time of 6 days. - 9 days: This would imply A and B have the same efficiency, which contradicts the requirement that together they are faster than A alone by a clear margin. Common Pitfalls: A common mistake is to attempt to average the times 6 and 9 instead of working with rates. Another error is to invert fractions incorrectly when converting from time to rate and back. Always convert times to rates before combining or subtracting, and only then convert back to the required time. Final Answer: B alone will complete the work in 18 days.

Question 19

A is twice as efficient a worker as B, and together they finish a piece of work in 16 days. In how many days will A alone finish the entire work?

Accepted Answer

Introduction / Context: Here we are given a relationship between the efficiencies of two workers, A and B, and the time they take together to finish the work. The phrase that A is twice as good as B means that A works at twice the rate of B. Using this relationship, we can express their rates in terms of a variable, find the combined rate using the given joint time, and then determine A’s individual time to complete the job. Given Data / Assumptions: - A is twice as efficient as B. - A and B together complete the work in 16 days. - Total work is assumed to be 1 unit. - Work rates are constant over time. Concept / Approach: Let the daily work rate of B be x units. Then the rate of A is 2x units per day. Together, their combined rate is 3x units per day. Since they finish the total work in 16 days, the combined rate is also 1 / 16 units per day. Equating 3x to 1 / 16 allows us to solve for x and then compute A’s individual time by taking the reciprocal of A’s rate, which is 2x. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Let the rate of B = x units per day. Step 3: Since A is twice as efficient, rate of A = 2x units per day. Step 4: Combined rate of A and B = 2x + x = 3x units per day. Step 5: They finish the work in 16 days, so 3x = 1 / 16. Step 6: Solve for x: x = 1 / (16 * 3) = 1 / 48. Step 7: Rate of A = 2x = 2 * (1 / 48) = 1 / 24 units per day. Step 8: Time taken by A alone = 1 / (1 / 24) = 24 days. Verification / Alternative check: If A alone takes 24 days, then B’s rate is half of A’s rate, so B alone would take 48 days. Combined rate should be 1 / 24 + 1 / 48 = 2 / 48 + 1 / 48 = 3 / 48 = 1 / 16, which matches the given joint time of 16 days. This confirms that A’s individual time is correctly computed as 24 days. Why Other Options Are Wrong: - 20, 21, 22 days: These values do not maintain the correct ratio between A and B’s times and fail to reproduce the combined time of 16 days. - 18 days: This would imply an even higher efficiency for A, leading to a combined rate larger than 1 / 16 and thus a shorter joint time than given. Common Pitfalls: Learners sometimes confuse the ratio of efficiency with the ratio of time. If A is twice as efficient as B, then A’s time is half of B’s time, not double. Another typical mistake is forgetting to express the combined work rate in terms of the defined variable and equating it to 1 divided by the given total time. Final Answer: A alone will complete the work in 24 days.

Question 20

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.

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