Question 1

Meghana alone would take 32 hours more than Meghana and Ganesh working together to complete a certain job. If Ganesh worked alone, he would take 12 1/2 hours more than their combined time to finish the same job. How many hours will it take for Megha…

Accepted Answer

Introduction / Context: This problem tests the standard time and work concept where two people can do a job together faster than either of them can do it alone. The question gives how much extra time Meghana and Ganesh would each take when working alone compared with their combined time, and asks us to find the time required when they work together from start to finish. Given Data / Assumptions: The combined time of Meghana and Ganesh working together is T hours. Meghana alone would take T + 32 hours to complete the job. Ganesh alone would take T + 12.5 hours (that is, 12 1/2 hours) to complete the job. Both work at constant rates and their work rates are additive when they work together. Concept / Approach: When people work together, their combined work rate is the sum of their individual work rates. If someone finishes a job in t hours, the work rate is 1 / t jobs per hour. Here, we convert each time into a rate, add the rates for Meghana and Ganesh, and equate this sum to the joint rate 1 / T. This gives us an equation in T that we can solve to get the required joint time. Step-by-Step Solution: Let T be the number of hours Meghana and Ganesh take together to finish the job. Meghana's time alone = T + 32, so her rate = 1 / (T + 32). Ganesh's time alone = T + 12.5, so his rate = 1 / (T + 12.5). Their combined rate when both work together is 1 / T. Therefore, we have: 1 / T = 1 / (T + 32) + 1 / (T + 12.5). Multiply through by T (T + 32) (T + 12.5) to clear denominators and simplify the resulting quadratic equation. Solving the equation gives T = 20 as the meaningful positive solution. Thus, Meghana and Ganesh together can complete the work in 20 hours. Verification / Alternative check: If T = 20, Meghana alone would take 52 hours and Ganesh alone would take 32.5 hours. Their work rates are 1 / 52 and 1 / 32.5 respectively. Adding these two rates gives a value that is very close to 1 / 20, confirming that the pair working together need about 20 hours for one complete job. This matches the equation and confirms consistency. Why Other Options Are Wrong: 18 hours is too small and would make the required single person times inconsistent with the given extra time conditions. 22 hours does not satisfy the rate equation formed from the information about extra 32 and 12.5 hours. 20.5 hours is close but still not an exact solution of the equation, so it is not correct. 24 hours is much larger than the real joint time and would not be compatible with the given solo times. Common Pitfalls: Many learners forget to convert the mixed fraction 12 1/2 hours into 12.5 hours and make arithmetic mistakes. Another common mistake is to add times directly instead of adding work rates, which is incorrect. Always remember that rates add, not times. Also, take care to correctly form and solve the quadratic equation without dropping terms. Final Answer: Meghana and Ganesh together will complete the job in 20 hours.

Question 2

A single reservoir supplies petrol to an entire city and is fed continuously by one pipeline that fills it with a uniform inflow rate. When the reservoir is full and 40,000 litres of petrol are used every day, the supply runs out in 90 days. If only…

Accepted Answer

Introduction / Context: This is a reservoir and uniform inflow problem based on time and work concepts. The reservoir is being filled continuously by a pipeline while petrol is simultaneously used up. We are given two different daily usage levels and the durations until the reservoir becomes empty, and we must determine the maximum sustainable daily usage that will never exhaust the supply in the long run. Given Data / Assumptions: Let the capacity of the reservoir be C litres. Let the daily inflow from the pipeline be R litres per day (constant). Case 1: Daily usage = 40000 litres; the reservoir lasts for 90 days. Case 2: Daily usage = 32000 litres; the reservoir lasts for 60 days. We assume the reservoir is full at the start of each scenario and that inflow and outflow are continuous and uniform. Concept / Approach: The change in the amount of petrol stored equals inflow minus outflow. Over the entire duration until the reservoir becomes empty, the net amount used equals the initial capacity plus any extra inflow. We can express this by linear equations in terms of C and R. Solving these simultaneous equations gives the daily inflow R. If the city uses exactly R litres per day, then inflow equals outflow, so the water level remains constant and the supply never fails. Step-by-Step Solution: For Case 1: C + 90 * R - 90 * 40000 = 0, so C + 90 R = 90 * 40000. For Case 2: C + 60 * R - 60 * 32000 = 0, so C + 60 R = 60 * 32000. Subtract the second equation from the first: (C + 90 R) - (C + 60 R) = 90 * 40000 - 60 * 32000. This simplifies to 30 R = 90 * 40000 - 60 * 32000. Compute the right side: 90 * 40000 = 3600000 and 60 * 32000 = 1920000, so 30 R = 1680000. Therefore, R = 1680000 / 30 = 56000 litres per day. If the city uses petrol at exactly 56000 litres per day, the daily outflow equals the inflow from the pipeline, so the reservoir level never decreases and supply does not fail. Verification / Alternative check: You can use the found inflow R = 56000 litres to recover C from either of the two equations and check that the reservoir indeed becomes empty at the stated times when daily usage is 40000 or 32000 litres. Both equations will yield the same value of C, confirming that the inflow value is consistent with both scenarios. This validates the correctness of R = 56000 litres per day. Why Other Options Are Wrong: 64000 litres is larger than the actual inflow, so long term usage at that rate would drain the reservoir. 78000 litres is even larger and clearly unsustainable compared with the inflow. 60000 litres also exceeds 56000 litres, so the reservoir would slowly empty. 52000 litres is less than the inflow, so it is safe, but the question asks for the maximum daily usage without failure, which is the inflow rate itself. Common Pitfalls: Students sometimes forget that inflow continues while the reservoir is being used, so they treat the problem as a simple capacity divided by consumption case. Another mistake is mixing up units or miscalculating the subtraction step when forming the simultaneous equations. Always clearly define C and R, write equations for each scenario, and subtract carefully to remove C and solve for R. Final Answer: The maximum petrol that can be used daily without the supply ever failing is 56000 litres per day.

Question 3

Nine girls working 7 hours every day can complete a certain piece of work in 15 days. If only six girls are available but they work for 9 hours each day, in how many days will they complete the same amount of work?

Accepted Answer

Introduction / Context: This problem is a classic time and work question involving different numbers of workers and different daily working hours. The total amount of work is the same in both situations, and we are asked to find how long a smaller group working longer hours will take to finish the same task. Given Data / Assumptions: Nine girls working 7 hours per day can complete the work in 15 days. We want to find the number of days required when six girls work 9 hours per day on the same work. All girls work at the same constant rate and productivity does not change with group size. Concept / Approach: Total work is the product of the number of workers, hours per day, days, and the work rate per worker. Because the nature of the work does not change, the same total work is performed in each scenario. Instead of computing absolute work units, we can equate the total girl hours from both cases and solve for the unknown number of days in the second scenario. Step-by-Step Solution: Let the total work be W and the work rate of each girl be r units per hour. In the first scenario, total work W = 9 girls * 7 hours per day * 15 days * r. So W = 9 * 7 * 15 * r = 945 r. In the second scenario, six girls work 9 hours per day for D days, so W = 6 * 9 * D * r. Equate the work: 6 * 9 * D * r = 945 r. Cancel r and simplify: 54 D = 945. Therefore, D = 945 / 54 = 17.5 days. Thus six girls working 9 hours per day need 17.5 days to complete the task. Verification / Alternative check: A quick reasonableness check is that we reduced the number of girls from nine to six (a factor of 2/3) but increased daily hours from seven to nine (a factor of 9/7). The combined change in daily girl hours is (6 * 9) / (9 * 7) = 54 / 63 = 6 / 7, so daily work capacity is 6/7 of what it was initially. Therefore, the number of days should increase by a factor of 7/6. Initial days are 15, so new days are 15 * 7 / 6 = 105 / 6 = 17.5, matching our earlier calculation. Why Other Options Are Wrong: 14 days is too small and would imply higher overall daily capacity, which conflicts with fewer girls even with longer hours. 19.5 days is greater than 17.5 and does not agree with the ratio 7/6 derived from comparing daily girl hours. 21 days also overestimates the needed time and contradicts the calculated ratio. 12.5 days is unrealistically small and clearly inconsistent with having fewer workers. Common Pitfalls: A frequent mistake is to compare only numbers of workers and ignore the change in working hours, or to simply add or subtract days. Another error is to assume the work rate per group is linear in days without accounting for hours per day. Always convert everything into total worker hours before equating the two scenarios. Final Answer: Six girls working 9 hours a day will complete the work in 17.5 days.

Question 4

One woman can fill a single bucket completely in 6 minutes. Starting at 8:00 a.m., a total of 1845 buckets must be filled by 9:30 a.m., which is a time span of 1 hour 30 minutes. How many women workers are required so that all 1845 buckets are fille…

Accepted Answer

Introduction / Context: This question is a direct application of time and work ideas where a fixed number of identical workers must complete a known quantity of identical tasks within a fixed time window. We are told how fast a single woman can fill one bucket and asked to determine how many women are required to fill a large number of buckets within 90 minutes. Given Data / Assumptions: One woman fills 1 bucket in 6 minutes. Total number of buckets to be filled = 1845. Available time from 8:00 a.m. to 9:30 a.m. = 1 hour 30 minutes = 90 minutes. All women work at the same constant rate and can work simultaneously without interfering with each other. Concept / Approach: The work unit here is one full bucket. The rate of one woman is buckets per minute. If there are N women, the total rate is N times the rate of one woman. The total work done in the allotted time equals total rate multiplied by time. We equate that to the required 1845 buckets and solve for N, rounding only if necessary. In this problem the number comes out exactly as an integer. Step-by-Step Solution: Time taken by one woman to fill 1 bucket = 6 minutes. So, rate of one woman = 1 / 6 buckets per minute. Let N be the required number of women. Total rate for N women = N * (1 / 6) = N / 6 buckets per minute. Available time = 90 minutes. Total buckets filled in 90 minutes = (N / 6) * 90 = 15 N. We need 15 N = 1845 buckets. Therefore, N = 1845 / 15 = 123 women. So 123 women must work together to finish 1845 buckets in the given time. Verification / Alternative check: If 123 women work for 90 minutes, each woman fills 90 / 6 = 15 buckets. Multiplying, the total buckets filled equals 123 * 15 = 1845, which is exactly the required number. This confirms that the calculation is correct and that no rounding or approximation is involved. Why Other Options Are Wrong: 111 women would fill only 111 * 15 = 1665 buckets, which is short of 1845. 117 women would fill 117 * 15 = 1755 buckets, still not enough. 139 women would produce 139 * 15 = 2085 buckets, which is more than required. 129 women would also give 129 * 15 = 1935 buckets, exceeding the target. Common Pitfalls: Learners sometimes confuse the time for one bucket with the total time window and try to divide buckets by minutes directly without considering the number of workers. Others may incorrectly convert the time from 1 hour 30 minutes into minutes. Always first convert the full time to minutes, compute the capacity of one worker over that period, then divide the required total work by that capacity to find the number of workers needed. Final Answer: A total of 123 women are required to fill 1845 buckets between 8:00 a.m. and 9:30 a.m.

Question 5

A contractor undertakes to complete a piece of work in 120 days and initially employs 140 men on the job. At the end of 66 days it is found that only half of the work has been completed, so the contractor hires 25 extra men. By how many days does th…

Accepted Answer

Introduction / Context: This problem involves a contractor who has underestimated the workforce needed to complete a job on time. The question tests understanding of work rate, total work in man days, and how adding more workers affects completion time. We are asked to determine by how many days the actual completion exceeds the originally agreed schedule. Given Data / Assumptions: Planned schedule: the job should be completed in 120 days with 140 men. After 66 days, only half of the work is complete. From day 67 onward, 25 additional men are employed, so total workers become 165. All men work at the same constant rate and work full days. Concept / Approach: The key is to treat the work rate as men days. Half of the total work is completed in 66 days by 140 men. That allows us to compute the total work in men days. Knowing how much work remains and the new workforce size, we compute the remaining time required. Adding this to the initial 66 days gives the actual total duration, which we compare to the originally planned 120 days. Step-by-Step Solution: Let the total work be W men days. Given that half the work is completed in 66 days by 140 men, we have W / 2 = 140 * 66. Therefore, W = 2 * 140 * 66 = 140 * 132 = 18480 men days. Remaining work after 66 days is W / 2 = 18480 / 2 = 9240 men days. After hiring extra men, total workers = 140 + 25 = 165 men. Time to complete the remaining work = 9240 / 165 days. Simplify 9240 / 165 = 56 days. So, actual total time taken = 66 days + 56 days = 122 days. Excess time beyond the specified 120 days = 122 - 120 = 2 days. Verification / Alternative check: Compute the effective average workforce if the entire 18480 men days of work are spread over 122 days: 18480 / 122 is approximately 151.5 men. The actual schedule uses 140 men for 66 days and 165 men for 56 days, which averages out to this value. This shows the calculation is consistent and the total men days tally correctly with the work done. Why Other Options Are Wrong: 3 days would correspond to an actual duration of 123 days, which would require a slightly different remaining work or strength, not supported by the given data. 4 days or 5 days excess time would imply 124 or 125 days total, which would contradict the computed remaining time of 56 days with 165 men. 1 day extra (121 days total) is too small and would not allow enough time for 165 men to complete the remaining 9240 men days of work. Common Pitfalls: A common mistake is to assume that the initial plan 140 men * 120 days equals the actual total work, even though the progress data shows that only half the work was done in 66 days. The actual total work must be recalculated from the given progress information. Another trap is arithmetic simplification errors while dividing 9240 by 165. Carefully factor both numbers or use cancellation to avoid mistakes. Final Answer: The contractor exceeds the specified time by 2 days.

Question 6

A garrison of 2000 soldiers has food provisions that will last for 54 days. After 15 days, a reinforcement of soldiers arrives, and it is then calculated that the remaining provisions will last only for 20 more days. How many soldiers are there in t…

Accepted Answer

Introduction / Context: This is a standard garrison provision problem that uses the idea that total food supply is fixed and daily consumption depends on the number of men. When reinforcements arrive, the daily consumption increases and the total duration for which the provisions last decreases. The task is to find how many additional men joined the original group. Given Data / Assumptions: Initial number of soldiers = 2000. Provisions initially last for 54 days at that strength. After 15 days, reinforcement arrives. From that point, the provisions last for only 20 more days. Food consumption per soldier per day is constant and the total supply does not change except by consumption. Concept / Approach: The total amount of food can be measured in men days, which is the product of the number of men and the number of days the food lasts. Initially, the total men days of food are calculated from the original garrison. After 15 days some of this food has been consumed. The remaining food must support the larger garrison for 20 more days. Using these facts, we can set up equations to find the size of the reinforcement. Step-by-Step Solution: Let F be the total food measured in men days. Initially, with 2000 men and 54 days of food, F = 2000 * 54. After 15 days, food consumed = 2000 * 15 men days. Remaining food = F - 2000 * 15 = 2000 * (54 - 15) = 2000 * 39 men days. Let N be the total number of soldiers after the reinforcement joins. The remaining food lasts 20 days for N men, so 2000 * 39 = N * 20. Compute N: N = (2000 * 39) / 20 = 200 * 39 / 2 = 100 * 39 = 3900 men. Reinforcement size = N - 2000 = 3900 - 2000 = 1900 men. Verification / Alternative check: Check that the food calculation is consistent: 2000 men for 15 days consume 30000 men days. Then 3900 men for 20 days consume 78000 men days. The total is 108000 men days, which matches the initial F = 2000 * 54 = 108000. This confirms the arithmetic and the logic. Why Other Options Are Wrong: 2100 as reinforcement would give 4100 total men, leading to inconsistent consumption compared with the remaining food. 1700 as reinforcement would give 3700 total men, which would require more days than 20 for the remaining food. 2000 as reinforcement suggests the number of men doubles, which would make the provisions last less than 20 days. 1500 reinforcement also does not satisfy the men days balance for the given durations. Common Pitfalls: Learners sometimes equate the days directly or try to use ratios of days without translating everything into men days. Another mistake is forgetting to subtract the food consumed during the first 15 days before the reinforcement arrives. Always compute total food in men days, subtract the used portion, and then equate the remaining food to the product of the new strength and the remaining days. Final Answer: The reinforcement consists of 1900 additional men.

Question 7

A mother can complete a certain job alone in x hours. Her daughter takes exactly twice as long as her mother to do the same job on her own. Working together, the mother and the daughter can finish the job in 6 hours. How many hours does the mother a…

Accepted Answer

Introduction / Context: This is a classic work rate problem involving two people with different individual times but a known combined time. The mother is faster, the daughter is slower, and together they complete the job in a given time. We are asked to find how long the mother alone would take to complete the job. Given Data / Assumptions: Mother alone takes x hours to complete the job. Daughter alone takes 2x hours to complete the same job. Together, they can complete the job in 6 hours. Work rates of both remain constant and add when they work together. Concept / Approach: If a person takes t hours to finish a job, the work rate is 1 / t jobs per hour. For two people working together, their combined rate is the sum of their individual rates. Here, we write the combined rate of mother and daughter in terms of x and then set it equal to the known combined rate 1 / 6. Solving that equation gives x, the mother's time alone. Step-by-Step Solution: Mother's time alone = x hours, so her rate = 1 / x jobs per hour. Daughter's time alone = 2x hours, so her rate = 1 / (2x) jobs per hour. Together their rate = 1 / x + 1 / (2x) = (2 + 1) / (2x) = 3 / (2x). It is given that together they finish the work in 6 hours, so combined rate = 1 / 6. Therefore, 3 / (2x) = 1 / 6. Cross multiply: 3 * 6 = 2x. So, 18 = 2x, which gives x = 9. Thus the mother alone will take 9 hours to complete the job. Verification / Alternative check: If the mother takes 9 hours, her rate is 1 / 9; the daughter then takes 18 hours, with rate 1 / 18. The combined rate is 1 / 9 + 1 / 18 = 2 / 18 + 1 / 18 = 3 / 18 = 1 / 6, meaning they finish in 6 hours as stated in the problem. This confirms that x = 9 hours is consistent. Why Other Options Are Wrong: 3 hours would make the mother much too fast and the combined time far less than 6 hours. 6 hours would imply mother and daughter have equal speeds and contradicts the condition that the daughter takes twice as long. 12 hours and 15 hours would both give combined rates that do not correspond to a 6 hour completion time. Common Pitfalls: A common mistake is to average the times directly (for example, taking (x + 2x) / 2) instead of working with rates. Another error is forgetting that the daughter takes twice the time of the mother, not twice the rate. Always set up the problem in terms of rates (reciprocals of time) when dealing with combined work scenarios. Final Answer: The mother alone takes 9 hours to complete the job.

Question 8

Raghu can complete a job in 12 days working alone, while Sam can finish the same job in 15 days working alone. A third person, Aru, has an efficiency equal to two thirds of the combined efficiency of Raghu and Sam together. In how many days can Aru …

Accepted Answer

Introduction / Context: This question tests the concept of combining work rates and then scaling them to find the time for one worker whose rate is a given fraction of a combined rate. Raghu and Sam have known individual completion times, and Aru's efficiency is defined relative to their combined efficiency. Given Data / Assumptions: Raghu alone can complete the work in 12 days. Sam alone can complete the same work in 15 days. Aru's efficiency is two thirds of the combined efficiency of Raghu and Sam working together. All work at constant rates and there is only one job to be completed. Concept / Approach: We first calculate the combined rate of Raghu and Sam by adding their individual rates. Then we take two thirds of this combined rate to get Aru's rate. Since time is the reciprocal of rate for a fixed amount of work, we can convert Aru's rate back into the number of days Aru would need to complete the job alone. Step-by-Step Solution: Raghu's rate = 1 / 12 jobs per day. Sam's rate = 1 / 15 jobs per day. Combined rate of Raghu and Sam = 1 / 12 + 1 / 15. Compute the sum: the least common multiple of 12 and 15 is 60, so 1 / 12 = 5 / 60 and 1 / 15 = 4 / 60. Therefore, combined rate = (5 + 4) / 60 = 9 / 60 = 3 / 20 jobs per day. Aru's rate is two thirds of this, so Aru's rate = (2 / 3) * (3 / 20) = 2 / 20 = 1 / 10 jobs per day. Hence, Aru alone would take 10 days to complete one job. Verification / Alternative check: As a check, note that Raghu and Sam together finish in 20 / 3 days, which is about 6.67 days. Aru's efficiency is two thirds of theirs, so his time should be 3 / 2 times their time: (3 / 2) * (20 / 3) = 10 days. This matches the computed time and confirms the result. Why Other Options Are Wrong: 12 days would correspond to a rate of 1 / 12, which is slower than Aru's defined efficiency. 13 days and 15 days would give even lower rates and are inconsistent with Aru being two thirds as efficient as the pair. 9 days would give a slightly higher rate than 1 / 10 and does not match exactly two thirds of the combined rate. Common Pitfalls: Students sometimes misinterpret "two thirds of the efficiency" as "two thirds of the time," which is incorrect because time and efficiency are inversely related. Another frequent error is adding the times instead of the rates. Always convert completion times into rates before combining or scaling efficiencies. Final Answer: Aru alone would complete the work in 10 days.

Question 9

Worker A can finish a piece of work in 15 days, and worker B can complete the same work in 12 days. B works on the job alone for 8 days and then leaves. For how many additional days must A work alone to finish the remaining part of the work?

Accepted Answer

Introduction / Context: This is a straightforward time and work problem involving two workers with different individual completion times. One worker, B, works for a certain period and then leaves, and we must find how long the other worker, A, will take to complete the remaining portion of the job. Given Data / Assumptions: Worker A can finish the work in 15 days working alone. Worker B can finish the work in 12 days working alone. B works alone for 8 days and then stops working. A finishes the remaining work alone at a constant rate. Concept / Approach: We interpret the total work as a fixed quantity that can be divided into units. It is often convenient to take the total work as the least common multiple of the given times so that daily work units become integers. We then compute how much of the work B completes in 8 days and subtract that from the total. The remainder is completed by A, and we divide the remaining work by A's daily rate to get the number of days A needs. Step-by-Step Solution: Let the total work be 60 units, which is the least common multiple of 12 and 15. A's rate = 60 / 15 = 4 units per day. B's rate = 60 / 12 = 5 units per day. B works alone for 8 days, so work done by B = 8 * 5 = 40 units. Remaining work = 60 - 40 = 20 units. A now works alone at 4 units per day. Time required for A to finish = 20 / 4 = 5 days. Therefore, A will need 5 additional days to complete the work. Verification / Alternative check: You can view the fractions directly: B's daily rate is 1 / 12 of the work. In 8 days B does 8 / 12 = 2 / 3 of the work. The remaining work is 1 - 2 / 3 = 1 / 3 of the job. A completes 1 / 15 of the job per day, so the time for A is (1 / 3) / (1 / 15) = 5 days, consistent with the unit based approach. Why Other Options Are Wrong: 6 days would imply that A completes 6 / 15 = 2 / 5 of the job, which is more than the remaining 1 / 3. 4 days would give only 4 / 15 of the job, which is less than the 1 / 3 needed. 3 days or 7 days also do not produce the fraction 1 / 3 of the remaining work when multiplied by A's daily rate. Common Pitfalls: Some learners mistakenly average the times or mix up which worker continues and which stops. Others forget to compute how much work B has already completed before bringing in A's contribution. Always track the fraction of work done at each stage and ensure that the sum of completed portions equals the total work. Final Answer: A alone will take 5 days to finish the remaining work.

Question 10

In a public bathroom there are taps numbered 1, 2, 3, ..., n. Tap 1 and Tap 2 each take the same time to fill a tank when they run alone. Tap 3 takes half the time taken by Tap 2, Tap 4 takes half the time taken by Tap 3, and in general, every subse…

Accepted Answer

Introduction / Context: This puzzle style question is about relative efficiency of taps that fill a tank. The times taken by successive taps form a geometric progression, and we are given the filling time for the tenth tap. The objective is to compare the efficiencies of the eighth and twelfth taps, where efficiency means the rate of filling the tank. Given Data / Assumptions: Tap 1 and Tap 2 each take the same time to fill the tank. Tap 3 takes half the time taken by Tap 2. Tap 4 takes half the time taken by Tap 3, and so on. Each Kth tap takes half the time taken by the (K - 1)th tap. Tap 10 alone takes 2 hours to fill the tank. Efficiency of a tap is defined as work rate, equal to 1 divided by the time taken to fill the tank. Concept / Approach: If time decreases by a factor of 2 from one tap to the next, then going forward through the taps halves the time, while going backward doubles the time. First we find the times for Tap 8 and Tap 12 by stepping back or forward from Tap 10. Then we convert these times into rates and form the ratio of efficiencies (rates) of Tap 8 to Tap 12. Step-by-Step Solution: Tap 10 takes 2 hours. Working backward: Tap 9 takes twice the time of Tap 10, so Tap 9 takes 2 * 2 = 4 hours. Tap 8 takes twice the time of Tap 9, so Tap 8 takes 2 * 4 = 8 hours. Working forward: Tap 11 takes half the time of Tap 10, so Tap 11 takes 2 / 2 = 1 hour. Tap 12 takes half the time of Tap 11, so Tap 12 takes 1 / 2 = 0.5 hours. Efficiency is rate = 1 / time. Efficiency of Tap 8 = 1 / 8. Efficiency of Tap 12 = 1 / 0.5 = 2. Ratio of efficiency of Tap 8 to Tap 12 = (1 / 8) : 2 = 1 : 16. Verification / Alternative check: We can express all times relative to Tap 10. Since each step multiplies time by 2 or 1/2, Tap 8 is two steps back, so time factor is 2^2 = 4, giving 2 * 4 = 8 hours. Tap 12 is two steps forward, so factor is (1 / 2)^2 = 1 / 4, giving 2 * 1 / 4 = 0.5 hours. The ratio of times is 8 : 0.5 = 16 : 1, so the ratio of efficiencies is the inverse, 1 : 16. This matches our earlier result. Why Other Options Are Wrong: 4:1 and 16:1 both put Tap 8 as more efficient than Tap 12, which is impossible since Tap 12 fills the tank much faster. 5:3 does not follow from the geometric halving pattern of times. 8:1 also implies Tap 8 is faster than Tap 12 and is inconsistent with the computed times. Common Pitfalls: The main trap is going in the wrong direction with the halving rule, for example dividing times when moving to earlier taps instead of multiplying by 2. Another mistake is to compare times directly instead of rates. Remember that efficiency is inversely proportional to time, so the ratio of efficiencies is the inverse of the ratio of times. Final Answer: The ratio of the efficiency of Tap 8 to Tap 12 is 1 : 16.

Question 11

Two persons, Shyam and Rahim, can complete a certain job together in 32 days. Rahim alone can finish the same job in 48 days. They start working together and after 8 days Rahim is replaced by a third person, Ram, whose efficiency is exactly half tha…

Accepted Answer

Introduction / Context: This question involves three workers with different efficiencies, where one person is later replaced by someone less efficient. We know how long Shyam and Rahim together take to finish a job, and how long Rahim alone needs. After some days, Rahim is replaced by Ram, whose efficiency is half that of Rahim. We must find how many days Shyam and Ram will require to complete the remaining work. Given Data / Assumptions: Shyam and Rahim together can complete the job in 32 days. Rahim alone can complete the job in 48 days. They work together for the first 8 days. After 8 days, Rahim is replaced by Ram, whose efficiency is half that of Rahim. All workers have constant work rates and the job size is fixed at one unit. Concept / Approach: We first compute the joint rate of Shyam and Rahim from the given completion time, and then isolate Shyam's rate using Rahim's solo time. Once we know how much work is done in the first 8 days, we compute the remaining fraction. Then we determine Ram's rate as half of Rahim's rate, add it to Shyam's rate to get the new combined rate, and finally find the time required to finish the remaining work. Step-by-Step Solution: Combined rate of Shyam and Rahim = 1 / 32 job per day. Rahim alone completes the job in 48 days, so Rahim's rate = 1 / 48 job per day. Shyam's rate = (1 / 32) - (1 / 48). Compute Shyam's rate: with common denominator 96, 1 / 32 = 3 / 96 and 1 / 48 = 2 / 96, so Shyam's rate = 1 / 96 job per day. In the first 8 days, Shyam and Rahim together do 8 * (1 / 32) = 8 / 32 = 1 / 4 of the work. Remaining work = 1 - 1 / 4 = 3 / 4 of the job. Ram's efficiency is half of Rahim's, so Ram's rate = (1 / 2) * (1 / 48) = 1 / 96 job per day. Now Shyam and Ram together have rate = 1 / 96 + 1 / 96 = 1 / 48 job per day. Time to complete remaining 3 / 4 job = (3 / 4) / (1 / 48) = (3 / 4) * 48 = 36 days. Thus, Shyam and Ram together need 36 days to complete the remaining work. Verification / Alternative check: We can calculate the total equivalent work done in men days as a consistency check. The initial 8 days give 8 / 32 = 0.25 of the work. The remaining 0.75 divided by rate 1 / 48 gives 36 days. Total calendar days spent on the project is 8 + 36 = 44 days, which is sensible because the replacement worker is less efficient than Rahim, so the job takes longer than 32 days overall. Why Other Options Are Wrong: 24 days and 28 days would require a much higher combined rate than 1 / 48 and are inconsistent with Ram being slower than Rahim. 32 days would imply the replacement does not slow the work at all, which contradicts the lower efficiency of Ram. 40 days is close but does not match the exact ratio derived from the rates 1 / 96 and 1 / 96. Common Pitfalls: Learners often confuse time and efficiency when the phrase "half as efficient" appears, mistakenly halving the time instead of the rate. Another common error is miscomputing Shyam's rate by subtracting times instead of rates. Always work in terms of rates (jobs per day), not times, and be careful to apply the half efficiency relation to the rate of Rahim, not to his time. Final Answer: The remaining work will be completed by Shyam and Ram together in 36 days.

Question 12

A soldier fires 7 shots from a gun in 14 minutes at a constant rate. If he continues firing at the same rate, how many shots will he fire in 3/2 hours (that is, in 1 hour 30 minutes)?

Accepted Answer

Introduction / Context: This is a simple proportionality problem involving a constant rate of firing shots from a gun. The soldier fires at a steady rate and we scale that rate up to a longer time interval to determine how many shots he will fire in total. Given Data / Assumptions: The soldier fires 7 shots in 14 minutes. His rate of firing is constant over time. We are asked to find how many shots he fires in 3/2 hours. 3/2 hours is equivalent to 1.5 hours or 90 minutes. Concept / Approach: When a rate is constant, the quantity produced is directly proportional to time. We first calculate the soldier's rate in shots per minute from the given 14 minute interval. Then we convert 3/2 hours into minutes and multiply the rate by this total time to get the total number of shots fired in that period. Step-by-Step Solution: The soldier fires 7 shots in 14 minutes. Rate of firing = 7 shots / 14 minutes = 0.5 shot per minute. Convert 3/2 hours to minutes: 3/2 hours = 1.5 hours. 1.5 hours = 1.5 * 60 = 90 minutes. Total shots in 90 minutes = rate * time = 0.5 shot per minute * 90 minutes. This equals 45 shots. Therefore, the soldier will fire 45 shots in 3/2 hours. Verification / Alternative check: Another way to see this is to note that 90 minutes contains 90 / 14 blocks of 14 minutes. In each 14 minute block, the soldier fires 7 shots. So total shots = 7 * (90 / 14) = (7 * 90) / 14 = 630 / 14 = 45 shots. This matches the earlier calculation and confirms the result. Why Other Options Are Wrong: 44 shots would require a slightly smaller rate than 0.5 shot per minute, which contradicts the given data. 46 and 47 shots would require a slightly higher firing rate than stated. 42 shots is based on a rate of 0.4667 shots per minute, which does not match the basic proportion of 7 shots in 14 minutes. Common Pitfalls: A frequent error is misinterpreting 3/2 hours as 3 minutes or 2 hours instead of 1.5 hours. Another trap is in the unit conversion, where some learners use 3/2 * 14 directly without recognising that the reference interval is 14 minutes and the target interval is in hours. Always convert all times to the same unit, here minutes, before applying proportions. Final Answer: The soldier will fire 45 shots in 3/2 hours.

Question 13

A certain job was assigned to a group of women with the plan that it would be finished in 20 days. However, 12 women did not turn up for the job, and the remaining women completed the work in 32 days. What was the original number of women in the gro…

Accepted Answer

Introduction / Context: This question deals with a work assignment where the planned number of workers does not all show up. Because fewer women participate, the job takes longer. Using the relationship between the number of workers, time, and total work, we need to find how many women were originally planned for the job. Given Data / Assumptions: Original plan: a group of women was supposed to complete the work in 20 days. 12 women did not turn up, so the actual number of women working was reduced by 12. The reduced group completed the work in 32 days. All women work at the same constant rate and the job size is fixed. Concept / Approach: Total work can be expressed as the product of the number of workers, days, and an individual work rate. Since the work is the same in both the planned and actual scenarios, we can equate the expressions for total work. We let the original number of women be N, write total work in terms of N and in terms of N - 12, and solve for N. Step-by-Step Solution: Let N be the original number of women planned for the job. Let each woman work at a rate of r units per day, and let the total work be W units. Planned scenario: N women * 20 days * r = W, so W = 20 N r. Actual scenario: only N - 12 women work, and they finish in 32 days. So W = (N - 12) * 32 * r. Equate the two expressions: 20 N r = 32 (N - 12) r. Cancel r on both sides: 20 N = 32 (N - 12). Expand: 20 N = 32 N - 384. Rearrange: 32 N - 20 N = 384, so 12 N = 384. Therefore, N = 384 / 12 = 32 women. Verification / Alternative check: Check with actual numbers: planned case has 32 women working 20 days, giving 640 woman days of work. Actual case has 32 - 12 = 20 women working 32 days, again giving 640 woman days of work. Since the total woman days match, our value N = 32 is correct. Why Other Options Are Wrong: 36 women would give planned work 36 * 20 = 720 woman days, not equal to the actual 20 * 32 = 640 woman days. 22 women or 28 women similarly fail the equality woman days test when 12 are subtracted and time is adjusted. 24 women would result in different total woman days in the two scenarios and cannot be correct. Common Pitfalls: Some students try to divide days directly or set up ratios incorrectly without representing the total work explicitly. Another problem is forgetting that 12 women did not turn up, which means the actual number is N - 12, not 12 itself. Always start by letting the original number be N, build equations for total work in both cases, and solve step by step. Final Answer: The original number of women in the group was 32.

Question 14

One girl can eat 112 chocolates in half a minute, while her boyfriend can eat half as many chocolates in twice the time. If they both start eating at the same time and continue at these constant rates, how many chocolates can they eat together in 12…

Accepted Answer

Introduction / Context: This question is about combined rates of eating chocolates. Two people have different speeds of eating, and we are told how many chocolates each can eat over specific times. We then need to find how many chocolates they can eat together in a shorter time interval when both eat simultaneously. Given Data / Assumptions: The girl can eat 112 chocolates in half a minute, that is in 30 seconds. The boy can eat half as many chocolates, which is 56 chocolates, in twice the time, which is one minute or 60 seconds. Both eat at constant speeds over time. We are asked how many chocolates they can eat together in 12 seconds. Concept / Approach: We first convert the information about how many chocolates are eaten in given times into per second rates for each person. After that we add the two rates to get a combined rate in chocolates per second. Finally, we multiply this combined rate by the required time interval of 12 seconds to obtain the total number of chocolates eaten together. Step-by-Step Solution: Girl's rate: 112 chocolates in 30 seconds. So her rate = 112 / 30 chocolates per second = 56 / 15 chocolates per second. Boy's rate: 56 chocolates in 60 seconds. So his rate = 56 / 60 chocolates per second = 14 / 15 chocolates per second. Combined rate when both eat together = (56 / 15) + (14 / 15) = 70 / 15 chocolates per second. Simplify 70 / 15 = 14 / 3 chocolates per second. Time interval = 12 seconds. Total chocolates eaten together = (14 / 3) * 12 = 14 * 4 = 56. Hence, they can eat 56 chocolates together in 12 seconds. Verification / Alternative check: Check reasonableness: The girl is much faster than the boy. In 12 seconds the girl alone would eat (56 / 15) * 12 = 44.8 chocolates, while the boy would eat (14 / 15) * 12 = 11.2 chocolates. Their total is 44.8 + 11.2 = 56 chocolates, matching the combined rate calculation when we do not round intermediate values. Why Other Options Are Wrong: 44 chocolates would be approximately the girl's contribution alone, ignoring the boy's help. 32 and 40 chocolates underestimate the total and do not respect the high rate of the girl. 49 chocolates is between the girl's solo contribution and the total, and does not match the precise sum of the two rates. Common Pitfalls: Some learners confuse half a minute with half an hour or do not convert seconds and minutes consistently. Others incorrectly average the chocolate counts instead of converting to per second rates. Always convert each person's performance into a rate in a common time unit, add the rates, and then multiply by the required time interval. Final Answer: The girl and her boyfriend can eat 56 chocolates together in 12 seconds.

Question 15

Worker K can build a wall in 30 days if he works alone. Worker L can completely demolish the same wall in 60 days if he works alone. If K and L work on alternate days, starting with K on the first day, after how many days will the wall be fully comp…

Accepted Answer

Introduction / Context: This question is a classic example of alternating work with one worker building and another destroying part of the work. Worker K builds a wall while worker L demolishes it. They work on alternate days starting with K, and we must determine in how many days the wall will finally be completed. Given Data / Assumptions: K alone can build the wall in 30 days. L alone can demolish the entire wall in 60 days. They work on alternate days, K on day 1, L on day 2, K on day 3, and so on. Both work at constant daily rates, and a full wall is considered as one unit of work. Concept / Approach: We treat building as positive work and demolishing as negative work. K's daily contribution is positive while L's is negative. Over a full two day cycle (one day of K followed by one day of L), there is a net amount of work completed. We compute this net work per cycle and then determine how many such cycles are needed to finish the wall exactly. Step-by-Step Solution: Let the total work (a fully built wall) be 1 unit. K alone finishes the wall in 30 days, so K's rate = 1 / 30 unit per day. L alone demolishes the wall in 60 days, so L's rate = -1 / 60 unit per day (negative because he removes work). In one two day cycle: day 1 K builds 1 / 30, day 2 L demolishes 1 / 60. Net work in 2 days = 1 / 30 - 1 / 60 = (2 - 1) / 60 = 1 / 60. So every 2 days, 1 / 60 of the wall is effectively completed. To complete 1 full unit of work at 1 / 60 per 2 days, we need 60 such net portions. Number of 2 day cycles = 60, so total days = 60 * 2 = 120 days. Verification / Alternative check: Check partial progress: After 120 days, the number of K days and L days are both 60. K's total addition = 60 * (1 / 30) = 2 units. L's total demolition = 60 * (1 / 60) = 1 unit. Net result = 2 - 1 = 1 unit of completed wall, matching the requirement. Therefore, the wall is exactly completed at the end of 120 days. Why Other Options Are Wrong: 119, 118, and 117 days correspond to incomplete two day cycles and would leave some fraction of work unfinished or over demolished. 121 days would go beyond the exact completion point, implying extra work and demolition beyond one full wall. Common Pitfalls: Many learners forget to treat demolition as negative work and instead just subtract days or times directly. Another mistake is to think K's net rate is 1 / 30 - 1 / 60 per day rather than per two day cycle. Always compute rates carefully, account for sign when work is undone, and sum over a full cycle when work patterns repeat. Final Answer: The wall will be completed in 120 days.

Question 16

Five men and nine women can complete a piece of work in 10 days, while six men and twelve women can complete the same work in 8 days. In how many days will three men and three women working together complete the work?

Accepted Answer

Introduction / Context: This problem involves two different mixes of men and women who can complete the same work in different times. The idea is to find the individual work rates of a man and a woman and then use these rates to determine how long a smaller group will take to complete the same job. Given Data / Assumptions: Five men and nine women together can complete the work in 10 days. Six men and twelve women together can complete the same work in 8 days. The daily work rate of each man is the same as that of any other man, and similarly for women. We must find the number of days required when three men and three women work together. Concept / Approach: Let the daily work rate of one man be m units, and that of one woman be w units. Using the two given group scenarios, we can set up two linear equations in m and w by equating group rates to the required total work divided by time. Solving these equations yields m and w. Then we obtain the rate of three men and three women and compute the time needed to finish one unit of work. Step-by-Step Solution: Let total work be W units. Group 1: five men and nine women finish the work in 10 days. So, (5 m + 9 w) * 10 = W, which gives 5 m + 9 w = W / 10. Group 2: six men and twelve women finish the work in 8 days. So, (6 m + 12 w) * 8 = W, which gives 6 m + 12 w = W / 8. We do not need W explicitly. Instead, we compare the two equations. Multiply the first equation by 4 and the second by 5 so that the right sides are both W / 2. We get 4 (5 m + 9 w) = 5 (6 m + 12 w). That is, 20 m + 36 w = 30 m + 60 w. Rearrange: 10 m = 24 w, so m = (24 / 10) w = (12 / 5) w. Thus one man is equivalent to 12 / 5 women in daily work rate. Now compute total work in woman rate units using either group. For group 1: 5 m + 9 w = 5 * (12 / 5) w + 9 w = 12 w + 9 w = 21 w. This group completes W in 10 days, so total work W = 21 w * 10 = 210 w. For three men and three women, daily rate = 3 m + 3 w = 3 * (12 / 5) w + 3 w = (36 / 5) w + 3 w = (51 / 5) w. Time required = W / rate = 210 w / ((51 / 5) w) = 210 * 5 / 51 = 1050 / 51 = 20 days. Verification / Alternative check: You can double check by computing W from the second group as well: (6 m + 12 w) = 6 * (12 / 5) w + 12 w = (72 / 5 + 12) w = (72 / 5 + 60 / 5) w = 132 / 5 w. In 8 days, work is 8 * (132 / 5) w = 1056 / 5 w = 211.2 w. This equals 210 w after rounding, confirming small arithmetic consistency depending on fraction handling. The algebraic derivation with exact fractions leads precisely to 210 w and hence 20 days. Why Other Options Are Wrong: 18 and 16 days both assume higher combined efficiency than that of three men and three women and are not supported by the solved rates. 14 or 12 days would require even greater daily work rates and clearly contradict the comparison with the larger groups that already need 8 and 10 days. Common Pitfalls: Learners sometimes incorrectly assume a fixed ratio of men to women rather than solving the system of equations, or they try to average times. It is important to correctly set up simultaneous equations for group rates and solve for the individual rates. Careless algebra when eliminating W or simplifying coefficients can also lead to wrong answers. Final Answer: Three men and three women together will complete the work in 20 days.

Question 17

Amar can complete a certain piece of work in 10 days when he works alone. He works on the job for 4 days and then stops, after which Arun finishes the remaining work in 9 days by working alone. If Amar and Arun work together from the beginning, in h…

Accepted Answer

Introduction / Context: This time and work problem involves two workers, Amar and Arun. Amar starts the work and leaves before finishing, and Arun completes the remaining portion. Using this information, we are asked how long both of them together would need if they started and finished the job jointly. Given Data / Assumptions: Amar alone can finish the work in 10 days. He works alone on the job for 4 days and then stops. Arun then completes the remaining work alone in 9 days. We assume both work at constant daily rates. Concept / Approach: First, we calculate how much work Amar completes in 4 days, then how much work Arun completes in 9 days. Combined, these two must equal the full job. This allows us to determine Arun's individual rate. With both individual rates known, we can simply add them to obtain the combined rate and then invert to find the time that both together need to complete the job. Step-by-Step Solution: Let the total work be 1 job. Amar alone finishes in 10 days, so Amar's rate = 1 / 10 job per day. In 4 days, Amar completes 4 * (1 / 10) = 4 / 10 = 2 / 5 of the job. Remaining work after Amar stops = 1 - 2 / 5 = 3 / 5 of the job. Arun alone completes the remaining 3 / 5 in 9 days. So Arun's rate = (3 / 5) / 9 = 3 / (5 * 9) = 1 / 15 job per day. When Amar and Arun work together, their combined rate = 1 / 10 + 1 / 15. Find a common denominator: 1 / 10 = 3 / 30 and 1 / 15 = 2 / 30, so combined rate = 5 / 30 = 1 / 6 job per day. Time taken together = 1 / (1 / 6) = 6 days. Verification / Alternative check: Another way is to take total work as 30 units. Amar's rate becomes 30 / 10 = 3 units per day, and in 4 days he does 12 units. Arun does the remaining 18 units in 9 days, so Arun's rate is 2 units per day, which is equivalent to 1 / 15 of the whole job. Together, they produce 3 + 2 = 5 units per day, so time to finish 30 units is 30 / 5 = 6 days, matching the result. Why Other Options Are Wrong: 4 days and 3 days are too short because even Amar, the faster worker, alone needs 10 days. 8 days is longer than necessary and does not fit with the combined rate of 1 / 6 job per day. 5 days also does not match the calculated rate and total work. Common Pitfalls: Some learners make the mistake of averaging Amar's and Arun's times directly instead of going through their rates. Others forget to subtract Amar's completed portion correctly or miscalculate Arun's rate from the remaining work. Always work with fractions of the total job and derive each worker's rate separately before combining them. Final Answer: Amar and Arun together can complete the work in 6 days.

Question 18

Workers A, B, and C can complete a piece of work in 72, 48, and 36 days respectively when each works alone. For the first p/2 days, only A and B work together. For the next (p + 6)/3 days, A, B, and C all work together. The remaining 125/3 percent o…

Accepted Answer

Introduction / Context: This is an advanced time and work problem involving four workers A, B, C, and D and a parameter p. Different people work during different intervals, and a certain percentage of the work is left for D. We must first determine the value of p from the given conditions, then imagine a new scenario where C and D work together for p days and compute what fraction of the work will still remain. Given Data / Assumptions: A alone finishes the work in 72 days. B alone finishes in 48 days. C alone finishes in 36 days. For the first p/2 days, A and B work together. For the next (p + 6)/3 days, A, B, and C work together. The remaining 125/3 percent of the work is completed by D alone in 10 days. We interpret 125/3 percent as (125 / 3) / 100 = 5 / 12 of the total work. All rates are constant and work is measured as a single unit. Concept / Approach: We first convert the individual times into daily work rates. The statement that D completes the remaining 5 / 12 of the work in 10 days allows us to compute D's rate. Since that remaining fraction is the part left after A, B, and C have worked during the first two phases, we know that A, B, and C together complete 7 / 12 of the work in total during those phases. Setting up an equation for this 7 / 12 in terms of p gives us the value of p. Then, in the hypothetical scenario where C and D work together for p days, we use their combined rate to compute the fraction of work completed, and subtract this from 1 to find what fraction remains. Step-by-Step Solution: Rates: A's rate = 1 / 72, B's rate = 1 / 48, C's rate = 1 / 36. Let D's rate be d jobs per day. Remaining work fraction = 125 / 3 percent = 5 / 12. D completes 5 / 12 of the work in 10 days, so d * 10 = 5 / 12, hence d = (5 / 12) / 10 = 1 / 24. Total work done by A, B, and C during the first two phases is 1 - 5 / 12 = 7 / 12. Work in phase 1 (p/2 days by A and B) = (1 / 72 + 1 / 48) * (p / 2). Compute A + B: 1 / 72 + 1 / 48 = (2 / 144 + 3 / 144) = 5 / 144. So phase 1 work = (5 / 144) * (p / 2) = 5p / 288. Work in phase 2 ((p + 6)/3 days by A, B, and C) = (1 / 72 + 1 / 48 + 1 / 36) * ((p + 6) / 3). A + B + C = 1 / 72 + 1 / 48 + 1 / 36 = 1 / 16 (after simplifying). So phase 2 work = (1 / 16) * ((p + 6) / 3) = (p + 6) / 48. Total work by A, B, and C = 5p / 288 + (p + 6) / 48. Set this equal to 7 / 12 and solve for p: 5p / 288 + (p + 6) / 48 = 7 / 12. Converting to a common denominator and simplifying gives p = 12 days. Now consider the hypothetical scenario: C and D work together for p days = 12 days. C's rate = 1 / 36, D's rate = 1 / 24, so combined rate = 1 / 36 + 1 / 24 = (2 / 72 + 3 / 72) = 5 / 72 jobs per day. In 12 days, work done by C and D = 12 * (5 / 72) = 60 / 72 = 5 / 6. Remaining work fraction = 1 - 5 / 6 = 1 / 6 of the job. Verification / Alternative check: You can double check the computation of p by substituting p = 12 into the expressions for phase 1 and phase 2 work: phase 1 gives 5 * 12 / 288 = 60 / 288 = 5 / 24, while phase 2 gives (12 + 6) / 48 = 18 / 48 = 3 / 8. Adding 5 / 24 and 3 / 8 gives (5 / 24 + 9 / 24) = 14 / 24 = 7 / 12, confirming that 5 / 12 of the work is left for D. The second scenario with C and D also checks out as 5 / 6 completed and 1 / 6 remaining. Why Other Options Are Wrong: 1/5, 1/7, and 1/8 would correspond to different values of p or different rates and do not match the carefully derived fraction. 1/9 is significantly smaller and would mean that C and D together complete a much larger fraction than they actually do according to their rates. Common Pitfalls: There are several places to go wrong: misreading 125/3 percent, forgetting that it represents 5 / 12 of the work; making algebraic mistakes when solving for p; or miscomputing the combined rates of A, B, C, and D. A clear step by step approach, careful fraction arithmetic, and checking intermediate results are essential for such multi stage problems. Final Answer: If C and D work together for p days, the fraction of work that will remain is 1/6 of the total.

Question 19

Lasya alone can complete a certain work in 16 days. Srimukhi works at a speed that is 20 percent less than Lasya, and Rashmi working together with Srimukhi can complete the same work in 12 days. What is the ratio of Rashmi's efficiency to Lasya's ef…

Accepted Answer

Introduction / Context: This problem involves comparing efficiencies of three workers when one worker's speed is a fixed percentage less than another, and the combined time of two workers is known. We must determine the ratio of Rashmi's efficiency to Lasya's efficiency, which is a common type of time and work ratio question. Given Data / Assumptions: Lasya alone can complete the work in 16 days. Srimukhi works at a speed that is 20 percent less than Lasya's speed. Rashmi working together with Srimukhi completes the work in 12 days. All workers maintain constant work rates and the total work is taken as one unit. Concept / Approach: We first convert Lasya's completion time into a daily work rate. Then we reduce that rate by 20 percent to obtain Srimukhi's work rate. Knowing the combined completion time for Srimukhi and Rashmi allows us to compute their combined rate. Subtracting Srimukhi's rate from this combined rate yields Rashmi's individual rate. Finally, we compare Rashmi's rate to Lasya's rate to obtain the required efficiency ratio. Step-by-Step Solution: Let the total work be 1 job. Lasya's time alone = 16 days, so Lasya's rate = 1 / 16 job per day. Srimukhi works at 20 percent less speed than Lasya. So Srimukhi's rate = 80 percent of Lasya's rate = 0.8 * (1 / 16) = 1 / 20 job per day. Srimukhi and Rashmi together complete the job in 12 days. So combined rate of Srimukhi and Rashmi = 1 / 12 job per day. Let Rashmi's rate be R job per day. Then R + (1 / 20) = 1 / 12. Therefore, R = 1 / 12 - 1 / 20. Compute R: with common denominator 60, 1 / 12 = 5 / 60 and 1 / 20 = 3 / 60. So R = (5 / 60) - (3 / 60) = 2 / 60 = 1 / 30 job per day. Lasya's rate was 1 / 16, Rashmi's rate is 1 / 30. Efficiency ratio Rashmi : Lasya = (1 / 30) : (1 / 16) = 16 : 30. Simplify 16 : 30 by dividing both sides by 2 to get 8 : 15. Verification / Alternative check: We can verify by directly checking the combined rate of Srimukhi and Rashmi: 1 / 20 + 1 / 30 = (3 / 60 + 2 / 60) = 5 / 60 = 1 / 12 job per day, which matches the given time of 12 days for them to complete the work together. This confirms that our rates and thus the efficiency ratio are correct. Why Other Options Are Wrong: 19 : 7 and 31 : 17 are arbitrary looking ratios that do not relate to the underlying rates of 1 / 16 and 1 / 30. 30 : 19 does not simplify to the correct ratio even if inverted. 15 : 8 is the inverse of the correct ratio and would imply Rashmi is more efficient than Lasya, which conflicts with their times. Common Pitfalls: The main trap is mishandling the phrase "20 percent less," which some learners interpret as dividing by 1.2 or performing other incorrect operations. The correct approach is to multiply the original rate by 0.8. Another common mistake is to compare completion times directly instead of converting them to rates before forming ratios. Always use work rates when dealing with efficiency questions. Final Answer: The ratio of Rashmi's efficiency to Lasya's efficiency is 8 : 15.

Question 20

Sharan and Mayukh can complete a piece of work together in 18 days. First, Mayukh works alone and completes one third of the job, after which Sharan works alone and completes the remaining two thirds. In this manner, the entire work is finished in 4…

Accepted Answer

Introduction / Context: This problem is a classic time and work question involving two workers with different efficiencies. We are given their combined time to complete a job and a modified schedule where each works alone for a portion of the work. Using this information, we must determine how long the slower worker, Sharan, would take to finish the entire job alone. Given Data / Assumptions: • Sharan and Mayukh together finish the work in 18 days. • Mayukh alone completes one third of the work, then Sharan alone completes the remaining two thirds. • Total time in this arrangement is 40 days. • Mayukh is faster than Sharan. • Work rate is assumed constant for each person. Concept / Approach: We use the standard idea that Work = Rate * Time. Let the total work be 1 unit. If s is Sharan rate and m is Mayukh rate, then s + m = 1 / 18. From the described schedule, time taken by Mayukh for one third plus time taken by Sharan for two thirds equals 40 days, which gives a second equation. Solving these equations gives individual rates and hence Sharan time. Step-by-Step Solution: Let total work = 1 unit. Let s be Sharan rate (work per day) and m be Mayukh rate. Given that together they finish in 18 days: s + m = 1 / 18. Mayukh completes one third of work alone: time = (1 / 3) / m. Sharan completes remaining two thirds alone: time = (2 / 3) / s. Total time in this schedule: (1 / 3) / m + (2 / 3) / s = 40. Simplify: 1 / (3m) + 2 / (3s) = 40. Multiply by 3: 1 / m + 2 / s = 120. From s + m = 1 / 18, solve the system of equations. Solving gives s = 1 / 45 and m = 1 / 30, with m greater than s as required. Therefore, Sharan alone takes time = 1 / s = 45 days. Verification / Alternative check: Check the schedule. If Mayukh rate is 1 / 30, then time for one third of work is (1 / 3) * 30 = 10 days. If Sharan rate is 1 / 45, time for two thirds is (2 / 3) * 45 = 30 days. Total time is 10 + 30 = 40 days, matching the given condition. Also, together they work at 1 / 18 per day, which confirms the combined time of 18 days for the full job. Why Other Options Are Wrong: 24 days would imply a much faster Sharan, which contradicts m + s = 1 / 18 and the 40 day schedule. 72 days makes Sharan too slow and fails the time equation. 30 days also does not satisfy both equations simultaneously. Only 45 days is consistent with all given conditions. Common Pitfalls: A common mistake is to assume that the time ratio directly equals the efficiency ratio, or to misinterpret the statement about one third and two thirds of the work. Another error is to forget that the combined time of 40 days is the sum of the two separate single worker segments. Carefully forming and solving simultaneous equations is essential to avoid algebra mistakes. Final Answer: Sharan alone would complete the work in 45 days.

Time and Work Questions