Question 1

P can do a piece of work in 5 days less than Q. If both P and Q working together can complete the work in 11 1/9 days, in how many days will Q alone complete the same work?

Accepted Answer

Introduction / Context: This is a classic two person time and work problem where there is a given difference in the individual completion times and a given combined completion time. From this information we need to find one person's individual time. The data includes a mixed fraction for the combined time, which must be interpreted correctly. Questions of this type check understanding of reciprocals and the method of expressing rates in terms of unknown times and then solving a quadratic equation that arises naturally. Given Data / Assumptions: P alone takes 5 days less than Q to complete the work. P and Q together complete the work in 11 1/9 days, which is 100/9 days. All work rates are constant and linear with time. We must find the number of days Q alone will take. Concept / Approach: Let Q's time to complete the work alone be x days. Then P's time is x minus 5 days. Their daily work rates are the reciprocals of these times. Since together they take 100/9 days, their combined daily rate is 9/100 of the work per day. Setting the sum of the individual rates equal to this combined rate leads to a rational equation which simplifies to a quadratic in x. Solving that quadratic and selecting the physically meaningful positive root gives Q's time. Step-by-Step Solution: Step 1: Let Q alone take x days, so P alone takes x - 5 days. Step 2: Daily rate of Q is 1 / x and of P is 1 / (x - 5). Step 3: Together they take 100/9 days, so their combined rate is 1 divided by 100/9 = 9 / 100 per day. Step 4: Set up the equation 1 / x + 1 / (x - 5) = 9 / 100. Step 5: Combine fractions: (2x - 5) / [x(x - 5)] = 9 / 100. Step 6: Cross multiply to get 100(2x - 5) = 9x(x - 5). Step 7: Simplify: 200x - 500 = 9x^2 - 45x, giving 9x^2 - 245x + 500 = 0. Step 8: Solve this quadratic to get x = 25 or x = 20/9. The meaningful positive integer time is x = 25 days. Verification / Alternative check: If Q takes 25 days, P takes 25 - 5 = 20 days. P's daily rate is 1/20 and Q's daily rate is 1/25. Their combined daily rate is 1/20 + 1/25 = (5 + 4) / 100 = 9/100 of the work per day. The time taken together is therefore 1 divided by 9/100 which equals 100/9 days, the same as the given 11 1/9 days. This confirms that Q needing 25 days is consistent with all the data in the problem. Why Other Options Are Wrong: 24 days and 19 days would not produce a combined time of 11 1/9 days when used with P's time defined as x - 5. 20 days would imply P takes 15 days, and the resulting combined rate would not match 9/100 per day. Only 25 days correctly reproduces the given combined time when the implied value for P's time is used. Common Pitfalls: A common error is to misread 11 1/9 days and treat it as 11.9 days rather than 100/9 days. Another mistake is to forget that P's time is x minus 5 and accidentally use x plus 5. Finally, algebraic mistakes in solving the quadratic may lead to discarding the correct root. Careful conversion to improper fractions and careful algebra prevent these issues. Final Answer: Q alone will complete the work in 25 days.

Question 2

X can complete a work in 10 days and Y can complete the same work in 15 days. If both of them are hired together for 5 days, what fraction of the work will remain unfinished after those 5 days?

Accepted Answer

Introduction / Context: This is a basic work and time question involving two workers, X and Y, who have different completion times for the same job. We are asked to find the fraction of work unfinished after they work together for a given number of days. Such questions are often used early in aptitude tests to check whether candidates can handle simple rate addition and fraction calculations correctly. Given Data / Assumptions: X can complete the work alone in 10 days. Y can complete the same work alone in 15 days. X and Y work together for 5 days. Work rates are constant, and there is no change in efficiency over time. We need the fraction of the work that remains unfinished after 5 days. Concept / Approach: The core idea is to convert individual completion times into daily work rates and then add these rates to obtain the combined rate. Once we know how much of the work is completed per day by both workers together, multiplying that rate by the number of days worked gives the total fraction of the work completed. The unfinished fraction is then one minus the completed fraction. This straightforward method avoids any unnecessary complexity. Step-by-Step Solution: Step 1: Rate of X is 1 / 10 of the work per day. Step 2: Rate of Y is 1 / 15 of the work per day. Step 3: Combined daily rate of X and Y is 1/10 + 1/15. Step 4: Compute 1/10 + 1/15 = (3 + 2) / 30 = 5 / 30 = 1 / 6 of the work per day. Step 5: In 5 days, X and Y together complete 5 times 1/6 = 5/6 of the work. Step 6: Fraction of work remaining unfinished is 1 - 5/6 = 1/6 of the work. Verification / Alternative check: We can assign a convenient total work amount, such as 30 units. X then completes 3 units per day and Y completes 2 units per day, together completing 5 units per day. In 5 days they complete 25 units, leaving 5 units out of 30. The remaining fraction is 5/30 which simplifies to 1/6. This unit method exactly matches the fraction based solution and confirms the answer. Why Other Options Are Wrong: 1/3 of the work remaining would mean they completed only 2/3, which does not agree with the calculated 5/6 completion. 2/3 of the work remaining would mean very little progress, which is unrealistic given their combined rate. 5/6 of the work remaining would mean almost no work done, which is clearly inconsistent with working together for 5 days. Common Pitfalls: Some learners mistakenly add the times instead of the rates or forget to multiply the combined rate by the number of days worked. Others confuse the completed fraction with the remaining fraction and answer 5/6 instead of 1/6. Always compute what is completed first, then subtract from 1 to obtain what remains. Final Answer: After working together for 5 days, the fraction of work that remains unfinished is 1/6 of the work.

Question 3

Three men, four women and six children can together complete a piece of work in 9 days. Each woman does double the work that one man does, and each child does half the work that one man does. How many women working alone can complete the same work i…

Accepted Answer

Introduction / Context: This time and work question checks your understanding of equivalent work rates and how to convert a mixed group of men, women and children into a single type of worker. Such problems are very common in competitive exams, and the key idea is to express the work done by every category of worker in terms of one common base unit so that comparisons and calculations become simple. Given Data / Assumptions: Three men, four women and six children can complete the work in 9 days. One woman does double the work that one man does. One child does half the work that one man does. We assume constant efficiency for all workers throughout the work period. We need the number of women who can alone complete the same work in 9 days. Concept / Approach: The standard approach is to choose one worker type as the base unit of efficiency. Here, we take one man's daily work as 1 unit. Then we convert the work done by a woman and a child into equivalent man units. Once we know the total work in man units, we convert it into woman units and compute how many women working together for 9 days can finish the job. Step-by-Step Solution: Let the daily work done by one man = 1 unit. Then daily work done by one woman = 2 units (given as double a man). Daily work done by one child = 0.5 units (half the work of a man). Group daily work = 3 men + 4 women + 6 children. = 3 * 1 + 4 * 2 + 6 * 0.5 = 3 + 8 + 3 = 14 units per day. Total work in units = 14 * 9 = 126 units. Let the number of women needed = n. Daily work of n women = n * 2 units. In 9 days, work done by n women = n * 2 * 9 = 18n units. This must equal total work: 18n = 126. So, n = 126 / 18 = 7. Verification / Alternative check: We can check quickly: Seven women do 7 * 2 = 14 units per day, exactly the same daily work as the original group of 3 men, 4 women and 6 children. Since both do 14 units per day, and the original group completed the work in 9 days, seven women will also complete the work in 9 days. This confirms that our calculation is consistent and correct. Why Other Options Are Wrong: Each of 5, 6 and 8 women working alone would produce a different daily output from 14 units per day. For example, 6 women would do 12 units per day and take more than 9 days, while 8 women would do 16 units per day and finish in fewer than 9 days. Hence these options do not match the given condition of finishing in 9 days. Common Pitfalls: A common mistake is to forget that a woman and a child have different efficiencies from a man and to simply add the headcount. Another frequent error is to ignore the requirement that the work must still be completed in exactly 9 days, which is crucial for forming the equation. Carefully converting all workers into equivalent units avoids these errors. Final Answer: The number of women who can complete the work in 9 days is 7.

Question 4

Twenty men can complete a piece of work in 18 days, while eighteen women can complete the same work in 15 days. What is the ratio between the working capacity per day of a woman and that of a man?

Accepted Answer

Introduction / Context: This question tests your understanding of work rate comparison between two different types of workers, here men and women. Instead of asking for the total time to complete a job, the problem asks for the ratio of the capacities of individual workers. Such questions are very important in aptitude exams because they check whether you can convert group work information into per person efficiencies and then compare them properly. Given Data / Assumptions: Twenty men can complete the work in 18 days. Eighteen women can complete the same work in 15 days. All men have equal efficiency and all women have equal efficiency. The total work is the same in both situations. We need the ratio: (work done by one woman in one day) : (work done by one man in one day). Concept / Approach: Use the basic work formula: Work = Rate * Time. If a group completes a job, the total work can be expressed as (number of workers * individual rate * time). Since both groups complete the same work, we can equate the total work in man units and woman units and compare their individual rates. The unknowns here are the daily work rates of one man and one woman, but only the ratio is needed, which simplifies the calculation. Step-by-Step Solution: Let the work done by one man per day be m units. Let the work done by one woman per day be w units. Total work using men: 20 men * 18 days * m = 360m units. Total work using women: 18 women * 15 days * w = 270w units. Since both complete the same work, 360m = 270w. So, w / m = 360 / 270. Simplify: 360 / 270 = 36 / 27 = 4 / 3. Therefore, the ratio woman : man = 4 : 3. Verification / Alternative check: If you assume a convenient total work value, for example 360 units, then one man's rate becomes 360 / (20 * 18) = 1 unit per day, and one woman's rate becomes 360 / (18 * 15) = 4 / 3 units per day. This again gives the ratio woman : man = 4 : 3, confirming the previous result. Why Other Options Are Wrong: Ratios such as 3:4, 5:4 and 2:3 arise if you incorrectly invert the ratio or mix up the roles of men and women when equating the work. These options represent common handling mistakes in algebra or simplification but do not satisfy the equation 360m = 270w with correct orientation for woman : man. Common Pitfalls: Students sometimes compare total days directly rather than total work. Another frequent error is to compute m / w instead of w / m and then forget to invert the ratio. Careful interpretation of “capacity of a woman to that of a man” is essential: it clearly asks for woman : man, not man : woman. Final Answer: The ratio of the working capacity of a woman to that of a man is 4 : 3.

Question 5

Sixteen men and twelve women can together complete a piece of work in 20 days, and eighteen women alone can complete the same work in 40 days. In how many days will twelve men and twenty-seven women working together complete the work?

Accepted Answer

Introduction / Context: This is a classic combined time and work question involving two types of workers, men and women. The problem gives the time taken by a mixed group and by women alone, and then asks for the time taken by another mixed group. The key idea is to determine the relative efficiency of men in terms of women and then find how fast the new group can work together. Given Data / Assumptions: Sixteen men and twelve women finish the work in 20 days. Eighteen women alone finish the same work in 40 days. Efficiency of each man is the same among all men. Efficiency of each woman is the same among all women. All workers maintain constant efficiency throughout. We must find the time for twelve men and twenty-seven women together. Concept / Approach: We let the daily work of one woman be w units and that of one man be m units. Using the women-only scenario, we can express the total work in terms of w. Then we use the mixed scenario to find the relation between m and w. Once we know that, we can find the combined daily work of 12 men and 27 women and compute the required time using the formula Time = Total work / Daily work. Step-by-Step Solution: Let one woman do w units of work per day. Then total work W = 18 women * 40 days * w = 720w units. For 16 men and 12 women in 20 days: (16m + 12w) * 20 = W. So (16m + 12w) * 20 = 720w. Divide both sides by 20: 16m + 12w = 36w. Thus 16m = 24w, so m = (24 / 16)w = (3 / 2)w. Now for 12 men and 27 women: daily work = 12m + 27w. Substitute m = 1.5w: 12 * 1.5w + 27w = 18w + 27w = 45w. Total work W is still 720w, so time T = W / daily work = 720w / 45w = 16 days. Verification / Alternative check: You can quickly verify ratios: the group 16 men + 12 women is equivalent to 16 * 1.5 + 12 = 36 women. Our final mixed group 12 men + 27 women is equivalent to 12 * 1.5 + 27 = 45 women. Since 18 women finish in 40 days, total work in woman-days is 18 * 40 = 720 woman-days. For an equivalent of 45 women, time = 720 / 45 = 16 days, which matches our result. Why Other Options Are Wrong: Values such as 18 or 20 days arise if you miscalculate the ratio m : w or mistakenly treat 16 men and 12 women as equivalent to 28 workers without adjusting for differing efficiencies. 24 days is too large because 12 men and 27 women are more powerful than 16 men and 12 women, so they must take less than 20 days, not more. Common Pitfalls: A typical error is to assume that men and women are equally efficient and simply add the counts, which is incorrect here. Another mistake is to forget that total work remains constant, so once you express it in terms of one type of worker, you must consistently use that in all calculations. Final Answer: Twelve men and twenty-seven women together will complete the work in 16 days.

Question 6

A water tank has one inlet pipe and one outlet pipe. The inlet pipe can fill the tank completely in 2 hours when the outlet pipe is closed, and the outlet pipe can empty the full tank in 6 hours when the inlet pipe is closed. There is also a leakage…

Accepted Answer

Introduction / Context: This is a pipes and cisterns problem, a standard variation of time and work questions. Here, filling and emptying actions happen simultaneously because the inlet pipe, outlet pipe, and a leakage are all functioning together. You must convert each pipe and the leakage into a rate of work expressed as a fraction of the tank filled or emptied per hour and then combine these rates algebraically to find the net effect. Given Data / Assumptions: The inlet pipe fills the tank in 2 hours when working alone. The outlet pipe empties a full tank in 6 hours when working alone. The leakage drains water at half the rate of the outlet pipe. All rates are constant over time. The tank is initially empty, and all three are open together. Concept / Approach: When multiple pipes and leakages act on the same tank, the net rate is the algebraic sum of their individual rates. Filling rates are taken as positive, and emptying rates are taken as negative. Once the net rate of change of the volume of water is found, we use the formula Time = Total work / Net rate. Here, total work is one full tank, which we treat as 1 unit of work. Step-by-Step Solution: Let the capacity of the tank be 1 unit. Inlet pipe rate = 1 / 2 tank per hour (filling). Outlet pipe rate = 1 / 6 tank per hour (emptying). Leakage rate = half of outlet rate = (1 / 6) / 2 = 1 / 12 tank per hour (emptying). Net rate when all three act together = filling rate − outlet rate − leakage rate. Net rate = 1 / 2 − 1 / 6 − 1 / 12. Convert to common denominator 12: 1 / 2 = 6 / 12, 1 / 6 = 2 / 12, 1 / 12 = 1 / 12. Net rate = (6 / 12) − (2 / 12) − (1 / 12) = 3 / 12 = 1 / 4 tank per hour. Time taken to fill 1 tank = 1 / (1 / 4) = 4 hours. Verification / Alternative check: You can think in terms of actual tank fractions per hour: in 4 hours the inlet alone would fill 2 tanks, but during these 4 hours the outlet alone would empty 4/6 = 2/3 of a tank, and the leakage alone would empty 4 * (1 / 12) = 1/3 of a tank. Total emptied is 2/3 + 1/3 = 1 tank. Net effect is 2 tanks filled minus 1 tank emptied = 1 full tank effectively filled in 4 hours, which matches our answer. Why Other Options Are Wrong: 3 hours corresponds to a net rate of 1/3 tank per hour, which would imply the combined emptying effect is weaker than it actually is. 5 or 6 hours would mean the net rate is slower than 1/4 tank per hour, which contradicts the given strong inlet pipe of 1/2 tank per hour even after considering the outlet and leakage. Common Pitfalls: Students often forget to include the leakage as a separate emptying agent or misunderstand “half of the rate” and use an incorrect value. Another mistake is treating both the outlet and the leakage as positive when they actually reduce the water level. Always remember to subtract emptying rates from filling rates when calculating net effect. Final Answer: The empty tank will be completely filled in 4 hours when all three are operating together.

Question 7

Twenty-eight men and fifty-two women working together can complete a piece of work in 22.5 days. If thirty-five men and sixty-five women work together on the same work, in how many days will they complete it?

Accepted Answer

Introduction / Context: This is another mixed time and work problem involving men and women, but here you do not need to find individual efficiencies. Instead, the two groups of workers are in a fixed proportion, so their combined efficiencies scale linearly. This allows you to answer the question purely by reasoning with proportionality, which is a powerful shortcut in many aptitude problems. Given Data / Assumptions: Group 1: 28 men and 52 women finish the work in 22.5 days. Group 2: 35 men and 65 women will work on the same job. All men are equally efficient; all women are equally efficient. Efficiency of a man may differ from that of a woman, but this ratio is constant. We must find the number of days taken by Group 2. Concept / Approach: Instead of solving separate equations for men and women, notice that the composition of the two groups is proportional. If every component of Group 2 is a fixed multiple of the corresponding component of Group 1, then the total efficiency of Group 2 is the same multiple of the total efficiency of Group 1. This means the time taken is inversely proportional to that multiple, which makes the calculation quick and elegant. Step-by-Step Solution: Consider Group 1: 28 men and 52 women. Consider Group 2: 35 men and 65 women. Check proportionality: 35 / 28 = 5 / 4 = 1.25. Similarly, 65 / 52 = 5 / 4 = 1.25. So each count in Group 2 is 1.25 times the count in Group 1. Total efficiency is proportional to total workers with same efficiency pattern. Hence, Group 2 has 1.25 times the total efficiency of Group 1. Time is inversely proportional to efficiency, so T2 = T1 / 1.25. Given T1 = 22.5 days, T2 = 22.5 / 1.25. Compute: 22.5 / 1.25 = 22.5 * (4 / 5) = 90 / 5 = 18 days. Verification / Alternative check: If you wish, assign a combined rate R to Group 1. Then total work W = R * 22.5. Group 2 works at 1.25R, so time taken = W / (1.25R) = (R * 22.5) / (1.25R) = 22.5 / 1.25 = 18 days. This confirms the proportionality logic is sound. Why Other Options Are Wrong: 16 or 20 days might come from guessing or from incorrectly scaling only one of the worker types. 21 days would correspond to a smaller increase in efficiency than actually exists. Since every worker count is increased by a factor of 5/4, the total efficiency is exactly 1.25 times, which uniquely leads to 18 days. Common Pitfalls: A common mistake is to set up full equations for men and women and then incorrectly solve them. Another is to not notice the proportional relationship and end up overcomplicating the problem. Training yourself to spot proportional patterns saves time and reduces calculation errors in time and work questions. Final Answer: Thirty-five men and sixty-five women together will complete the work in 18 days.

Question 8

In a hostel there was food sufficient for 1000 students for one full month of 30 days. After 10 days, 1000 more students joined the hostel. From that point onward, for how many additional days will the students be able to manage with the remaining f…

Accepted Answer

Introduction / Context: This is a typical problem on the concept of “man-days” or in this case “student-days”. The idea is that the total quantity of food is fixed initially and can be expressed as the product of the number of consumers and the number of days that the food lasts. When the number of consumers changes, the duration for which the remaining food will last must be recalculated using the same total food concept. Given Data / Assumptions: Food is initially sufficient for 1000 students for 30 days. After 10 days, 1000 more students join the hostel. All students consume food at the same constant rate. We assume no wastage and no additional supply of food. We must find the number of days the remaining food will last after the extra students join. Concept / Approach: We use the concept of total student-days of food. Total food initially is equal to 1000 students * 30 days. Some of this food is consumed in the first 10 days by 1000 students. The remaining food is then divided among 2000 students. Using the formula Total food = number of students * number of days, we can compute how many more days the remaining food will last after the change in the number of students. Step-by-Step Solution: Total food in terms of student-days = 1000 students * 30 days = 30,000 student-days. In the first 10 days, 1000 students consume = 1000 * 10 = 10,000 student-days of food. Remaining food = 30,000 − 10,000 = 20,000 student-days. After 10 days, total students = 1000 original + 1000 new = 2000 students. Let the remaining number of days be D. Then 2000 * D = 20,000 student-days. So D = 20,000 / 2000 = 10 days. Verification / Alternative check: If the additional students had not joined, 1000 students with 20,000 student-days remaining would have managed for 20 more days, giving a total of 30 days as originally planned. Doubling the number of students halves the remaining duration, so the 20 days would become 10 days, which matches our calculation. Why Other Options Are Wrong: 15 or 20 days ignore the fact that the number of students doubles. 5 days is only one fourth of the original remaining time and would correspond to more than 4000 students, which is not the case. Only 10 days correctly respects the conservation of total student-days of food. Common Pitfalls: A frequent mistake is to subtract 10 from 30 and then randomly divide by 2, or to treat the 60 km style reasoning rather than student-days. Always remember that when population changes, the only invariant is total food measured in person-days, and you should form equations around that quantity. Final Answer: From the moment the new students join, the remaining food will last for 10 days.

Question 9

Fifty men can build a tank in 40 days. They all start the work together, but every 10 days, 5 men leave the job permanently. After how many total days will the tank be completely built?

Accepted Answer

Introduction / Context: This question is a variation of time and work problems where the workforce is not constant over time. Instead, workers leave periodically, and hence the effective daily work done decreases in steps. You must carefully account for the work done in each period with a different number of workers and sum up their contributions until the total work is completed. Given Data / Assumptions: Fifty men can complete the tank construction in 40 days if all remain. All 50 start together. Every 10 days, 5 men leave and do not return. All men have the same constant working efficiency. We must find the actual time required to finish the work with this decreasing workforce. Concept / Approach: First, calculate the total amount of work in man-days using the constant workforce scenario. Then, break the actual work period into segments of 10 days, during which the workforce is constant. For each segment, compute the work done using the number of men in that segment. Subtract these from the total work until the remaining work becomes zero. The sum of all segment lengths gives the total time required to complete the job. Step-by-Step Solution: Total work in man-days = 50 men * 40 days = 2000 man-days. First 10 days: 50 men work, so work done = 50 * 10 = 500 man-days. Men left = 50 − 5 = 45. Next 10 days: 45 men work, so work done = 45 * 10 = 450 man-days. Total = 950. Men left = 45 − 5 = 40. Next 10 days: 40 men work, so work done = 40 * 10 = 400 man-days. Total = 1350. Men left = 40 − 5 = 35. Next 10 days: 35 men work, so work done = 35 * 10 = 350 man-days. Total = 1700. Men left = 35 − 5 = 30. Remaining work = 2000 − 1700 = 300 man-days. With 30 men working, time = 300 / 30 = 10 days. Total time = 10 + 10 + 10 + 10 + 10 = 50 days. Verification / Alternative check: You can think of the pattern as five consecutive 10 day blocks with workforces 50, 45, 40, 35 and 30 men. The total man-days = 10 * (50 + 45 + 40 + 35 + 30) = 10 * 200 = 2000 man-days, exactly the required amount of work. This confirms that the work is completed right at the end of the fifth block, which is at 50 days. Why Other Options Are Wrong: 48 or 49 days would produce fewer than 2000 man-days, so the tank would still be incomplete. 47.5 days is not even aligned with the 10 day step pattern and would require fractional days in the middle of a segment, which does not match the way men leave the work (exactly every 10 days). Common Pitfalls: A common mistake is to assume the average workforce and then divide total work by this average, which does not always produce the correct result when the pattern of leaving is in discrete steps. Another mistake is to stop one step too early by miscalculating the remaining work after each segment. Final Answer: The tank will be completely built in 50 days.

Question 10

Adam and Smith are typing a project. Adam takes 6 hours to type 36 pages on a computer, while Smith takes 5 hours to type 40 pages on another computer. If both work together on two different computers, how much time will they take to type a 120-page…

Accepted Answer

Introduction / Context: This question is a time and work problem framed in terms of typing pages. Each person works on a separate computer, so their typing rates add together when they work at the same time. The key is to calculate each person's pages per hour, sum the rates, and then use the combined rate to compute the time required for the given total number of pages. Given Data / Assumptions: Adam types 36 pages in 6 hours. Smith types 40 pages in 5 hours. Both work simultaneously on separate computers. They must type a total of 120 pages. Typing speeds remain constant over time. Concept / Approach: First, calculate the individual rates in pages per hour. Then, since they are working together on the same project but on different computers, the total rate is the sum of the two rates. Finally, use the formula Time = Total work / Combined rate. Some fraction manipulation is needed to convert the result into hours and minutes for matching with the answer options. Step-by-Step Solution: Adam's rate = 36 pages / 6 hours = 6 pages per hour. Smith's rate = 40 pages / 5 hours = 8 pages per hour. Combined typing rate = 6 + 8 = 14 pages per hour. Total work = 120 pages. Time required T = Total work / Combined rate = 120 / 14 hours. Simplify 120 / 14 = 60 / 7 hours. Express 60 / 7 as a mixed number: 60 / 7 = 8 + 4 / 7. Now convert 4 / 7 hours to minutes: 4 / 7 * 60 minutes ≈ 34.2857 minutes. So T ≈ 8 hours and 34 minutes (a bit more than 34 minutes). Verification / Alternative check: To verify, convert 8 hours 34 minutes back to hours. 34 minutes is 34 / 60 ≈ 0.5667 hours. Total time ≈ 8.5667 hours. At 14 pages per hour, the total pages typed = 14 * 8.5667 ≈ 119.93 pages, which is very close to 120 pages, the minor difference being due to rounding the minutes. Hence, the nearest precise representation in the options is 8 hours 34 minutes. Why Other Options Are Wrong: 8 hours corresponds to typing only 14 * 8 = 112 pages. 8 hours 42 minutes is 8.7 hours, which yields 14 * 8.7 = 121.8 pages, more than needed. 8 hours 45 minutes is even longer. Therefore, among the given choices, only 8 hours 34 minutes correctly approximates 60 / 7 hours. Common Pitfalls: Many students mis-handle the fractional part 60 / 7 and incorrectly convert it into minutes, or they round too early, leading to an answer that does not match any option. It is better to keep the fraction until the last step and then convert the remainder separately into minutes. Final Answer: Working together on two different computers, Adam and Smith will complete the 120-page project in about 8 hours 34 minutes.

Question 11

Six boys and eight girls can complete a certain job in 10 days, and twenty-six boys and forty-eight women can complete the same job in 2 days. In how many days will fifteen boys and twenty girls together complete the job?

Accepted Answer

Introduction / Context: This is a mixed time and work problem involving boys and girls as separate worker types. You are given two scenarios where combinations of boys and girls complete the same job in different times. The goal is to determine how long another combination will take. The idea is to express total work in terms of daily efficiencies of boys and girls and then use both scenarios to find their relationship. Given Data / Assumptions: Six boys and eight girls finish the job in 10 days. Twenty-six boys and forty-eight women (treated as equivalent to girls in efficiency) finish the same job in 2 days. Each boy has the same efficiency; each girl has the same efficiency. Total work is the same in all scenarios. We must find the time taken by fifteen boys and twenty girls working together. Concept / Approach: Let the daily work done by one boy be b units and by one girl be g units. From each scenario, we can write an equation for the total work. Since the work is the same, we can equate these expressions and solve for the relationship between b and g. Then we calculate the combined daily work for fifteen boys and twenty girls and use Time = Work / Rate to get the required number of days. Step-by-Step Solution: Let one boy's daily work = b units and one girl's daily work = g units. From the first group: (6b + 8g) * 10 = W. So W = 60b + 80g. From the second group: (26b + 48g) * 2 = W. So W = 52b + 96g. Equate both expressions for W: 60b + 80g = 52b + 96g. Rearrange: 60b − 52b = 96g − 80g ⇒ 8b = 16g ⇒ b = 2g. Substitute g = b / 2 into W = 60b + 80g ⇒ W = 60b + 80 * (b / 2) = 60b + 40b = 100b. Now consider 15 boys and 20 girls: daily work = 15b + 20g. Replace g: 15b + 20 * (b / 2) = 15b + 10b = 25b. Thus time T = W / daily work = 100b / 25b = 4 days. Verification / Alternative check: You can think of each girl as half a boy in efficiency, since b = 2g implies g = 0.5b. Then in the final group, fifteen boys plus twenty girls is equivalent to 15 + 20 * 0.5 = 15 + 10 = 25 boys. Total work W was 100 boy-days, so with 25 equivalent boys, the time is 100 / 25 = 4 days, which matches our result. Why Other Options Are Wrong: 2 days or 3 days would require a much more powerful team than we have. 5 days would correspond to a lower effective workforce than 20 boys, which is not the case since fifteen boys and twenty girls are equivalent to 25 boys. Hence only 4 days is consistent with the computed efficiencies. Common Pitfalls: Students sometimes mix up the second scenario by reading “women” as a new worker type instead of treating them as girls with the same efficiency. Another pitfall is to equate only the brackets (6b + 8g) and (26b + 48g) instead of total work, which ignores the different durations and leads to incorrect relations. Final Answer: Fifteen boys and twenty girls together will complete the job in 4 days.

Question 12

Hari Ram works alone for 8 days on a job and finds that only one-third of the total work has been completed. He then employs Satya, who is 60% as efficient as Hari Ram. If Satya alone has to complete the remaining work, for how many additional days …

Accepted Answer

Introduction / Context: This problem combines the idea of partial work completed by one person and the remaining work completed by another person with a different efficiency. It tests your ability to translate verbal information about fractions of work and relative efficiencies into algebraic equations. The question specifically asks for the extra time taken by the second worker, Satya, working alone to complete the remaining part of the job. Given Data / Assumptions: Hari Ram works alone for 8 days. After 8 days, exactly one-third of the total work is completed. Satya is 60 percent as efficient as Hari Ram. Satya then works alone on the remaining work. We must find how many additional days Satya needs to finish the job. Concept / Approach: First, we deduce Hari Ram's daily work rate from the information that he completes one-third of the work in 8 days. Then we use the given efficiency comparison to find Satya's daily work rate. The remaining work is two-thirds of the total. Dividing this remaining work by Satya's rate gives the time he will take alone. Mixed numbers will appear naturally in this calculation, so they must be converted and simplified carefully. Step-by-Step Solution: Let the total work be 1 unit. Hari Ram completes 1 / 3 of the work in 8 days. So Hari Ram's daily work rate = (1 / 3) / 8 = 1 / 24 of the work per day. Satya is 60 percent as efficient as Hari Ram. So Satya's daily work rate = 60 percent * (1 / 24) = (60 / 100) * (1 / 24) = 3 / 5 * 1 / 24 = 1 / 40 of the work per day. Remaining work after Hari Ram's 8 days = 1 − 1 / 3 = 2 / 3 of the work. Let T be the number of days Satya needs to finish this remainder. Then T * (1 / 40) = 2 / 3. So T = (2 / 3) * 40 = 80 / 3 days. Convert 80 / 3 to a mixed number: 80 / 3 = 26 2 / 3 days. Verification / Alternative check: You can check by recombining both parts. Hari Ram's 8 days produce 8 * (1 / 24) = 1 / 3 of the work. Satya's 26 2 / 3 days produce (80 / 3) * (1 / 40) = 80 / 120 = 2 / 3 of the work. Total work = 1 / 3 + 2 / 3 = 1 unit, confirming that the entire job is completed. Why Other Options Are Wrong: Values like 24 1/2 or 24 2/3 days correspond to solving with an incorrect rate for Satya or miscomputing the remaining work fraction. 25 3/2 days is also an incorrect simplification of the fraction 80 / 3. Only 26 2/3 days satisfies the exact equation T / 40 = 2 / 3. Common Pitfalls: A common error is to interpret “60 percent as efficient” as meaning Satya completes 60 percent of the job in the same time that Hari Ram completes 100 percent, but then miscalculate the resulting rate. Another pitfall is to treat one-third as 0.3 instead of 1 / 3, which introduces rounding errors and wrong answers. Always work with exact fractions in such problems. Final Answer: Satya alone will need 26 2/3 days to complete the remaining work.

Question 13

Pavan and Sravan can complete a piece of work under two different efficiency arrangements. If Pavan works with his usual efficiency and Sravan works at twice his usual efficiency, they complete the work in 25 days. If Pavan works at twice his usual …

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Introduction / Context: Here we have a time and work problem involving variable efficiencies. The same two workers, Pavan and Sravan, work under two different efficiency arrangements, leading to two different completion times for the same job. Using these two conditions, we can solve for the individual daily work rates of each person and then determine how long Sravan alone would take to finish the work at his normal efficiency. Given Data / Assumptions: Case 1: Pavan at normal efficiency, Sravan at twice his normal efficiency, time = 25 days. Case 2: Pavan at twice his normal efficiency, Sravan at normal efficiency, time = 20 days. Both arrangements complete the same amount of work. Efficiencies remain constant within each scenario. We need Sravan's time working alone at his normal efficiency. Concept / Approach: Let P and S be the normal daily work rates of Pavan and Sravan respectively. In the first scenario, the combined rate is P + 2S, and in the second scenario, it is 2P + S. Using Work = Rate * Time, we express the same total work using both scenarios and form two equations. After solving these simultaneous equations for P and S, we can find the time taken by Sravan alone as 1 / S. Step-by-Step Solution: Let Pavan's normal daily work rate be p units. Let Sravan's normal daily work rate be s units. Total work W is the same in all cases. Case 1: (p + 2s) * 25 = W ⇒ p + 2s = 1 / 25. Case 2: (2p + s) * 20 = W ⇒ 2p + s = 1 / 20. Solve the system: p + 2s = 1 / 25 …(1) 2p + s = 1 / 20 …(2) From (1), p = 1 / 25 − 2s. Substitute in (2): 2(1 / 25 − 2s) + s = 1 / 20. ⇒ 2 / 25 − 4s + s = 1 / 20. ⇒ 2 / 25 − 3s = 1 / 20. ⇒ −3s = 1 / 20 − 2 / 25. Compute 1 / 20 − 2 / 25 = (5 / 100 − 8 / 100) = −3 / 100. So −3s = −3 / 100 ⇒ s = 1 / 100. Therefore Sravan's rate = 1 / 100 of the work per day. Time by Sravan alone = 1 / s = 100 days. Verification / Alternative check: Using s = 1 / 100, we can compute p from (1): p + 2 * (1 / 100) = 1 / 25 ⇒ p + 1 / 50 = 1 / 25 ⇒ p = 1 / 25 − 1 / 50 = 1 / 50. Check Case 2: 2p + s = 2 * (1 / 50) + 1 / 100 = 1 / 25 + 1 / 100 = 4 / 100 + 1 / 100 = 1 / 20, which matches equation (2). This confirms that the values of p and s are correct. Why Other Options Are Wrong: 50 days would require Sravan's rate to be 1 / 50, which would not satisfy the given relations. 70 or 35 days correspond to other incorrect combinations of p and s. Only 100 days is consistent with both scenarios and the simultaneous equations. Common Pitfalls: The main pitfalls include miswriting the equations, interchanging the coefficients of p and s, or making small errors when subtracting the fractions 1 / 20 and 2 / 25. Systematic use of fractional arithmetic or a common denominator avoids these mistakes. Final Answer: Sravan, working alone at his normal efficiency, will complete the work in 100 days.

Question 14

Priya can complete a piece of work in 16 days. Sai is 60 percent more efficient than Priya. In how many days will Sai alone complete the same work?

Accepted Answer

Introduction / Context: This question is a straightforward application of the relationship between efficiency and time. When a worker is more efficient, they can complete the same work in proportionally less time. The key here is to interpret “60 percent more efficient” correctly and then adjust the time needed for Sai using this efficiency factor. Given Data / Assumptions: Priya completes the work in 16 days. Sai is 60 percent more efficient than Priya. Both workers are assumed to have constant efficiency. The amount of work is the same in both cases. Concept / Approach: Efficiency is inversely proportional to time when the amount of work is fixed. If Sai is 60 percent more efficient than Priya, then Sai's efficiency is multiplied by a factor of 1.6 relative to Priya. Therefore, Sai's time will be Priya's time divided by 1.6. This proportional reasoning is much quicker than setting up detailed equations, though you can also use a rate and work approach if you prefer. Step-by-Step Solution: Let the total work be W units. Priya does W units in 16 days ⇒ Priya's daily work rate = W / 16. Sai is 60 percent more efficient ⇒ Sai's efficiency = 1.6 times Priya's. So Sai's daily work rate = 1.6 * (W / 16) = (1.6W) / 16. Simplify: 1.6 / 16 = 0.1 ⇒ Sai's rate = 0.1W per day = W / 10. Time taken by Sai alone = Total work / Sai's rate = W / (W / 10) = 10 days. Verification / Alternative check: Instead of using W, think in terms of ratios. Let Priya's efficiency be 1 unit. Then Sai's efficiency is 1 + 0.6 = 1.6 units. Time is inversely proportional to efficiency, so Sai's time : Priya's time = 1 : 1.6. Therefore Sai's time = 16 / 1.6 = 10 days, which matches the earlier calculation. Why Other Options Are Wrong: If Sai took 8 days, he would be exactly twice as efficient as Priya, that is 100 percent more efficient, not 60 percent. 12 or 14 days would mean his efficiency increase is far less than 60 percent. Only 10 days is consistent with the specified increase of 60 percent in efficiency. Common Pitfalls: A typical error is to treat 60 percent more efficient as meaning that Sai's time is 60 percent of Priya's time, which would give 16 * 0.6 = 9.6 days and then rounded incorrectly. The correct way is to scale efficiency, not time, and then use the inverse relationship between time and efficiency. Final Answer: Sai alone will finish the work in 10 days.

Question 15

Kaushalya can complete a piece of work in 20 days, while Kaikeyi can complete the same work in 25 days. They start the work together. After some time, Sumitra also joins them and together all three complete the entire work in a total of 10 days. If …

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Introduction / Context: This is a time and work question combined with a profit sharing or wage sharing concept. The workers are paid in proportion to the work they have done. Even though Sumitra joins later and we are not told exactly when, we can still find how much work must have been done by her based on the fact that the total job finishes in 10 days and the capacities of Kaushalya and Kaikeyi are known. Given Data / Assumptions: Kaushalya alone can complete the work in 20 days. Kaikeyi alone can complete the work in 25 days. They start together and work for the entire 10 days. Sumitra joins after some unknown number of days but stays until completion on the 10th day. Total payment is Rs. 700, distributed in proportion to each person's work. We must find Sumitra's share. Concept / Approach: The key idea is that we do not actually need to know the exact day Sumitra joined. Over 10 days, Kaushalya and Kaikeyi will each have done a fixed fraction of the total job because their rates and days of work are fully known. The remaining fraction of the work must then belong to Sumitra. Once we know the fraction of the total work done by Sumitra, we can multiply this fraction by the total payment to get her share. Step-by-Step Solution: Let total work be 1 unit. Kaushalya's daily work rate = 1 / 20. Kaikeyi's daily work rate = 1 / 25. They both work for the full 10 days. Work done by Kaushalya in 10 days = 10 * (1 / 20) = 1 / 2. Work done by Kaikeyi in 10 days = 10 * (1 / 25) = 2 / 5. Total work done by Kaushalya and Kaikeyi together = 1 / 2 + 2 / 5. Compute: 1 / 2 = 5 / 10 and 2 / 5 = 4 / 10, so sum = 9 / 10. Hence, remaining work done by Sumitra = 1 − 9 / 10 = 1 / 10. Sumitra's share of the work = 1 / 10 of total job. Total payment = Rs. 700, so Sumitra's share = 1 / 10 of 700 = Rs. 70. Verification / Alternative check: You can quickly verify that the other two shares sum to Rs. 630. Since their combined work is 9 / 10 of the job, the ratio of their combined payment to Sumitra's payment must be 9 : 1, i.e. 630 : 70 = 9 : 1, which is correct. This consistency check confirms that our approach is valid. Why Other Options Are Wrong: Rs. 130 or Rs. 185 would correspond to larger fractions of the work than 1 / 10, which contradicts the computed contributions of Kaushalya and Kaikeyi over 10 days. The option “can't be determined” is tempting but incorrect because the missing information about the exact joining day of Sumitra is not actually required for determining her share of work. Common Pitfalls: Many students try to first find the exact time when Sumitra joined the work, which is not necessary and can lead to algebraic confusion. Focusing instead on the total work done by those whose full working period is known (here Kaushalya and Kaikeyi) simplifies the problem significantly. Final Answer: Sumitra's share of the Rs. 700 is Rs. 70.

Question 16

Three taps P, Q and R can fill a tank in 12 hours, 15 hours and 20 hours respectively. Tap P is kept open all the time, while taps Q and R are opened for one hour each alternately, starting with tap Q. In how many hours will the tank be completely f…

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Introduction / Context: This is a pipes and cisterns problem with an alternating pattern. Tap P is always on, while taps Q and R are turned on alternately for one hour each. You must carefully consider the work done in each two hour cycle and then determine how many such cycles plus any extra time are needed to fill the tank fully. Given Data / Assumptions: Tap P alone fills the tank in 12 hours. Tap Q alone fills the tank in 15 hours. Tap R alone fills the tank in 20 hours. P is open continuously. Q and R are open alternately, each for one hour, starting with Q. The tank is initially empty. Concept / Approach: We first compute the filling rates of each tap as a fraction of the tank per hour. Then we analyse the pattern over a 2 hour cycle: in the first hour, taps P and Q are open; in the second hour, taps P and R are open. This gives a fixed amount of work per 2 hour cycle. We then find how many whole cycles are needed to approach full capacity and how much additional time is required in the next hour of the sequence to complete the tank. Step-by-Step Solution: Let tank capacity be 1 unit. Rate of P = 1 / 12 tank per hour. Rate of Q = 1 / 15 tank per hour. Rate of R = 1 / 20 tank per hour. Hour 1 (P + Q): work done = 1 / 12 + 1 / 15. Compute: LCM of 12 and 15 is 60, so 1 / 12 = 5 / 60, 1 / 15 = 4 / 60, total = 9 / 60 = 3 / 20. Hour 2 (P + R): work done = 1 / 12 + 1 / 20. LCM of 12 and 20 is 60, so 1 / 12 = 5 / 60, 1 / 20 = 3 / 60, total = 8 / 60 = 2 / 15. Thus, in each 2 hour cycle, work done = 3 / 20 + 2 / 15. Compute with LCM 60: 3 / 20 = 9 / 60 and 2 / 15 = 8 / 60, sum = 17 / 60. After 3 full cycles (6 hours), work done = 3 * 17 / 60 = 51 / 60 of the tank. Remaining work = 1 − 51 / 60 = 9 / 60 = 3 / 20. Next hour in sequence is again P + Q, whose rate is 3 / 20 per hour. Time required to do remaining 3 / 20 with rate 3 / 20 is exactly 1 hour. Total time = 6 hours + 1 hour = 7 hours. Verification / Alternative check: You can list the work hour by hour: hour 1 (P+Q) adds 3 / 20, hour 2 (P+R) adds 2 / 15, and so on. Summing after 7 hours gives exactly 1 tank full, confirming that there is no overfilling or shortfall. The cycle based approach is usually simpler and less error prone. Why Other Options Are Wrong: 9 hours and 11 hours are too long given the strong combined rates, while 13 hours is clearly excessive. Any time greater than 7 hours would exceed the amount of work required to fill one tank. On the other hand, less than 7 hours would not accumulate enough fraction of the tank when summing the contributions of each hour following the prescribed pattern. Common Pitfalls: A common mistake is to average the rates of Q and R instead of analysing the actual alternating pattern. Another error is to mistakenly assume that all three taps are open all the time, which would give a different time altogether. Always track the hour by hour pattern when taps or workers are operating alternately. Final Answer: The tank will be completely filled in 7 hours.

Question 17

P and Q can complete a piece of work in 4 hours and 12 hours respectively. P starts the work alone at 9:00 a.m., and they then work alternately for one hour each (first P, then Q, then P, and so on). At what time will the work be completed?

Accepted Answer

Introduction / Context: This problem combines time and work with an alternating work schedule. Workers P and Q have different efficiencies and work one hour each in turns. The problem asks you to track the cumulative work done and determine the exact time when the job finishes, starting from a known time (9:00 a.m.). Given Data / Assumptions: P alone can complete the work in 4 hours. Q alone can complete the work in 12 hours. P starts at 9:00 a.m., working for the first hour. Then Q works for the next hour, then P again, and so on, alternately. We assume constant efficiency for both P and Q. Concept / Approach: We first compute the hourly work rates of P and Q. Next, we determine the total work done in each 2 hour cycle (one hour by P and one hour by Q). Then we calculate how many such full cycles are needed to approach completion, and finally we see exactly at which hour the remaining work is completed. Since the cycle repeats every 2 hours, this approach is systematic and avoids confusion. Step-by-Step Solution: Let total work be 1 unit. P's rate = 1 / 4 work per hour. Q's rate = 1 / 12 work per hour. In the first hour (9 to 10 a.m.), P works: work done = 1 / 4. Second hour (10 to 11 a.m.), Q works: work done = 1 / 12. Total work in 2 hours = 1 / 4 + 1 / 12. LCM of 4 and 12 is 12, so 1 / 4 = 3 / 12, and total = 3 / 12 + 1 / 12 = 4 / 12 = 1 / 3. Thus, every 2 hours they finish 1 / 3 of the work. After 6 hours (three cycles), total work done = 3 * (1 / 3) = 1 full job. Duration from 9:00 a.m. to completion = 6 hours. Completion time = 9:00 a.m. + 6 hours = 3:00 p.m. Verification / Alternative check: You can check hour by hour: 9–10 a.m. P adds 1 / 4, 10–11 a.m. Q adds 1 / 12 (total 1 / 3), 11–12 a.m. P adds 1 / 4 (7 / 12), 12–1 p.m. Q adds 1 / 12 (2 / 3), 1–2 p.m. P adds 1 / 4 (11 / 12), and 2–3 p.m. Q adds 1 / 12, reaching exactly 1. This confirms that the work is complete at 3 p.m. Why Other Options Are Wrong: 12 p.m. corresponds to only 3 hours of work and yields 7 / 12 of the job, not complete. 1 p.m. yields only 2 / 3 of the work. Any time after 3 p.m. is too late and would imply doing more than the required amount of work, which is not considered in such problems. Common Pitfalls: Students sometimes average the times 4 hours and 12 hours, or they misinterpret the phrase “alternately for one hour each” and let both work together all the time. Another pitfall is losing track of hours and fractions of work as you progress through the alternating pattern. Writing out a small table of hours versus work done often prevents mistakes. Final Answer: The work will be completed at 3 p.m.

Question 18

Contractor K can lay a highway road between two cities in 16 days, and contractor L can lay the same road in 12 days. With the help of contractor M, all three together complete the job in 4 days. In how many days would M alone complete the work?

Accepted Answer

Introduction / Context: This is a standard time and work problem involving three workers. We know the individual times for two workers and the combined time when all three work together. The goal is to find the individual time for the third worker. This requires translating times into daily work rates, using the total work equation, and solving for the unknown rate of worker M. Given Data / Assumptions: K alone can complete the work in 16 days. L alone can complete the work in 12 days. K, L and M together complete the work in 4 days. All workers have constant efficiencies over time. The work is identical in all scenarios. Concept / Approach: Let the total work be 1 unit. The daily work rates of K and L can be written as the reciprocals of their times. The combined rate of K, L and M is also the reciprocal of 4 days. Using the fact that combined rate equals the sum of individual rates, we can solve for M's rate and then take the reciprocal to find his individual time. Step-by-Step Solution: Let total work = 1 unit. K's daily rate = 1 / 16 of the work. L's daily rate = 1 / 12 of the work. Combined rate of K, L and M = 1 / 4 of the work per day. Let M's daily rate be m units of work per day. Then 1 / 16 + 1 / 12 + m = 1 / 4. First, compute 1 / 16 + 1 / 12. LCM of 16 and 12 is 48, so 1 / 16 = 3 / 48 and 1 / 12 = 4 / 48. Sum = 3 / 48 + 4 / 48 = 7 / 48. So 7 / 48 + m = 1 / 4. Convert 1 / 4 to denominator 48: 1 / 4 = 12 / 48. Thus m = 12 / 48 − 7 / 48 = 5 / 48. Therefore M's daily rate is 5 / 48 of the work per day. Time taken by M alone = 1 / m = 1 / (5 / 48) = 48 / 5 days. Verification / Alternative check: Check the sum of the three rates: 1 / 16 + 1 / 12 + 5 / 48. Using denominator 48, this is 3 / 48 + 4 / 48 + 5 / 48 = 12 / 48 = 1 / 4, which corresponds exactly to 4 days for all three together. This confirms that the computed rate of M is correct. Why Other Options Are Wrong: 47/7 days, 59/6 days or 57/5 days are simply other fractions that do not satisfy the rate equation. Using any of those times for M would give a different value for m and the combined rate would no longer be 1 / 4 of the work per day. Common Pitfalls: Common mistakes include forgetting to convert all fractions to a common denominator when adding rates or misinterpreting “4 days together” as meaning something other than the reciprocal rate. It is crucial to do all fraction operations carefully and to remember that the total work has been assumed to be 1 unit for simplicity. Final Answer: M alone will complete the work in 48/5 days (which is 9.6 days).

Question 19

P, Q and R can together complete a job in 12 days. Their efficiencies are in the ratio 3 : 8 : 5 respectively. In how many days can Q alone complete the same job?

Accepted Answer

Introduction / Context: This question is a time and work problem where workers have efficiencies in a given ratio. Knowing the total time taken by all of them together, we can deduce the total combined rate and then extract the individual rate for Q using the ratio. Finally, we take the reciprocal of Q's rate to obtain the time taken by Q alone. Given Data / Assumptions: P, Q and R together complete the job in 12 days. Their efficiencies are in the ratio 3 : 8 : 5. All three work at constant efficiencies. We need the time taken by Q alone. Concept / Approach: If efficiencies are in the ratio 3 : 8 : 5, we can assume that P, Q and R do 3k, 8k and 5k units of work per day respectively for some constant k. Together their daily work rate is (3k + 8k + 5k) = 16k. Since they finish the job in 12 days, the total work equals 12 * 16k units. The daily work rate of Q is 8k, so Q's time alone is total work divided by 8k. This gives a simple ratio based calculation. Step-by-Step Solution: Assume P's daily rate = 3k, Q's daily rate = 8k, R's daily rate = 5k. Together, their daily rate = 3k + 8k + 5k = 16k. They complete the work in 12 days ⇒ Total work W = 16k * 12 = 192k units. Q's daily rate = 8k. Time taken by Q alone, T = W / Q's rate = 192k / (8k). Simplify: 192k / 8k = 24 days. Verification / Alternative check: We can check that P and R alone would take W / (3k) = 192k / 3k = 64 days and W / (5k) = 192k / 5k = 38.4 days respectively, which are longer than Q's 24 days, as expected because Q has the highest efficiency (8k) among the three. The numbers are consistent and reflect the efficiency ratio correctly. Why Other Options Are Wrong: 36 days and 30 days would correspond to smaller daily rates for Q than 8k and thus would not maintain the 3 : 8 : 5 ratio while still allowing all three together to finish in 12 days. 22 days is not a proper multiple stemming from the ratio and the total work calculation. Only 24 days fits the calculations based strictly on the given ratio and combined time. Common Pitfalls: Students sometimes mistakenly divide 12 by 8 or attempt to assign direct times from the ratio instead of working with rates. Always remember that efficiency ratios refer to rates (work per day), and total time must be derived by dividing total work by an individual rate. Final Answer: Q alone can complete the job in 24 days.

Question 20

Five boys and three girls together can cultivate a 23-acre field in 4 days, while three boys and two girls together can cultivate a 7-acre field in 2 days. How many girls must work together with seven boys so that they can cultivate a 45-acre field …

Accepted Answer

Introduction / Context: This is a time and work style problem framed in terms of area of field cultivated by boys and girls. You are given two scenarios involving different combinations of boys and girls and the corresponding areas they cultivate in given times. From this information, you must deduce the individual daily productivity of a boy and a girl and then determine how many girls are needed with seven boys to achieve a particular target area in a specified number of days. Given Data / Assumptions: Five boys and three girls cultivate 23 acres in 4 days. Three boys and two girls cultivate 7 acres in 2 days. All boys have equal efficiency; all girls have equal efficiency. We need to find how many girls must work with seven boys to cultivate 45 acres in 6 days. Work rates are constant. Concept / Approach: Let the daily work of one boy be b acres per day and that of one girl be g acres per day. Convert each scenario into an equation involving b and g by using the formula Work done = (total daily rate) * days. With two equations from the two scenarios, we can solve for b and g. Then apply these values to the final scenario, where seven boys and x girls work for six days to cultivate 45 acres, and solve for x. Step-by-Step Solution: Let one boy cultivate b acres per day. Let one girl cultivate g acres per day. Scenario 1: (5b + 3g) * 4 = 23 ⇒ 5b + 3g = 23 / 4. Scenario 2: (3b + 2g) * 2 = 7 ⇒ 3b + 2g = 7 / 2. We now solve the simultaneous equations: 5b + 3g = 23 / 4 …(1) 3b + 2g = 7 / 2 …(2) Multiply (2) by 2 to ease elimination: 6b + 4g = 7. Multiply (1) by 4: 20b + 12g = 23. Now solve directly with a simpler method: from the solution it turns out b = 1 acre per day and g = 0.25 acre per day (you can verify by substitution). Check: 5 * 1 + 3 * 0.25 = 5 + 0.75 = 5.75, and 5.75 * 4 = 23 acres, correct. Also 3 * 1 + 2 * 0.25 = 3 + 0.5 = 3.5, and 3.5 * 2 = 7 acres, correct. Now let the required number of girls be x working with 7 boys for 6 days to cultivate 45 acres. Daily work in this scenario = 7b + xg = 7 * 1 + x * 0.25 = 7 + 0.25x acres per day. Total work = (7 + 0.25x) * 6 = 45. So 42 + 1.5x = 45 ⇒ 1.5x = 3 ⇒ x = 2. Verification / Alternative check: With 7 boys and 2 girls, daily work = 7 * 1 + 2 * 0.25 = 7 + 0.5 = 7.5 acres per day. Over 6 days, they would cultivate 7.5 * 6 = 45 acres, exactly matching the requirement. This confirms that x = 2 is correct. Why Other Options Are Wrong: If x = 1, daily work = 7.25 acres, so in 6 days only 43.5 acres are cultivated. If x = 3, daily work = 7.75 acres, so in 6 days they cultivate 46.5 acres, overshooting the target. Options 3 and 4 therefore do not satisfy the equation, and so only x = 2 is valid. Common Pitfalls: Typical errors include incorrect algebra when solving the system for b and g, or mixing up acres with days. Some students assume boys and girls have equal efficiency, which is not supported by the data and leads to contradictions. Always derive the individual rates from the given scenarios rather than assuming them. Final Answer: Seven boys must work together with 2 girls to cultivate 45 acres in 6 days.

Time and Work Questions