Question 1

Twenty eight women can complete a certain piece of work in 36 days. If 4 women leave after working for 18 days, how many additional days will be needed to finish the remaining work?

Accepted Answer

Introduction / Context: This time and work problem involves a change in the size of the workforce halfway through a job. After a certain period some workers leave, and we are asked to compute the extra time needed to complete the remaining work with fewer workers. It tests understanding of total work in man days or woman days and the effect of changing the number of workers. Given Data / Assumptions: 28 women can complete the work in 36 days. All 28 women initially work for 18 days. After 18 days, 4 women leave, so 24 women continue. The work rate of each woman is constant throughout. We must find how many days the remaining 24 women need to complete the remaining work. Concept / Approach: We treat the total work in terms of woman days. If W is the total work, then W = number of women multiplied by number of days, assuming one unit of work per woman per day. After computing W, we calculate how much work is done in the first phase with 28 women for 18 days, subtract that from the total work and then divide the remainder by the daily capacity of 24 women to get the additional days required. Step-by-Step Solution: Step 1: Let work done per woman per day be 1 unit. Then total work W = 28 women * 36 days = 1008 woman days.Step 2: Work completed in the first 18 days with 28 women = 28 * 18 = 504 woman days.Step 3: Remaining work = total work - work already done = 1008 - 504 = 504 woman days.Step 4: After 4 women leave, number of women remaining = 28 - 4 = 24.Step 5: Daily work capacity of the remaining group = 24 woman days per day.Step 6: Days required to finish remaining work = remaining work / daily capacity = 504 / 24.Step 7: Compute 504 / 24. Divide numerator and denominator by 8: 504 / 24 = 63 / 3 = 21 days. Verification / Alternative check: We can check that the total calendar days are 18 (before women leave) plus 21 (after some leave) = 39 days. If 24 women had worked alone from the beginning at the same rate, time would have been total work divided by 24, that is 1008 / 24 = 42 days, which is longer. This makes sense because initially there were more workers. The internal calculation remains consistent, confirming that 21 days of additional work is correct. Why Other Options Are Wrong: 24 days: This would mean total woman days after reduction is 24 * 24 = 576, which exceeds the remaining work of 504. 27 days: This would give 24 * 27 = 648 woman days, also more than needed. 25 days: This implies 600 woman days, still more than 504, making it inconsistent with the original total work. Common Pitfalls: Some students miscalculate the total work or forget that the number of workers changes midway. Another frequent mistake is to average the number of workers over the entire duration instead of correctly splitting the work into phases. Always compute total work first, then account for each phase of the work with the correct number of workers. Final Answer: The remaining work will be completed in 21 additional days.

Question 2

A man travels from place A to place B in 26 hours. He covers one third of the distance at a speed of 4 km per hour and the remaining distance at a speed of 5 km per hour. What is the distance between A and B in kilometres?

Accepted Answer

Introduction / Context: This problem belongs to the time and distance category, where the journey is divided into segments with different speeds. The total travel time is given and we are asked to determine the total distance between two points. The important idea is that time equals distance divided by speed, and we must sum the times of each segment to match the total time. Given Data / Assumptions: Total time taken from A to B is 26 hours. One third of the distance is travelled at 4 km per hour. The remaining two thirds of the distance is travelled at 5 km per hour. Speeds are constant on each segment of the trip. Concept / Approach: Let the total distance between A and B be D kilometres. Then the first segment distance is D / 3 and the second segment distance is 2D / 3. Time taken for each segment is segment distance divided by segment speed. The sum of these two times must equal 26 hours. This gives a linear equation in D, which we can solve to obtain the total distance. Step-by-Step Solution: Step 1: Let total distance between A and B be D kilometres.Step 2: Distance of the first part = D / 3 kilometres, speed = 4 km per hour.Step 3: Time for first part = (D / 3) / 4 = D / 12 hours.Step 4: Distance of the second part = 2D / 3 kilometres, speed = 5 km per hour.Step 5: Time for second part = (2D / 3) / 5 = 2D / 15 hours.Step 6: Total time = time for first part + time for second part = D / 12 + 2D / 15.Step 7: This total time is given as 26 hours, so D / 12 + 2D / 15 = 26.Step 8: Use common denominator 60 to combine fractions: D / 12 = 5D / 60 and 2D / 15 = 8D / 60, total = 13D / 60.Step 9: So 13D / 60 = 26. Multiply both sides by 60: 13D = 26 * 60.Step 10: Compute 26 * 60 = 1560. Thus 13D = 1560, so D = 1560 / 13 = 120 kilometres. Verification / Alternative check: Check the total time with D = 120 kilometres. First segment distance = 120 / 3 = 40 km at 4 km per hour gives 40 / 4 = 10 hours. Second segment distance = 80 km at 5 km per hour gives 80 / 5 = 16 hours. Total time = 10 + 16 = 26 hours, exactly matching the given total. This confirms that the distance of 120 km is correct. Why Other Options Are Wrong: 45 km: With this distance, the computed time would be much less than 26 hours. 80 km: This gives first segment time 80 / 3 divided by 4 and second segment at 5 km per hour, which does not sum to 26 hours. 90 km: This also results in a total time that does not equal 26 hours when calculated segment by segment. Common Pitfalls: Some students mistakenly average the two speeds without weighting by distance or time. Others may incorrectly assume half the distance is travelled at each speed. It is important to respect the stated fractions of the distance and always write expressions for time as distance divided by speed, then sum them to match the given total time. Final Answer: The distance between A and B is 120 km.

Question 3

A alone completes 75 percent of a certain work in 50 days. He then calls in B, and together they finish the remaining work in 10 more days. How long would B alone take to complete the entire work?

Accepted Answer

Introduction / Context: This problem involves fractional completion of work by one person and then joint completion by two people. We are given the time taken by A to finish three quarters of the job and the time taken by A and B together to finish the remaining one quarter. Based on this, we are asked to find how long B alone would need to complete the entire job. Given Data / Assumptions: A completes 75 percent (that is 3/4) of the work in 50 days. The remaining 25 percent (that is 1/4) of the work is finished by A and B together in 10 days. Work rates of A and B are constant. We want the time B alone would require to complete 100 percent of the work. Concept / Approach: First, we determine A's work rate from the information that A finishes 3/4 of the work in 50 days. This gives us A's rate in terms of work per day. Next, we use the joint completion of the remaining 1/4 in 10 days to determine the combined rate of A and B. Subtracting A's rate from this combined rate yields B's rate. The reciprocal of B's rate then gives the time B would take alone to complete the entire job. Step-by-Step Solution: Step 1: Let the total work be 1 unit.Step 2: A completes 3/4 of the work in 50 days. So A's daily work rate = (3 / 4) / 50 = 3 / 200 of the work per day.Step 3: Remaining work = 1 - 3 / 4 = 1 / 4 of the job.Step 4: A and B together complete this remaining 1 / 4 in 10 days.Step 5: Combined daily rate of A and B = (1 / 4) / 10 = 1 / 40 of the work per day.Step 6: Rate of B alone = combined rate of A and B minus rate of A alone = 1 / 40 - 3 / 200.Step 7: Use denominator 200: 1 / 40 = 5 / 200, so rate of B = 5 / 200 - 3 / 200 = 2 / 200 = 1 / 100 of the work per day.Step 8: Time taken by B alone to finish the whole work = 1 divided by B's rate = 1 / (1 / 100) = 100 days. Verification / Alternative check: We can check by recomputing the phases. In 50 days, A does 50 * (3 / 200) = 150 / 200 = 3 / 4 of the work. In the next 10 days, A adds 10 * (3 / 200) = 30 / 200 = 3 / 20, and B adds 10 * (1 / 100) = 10 / 100 = 1 / 10. Together in the last 10 days they complete 3 / 20 + 1 / 10 = 3 / 20 + 2 / 20 = 5 / 20 = 1 / 4. So total work = 3 / 4 + 1 / 4 = 1, confirming that the rates are consistent and that B alone requires 100 days for the full job. Why Other Options Are Wrong: 200 days: This would imply a rate of 1 / 200 per day, half the correct rate, and would not match the given joint completion of the last quarter in 10 days. 50 days: This is the time A spends on the first 3/4 of the work and cannot be B's time for the full job. 125 days: This is slower than the correct value and would give too small a contribution from B in the final 10 days. Common Pitfalls: Many candidates incorrectly assume that the fraction completed by A in 50 days applies to the whole work done together with B, which is not true. Another error is to forget to convert percentages into fractions correctly. Always translate percentages (here 75 percent and 25 percent) into fractions, compute rates carefully, and keep track of each phase of the work. Final Answer: B alone would take 100 days to complete the entire work.

Question 4

Mehdi can complete a piece of work in 25 hours. Jahnavi is 50 percent more efficient than Mehdi. If they work together on the same work, in how many hours will they finish it?

Accepted Answer

Introduction / Context: This time and work problem introduces an efficiency comparison between two workers. One worker is said to be 50 percent more efficient than the other. We are given the time taken by the less efficient person alone and are asked to compute the time required when both work together. Given Data / Assumptions: Mehdi alone completes the work in 25 hours. Jahnavi is 50 percent more efficient than Mehdi. Both workers maintain constant efficiency and work rates. We need to find the time taken to complete the work when Mehdi and Jahnavi work together. Concept / Approach: If Jahnavi is 50 percent more efficient, it means her rate of working is 1.5 times Mehdi's rate. Since Mehdi's time for the full work is known, we can find his rate and then multiply by 1.5 to find Jahnavi's rate. The combined rate is the sum of the two rates, and the reciprocal of this combined rate gives the total time when they work together. Step-by-Step Solution: Step 1: Let total work be 1 unit.Step 2: Mehdi completes the work in 25 hours, so Mehdi's rate = 1 / 25 of the work per hour.Step 3: Jahnavi is 50 percent more efficient than Mehdi. That means her rate = 1.5 times Mehdi's rate.Step 4: Compute Jahnavi's rate = 1.5 * (1 / 25) = 3 / 50 of the work per hour.Step 5: Combined rate of Mehdi and Jahnavi = Mehdi's rate + Jahnavi's rate = 1 / 25 + 3 / 50.Step 6: Convert to common denominator 50: 1 / 25 = 2 / 50, so combined rate = 2 / 50 + 3 / 50 = 5 / 50 = 1 / 10 of the work per hour.Step 7: Time taken when both work together = 1 divided by combined rate = 1 / (1 / 10) = 10 hours. Verification / Alternative check: To verify, in 10 hours Mehdi completes 10 * (1 / 25) = 10 / 25 = 2 / 5 of the work. In the same 10 hours, Jahnavi completes 10 * (3 / 50) = 30 / 50 = 3 / 5 of the work. The total completed work is 2 / 5 + 3 / 5 = 1 full job, confirming that 10 hours is correct when both work together. Why Other Options Are Wrong: 12 hours: In 12 hours their total work would exceed 1, making this time too long relative to their combined rate. 3 hours: This is far too short for the work rates given and would require a much higher combined rate than 1 / 10. 9 hours: Their combined rate applied for 9 hours would produce only 9 / 10 of the work, not the full job. Common Pitfalls: Students sometimes misinterpret 50 percent more efficient as 50 percent of Mehdi's rate instead of 150 percent. The correct multiplier for 50 percent more is 1.5, not 0.5. Another pitfall is adding the times instead of the rates, which is incorrect. Always work with rates when multiple workers are involved simultaneously. Final Answer: Mehdi and Jahnavi together can complete the work in 10 hours.

Question 5

Twenty women can complete a piece of work in 16 days, while sixteen men can complete the same work in 15 days. What is the ratio of the working capacity (efficiency) of a man to that of a woman?

Accepted Answer

Introduction / Context: This question compares the efficiencies of men and women workers based on how many of each are needed to complete the same job in different numbers of days. It requires forming equations using the concept of total work as a product of number of workers, time and individual efficiency, and then finding the ratio of efficiencies. Given Data / Assumptions: 20 women can complete the work in 16 days. 16 men can complete the same work in 15 days. All women have the same efficiency as one another. All men have the same efficiency as one another. We are asked to find the ratio of the efficiency of a man to the efficiency of a woman. Concept / Approach: We denote the daily work rate of a woman by w and that of a man by m. The total work can be expressed in two different ways: once using women and once using men. Since both expressions represent the same amount of work, we can equate them and solve for the ratio m : w. The ratio of efficiencies is simply m divided by w, and we finally express it as a ratio of two integers. Step-by-Step Solution: Step 1: Let work done per woman per day be w units. With 20 women working for 16 days, total work W = 20 * 16 * w.Step 2: Let work done per man per day be m units. With 16 men working for 15 days, the same work W = 16 * 15 * m.Step 3: Equate the two expressions for total work: 20 * 16 * w = 16 * 15 * m.Step 4: Simplify the equation. First divide both sides by 16: 20 * w = 15 * m.Step 5: Rearranging for m in terms of w gives m = (20 / 15) * w = (4 / 3) * w.Step 6: Therefore the ratio of man's efficiency to woman's efficiency is m : w = (4 / 3) * w : w = 4 : 3. Verification / Alternative check: We can check this ratio by plugging back. If a woman does 3 units per day and a man does 4 units per day, then in 16 days, 20 women do 20 * 16 * 3 = 960 units. In 15 days, 16 men do 16 * 15 * 4 = 960 units. The total work is the same, which confirms that the ratio 4 : 3 is correct. Why Other Options Are Wrong: 3:4: This is the inverse ratio and would claim that women are more efficient than men in this setting, which does not match the calculated relation. 5:3: This suggests men are much more efficient than indicated by the total work comparison and would not balance the two work expressions. 5:7: This ratio would not equalize the total work; substituting it back would give different amounts of work done in the two scenarios. Common Pitfalls: A common mistake is to compare only the numbers of workers and the days without introducing the efficiency variables, or to attempt to directly divide 20 by 16 and 16 by 15 in an incorrect way. Another error is inverting the ratio at the end, giving the ratio of women to men instead of men to women. Always double check which order the ratio is requested in the question. Final Answer: The ratio of the efficiency of a man to that of a woman is 4:3.

Question 6

A mason can build a wall in 70 hours. After working continuously for 7 hours he takes a break. What fraction of the wall is still left to be built at that moment?

Accepted Answer

Introduction / Context: This is a straightforward work and time question involving a single worker. We are told how long the worker takes to complete the entire job and asked what fraction of the job remains after working for a shorter number of hours. It directly uses the concept of proportional completion of work over time for a constant rate worker. Given Data / Assumptions: The mason can complete the whole wall in 70 hours. He works for 7 hours and then stops. His working speed or rate remains constant. We are asked to find the fraction of the wall that is not yet built after these 7 hours. Concept / Approach: When a worker completes a job in T hours, in one hour the worker completes 1 / T of the job. Therefore, in t hours, the worker will complete t / T of the job. For this problem, we find the fraction of the job done in 7 hours and subtract this from 1 to obtain the incomplete fraction. The result is expressed as a decimal fraction, as shown in the options. Step-by-Step Solution: Step 1: Let the total work of building the wall be 1 unit.Step 2: Time required to complete the wall = 70 hours.Step 3: Therefore, in 1 hour, the mason completes 1 / 70 of the wall.Step 4: In 7 hours, the fraction of the wall completed = 7 * (1 / 70) = 7 / 70 = 1 / 10.Step 5: Fraction of the wall remaining = 1 - 1 / 10 = 9 / 10.Step 6: Express 9 / 10 as a decimal: 9 / 10 = 0.9. Verification / Alternative check: We can also note that 7 hours is exactly one tenth of the total required time, since 7 is 1 / 10 of 70. With a constant rate, in one tenth of the total time, the worker completes one tenth of the job. Thus 9 tenths remain, confirming that the fraction 0.9 is correct. Why Other Options Are Wrong: 0.8: This suggests that only 0.2 of the work has been completed, which would correspond to 14 hours rather than 7 hours. 0.5: This would mean half the wall is completed, which would require 35 hours of work, not 7 hours. 0.75: This suggests only one quarter of the wall has been completed, which would be 17.5 hours of work, again inconsistent with 7 hours. Common Pitfalls: Some learners mistakenly divide the completed fraction as 7 / 70 and then forget to subtract from 1 to find what is left, or they confuse the fraction completed with the fraction remaining. Others might try to convert hours in an unnecessary way that complicates the calculation. The best approach is to work directly with fractions of the total time. Final Answer: The fraction of the wall still left to be built is 0.9.

Question 7

A can paint a house in 42 days and B can paint the same house in 21 days. Along with C, they finish painting the house in 7 days. In how many days can C alone paint the house?

Accepted Answer

Introduction / Context: This is a classic three person time and work question. We are given the individual times for A and B, as well as the joint time for A, B and C together. The objective is to deduce the time required by C alone to paint the house, which is done by using the relationship between their individual and combined work rates. Given Data / Assumptions: A alone can paint the house in 42 days. B alone can paint the house in 21 days. A, B and C together paint the house in 7 days. Each painter works at a constant rate with no interruptions. Concept / Approach: As usual, we express each worker's efficiency as a rate equal to 1 divided by the time taken to complete the job. We know the combined rate of A, B and C together and subtract the known individual rates of A and B to isolate C's rate. The reciprocal of C's rate then gives the time required for C alone to complete the work. Step-by-Step Solution: Step 1: Let the total painting work be 1 unit.Step 2: Rate of A = 1 / 42 of the house per day.Step 3: Rate of B = 1 / 21 of the house per day.Step 4: Rate of A, B and C together = 1 / 7 of the house per day.Step 5: Combined rate of A and B = 1 / 42 + 1 / 21.Step 6: Convert to common denominator 42: 1 / 42 = 1 / 42 and 1 / 21 = 2 / 42, so combined rate of A and B = 3 / 42 = 1 / 14.Step 7: Rate of C alone = combined rate of A, B and C minus rate of A and B.Step 8: Rate of C = 1 / 7 - 1 / 14.Step 9: Convert 1 / 7 to denominator 14: 1 / 7 = 2 / 14, so rate of C = 2 / 14 - 1 / 14 = 1 / 14 of the house per day.Step 10: Time taken by C alone to complete the painting = 1 divided by (1 / 14) = 14 days. Verification / Alternative check: We can verify that the combined work in 7 days equals the whole job. In one day, A does 1 / 42, B does 1 / 21 and C does 1 / 14 of the work. Their total daily rate is 1 / 42 + 1 / 21 + 1 / 14. Using denominator 42, this is 1 / 42 + 2 / 42 + 3 / 42 = 6 / 42 = 1 / 7. Therefore, in 7 days they do 7 * (1 / 7) = 1 full job, confirming that the calculation is consistent and C alone needs 14 days. Why Other Options Are Wrong: 9 days: This would imply a faster rate and would change the combined rate of A, B and C, so the total time would not be 7 days. 12 days: This gives rate 1 / 12 which is different from 1 / 14 and does not fit the given combined completion time. 15 days: This rate 1 / 15 makes the combined rate smaller, leading to a total time greater than 7 days for the three workers. Common Pitfalls: A frequent mistake is to average the individual times instead of working with rates. Another error is failing to convert fractions to a common denominator when adding or subtracting. Always remember that for multiple workers, it is the rates that add and subtract, and careful fraction handling is essential. Final Answer: C alone can paint the house in 14 days.

Question 8

Sashi can do a piece of work in 20 days. Tanya is 25 percent more efficient than Sashi. How many days will Tanya alone take to complete the same work?

Accepted Answer

Introduction / Context: This question deals with comparative efficiency. We are told that one worker is 25 percent more efficient than another and given the time taken by the less efficient worker. The aim is to find how long the more efficient worker will take to complete the same job alone. Given Data / Assumptions: Sashi can complete the work in 20 days. Tanya is 25 percent more efficient than Sashi. Both maintain a constant working rate. We must determine the time required by Tanya alone to complete the entire work. Concept / Approach: If Tanya is 25 percent more efficient, her work rate is 1.25 times the work rate of Sashi. Efficiency is directly proportional to work rate and inversely proportional to the time needed for the job. Therefore, if the work rate increases by a factor, the time taken decreases by the same factor. We first calculate Tanya's rate as a multiple of Sashi's rate and then obtain her time as total work divided by her rate. Step-by-Step Solution: Step 1: Let the total work be 1 unit.Step 2: Sashi completes the work in 20 days, so Sashi's rate = 1 / 20 of the work per day.Step 3: Tanya is 25 percent more efficient than Sashi. This means Tanya's rate = 1.25 times Sashi's rate = 1.25 * (1 / 20).Step 4: Convert 1.25 to fraction 5 / 4. So Tanya's rate = (5 / 4) * (1 / 20) = 5 / 80 = 1 / 16 of the work per day.Step 5: Time taken by Tanya alone = 1 divided by her rate = 1 / (1 / 16) = 16 days. Verification / Alternative check: We can compare the times directly using the efficiency ratio. If Sashi takes 20 days, and Tanya is 1.25 times as efficient, Tanya should take 20 / 1.25 days. Compute 20 / 1.25 = 20 * (4 / 5) = 16, which matches the rate based calculation. This confirms that 16 days is correct. Why Other Options Are Wrong: 15 days: This would imply a work rate greater than 1 / 16 and therefore an efficiency gain larger than 25 percent. 18 days: This corresponds to a rate of 1 / 18, which is less efficient than 1 / 16, so it does not match the stated 25 percent increase. 25 days: This is longer than Sashi's time, which contradicts Tanya being more efficient. Common Pitfalls: Some students treat 25 percent more efficient as meaning Tanya's time is 25 percent less and then subtract 25 percent of 20 from 20 without checking the correct factor, which would give 15 days. While this sometimes works, it can lead to confusion. It is safer to work through rates and use multiplicative factors, especially for more complicated percentage changes. Final Answer: Tanya alone will take 16 days to complete the work.

Question 9

Eighteen men or thirty six boys working 6 hours per day can plough a field in 24 days. In how many days will 24 men and 24 boys working 9 hours per day plough the same field?

Accepted Answer

Introduction / Context: This time and work question compares the work capacities of men and boys. We are given that either a group of men or a group of boys can complete a task in the same number of days. From this, we can determine the relative efficiency of a man and a boy. Then we use that relationship to find how long a mixed group of men and boys will take to complete the same job when working different hours per day. Given Data / Assumptions: 18 men working 6 hours per day can plough the field in 24 days. 36 boys working 6 hours per day can also plough the field in 24 days. We assume all men have the same efficiency and all boys have the same efficiency. We need to find the time for 24 men and 24 boys working 9 hours per day to plough the same field. Concept / Approach: First, we express the total work in terms of man hours or boy hours. Using the equality of total work from the two given scenarios, we deduce the efficiency ratio of a man to a boy. Then, we compute the total daily work capacity of the mixed group of 24 men and 24 boys working 9 hours per day and divide the total work by this daily capacity to obtain the required number of days. Step-by-Step Solution: Step 1: Let the work done per man per hour be m units and per boy per hour be b units.Step 2: From the first case, total work W = 18 men * 6 hours per day * 24 days * m = 18 * 6 * 24 * m.Step 3: From the second case, W = 36 boys * 6 hours per day * 24 days * b = 36 * 6 * 24 * b.Step 4: Equating the two expressions for W: 18 * 6 * 24 * m = 36 * 6 * 24 * b.Step 5: Cancel the common factor 6 * 24 on both sides to get 18m = 36b.Step 6: Therefore m = 2b, so one man is equivalent to two boys in hourly efficiency.Step 7: Now compute total work W explicitly in terms of b. Using the boys expression: W = 36 * 6 * 24 * b = 36 * 144 * b = 5184b units.Step 8: For the new group, we have 24 men and 24 boys working 9 hours per day.Step 9: Convert men to equivalent boys. Since m = 2b, 24 men contribute the same as 24 * 2 = 48 boys per hour.Step 10: Total workers in equivalent boy units = 48 (from men) + 24 (actual boys) = 72 boy units per hour.Step 11: Working 9 hours per day, total daily work capacity = 72 * 9 * b = 648b units per day.Step 12: Number of days required = total work / daily work capacity = 5184b / 648b = 8 days. Verification / Alternative check: We can check the proportionality. In the original scenario, if we convert 18 men to equivalent boys, that is 18 * 2 = 36 boys, which matches the second given scenario. The mixed group is larger in equivalent capacity than either original group, and also works more hours per day, so fewer days are expected. Eight days is a reasonable and consistent result after the rate calculations. Why Other Options Are Wrong: 9 days: This would require a slightly smaller daily capacity or a larger total work, which is not consistent with the given data. 10 days: This would suggest the mixed group is less efficient than computed and contradict the equality of the original scenarios. 6 days: This time is too short for the computed daily capacity and would imply total work smaller than 5184b. Common Pitfalls: A common error is to ignore the hours per day and compare only numbers of workers and days. Another mistake is misinterpreting the equality of work done by 18 men and 36 boys, leading to an incorrect ratio of man efficiency to boy efficiency. Always include the hours per day and carefully equate total work for the different groups before moving on to the final calculation. Final Answer: The field will be ploughed in 8 days by 24 men and 24 boys working 9 hours per day.

Question 10

Anil is twice as efficient a workman as Bimal and together they finish a piece of work in 9 days. In how many days will Anil alone finish the work?

Accepted Answer

Introduction / Context: This is a typical work and time question involving relative efficiency. One worker is twice as efficient as another, and together they complete a job in a known number of days. The problem is to find the time required by the more efficient worker alone to complete the entire job. Given Data / Assumptions: Anil is twice as efficient as Bimal. Anil and Bimal together complete the work in 9 days. Both have constant work rates during the entire period. We are required to find the number of days taken by Anil alone to do the work. Concept / Approach: If Anil is twice as efficient as Bimal, then Anil's rate is 2 times Bimal's rate. Let us denote Bimal's rate as x units of work per day and Anil's rate as 2x units of work per day. Together their rate is 3x units per day. Since they complete the entire work in 9 days, we can determine the total work in terms of x and then find the time Anil would need alone using his rate. Step-by-Step Solution: Step 1: Let Bimal's work rate be x units per day.Step 2: Since Anil is twice as efficient, his rate = 2x units per day.Step 3: Combined rate of Anil and Bimal = x + 2x = 3x units per day.Step 4: Time taken together to finish the work = 9 days. Therefore total work W = 3x * 9 = 27x units.Step 5: Time taken by Anil alone = total work / Anil's rate = 27x / (2x) days.Step 6: Simplify 27x / (2x) = 27 / 2 = 13.5 days. Verification / Alternative check: We can check by computing daily contributions. In 13.5 days, Anil alone working at 2x units per day would complete 13.5 * 2x = 27x units of work, which matches the total work computed earlier. Also, working together at 3x units per day for 9 days gives 9 * 3x = 27x units, again matching. This consistency confirms that Anil needs 13.5 days alone to finish the same work. Why Other Options Are Wrong: 12.5 days: This would mean total work is 25x, but our combined work in 9 days would then not match 25x, contradicting the given condition. 11.5 days: This gives total work 23x, inconsistent with the requirement that together they finish in 9 days at rate 3x per day. 10.5 days: This yields 21x total work for Anil, again inconsistent with the 27x units required when both work for 9 days. Common Pitfalls: Some students invert the efficiency relationship and assume that if Anil is twice as efficient, he will take twice as long, which is incorrect. Efficiency and time for the same job are inversely related. Others attempt to average times directly without considering rates. Carefully define rates, use them to find total work, and then derive the required time. Final Answer: Anil alone will finish the work in 13.5 days.

Question 11

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

Accepted Answer

Introduction / Context: This is a classic three worker time and work problem where the pairwise times for different pairs of workers are given. The goal is to determine the time required when all three workers collaborate. It involves solving a small system of equations based on work rates rather than times. Given Data / Assumptions: A and B together complete the work in 36 days. B and C together complete the work in 24 days. A and C together complete the work in 18 days. Each worker maintains a constant rate of work. We must find the time taken if A, B and C all work together. Concept / Approach: Let the individual daily work rates of A, B and C be a, b and c units per day respectively. Then the combined rates for the given pairs are a + b, b + c and a + c. These three expressions correspond to 1 / 36, 1 / 24 and 1 / 18 respectively. By adding these three equations and simplifying, we can find the total rate a + b + c, and then compute the time taken by all three together as the reciprocal of this total rate. Step-by-Step Solution: Step 1: Let total work be 1 unit. Let daily work rates of A, B and C be a, b and c respectively.Step 2: From the data, we have three equations based on pairwise work:(i) a + b = 1 / 36(ii) b + c = 1 / 24(iii) a + c = 1 / 18Step 3: Add all three equations: (a + b) + (b + c) + (a + c) = 1 / 36 + 1 / 24 + 1 / 18.Step 4: On the left side, this becomes 2(a + b + c). So 2(a + b + c) = 1 / 36 + 1 / 24 + 1 / 18.Step 5: Find the sum on the right. The common denominator of 36, 24 and 18 is 72 or 360, but we can compute directly. For convenience, use 72: 1 / 36 = 2 / 72, 1 / 24 = 3 / 72, and 1 / 18 = 4 / 72.Step 6: Sum = 2 / 72 + 3 / 72 + 4 / 72 = 9 / 72 = 1 / 8.Step 7: Therefore 2(a + b + c) = 1 / 8, giving a + b + c = 1 / 16.Step 8: Thus the combined daily rate of A, B and C together is 1 / 16 of the work per day.Step 9: Time taken when A, B and C work together = 1 divided by 1 / 16 = 16 days. Verification / Alternative check: We can quickly verify the arithmetic using another denominator like 360 to ensure the fraction sum is correct. Using 360, 1 / 36 = 10 / 360, 1 / 24 = 15 / 360, 1 / 18 = 20 / 360. Their sum is 45 / 360 = 1 / 8, which matches our earlier calculation. Therefore a + b + c = 1 / 16 is reliable, and 16 days is the correct combined time. Why Other Options Are Wrong: 8 days: This corresponds to a rate of 1 / 8 per day, which is double the actual combined rate and inconsistent with the pairwise times. 30 days: This implies a very small combined rate, which would contradict the relatively short pairwise completion times. 32 days: This is also too long and not supported by the combined rate deduced from the three pairwise values. Common Pitfalls: Students may try to average the given times instead of working with rates, or they may incorrectly subtract equations, leading to errors in the total rate. Another common issue is adding fractions carelessly without a correct common denominator. Always use rates, not times, and handle fractions with care to avoid arithmetic mistakes. Final Answer: All three working together will finish the work in 16 days.

Question 12

A and B together can complete a piece of work in 24 days. B alone completes one-third of the same work in 12 days. In how many days will A alone complete the remaining two-thirds of the work?

Accepted Answer

Introduction / Context: This time and work question compares the individual efficiency of two workers, A and B, using information about how long they take together and how much of the work B can do alone. Questions of this type are very common in aptitude exams because they test your ability to convert verbal statements into work rates and then combine or separate those rates correctly. Given Data / Assumptions: A and B together complete the entire work in 24 days. B alone completes one-third of the same work in 12 days. Work rate is assumed to be constant over time. We are asked to find the number of days A alone will take to complete the remaining two-thirds of the work. Concept / Approach: The main ideas are: total work is treated as one unit, each person has a steady work rate, and time is calculated as work divided by rate. First we convert the information about B into a daily work rate. Then we compute the combined rate of A and B using their joint time. Finally we subtract B's rate from the combined rate to get A's rate, and use that to find how long A takes to finish the remaining part of the work. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: B completes one-third of the work in 12 days, so B's daily rate = (1/3) / 12 = 1/36 of the work per day. Step 3: A and B together complete the entire work in 24 days, so their combined daily rate = 1 / 24. Step 4: A's rate = (A + B) rate − B's rate = 1/24 − 1/36. Step 5: Compute A's rate: 1/24 − 1/36 = (3 − 2) / 72 = 1/72 of the work per day. Step 6: After B finishes one-third of the work, the remaining work = 1 − 1/3 = 2/3 of the work. Step 7: Time for A to finish the remaining 2/3 = (2/3) / (1/72) = (2/3) × 72 = 48 days. Verification / Alternative check: We can check consistency by working backward. If A's rate is 1/72 per day, in 48 days A completes (1/72) × 48 = 48/72 = 2/3 of the work. B had already done 1/3, so together that makes 1 full work. The given data and computed rates are consistent, so the answer 48 days is correct. Why Other Options Are Wrong: 36 days: With A's rate of 1/72, in 36 days A would complete only 36/72 = 1/2 of the work, not two-thirds. 54 days: This would give (1/72) × 54 = 3/4 of the work, which is too much. 60 and 72 days would both make A extremely slow relative to B and contradict the given combined time of 24 days. Therefore, only 48 days fits all the given information. Common Pitfalls: A frequent mistake is to assume that since B does one-third in 12 days, the remaining two-thirds should be done in simply double that time. However, the remaining work is done by A alone, whose rate is different from B's rate. Another common error is to treat 24 days as if A and B each took 24 days individually, which is not stated. Always convert everything into daily rates and use proper addition or subtraction of those rates. Final Answer: A alone will take 48 days to complete the remaining two-thirds of the work.

Question 13

A school currently has 8 periods of 45 minutes each every day. If the total working hours of the school remain the same but the number of periods is increased to 9 per day, what will be the duration of each period?

Accepted Answer

Introduction / Context: This is a classic time and work style question framed around school periods. It tests the concept that when total work or total time is fixed, changing the number of equal parts changes the size of each part. Here, the total school time per day is fixed, but the number of periods increases, so each period must become shorter. Given Data / Assumptions: Initially there are 8 periods each day. Each of these 8 periods is 45 minutes long. The total working time of the school per day remains unchanged. We change the timetable to have 9 periods in the same total time. Each of the 9 periods is assumed to have equal duration. Concept / Approach: The total daily school time is the product of the number of periods and the length of each period. Since that total is fixed, we compute it using the original structure and then divide by the new number of periods to get the new duration. This involves basic multiplication and division of time, without any advanced algebra. Step-by-Step Solution: Step 1: Compute total school time with the original schedule: 8 periods × 45 minutes per period. Step 2: 8 × 45 = 360 minutes of school time per day. Step 3: With the new schedule, this same 360 minutes will be split into 9 equal periods. Step 4: New period length = total minutes / number of periods = 360 / 9 minutes. Step 5: 360 ÷ 9 = 40 minutes per period. Step 6: So each of the 9 periods must be 40 minutes long to keep the total time unchanged. Verification / Alternative check: We can reverse-check the answer. If each of the 9 periods is 40 minutes, then total time = 9 × 40 = 360 minutes. This matches the original total of 8 × 45 = 360 minutes. Since total time is the same in both arrangements, the calculation is verified and consistent with the condition that working hours remain unchanged. Why Other Options Are Wrong: 30 minutes would give a total of 9 × 30 = 270 minutes, which is too short. 35 minutes would give 9 × 35 = 315 minutes, still less than required. 45 or 50 minutes would result in total times greater than or equal to 405 or 450 minutes, which contradict the fixed total time of 360 minutes. Therefore, only 40 minutes satisfies the constraint. Common Pitfalls: Students sometimes try to proportion the change in period length directly with the ratio 8:9 without first computing total time, which can lead to confusion. Others mix minutes and hours incorrectly. The safest approach is always to first compute the total time in minutes, because minutes avoid fractional hours. Then divide by the new number of periods to obtain the new duration. Final Answer: Each period will be 40 minutes long.

Question 14

A, B and C can finish a job working alone in 20 days, 30 days and 60 days respectively. They all work together for 1 full day, and then only C continues working alone. How many days will C alone take to finish the remaining part of the job?

Accepted Answer

Introduction / Context: This problem combines individual work rates with a short initial period of teamwork followed by one person completing the remaining job. It is a standard pattern in time and work questions that teaches you how to account for partial completion by a group and then compute how long one worker, here C, will take to complete what is left. Given Data / Assumptions: A alone can finish the job in 20 days. B alone can finish the job in 30 days. C alone can finish the job in 60 days. All three work together for exactly 1 day. After that, A and B stop working and only C continues until the job is finished. Work rate is constant for each person. Concept / Approach: We treat the total job as 1 unit of work. Each person has a daily work rate equal to 1 divided by the number of days they take. First, we calculate the combined rate of A, B, and C to see how much of the job is done in that one day. Then we subtract that from 1 to find the fraction that remains. Finally, using C's individual rate, we calculate the extra days needed for C to finish the remaining work alone. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Daily rate of A = 1/20, of B = 1/30, and of C = 1/60. Step 3: Combined rate of A + B + C = 1/20 + 1/30 + 1/60. Step 4: Compute this sum: 1/20 + 1/30 + 1/60 = 3/60 + 2/60 + 1/60 = 6/60 = 1/10. Step 5: In one day they complete 1/10 of the work, so remaining work = 1 − 1/10 = 9/10. Step 6: C alone works at 1/60 of the job per day. Step 7: Time taken by C to complete 9/10 = (9/10) / (1/60) = (9/10) × 60 = 54 days. Verification / Alternative check: We can estimate: C is the slowest worker, taking 60 days for the full job. Since about 90 percent of the job is still left (9/10), a time very close to 0.9 × 60 = 54 days is expected. Our exact calculation also gives 54 days, so the result is consistent. Nothing in the calculation conflicts with the original data, so the answer is verified. Why Other Options Are Wrong: 48 days would mean C completes 48/60 = 4/5 of the job, which is less than the 9/10 that remains. 45 and 36 days similarly correspond to even smaller fractions of the work. 60 days would be the time for C to do the entire work alone, not just the remaining 9/10. Therefore, none of these matches the required remaining fraction, and only 54 days is correct. Common Pitfalls: A common mistake is to miscompute the combined rate of the three workers, especially when converting to a common denominator. Another is to forget that the team works for only 1 day, not until the work is half or fully done. Some learners also incorrectly subtract C's work twice when calculating the remaining fraction. Keeping a clear structure of total work, work done, and work left avoids these errors. Final Answer: C will take 54 days to finish the remaining work alone.

Question 15

A can complete one-third of a piece of work in 5 days, while B can complete two-fifths of the same work in 10 days. If both A and B work together from the beginning, in how many days will they complete the entire work?

Accepted Answer

Introduction / Context: This question explores partial work information for two workers and asks for the total time when they work together. Instead of directly giving complete-time values for A and B, it provides fractions of work completed in specific times. This is a common technique in aptitude tests to check if you can convert fractional work into individual work rates and then combine them correctly. Given Data / Assumptions: A completes one-third (1/3) of the work in 5 days. B completes two-fifths (2/5) of the work in 10 days. They both start together and work simultaneously. Work rate of each person is constant over time. We need the total time required to complete the full work when they work together. Concept / Approach: We use the concept of work rate, defined as fraction of work done per day. From the partial work information, we find A's daily rate and B's daily rate. Then we add the two rates to get the combined daily rate. Finally, we compute total time as the total work (taken as 1 unit) divided by this combined rate. The concept of fractional work is central here. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: A completes 1/3 of the work in 5 days, so A's rate = (1/3) / 5 = 1/15 of the work per day. Step 3: B completes 2/5 of the work in 10 days, so B's rate = (2/5) / 10 = 1/25 of the work per day. Step 4: Combined rate of A and B = 1/15 + 1/25. Step 5: Compute the sum: 1/15 + 1/25 = (5/75) + (3/75) = 8/75. Step 6: Time to complete 1 unit of work together = 1 / (8/75) = 75/8 days. Step 7: 75/8 days = 9.375 days, or 9 3/8 days. Verification / Alternative check: To verify, we can check how much work they would complete in 9.375 days. Combined rate is 8/75 per day, so work done = (8/75) × (75/8) = 1 unit, which matches the total job. Also, A alone would take 15 days to do the whole work and B alone would take 25 days. Since they work together, the combined time should be less than 15 days, and 9.375 days fits this expectation well. Why Other Options Are Wrong: 7.375 days and 8.5 days are too small and would imply a combined rate larger than what is actually possible from the individual rates. 10 and 12.5 days are larger than 9.375; 12.5 is even greater than the time that B would need alone, which is illogical for combined work. Only 9.375 days satisfies the exact calculation based on their rates. Common Pitfalls: A typical mistake is to directly average the partial times or to convert fractions incorrectly, such as treating 2/5 in 10 days as equivalent to 1/5 in 5 days without proper calculation. Another frequent error is not reducing the fractions to a common denominator when adding the rates. Carefully computing each person's rate and then summing them avoids these issues. Final Answer: A and B together will complete the entire work in 9.375 days (that is, 9 3/8 days).

Question 16

A can complete a piece of work in 9 days and B can complete the same work in 12 days. If A and B work together from the start, in how many days will they finish the entire work?

Accepted Answer

Introduction / Context: This is a straightforward time and work problem that involves two workers, A and B, who can complete a task individually in different numbers of days. The question asks for the combined time when they work together from the beginning. Such problems test your understanding of how to add work rates and then use the result to compute the time required for completion. Given Data / Assumptions: A alone can finish the work in 9 days. B alone can finish the same work in 12 days. A and B work together on the entire job from start to finish. The work rate of each person is constant over time. Total work is assumed to be a single unit. Concept / Approach: We use the standard concept that if a worker finishes a job in N days, the worker's rate is 1/N of the job per day. For two workers, the combined rate is the sum of their individual rates. After we obtain the combined daily rate, we compute the total time as total work divided by this rate. This method always works for such basic work problems. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: A's daily rate = 1/9 of the work per day. Step 3: B's daily rate = 1/12 of the work per day. Step 4: Combined daily rate of A and B = 1/9 + 1/12. Step 5: Compute the sum using common denominator 36: 1/9 = 4/36 and 1/12 = 3/36, so combined rate = (4 + 3)/36 = 7/36. Step 6: Time to finish the entire work together = 1 / (7/36) = 36/7 days. Step 7: 36/7 days = 5 and 1/7 days, which is the required time. Verification / Alternative check: We can verify by approximate reasoning. A alone takes 9 days, and B alone takes 12 days. The combined time must be less than 9 days (the fastest individual time). The exact result 36/7 is approximately 5.14 days, which is comfortably less than 9 days. Also, if they worked for 5 1/7 days at the rate of 7/36 per day, they would complete exactly 1 unit of work, confirming the calculation. Why Other Options Are Wrong: 5 2/7, 6 1/7, 6 2/7, and 7 days all correspond to times different from 36/7 days. Some of them are too large and would imply that working together is slower than one of them alone, which is impossible if both are contributing positively. None of these matches the precise reciprocal of the combined rate 7/36, so they cannot be correct. Common Pitfalls: Common mistakes include averaging the times (such as taking (9 + 12)/2 = 10.5 days) or adding the days directly without converting to rates, which is incorrect. Another frequent error is to invert the combined rate wrongly. Always remember: time = 1 / (sum of individual rates) when total work is taken as 1 unit. Final Answer: A and B together can complete the work in 5 1/7 days.

Question 17

If 7 spiders together make 7 webs in 7 days, assuming all spiders work at the same constant rate and all webs are identical, then in how many days will 1 spider make 1 web?

Accepted Answer

Introduction / Context: This is a simple but slightly tricky work and rate puzzle involving spiders and webs. It is designed to see if you understand the concept of uniform work rates and how to convert a group's total work over time into an individual rate. The numbers are deliberately all sevens to tempt quick but careless answers, so careful reasoning is important. Given Data / Assumptions: 7 spiders make 7 webs in 7 days. All spiders are assumed to work at the same constant rate. Each web is considered to require the same amount of work. We are asked for the time taken by 1 spider to make 1 web. Concept / Approach: We treat "web" as the unit of work. The key idea is to find the overall work rate of the group (spiders per day per web) and then divide by the number of spiders to find the rate per spider. Once we know the daily rate of a single spider, we can compute the time required for that spider to produce one web. This is exactly analogous to workers completing units of work. Step-by-Step Solution: Step 1: Total work is 7 webs. Step 2: Total time for 7 spiders together to make 7 webs is 7 days. Step 3: Combined rate of 7 spiders = total webs / total days = 7 webs / 7 days = 1 web per day. Step 4: This 1 web per day is produced by 7 spiders working together. Step 5: Therefore, rate of 1 spider = (1 web per day) / 7 = 1/7 web per day. Step 6: Time required for 1 spider to make 1 web = 1 web / (1/7 web per day) = 7 days. Verification / Alternative check: We can also reason directly: if 7 spiders make 7 webs in 7 days, then each day the group creates 1 web. It takes 7 spiders to create that 1 web in a day. To get 1 web from just a single spider, you need 7 times as long, since the workforce is 7 times smaller. That leads to 7 days, which matches our rate-based calculation. Why Other Options Are Wrong: 1 day is wrong because that would imply 1 spider makes 1 web per day, so 7 spiders would make 7 webs per day, meaning 7 webs would be completed in just 1 day, not 7. 3.5 days would mean the group could produce more than 7 webs in 7 days, which contradicts the original statement. 14 and 49 days are both far too large; they would make the combined rate inconsistent with the given data. Common Pitfalls: A typical error is to assume that because the numbers are all sevens, 1 spider should take 1 day, without considering the work rate distribution. Another pitfall is mixing up who is doing the work (the whole group versus one individual). Always translate the word problem into total work, group rate, individual rate, and then time, in that order. Final Answer: One spider will take 7 days to make one web.

Question 18

A alone can paint a house in 25 days and B alone can paint the same house in 10 days. Working together with a third painter C, they finish the job in 6.25 days. In how many days can C alone complete the job if working at the same constant rate?

Accepted Answer

Introduction / Context: This problem involves three workers, A, B, and C, who together complete a painting job in a given time. We know the individual times of A and B, and the combined time of all three, and we must find C's individual time. This is a classic application of the concept of work rate addition and subtraction, where we isolate the unknown worker's rate by subtracting the known rates from the combined rate. Given Data / Assumptions: A alone can paint the house in 25 days. B alone can paint the house in 10 days. A, B, and C together complete the house in 6.25 days (which is 25/4 days). All painters work at constant rates. We need the time taken by C alone to finish the house. Concept / Approach: We treat the entire house as 1 unit of work. Each painter has a rate equal to 1 divided by their respective time. The combined rate of A, B, and C is 1 divided by 6.25 days. We first compute the individual rates of A and B, sum them, and subtract that sum from the combined rate to find C's rate. Finally, we take the reciprocal of C's rate to obtain the time C needs alone. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: A's rate = 1/25 of the work per day. Step 3: B's rate = 1/10 of the work per day. Step 4: Time of A + B + C together = 6.25 days = 25/4 days, so combined rate = 1 / (25/4) = 4/25 per day. Step 5: Sum of A and B's rates = 1/25 + 1/10 = (2/50 + 5/50) = 7/50 per day. Step 6: C's rate = combined rate − (A + B) rate = 4/25 − 7/50. Step 7: Convert 4/25 to 8/50, so C's rate = 8/50 − 7/50 = 1/50 of the work per day. Step 8: Time taken by C alone = 1 / (1/50) = 50 days. Verification / Alternative check: We can verify by recombining the rates. A's rate is 1/25, B's rate is 1/10, and C's rate we found as 1/50. Their combined rate should be 1/25 + 1/10 + 1/50 = (2/50 + 5/50 + 1/50) = 8/50 = 4/25. This is exactly the rate corresponding to 25/4 days, or 6.25 days, which matches the given data. Hence our calculation for C is consistent and correct. Why Other Options Are Wrong: 40 and 30 days would correspond to higher rates for C, which would make the combined time less than 6.25 days, contradicting the question. 60 and 25 days either make C too slow or duplicate A's speed, which is not implied. Only 50 days gives a rate that, when combined with A and B, reproduces the correct total time of 6.25 days. Common Pitfalls: A common mistake is to average the times (25, 10, and 6.25) directly instead of working with rates. Another error is to mishandle fractional time like 6.25 days when converting it to a fraction. Writing 6.25 as 25/4 is more precise and simplifies the calculation. Carefully subtracting known rates from the total combined rate is essential to isolate the unknown worker's rate correctly. Final Answer: C alone can complete the painting job in 50 days.

Question 19

A, B and C can complete a job working alone in 6 days, 9 days and 18 days respectively. They all work together for 1 day and then only C continues working alone. How many additional days will C take to finish the remaining work?

Accepted Answer

Introduction / Context: This question is another variation on combined work followed by a single worker completing the remaining part. Knowing the individual times for A, B, and C, we first find how much work is completed when all three work together for one day, and then use C's individual rate to determine the extra time needed. Problems of this type reinforce the idea of breaking a job into fractions completed per day. Given Data / Assumptions: A alone can complete the job in 6 days. B alone can complete the job in 9 days. C alone can complete the job in 18 days. All three work together for 1 day from the start. After that, A and B stop, and C finishes the rest alone. All work rates are constant. Concept / Approach: As usual, we consider the total work to be 1 unit. We convert each worker's time into a daily work rate. Adding the three rates gives the joint work done in a day when they are all working. Subtracting that from 1 yields the remaining fraction of the work. Dividing this remainder by C's individual rate gives the extra days C needs to complete the job alone. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: A's rate = 1/6 of the work per day. Step 3: B's rate = 1/9 of the work per day. Step 4: C's rate = 1/18 of the work per day. Step 5: Combined rate of A + B + C = 1/6 + 1/9 + 1/18. Step 6: Using common denominator 18, this becomes 3/18 + 2/18 + 1/18 = 6/18 = 1/3. Step 7: Work done in the first day by all three = 1/3 of the job. Step 8: Remaining work = 1 − 1/3 = 2/3 of the job. Step 9: C alone works at 1/18 per day, so time required = (2/3) / (1/18) = (2/3) × 18 = 12 days. Verification / Alternative check: We can check approximate logic. C alone would need 18 days to complete the full job. Here, two-thirds of the job remains when C starts alone. Two-thirds of 18 is 12, which matches the exact calculation. Also, the total work done is 1/3 in the first day plus 2/3 over the next 12 days, summing to 1. Everything is consistent with the given data. Why Other Options Are Wrong: 9 and 10 days are too small; at 1/18 per day, in 9 days C would complete only half the job (9/18), not two-thirds. 6 days is even smaller. 15 days is larger than the two-thirds scaled time of 18, and would result in more than the remaining work being done. Thus only 12 days fits both the rate and the remaining fraction of the job. Common Pitfalls: One pitfall is forgetting that the first day of combined work is only 1 day, not the entire duration. Another is incorrectly adding the fractions for the combined rate. Some learners also try to directly use proportions of days instead of rates, which leads to incorrect conclusions. Keeping track of rates and the fraction of work completed at each step prevents these errors. Final Answer: C will take 12 additional days to finish the remaining work.

Question 20

Asif is twice as efficient a workman as Bashir. Together they finish a piece of work in 30 days. If they both work at their constant efficiencies, in how many days will Asif alone complete the entire work?

Accepted Answer

Introduction / Context: This is a standard work and efficiency problem, where we compare two workers, Asif and Bashir. We are told that one is twice as efficient as the other and given the time they take together to complete the job. The question asks how long the more efficient worker, Asif, would take alone. Such problems test your understanding of efficiency ratios and their relation to time taken. Given Data / Assumptions: Asif is twice as efficient as Bashir. Asif and Bashir together complete the work in 30 days. Both work at constant efficiencies. We need the time Asif alone would require to finish the job. Concept / Approach: Efficiency is directly proportional to work rate and inversely proportional to time for the same work. If Asif is twice as efficient as Bashir, then his rate is twice Bashir's rate. We set Bashir's rate as a base variable, express Asif's rate, add them to get the combined rate, and then solve for Bashir's rate using the given combined time. From this we compute Asif's rate and finally his time to complete the work alone. Step-by-Step Solution: Step 1: Let total work = 1 unit. Step 2: Let Bashir's daily rate = b units per day. Step 3: Asif is twice as efficient, so Asif's daily rate = 2b units per day. Step 4: Together, Asif and Bashir have a combined rate = b + 2b = 3b. Step 5: They complete the whole work in 30 days, so combined rate = 1/30 units per day. Step 6: Hence, 3b = 1/30, so b = 1/90. Step 7: Asif's rate = 2b = 2 × 1/90 = 1/45 of the work per day. Step 8: Time for Asif alone = 1 / (1/45) = 45 days. Verification / Alternative check: If Asif alone takes 45 days, then Bashir, being half as efficient, would take 90 days. The combined rate is then 1/45 + 1/90 = 2/90 + 1/90 = 3/90 = 1/30 of the work per day, which means 30 days to complete the job together. This matches the given condition perfectly, confirming that the answer is consistent. Why Other Options Are Wrong: 90 days would be the time for Bashir, not Asif, since he is less efficient. 60 and 75 days are larger than 45 and would imply Asif is slower than calculated from the efficiency ratio. 30 days is the time they take together, not Asif alone. Therefore, only 45 days correctly reflects the doubled efficiency relationship and the joint completion time. Common Pitfalls: Many students confuse the ratio of efficiencies with the ratio of times. If Asif is twice as efficient as Bashir, Asif's time should be half of Bashir's time, not double. Another mistake is trying to split the 30 days directly in the ratio 2:1 without first expressing rates. Always use variables for rates, set up equations correctly, and then derive the required time from the rate. Final Answer: Asif alone will complete the work in 45 days.

Time and Work Questions