Question 1

Directions: Read the information carefully and answer the questions.  A, B, C and D together can do a work 'X' in 14 days. The ratio of efficiency of A to that of C is $3 : 2$. C is $50\%$ less efficient than that of B and B is $33\frac{1}{3}\%$ mor…

Accepted Answer

Concept & Man-Days Work Equivalence: This problem builds on the previously found variables and uses the standard Man-Days formula where total work is conserved across different group sizes. M_1 × D_1 = M_2 × D_2 = Total Work Step-by-Step Solution: 1. Recall 'x' and 'y' from the directions: As derived in the previous question based on the efficiency ratios: x = 24 y = 28 2. Formulate the work equations: - Case 1: 'j' men complete work in (x + 4) = 24 + 4 = 28 days. Total Work = 28j - Case 2: '(y - 8)' men complete work in 'k' days. (28 - 8) = 20 men. Total Work = 20k - Case 3: '(y + 2)' men complete work in (0.5k + 3.5) days. (28 + 2) = 30 men. Total Work = 30(0.5k + 3.5) = 15k + 105 3. Solve for 'k': Since the work is the same in all cases, equate Case 2 and Case 3: 20k = 15k + 105 5k = 105 \implies k = 21 4. Solve for 'j': Equate Case 1 and Case 2: 28j = 20k 28j = 20(21) 28j = 420 \implies j = 15 5. Calculate the final requirement: - Total Work = 420 man-days. - We need the time for (k - j) men to complete it. - (k - j) = 21 - 15 = 6 men. - Number of days = \frac{Total Work}{Number of Men} = (420/6) = 70 days. Exam Strategy & Shortcut: You can solve for k directly by only comparing the cases with y values, since y is a known constant. Equating 20k = 30(0.5k + 3.5) bypasses the unknown j entirely until you actually need it to find the total work. Common Pitfall: A common mistake here is messing up the algebraic expansion of 30(0.5k + 3.5). Often, students will multiply the 30 by 0.5k but forget to multiply it by the 3.5 term. Final Answer: Therefore, the correct answer is 70 days.

Question 2

A man completes $\frac{5}{8}$ of a job in 10 days. At this rate, how many more days will it take him to finish the job?  (M.B.A., 2003)

Accepted Answer

Concept & Unitary Method for Work: When a fraction of work is tied to a specific time, you can determine the time for a single "unit" or "part" of the work and scale it to the remaining parts. Time for Remaining Work = Remaining Fraction × Total Time for 1 Job Step-by-Step Solution: • Given: The man completes 5 parts out of 8 total parts ((5/8)) in 10 days. • Time taken for 1 part of the job = (10/5) = 2 days. • Remaining parts of the job to be completed = 8 - 5 = 3 parts (or (3/8) of the job). • Time taken for remaining 3 parts = 3 × 2 = 6 days. Exam Strategy & Shortcut: Set up a direct proportion between work parts and days: 5 parts \rightarrow 10 days 1 part \rightarrow 2 days 3 parts \rightarrow 6 days. The answer is 6 immediately. Common Pitfall: Calculating the total number of days required to finish the *entire* job (16 days) instead of reading carefully that the question asks for how many *more* days it will take. Final Answer: Therefore, the correct answer is 6.

Question 3

56 men can complete a piece of work in 24 days. In how many days can 42 men complete the same piece of work?  (Bank P.O., 2008)

Accepted Answer

Concept & Man-Days Formula: The total amount of work can be measured in "man-days" (number of men multiplied by the number of days). For the same piece of work, the total man-days remains constant. M_1 × D_1 = M_2 × D_2 Step-by-Step Solution: • Given: M_1 = 56 men, D_1 = 24 days. • Given: M_2 = 42 men, D_2 = ? days. • Total work = 56 × 24 man-days. • Equate the total work to the second scenario: 56 × 24 = 42 × D_2. • Solve for D_2: D_2 = (56 × 24/42). • Simplify the fraction: Divide 56 and 42 by 14 to get (4 × 24/3). • Divide 24 by 3 to get 8. • D_2 = 4 × 8 = 32 days. Exam Strategy & Shortcut: Use the ratio method. The ratio of men is 56 : 42 = 4 : 3. Since men and days are inversely proportional, the ratio of days will be 3 : 4. If 3 parts = 24 days, then 1 part = 8 days. So, 4 parts = 4 × 8 = 32 days. Common Pitfall: Setting up a direct proportion (56/24 = 42/x) instead of an inverse proportion, forgetting that fewer men will take *more* time, not less. Final Answer: Therefore, the correct answer is 32.

Question 4

$4$ mat-weavers can weave $4$ mats in $4$ days. At the same rate, how many mats would be woven by $8$ mat-weavers in $8$ days?

Accepted Answer

Concept & Work Equivalence Formula: The fundamental chain rule equation relates men (M), days (D), and work (W): (M_1 × D_1/W_1) = (M_2 × D_2/W_2) Step-by-Step Solution: • Given: M_1 = 4, W_1 = 4, D_1 = 4. For the second scenario: M_2 = 8, D_2 = 8. Let the required work be W_2. • Calculation: Substitute the given values into the formula: (4 × 4/4) = (8 × 8/W_2) 4 = (64/W_2) W_2 = (64/4) = 16 Exam Strategy & Shortcut: Double the workers and double the time means the output is quadrupled. 2 (from workers) × 2 (from days) = 4. The original work was 4 mats, so the new work is 4 × 4 = 16 mats. Common Pitfall: Assuming that because 4 weavers in 4 days make 4 mats, 8 weavers in 8 days will make 8 mats. This ignores that both the workforce and the duration have doubled. Final Answer: Therefore, the correct answer is 16.

Question 5

Runnning at the same constant rate, $6$ identical machines can produce a total of $180$ bottles per hour. How many bottles could $15$ such machines produce in $30$ minutes?

Accepted Answer

Concept & Work Equivalence Formula: Production output (W) is proportional to the number of machines (M) and the operating time (T). (M_1 × T_1/W_1) = (M_2 × T_2/W_2) Step-by-Step Solution: • Given: - M_1 = 6 machines, T_1 = 60 minutes (1 hour), W_1 = 180 bottles. - M_2 = 15 machines, T_2 = 30 minutes. • Calculation: Plug values into the formula to find W_2: (6 × 60/180) = (15 × 30/W_2) Simplify the left side: (360/180) = (450/W_2) 2 = (450/W_2) W_2 = (450/2) = 225 Exam Strategy & Shortcut: First, find the rate of one machine. 6 machines make 180 bottles in an hour, so 1 machine makes 30 bottles in an hour, which means 15 bottles in 30 minutes. For 15 machines, the production in 30 minutes is 15 × 15 = 225 bottles. Common Pitfall: Failing to unify the time units (mixing hours and minutes). Always convert "per hour" and "in 30 minutes" to the same unit (e.g., minutes) before calculating. Final Answer: Therefore, the correct answer is 225.

Question 6

If $5$ workers can collect $60$ kg wheat in $3$ days, how many kilograms of wheat will $8$ workers collect in $5$ days?

Accepted Answer

Concept & Work Equivalence: The amount of work (or wheat collected) is proportional to the number of workers and the days worked. (M_1 × D_1/W_1) = (M_2 × D_2/W_2) Step-by-Step Solution: • Given: - M_1 = 5, W_1 = 60, D_1 = 3 - M_2 = 8, W_2 = x, D_2 = 5 • Calculation: (5 × 3/60) = (8 × 5/x) (15/60) = (40/x) (1/4) = (40/x) x = 40 × 4 = 160 Exam Strategy & Shortcut: Find the daily collection rate of one worker. 5 workers collect 60 kg in 3 days, meaning they collect 20 kg in 1 day. So, 1 worker collects 4 kg per day. Therefore, 8 workers collect 32 kg per day. In 5 days, they collect 32 × 5 = 160 kg. Common Pitfall: Mistakenly putting work (W) in the numerator, which flips the proportion and leads to incorrect calculations. Final Answer: Therefore, the correct answer is 160 kg.

Question 7

12 men working 8 hours per day complete a piece of work in 10 days. To complete the same work in 8 days, working 15 hours a day, the number of men required, is:

Accepted Answer

Concept & Work Equivalence Formula: When the total amount of work is constant, the product of the number of men (M), days (D), and hours per day (H) remains constant. M_1 × D_1 × H_1 = M_2 × D_2 × H_2 Step-by-Step Solution: • Given: - Scenario 1: M_1 = 12, H_1 = 8, D_1 = 10 - Scenario 2: D_2 = 8, H_2 = 15, M_2 = x • Calculation: 12 × 8 × 10 = x × 8 × 15 960 = 120x x = (960/120) = 8 Exam Strategy & Shortcut: Cancel common terms on both sides before multiplying. Notice that 8 appears on both sides of the equation (12 × 8 × 10 = x × 8 × 15), which simplifies to 12 × 10 = 15x, yielding x = (120/15) = 8. Common Pitfall: Forgetting to account for either the hours or days change, or incorrectly setting up the inverse relationships as direct ratios. Final Answer: Therefore, the correct answer is 8.

Question 8

2 men and 7 boys can do a piece of work in 14 days; 3 men and 8 boys can do the same in 11 days. Then, 8 men and 6 boys can do three times the amount of this work in :

Accepted Answer

Concept & Simultaneous Equations for Efficiency: When given two distinct scenarios of mixed groups completing the same work, equate the total "man-days" to find the relative efficiency between the worker types. (M_1 × D_1) = (M_2 × D_2) Step-by-Step Solution: • Equate Total Work: • (2M + 7B) × 14 = (3M + 8B) × 11 • Solve for Efficiency Ratio: • 28M + 98B = 33M + 88B • 98B - 88B = 33M - 28M • 10B = 5M \implies 1M = 2B (One man is twice as efficient as one boy). • Calculate Baseline Work: Substitute 1M = 2B into the first group to find total work in terms of boys. • Group 1 = 2(2B) + 7B = 4B + 7B = 11B. • Total Work = 11B × 14 days = 154 boy-days. • Target Work: We need to do "three times the amount of this work". • Target Work = 154 × 3 = 462 boy-days. • Evaluate New Group: 8 men and 6 boys. • New Group = 8(2B) + 6B = 16B + 6B = 22B. • Calculate Time: Time = Target Work / New Group Size. • Days = 462 / 22 = 21 days. Exam Strategy & Shortcut: Once you find 1M = 2B, evaluate the group strengths mentally: Group 1 acts like 11 boys taking 14 days (11 × 14). The target group (8 men + 6 boys) acts like 22 boys. Since the target group has exactly double the manpower (22 vs 11), they would take half the time (7 days) for normal work. For 3× the work, it takes 7 × 3 = 21 days. Common Pitfall: Failing to multiply the final required work by 3, missing the "three times the amount" clause in the question, resulting in an answer of 7 days (which isn't an option, but would cause panic). Final Answer: Therefore, the correct answer is 21 days.

Question 9

Ayesha can complete a piece of work in 16 days. Amita can complete the same piece of work in 8 days. If both of them work together in how many days can they complete the same piece of work?

Accepted Answer

Concept & Time and Work: The rate of work is inversely proportional to the time taken. If a person can complete a work in n days, their 1-day work is (1/n). When working together, their combined 1-day work is the sum of their individual 1-day works. Step-by-Step Solution: • Given: Ayesha takes 16 days to complete the work. Amita takes 8 days to complete the work. • Calculation: Ayesha's 1-day work = (1/16) Amita's 1-day work = (1/8) Combined 1-day work = (1/16) + (1/8) To add the fractions, find a common denominator (which is 16): Combined 1-day work = (1/16) + (2/16) = (3/16) • Therefore, the time taken to complete the work together is the reciprocal of their combined 1-day work, which is (16/3) days. • Converting (16/3) to a mixed fraction: 16 \div 3 = 5 with a remainder of 1. So, (16/3) = 5(1/3) days. Exam Strategy & Shortcut: Use the LCM (Total Work) method. Let Total Work = LCM of (16, 8) = 16 units. Ayesha's efficiency = 16 / 16 = 1 unit/day. Amita's efficiency = 16 / 8 = 2 units/day. Combined efficiency = 1 + 2 = 3 units/day. Total time = Total Work / Combined efficiency = 16 / 3 = 5(1/3) days. Common Pitfall: A common mistake is simply averaging the days (e.g., (16+8)/2 = 12 days). Remember that working together always takes less time than the fastest individual. Final Answer: Therefore, the correct answer is 5(1/3) days.

Question 10

A can lay railway track between two given stations in 16 days and B can do the same job in 12 days. With the help of C, they did the job in 4 days only. Then, C alone can do the job in :

Accepted Answer

Concept & Multi-Person Work Rates: When two people are joined by a third to complete a task faster, the third person's efficiency is the difference between the total combined efficiency and the sum of the original two people's efficiencies. Step-by-Step Solution: • Given: A's time alone = 16 days. B's time alone = 12 days. Combined time (A + B + C) = 4 days. • Calculation: Combined 1-day work = (1/4) A's 1-day work = (1/16) B's 1-day work = (1/12) • C's 1-day work = (Combined 1-day work) - (A's 1-day work + B's 1-day work) C's 1-day work = (1/4) - ((1/16) + (1/12)) • Find a common denominator to add the fractions inside the parenthesis. The LCM of 16 and 12 is 48. (1/16) + (1/12) = (3/48) + (4/48) = (7/48) • Now, subtract this sum from the combined 1-day work ((1/4) can be written as (12/48)): C's 1-day work = (12/48) - (7/48) = (5/48) • Time taken by C alone = Reciprocal of C's 1-day work = (48/5) days. • Convert to a mixed fraction: 48 \div 5 = 9 with a remainder of 3. (48/5) = 9(3/5) days. Exam Strategy & Shortcut: Use the Total Work (LCM) method. Total Work = LCM(16, 12, 4) = 48 units. A's efficiency = 48 / 16 = 3 units/day. B's efficiency = 48 / 12 = 4 units/day. (A + B + C)'s combined efficiency = 48 / 4 = 12 units/day. C's efficiency = Total - (A + B) = 12 - (3 + 4) = 12 - 7 = 5 units/day. Time for C = Total Work / C's efficiency = 48 / 5 = 9(3/5) days. Common Pitfall: A common mistake is adding the days 16 + 12 or calculating the combined time of A and B first, and then making an error when subtracting that combined entity from the 4-day total. Always convert to unit rates/efficiencies first. Final Answer: Therefore, the correct answer is 9(3/5) days.

Question 11

A can complete $\frac{1}{3}$ of a work in 5 days and B, $\frac{2}{5}$ of the work in 10 days. In how many days both A and B together can complete the work?

Accepted Answer

Concept & Extrapolating Total Time: When a fraction of the work and its completion time is given, the time required to complete the *entire* work is calculated by dividing the time taken by the fraction of the work completed. Once individual total times are found, use standard time and work formulas. Step-by-Step Solution: • Given: A completes (1/3) of the work in 5 days. B completes (2/5) of the work in 10 days. • Extrapolating Total Times: Time for A to complete the whole work = 5 × (3/1) = 15 days. Time for B to complete the whole work = 10 × (5/2) = 25 days. • Calculation: A's 1-day work = (1/15) B's 1-day work = (1/25) Combined 1-day work = (1/15) + (1/25) • Find the LCM of 15 and 25, which is 75. Combined 1-day work = (5/75) + (3/75) = (8/75) • Total time for both A and B together = Reciprocal of their combined 1-day work. Total Time = (75/8) days. • Convert to a mixed fraction: 75 \div 8 = 9 with a remainder of 3. Total Time = 9(3/8) days. Exam Strategy & Shortcut: First, quickly establish total days: A \rightarrow 5 × 3 = 15, B \rightarrow 10 × (5/2) = 25. Use the formula for two people working together: T = (xy/x + y). T = (15 × 25/15 + 25) = (375/40). Simplify by dividing numerator and denominator by 5: (75/8) = 9(3/8) days. Common Pitfall: A frequent error is directly using the given partial days (5 and 10) in the standard formula without first converting them into the time required for 100% of the work. Final Answer: Therefore, the correct answer is 9(3/8).

Question 12

X can do $\frac{1}{4}$ of a work in 10 days, Y can do 40% of the work in 40 days and Z can do $\frac{1}{3}$ of the work in 13 days. Who will complete the work first ?

Accepted Answer

Concept & Strategy: To find out who completes the work first, calculate the total time each person takes to complete 100% (or 1 whole) of the work. The person who takes the minimum total time will complete the work first. Step-by-Step Solution: • Time taken by X: X completes (1/4) of the work in 10 days. Time taken to complete the whole work = 10 × 4 = 40 days. • Time taken by Y: Y completes 40% ((40/100) or (2/5)) of the work in 40 days. Time taken to complete the whole work = 40 × (5/2) = 100 days. • Time taken by Z: Z completes (1/3) of the work in 13 days. Time taken to complete the whole work = 13 × 3 = 39 days. • Comparison: Comparing the total times (40 days, 100 days, 39 days), Z takes the least number of days (39 days). Exam Strategy & Shortcut: Rather than finding full work times, you can quickly estimate their pace to find the fastest worker: X does 25% in 10 days -> 100% in 40 days. Y does 40% in 40 days -> very slow. Z does 33.33% in 13 days -> 100% in 39 days. Clearly, Z is the fastest. Common Pitfall: A common mistake is directly comparing the given days (10, 40, 13) without adjusting for the fraction of work completed, which leads to incorrectly choosing X. Final Answer: Therefore, the correct answer is Z.

Question 13

A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 2 days. B can do the work alone in :

Accepted Answer

Concept & Ratio of Time: When multiple people have comparative times for completing work, define one person's time as a variable and express the others in terms of it. Formula for combined work: (1/A) + (1/B) + (1/C) = (1/T_{total)} Step-by-Step Solution: • Defining Individual Times: Let the time taken by A to finish the work be x days. Given: A takes twice as much time as B \Rightarrow B = (x/2) days. Given: A takes thrice as much time as C \Rightarrow C = (x/3) days. • Setting up the Equation: Working together, they take 2 days. Their 1-day work combined: (1/A) + (1/B) + (1/C) = (1/2) Substitute the values: (1/x) + (1/\frac{x){2}} + (1/\frac{x){3}} = (1/2) (1/x) + (2/x) + (3/x) = (1/2) • Solving for x: (1 + 2 + 3/x) = (1/2) (6/x) = (1/2) x = 12 days (This is A's time). • Finding B's Time: Time taken by B = (x/2) = (12/2) = 6 days. Exam Strategy & Shortcut: Assume the total work is the LCM of the comparative time ratios. Let A take 6t days. Then B takes 3t days and C takes 2t days. Efficiencies (work per day): A = (1/6t), B = (1/3t), C = (1/2t). Sum of efficiencies = (1+2+3/6t) = (6/6t) = (1/t). This combined efficiency equals (1/2) work/day. Thus t = 2. B takes 3t = 3(2) = 6 days. Common Pitfall: A frequent error is assigning the ratios directly to efficiencies rather than time. "Takes twice as much time" means half the efficiency. Final Answer: Therefore, the correct answer is 6 days.

Question 14

Work done by A in one day is half of the work done by B in one day. Work done by B is half of the work done by C in one day. If C alone can complete the work in 7 days, in how many days can A, B and C together complete the work?

Accepted Answer

Concept & Efficiency: This problem compares "work done in one day", which is exactly the definition of efficiency. We can trace the efficiencies relative to the known worker (C) to find the combined working rate. Step-by-Step Solution: • Calculate C's 1-day work: C completes the work in 7 days. Work done by C in 1 day = (1/7). • Calculate B's 1-day work: Work done by B in 1 day is half of C's 1-day work. B's 1 day work = (1/2) × (1/7) = (1/14). (This means B alone takes 14 days). • Calculate A's 1-day work: Work done by A in 1 day is half of B's 1-day work. A's 1 day work = (1/2) × (1/14) = (1/28). (This means A alone takes 28 days). • Calculate Combined 1-day work: (A + B + C)'s 1-day work = (1/28) + (1/14) + (1/7) LCM of 28, 14, 7 is 28. Combined 1-day work = (1 + 2 + 4/28) = (7/28) = (1/4). • Total Time: Since they complete (1/4) of the work in one day, they will finish the entire work in 4 days. Exam Strategy & Shortcut: Work backwards from C's efficiency using ratios. C's efficiency = 4x. B's efficiency = 2x. A's efficiency = x. Total efficiency = 7x. Since C (4x efficiency) takes 7 days, Total Work = 4x × 7 = 28x. Time taken by all together = \frac{Total Work}{Total Efficiency} = (28x/7x) = 4 days. Common Pitfall: Mixing up the ratios by assuming A is the fastest. "Work done by A is half" means A is less efficient and slower, not faster. Final Answer: Therefore, the correct answer is 4.

Question 15

A conveyor belt delivers baggage at the rate of 3 tons in 5 minutes and a second conveyor belt delivers baggage at the rate of 1 ton in 2 minutes. How much time will it take to get 33 tons of baggage delivered using both the conveyor belts together?

Accepted Answer

Concept & Additive Rates: When multiple machines work together simultaneously, their output rates (work per unit time) are additive. Total Time = \frac{Total Target Work}{Combined Output Rate} Step-by-Step Solution: • Calculate Rate of Belt 1: Belt 1 delivers 3 tons in 5 minutes. Rate of Belt 1 = (3/5) tons / minute. • Calculate Rate of Belt 2: Belt 2 delivers 1 ton in 2 minutes. Rate of Belt 2 = (1/2) tons / minute. • Calculate Combined Rate: Combined rate = (3/5) + (1/2) Take LCM of 5 and 2, which is 10. Combined rate = (6/10) + (5/10) = (11/10) tons / minute. • Calculate Total Time for 33 Tons: Total work needed = 33 tons. Time taken = \frac{Total Work}{Combined Rate} Time = (33/\frac{11){10}} = 33 × (10/11) = 3 × 10 = 30 minutes. Exam Strategy & Shortcut: Instead of fractions, you can equalize the time to find combined output in a fixed interval. In 10 minutes (LCM of 5 and 2): Belt 1 delivers 3 × 2 = 6 tons. Belt 2 delivers 1 × 5 = 5 tons. Together they deliver 6 + 5 = 11 tons in 10 minutes. To deliver 33 tons (3 times the amount), it will take 3 × 10 = 30 minutes. Common Pitfall: A common mistake is trying to add the times directly or averaging the rates rather than strictly summing the capacities (tons per minute). Final Answer: Therefore, the correct answer is 30 min.

Question 16

Bob and David are two typists. One afternoon, they were each given 40 pages for typing. They divided the work equally but David finished 20 minutes before Bob who took 2 hours for the same. The next afternoon, they were again given 77 pages to type.…

Accepted Answer

Concept & Proportional Work: When workers operate for the exact same duration (finishing simultaneously), the amount of work they each complete is directly proportional to their efficiencies (typing speeds). Step-by-Step Solution: • Analyze the First Task (Efficiency Ratio): They divided the work equally (doing 20 pages each, though the exact number doesn't matter, just that the work was identical). Bob took 2 hours (120 minutes). David finished 20 minutes before Bob, so David took 120 - 20 = 100 minutes. Ratio of time taken (Bob : David) = 120 : 100 = 6 : 5. Efficiency (speed) is inversely proportional to time. Ratio of their efficiencies (Bob : David) = 5 : 6. • Analyze the Second Task (Simultaneous Finish): Total pages to type = 77. Since they finish simultaneously, the time spent is equal. Therefore, the pages they type will be split exactly in the ratio of their efficiencies. Pages typed ratio (Bob : David) = 5 : 6. • Calculate Bob's Share: Bob's share of pages = (5/5 + 6) × Total Pages Bob's share = (5/11) × 77 Bob's share = 5 × 7 = 35 pages. Exam Strategy & Shortcut: Recognize that "finishing simultaneously" means dividing the total pool of work by their speed ratio. The speeds are inversely proportional to their times for identical tasks. Times are 120 vs 100 (6:5), so speeds are 5:6. Bob gets 5 parts out of 11. 77 × (5/11) = 35. Common Pitfall: A frequent mistake is applying the time ratio directly to the work distribution instead of the efficiency ratio, which would erroneously give Bob more pages (42) even though he is the slower typist. Final Answer: Therefore, the correct answer is 35.

Question 17

P, Q and R are three typists who working simultaneously can type 216 pages in 4 hours. In one hour, R can type as many pages more than Q as Q can type more than P. During a period of five hours, R can type as many pages as P can during seven hours. …

Accepted Answer

Concept & Linear Equations: Translate the descriptive relationships between their work rates into a system of linear equations. Step-by-Step Solution: • Establish Hourly Rates: Let the number of pages typed per hour by P, Q, and R be P, Q, and R respectively. • Equation 1 (Combined Rate): Together, they type 216 pages in 4 hours. Combined hourly rate: P + Q + R = (216/4) = 54. • Equation 2 (Rate Differences): "R can type as many pages more than Q as Q can type more than P." R - Q = Q - P Rearranging gives: P + R = 2Q. • Solve for Q: Substitute (P + R) with 2Q in Equation 1: (P + R) + Q = 54 2Q + Q = 54 3Q = 54 Q = 18 pages/hour. • Equation 3 (Time/Work Equivalency): "In 5 hours, R types as many pages as P does in 7 hours." 5R = 7P R = (7/5)P. • Solve for P and R: We know P + R = 2Q, and Q = 18, so P + R = 36. Substitute R = (7/5)P: P + (7/5)P = 36 (12P/5) = 36 P = (36 × 5/12) = 3 × 5 = 15 pages/hour. Now, calculate R: R = (7/5) × 15 = 7 × 3 = 21 pages/hour. The rates are P = 15, Q = 18, R = 21. Exam Strategy & Shortcut: Use Option Elimination based on the first extracted rule: R - Q = Q - P. This implies the rates must be in an arithmetic progression. Check differences in options: (a) 14, 17, 20 (diff is 3, valid) (b) 15, 17, 22 (diffs are 2 and 5, invalid) (c) 15, 18, 21 (diff is 3, valid) (d) 16, 18, 22 (diffs are 2 and 4, invalid) Now check the "5 hrs of R = 7 hrs of P" rule (5 × R = 7 × P) on the remaining valid options: Test (a): 5 × 20 = 100; 7 × 14 = 98 (Not equal). Test (c): 5 × 21 = 105; 7 × 15 = 105 (Equal!). Common Pitfall: Setting up the relative rate equation incorrectly. Interpreting "R types as many more than Q" as R + Q instead of R - Q will completely break the algebraic flow. Final Answer: Therefore, the correct answer is 15, 18, 21.

Question 18

A and B together can complete a work in 12 days, B and C together can complete the same work in 8 days and A and C together can complete it in 16 days. In total, how many days do A, B and C together take to complete the same work?

Accepted Answer

Concept & Formula / Logic: When the work rates of pairs of individuals are given, summing all the pair rates yields exactly twice the combined rate of all individuals working together. 2(A + B + C) = (A + B) + (B + C) + (A + C) Step-by-Step Solution: 1. Determine the 1-day work for each pair: (A + B)'s 1-day work = (1/12) (B + C)'s 1-day work = (1/8) (A + C)'s 1-day work = (1/16) 2. Sum the 1-day work of all pairs: (A + B) + (B + C) + (A + C) = (1/12) + (1/8) + (1/16) Find the LCM of 12, 8, and 16, which is 48. 2(A + B + C) = (4/48) + (6/48) + (3/48) = (13/48) 3. Calculate the combined 1-day work of A, B, and C: Since 2(A + B + C) = (13/48) (A + B + C) = (13/48 × 2) = (13/96) This means A, B, and C together can do (13/96) of the work in 1 day. 4. Find the total time taken: Total time = Reciprocal of combined 1-day work = (96/13) days. Convert to a mixed fraction: (96/13) = 7 (5/13) days. Exam Strategy & Shortcut: Use the LCM (Total Work) method. Let total work = LCM(12, 8, 16) = 48 units. Efficiencies: (A+B) = 4, (B+C) = 6, (A+C) = 3. Sum of efficiencies = 4 + 6 + 3 = 13. This is twice the efficiency of A+B+C. Efficiency of (A+B+C) = (13/2) = 6.5 units/day. Time taken = (48/6.5) = (96/13) = 7 (5/13) days. Common Pitfall: A frequent mistake is forgetting to divide the summed rates by 2. Summing (A+B), (B+C), and (A+C) double-counts each worker's contribution. Final Answer: Therefore, the correct answer is 7 (5/13).

Question 19

A can do a piece of work in 4 hours, B and C together in 3 hours, and A and C together in 2 hours. How long will B alone take to do it?

Accepted Answer

Concept & Formula / Logic: The problem requires finding an individual's work rate by strategically subtracting known work rates of individuals or combinations from other given combinations. Rate of B = Rate of (B+C) - Rate of C Rate of C = Rate of (A+C) - Rate of A Step-by-Step Solution: 1. Identify the given rates (1-hour work): A's 1-hour work = (1/4) (B + C)'s 1-hour work = (1/3) (A + C)'s 1-hour work = (1/2) 2. Find C's 1-hour work: We know A's rate and (A + C)'s rate. Subtracting them will give C's rate. C's 1-hour work = (A + C) - A C = (1/2) - (1/4) = (2/4) - (1/4) = (1/4) 3. Find B's 1-hour work: We now know C's rate and (B + C)'s rate. Subtracting C's rate from (B + C)'s rate gives B's rate. B's 1-hour work = (B + C) - C B = (1/3) - (1/4) Find the LCM of 3 and 4, which is 12. B = (4/12) - (3/12) = (1/12) Since B's 1-hour work is (1/12), B alone will take 12 hours to complete the work. Exam Strategy & Shortcut: Using the total work (LCM) method is much faster. Let Total Work = LCM of (4, 3, 2) = 12 units. Efficiency of A = (12/4) = 3 units/hr. Efficiency of (B+C) = (12/3) = 4 units/hr. Efficiency of (A+C) = (12/2) = 6 units/hr. Since A=3 and A+C=6, then C = 6-3=3 units/hr. Since B+C=4 and C=3, then B = 4-3=1 unit/hr. Time for B = \frac{Total Work}{B's Efficiency} = (12/1) = 12 hours. Common Pitfall: A common mistake is trying to solve for B directly without systematically finding C first. You must isolate the unknown variable step-by-step. Final Answer: Therefore, the correct answer is 12 hours.

Question 20

A and B can do a work in 8 days, B and C can do the same work in 12 days. A, B and C together can finish it in 6 days. A and C together will do it in

Accepted Answer

Concept & Formula / Logic: To find the work rate of a specific pair (A+C) when given the combined rate (A+B+C) and the rates of other pairs, you can isolate the individual rates by subtracting the given pairs from the total combined rate. Step-by-Step Solution: 1. Identify the given 1-day work rates: (A + B)'s 1-day work = (1/8) (B + C)'s 1-day work = (1/12) (A + B + C)'s 1-day work = (1/6) 2. Find A's 1-day work: Subtract (B + C)'s rate from (A + B + C)'s rate. A = (A + B + C) - (B + C) A = (1/6) - (1/12) = (2 - 1/12) = (1/12) 3. Find C's 1-day work: Subtract (A + B)'s rate from (A + B + C)'s rate. C = (A + B + C) - (A + B) C = (1/6) - (1/8) = (4 - 3/24) = (1/24) 4. Calculate (A + C)'s combined 1-day work: (A + C)'s 1-day work = A + C A + C = (1/12) + (1/24) = (2/24) + (1/24) = (3/24) = (1/8) Since their combined 1-day work is (1/8), A and C together will finish the work in 8 days. Exam Strategy & Shortcut: Use the LCM (Total Work) method. Let total work = LCM of 8, 12, and 6, which is 24 units. Efficiency of (A+B) = (24/8) = 3 units/day. Efficiency of (B+C) = (24/12) = 2 units/day. Efficiency of (A+B+C) = (24/6) = 4 units/day. Efficiency of A = (A+B+C) - (B+C) = 4 - 2 = 2 units/day. Efficiency of C = (A+B+C) - (A+B) = 4 - 3 = 1 unit/day. Combined efficiency of (A+C) = 2 + 1 = 3 units/day. Time taken by (A+C) = (24/3) = 8 days. Common Pitfall: A frequent error is trying to add the rates of (A+B) and (B+C) to find (A+C), but this gives A+2B+C. Subtracting variables strategically is required. Final Answer: Therefore, the correct answer is 8 days.

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