Question 1

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 2

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 3

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 4

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 5

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.

Question 6

A takes 5 days more than B to do a certain job and 9 days more than C; A and B together can do the job in the same time as C. How many days A would take to do it?

Accepted Answer

Concept & Logic: When multiple individuals' completion times are expressed relative to one person, define that person's time as a variable (e.g., t). Then, build a rational equation based on the condition that combining efficiencies adds up to a known total efficiency. Rate of A + Rate of B = Rate of C Step-by-Step Solution: 1. Define variables based on the text: Let A's time to complete the job be t days. A takes 5 days more than B \implies B's time = t - 5 days. A takes 9 days more than C \implies C's time = t - 9 days. 2. Set up the work equation: A and B together take the same time as C. Thus, their combined daily work rate equals C's daily work rate. (1/t) + (1/t - 5) = (1/t - 9) 3. Solve the rational equation: Find a common denominator for the left side: ((t - 5) + t/t(t - 5)) = (1/t - 9) (2t - 5/t^2 - 5t) = (1/t - 9) Cross-multiply: (2t - 5)(t - 9) = t^2 - 5t Expand the left side: 2t^2 - 18t - 5t + 45 = t^2 - 5t 2t^2 - 23t + 45 = t^2 - 5t Bring all terms to one side to form a quadratic equation: t^2 - 18t + 45 = 0 4. Factor the quadratic equation: We need two numbers that multiply to 45 and add to -18. These are -15 and -3. (t - 15)(t - 3) = 0 So, t = 15 or t = 3. If t = 3, B's time would be 3 - 5 = -2 days (impossible). Therefore, t = 15. A takes 15 days. Exam Strategy & Shortcut: For quadratic equations derived from work problems, checking the options is often the fastest route. Try Option (c) 15: If A = 15, then B = 10, and C = 6. Check if A + B = C: (1/15) + (1/10) = (2/30) + (3/30) = (5/30) = (1/6). Since (1/6) is exactly C's rate, 15 is the correct answer. Common Pitfall: A common mistake is setting up the algebraic relations backwards (e.g., making B's time t+5). Always double-check "who takes more time" to ensure the subtractions/additions go in the right direction. Final Answer: Therefore, the correct answer is 15.

Question 7

A is twice as good a workman as B. If they work together, they can complete a job in 18 days. If A alone does the job, in how many days he will complete the job?

Accepted Answer

Concept & Strategy: "Twice as good a workman" means twice the efficiency. By setting up an efficiency ratio, you can easily calculate the total units of work and divide it by an individual's efficiency. Total Work = Combined Efficiency × Combined Time Step-by-Step Solution: 1. Establish the efficiency ratio: Let B's efficiency (work units per day) be 1. Since A is twice as good, A's efficiency is 2. 2. Calculate combined efficiency: Combined efficiency (A + B) = 2 + 1 = 3 units per day. 3. Determine the total work: They complete the job together in 18 days. Total Work = Combined Efficiency × Time Total Work = 3 × 18 = 54 units. 4. Calculate time for A alone: Time taken by A = \frac{Total Work}{A's Efficiency} Time taken by A = (54/2) = 27 days. Exam Strategy & Shortcut: Using ratios is the fastest method here. Speed ratio A : B = 2 : 1. Therefore, Time ratio A : B = 1 : 2. Let A's time = x, B's time = 2x. Combined formula: (x \cdot 2x/x + 2x) = 18 (2x^2/3x) = 18 \implies (2x/3) = 18 \implies 2x = 54 \implies x = 27. A takes 27 days. Common Pitfall: A common mistake is misinterpreting the phrasing and applying the ratio incorrectly to time instead of efficiency, which would lead to calculating 54 days for A instead of B. Final Answer: Therefore, the correct answer is 27 days.

Question 8

David and Michael together can finish a job in 4 days 19 hrs 12 min. If David works at two-thirds Michael's speed, how long does it take Michael alone to finish the same job?

Accepted Answer

Concept & Strategy: When dealing with mixed time units (days, hours, minutes), always convert the total time into a single standard unit (like days or fractions of a day). Then, use efficiency ratios to find individual times. Total Work = Total Time × Total Efficiency Step-by-Step Solution: 1. Convert time into a single unit (Days): 19 hrs 12 min = 19 hours + (12/60) hours = 19 + 0.2 = 19.2 hours. Convert 19.2 hours into days: (19.2/24) = (192/240) = (4/5) = 0.8 days. Total time taken together = 4 days + 0.8 days = 4.8 days = (24/5) days. 2. Establish efficiency ratios: David works at (2/3) of Michael's speed. Let Michael's efficiency be 3 units/day. David's efficiency will be (2/3) × 3 = 2 units/day. 3. Calculate total work: Combined efficiency = 3 + 2 = 5 units/day. Total Work = Combined Efficiency × Total Time Total Work = 5 × (24/5) = 24 units. 4. Calculate Michael's individual time: Time taken by Michael alone = \frac{Total Work}{Michael's Efficiency} Time taken by Michael = (24/3) = 8 days. Exam Strategy & Shortcut: If David's speed is (2/3) of Michael's, their speed ratio is 2:3. Combined speed is 5 parts. Michael alone is 3 parts. Because time and speed are inversely proportional, the time taken by Michael compared to their combined time will be the inverse of their efficiency ratio. Time ratio (Michael alone : Combined) = Combined Speed : Michael's Speed = 5 : 3. Combined time is 4.8 days. Michael's time = (5/3) × 4.8 = 5 × 1.6 = 8 days. Common Pitfall: A frequent error occurs during the time conversion step. Dividing 12 minutes by 100 instead of 60, or miscalculating 19.2 hours into days, will completely derail the rest of the calculations. Final Answer: Therefore, the correct answer is 8 days.

Question 9

A and B can do a job together in 7 days. A is $1 \frac{3}{4}$ times as efficient as B. The same job can be done by A alone in :

Accepted Answer

Concept & Efficiency Ratio: When comparing efficiencies with mixed fractions, convert them to improper fractions to easily determine the ratio of units of work completed per day. Total Work = Combined Efficiency × Combined Time Step-by-Step Solution: 1. Determine the Efficiency Ratio: A is 1 (3/4) times as efficient as B. Convert this to an improper fraction: (7/4). Let B's efficiency be 4 units/day. A's efficiency is (7/4) × 4 = 7 units/day. The efficiency ratio of A : B is 7 : 4. 2. Calculate Total Work: Combined efficiency of A and B = 7 + 4 = 11 units/day. They complete the job together in 7 days. Total Work = Combined efficiency × Time taken Total Work = 11 × 7 = 77 units. 3. Calculate Time Taken by A Alone: Time = \frac{Total Work}{A's Efficiency} Time for A = (77/7) = 11 days. Exam Strategy & Shortcut: If A is (7/4) times as efficient as B, for every 4 units of work B does, A does 7. Total work units per day = 11 parts. Since it takes 7 days to finish 11 parts daily, total work is 77 parts. To find A's time, divide the total work (77 parts) by A's rate (7 parts/day). (77/7) = 11. Common Pitfall: Students often mix up the ratios when translating "times as efficient as," accidentally assigning 7 to B and 4 to A, leading to the wrong individual times. Final Answer: Therefore, the correct answer is 11 days.

Question 10

A is 30% more efficient than B. How much time will they, working together, take to complete a job which A alone could have done in 23 days ?

Accepted Answer

Concept & Percentage Efficiency: When combining the efforts of two workers where one is given as a percentage of the other, map their efficiencies to simple units (like 100 and 130) to easily find total work and combined rates. Total Work = Individual Efficiency × Individual Time Step-by-Step Solution: 1. Establish the Efficiency Ratio: Let B's efficiency be 100 units/day. A is 30% more efficient, so A's efficiency is 130 units/day. Efficiency ratio of A : B = 130 : 100 = 13 : 10. 2. Calculate Total Work: A alone completes the work in 23 days. Total Work = A's Efficiency × A's Time Total Work = 13 × 23 = 299 units. 3. Calculate Combined Time: Combined efficiency of A and B = 13 + 10 = 23 units/day. Time taken working together = \frac{Total Work}{Combined Efficiency} Time taken = (299/23) = 13 days. Exam Strategy & Shortcut: By using the unit method: A = 13 units/day, B = 10 units/day. Combined = 23 units/day. Total work based on A = 13 × 23. Time together = (13 × 23/23). The 23s cancel out immediately, leaving exactly 13 days. You don't even need to calculate 13 × 23 = 299. Common Pitfall: A common error is unnecessarily calculating the total work to 299 and then doing long division, which wastes time and increases the risk of arithmetic errors compared to simply canceling out the factors. Final Answer: Therefore, the correct answer is 13 days.

Question 11

A can do a piece of work in 10 days working 8 hours per day. If B is two-thirds as efficient as A, then in how many days can B alone do the same piece of work, working 5 hours per day?

Accepted Answer

Concept & Multi-Variable Work Equations: When dealing with varying days, hours per day, and efficiency, convert everything into a standard unit (like total hours) and apply the formula linking these variables. Days_1 × Hours_1 × Efficiency_1 = Days_2 × Hours_2 × Efficiency_2 Step-by-Step Solution: 1. Determine Total Hours for A: A takes 10 days, working 8 hours a day. Total hours taken by A = 10 × 8 = 80 hours. 2. Establish the Efficiency Ratio: B is (2/3) as efficient as A. If A's efficiency is 3 units/hour, then B's efficiency is 2 units/hour. 3. Calculate Total Work: Total Work = A's Total Hours × A's Efficiency Total Work = 80 × 3 = 240 units. 4. Calculate B's Time: B needs to complete 240 units of work. B's total hours needed = \frac{Total Work}{B's Efficiency} = (240/2) = 120 hours. B works 5 hours per day. Number of days required by B = (120/5) = 24 days. Exam Strategy & Shortcut: Use the direct inverse relationship equation: M_1 D_1 H_1 E_1 = M_2 D_2 H_2 E_2. Here, workers are just 1 person (so M_1 = M_2 = 1). D_1 = 10, H_1 = 8, E_1 = 3 (since B is 2/3 of A, A=3, B=2). D_2 = x, H_2 = 5, E_2 = 2. 10 × 8 × 3 = x × 5 × 2 240 = 10x \implies x = 24. Common Pitfall: Final Answer: Therefore, the correct answer is 24.

Question 12

A does half as much work as B in one-sixth of the time. If together they take 10 days to complete a work, how much time shall B alone take to do it? (S.S.C., 2005)

Accepted Answer

Concept & Efficiency Ratio: The core concept involves relating the amount of work done to the time taken to establish an efficiency ratio between two workers. Efficiency \propto \frac{Work}{Time} Step-by-Step Solution: • Given: A does 1/2 work in 1/6 time of B. A and B together complete the work in 10 days. • Calculation: Let's assume B takes x days to complete 1 unit of work. • Thus, B's 1-day work is 1/x. • A does 1/2 of the work in (1/6)x days. • To do the full work (1 unit), A will take 2 × (1/6)x = x/3 days. • So, A's 1-day work is 3/x. • Together, their 1-day work is (1/x) + (3/x) = 4/x. • We are given that together they take 10 days, so their combined 1-day work is 1/10. • Equating the two: 4/x = 1/10 \implies x = 40. • Therefore, B alone takes 40 days to complete the work. Exam Strategy & Shortcut: Instead of fractions, use efficiency ratios. If A does 1/2 work in 1/6 time, for the same time (1 full unit of time), A would do 1/2 × 6 = 3 times the work of B. So, Efficiency of A : B = 3 : 1. Together, they have an efficiency of 4 units/day. In 10 days, total work = 40 units. Time for B alone = Total Work / B's Efficiency = 40 / 1 = 40 days. Common Pitfall: Misinterpreting "half as much work in one-sixth of the time". Students often directly write the efficiency ratio as 1:2 or 1:6, forgetting to mathematically combine both the work and time modifiers. Final Answer: Therefore, the correct answer is 40 days.

Question 13

Two workers A and B working together completed a job in 5 days. If A worked twice as efficiently as he actually did and B worked $\frac{1}{3}$ as efficiently as he actually did, the work would have been completed in 3 days. A alone could complete th…

Accepted Answer

Concept & Simultaneous Equations: When work completion changes based on modified individual efficiencies, we can model the scenario using a system of linear equations representing the 1-day work of each individual. Step-by-Step Solution: • Given: A and B take 5 days normally. With A at 2 × efficiency and B at 1/3 × efficiency, they take 3 days. • Calculation: Let A's 1-day work be x and B's 1-day work be y. • From the first condition: x + y = 1/5 --- (Equation 1) • From the second condition: 2x + (1/3)y = 1/3 --- (Equation 2) • We need to find the time A alone takes, which requires finding x. Let's eliminate y. • Multiply Equation 1 by 1/3: (1/3)x + (1/3)y = 1/15 --- (Equation 3) • Subtract Equation 3 from Equation 2: • (2x - (1/3)x) + ((1/3)y - (1/3)y) = 1/3 - 1/15 • (5/3)x = (5 - 1)/15 • (5/3)x = 4/15 • x = (4/15) × (3/5) = 12/75 = 4/25 • Since A's 1-day work is 4/25, A alone will take 25/4 days. • Convert to mixed fraction: 25/4 = 6(1/4) days. Exam Strategy & Shortcut: Use algebraic manipulation effectively. Once the equations x + y = 1/5 and 6x + y = 1 (multiplying the second equation by 3 to clear fractions) are set up, subtracting the first from the modified second instantly yields 5x = 4/5 \implies x = 4/25. Thus, time is 25/4. Common Pitfall: A common mistake is directly substituting efficiency ratios as time. Students might write 2x + y/3 = 3 instead of using the reciprocal (1/3) for the one-day work equivalent. Final Answer: Therefore, the correct answer is 6(1/4) days.

Question 14

A can do a work in 15 days and B in 20 days. If they work on it together for 4 days, then the fraction of the work that is left is :

Accepted Answer

Concept & Fractional Work: To find the remaining work, calculate the total work completed by adding the individuals' one-day work capacities, multiplying by the number of days worked, and subtracting from 1 (the whole work). Work Left = 1 - (Combined 1-day work × Days worked) Step-by-Step Solution: • Given: A's time = 15 days, B's time = 20 days. They work together for 4 days. • Calculation: Find the 1-day work for each. • A's 1-day work = 1/15 • B's 1-day work = 1/20 • (A + B)'s 1-day work = 1/15 + 1/20 • Find a common denominator, which is 60: (4 + 3)/60 = 7/60 • So, A and B complete 7/60 of the work in 1 day. • Work done by them in 4 days = 4 × (7/60) = 28/60 = 7/15. • The fraction of the work left = Total work - Work completed. • Work left = 1 - (7/15) = (15 - 7)/15 = 8/15. Exam Strategy & Shortcut: Use the LCM method (Total Work Method). LCM of 15 and 20 is 60. Let total work be 60 units. A's efficiency = 60/15 = 4 units/day. B's efficiency = 60/20 = 3 units/day. Total efficiency = 7 units/day. Work done in 4 days = 7 × 4 = 28 units. Remaining work = 60 - 28 = 32 units. Fraction left = Remaining Work / Total Work = 32/60 = 8/15. Common Pitfall: Students often calculate the work completed (7/15) and mistakenly select it as the answer, forgetting the final step of subtracting it from 1 to find the work *left*. Always read the final requirement of the prompt carefully. Final Answer: Therefore, the correct answer is (8/15).

Question 15

A can do a piece of work in 15 days, which B can do in 10 days. B worked at it for 8 days. A can finish the remaining work in (R.R.B., 2004)

Accepted Answer

Concept & Remaining Work Calculation: When one person partially completes a job and leaves, determine the exact fraction of the work they accomplished. The remaining fraction is then multiplied by the total time the second person would need to complete the entire job alone. Step-by-Step Solution: • Given: A finishes in 15 days, B finishes in 10 days. B works alone for 8 days. • Calculation: Calculate B's 1-day work. • B's 1-day work = 1/10. • B works for 8 days. Work done by B = 8 × (1/10) = 8/10 = 4/5. • Determine the remaining work to be done by A. • Remaining work = 1 - 4/5 = 1/5. • We know A can do 1 complete job (the whole work) in 15 days. • Time taken by A to finish the remaining 1/5 of the work = (1/5) × 15 days. • Time = 3 days. Exam Strategy & Shortcut: Using the LCM method for speed: Let Total Work = LCM(15, 10) = 30 units. Efficiency of A = 30/15 = 2 units/day. Efficiency of B = 30/10 = 3 units/day. B worked for 8 days, doing 8 × 3 = 24 units. Remaining work = 30 - 24 = 6 units. Time A needs to finish 6 units = Remaining Work / A's Efficiency = 6 / 2 = 3 days. Common Pitfall: A classic error is calculating the remaining work accurately but then dividing it by A's one-day work improperly, or mistakenly applying B's efficiency to A's remaining calculation timeline. Keep track of whose efficiency belongs to whom. Final Answer: Therefore, the correct answer is 3 days.

Question 16

A and B can complete a work in 18 days and 15 days respectively. They started doing the work together but after 3 days A had to leave and B alone completed the remaining work. The whole work was completed in

Accepted Answer

Concept & Total Work Calculation: To solve problems involving workers leaving mid-way, find the total work by taking the Least Common Multiple (LCM) of their individual times. Then, calculate daily efficiencies to determine the work done before someone left and the remaining work. Total Work = Efficiency × Time Step-by-Step Solution: • Given: A can complete the work in 18 days, B in 15 days. They work together for 3 days. • Calculation: Let Total Work = LCM(18, 15) = 90 units. • Efficiency of A = 90 / 18 = 5 units/day. • Efficiency of B = 90 / 15 = 6 units/day. • Combined efficiency of A + B = 5 + 6 = 11 units/day. • Work done by A and B together in 3 days = 11 × 3 = 33 units. • Remaining work = 90 - 33 = 57 units. • After 3 days, A leaves. So, B completes the remaining 57 units alone. • Time taken by B to complete remaining work = 57 / 6 = 19/2 = 9(1/2) days. • The question asks for the total time to complete the *whole* work. • Total time = Time worked together + Time worked by B alone = 3 days + 9(1/2) days = 12(1/2) days. Exam Strategy & Shortcut: Instead of fractional methods which cause messy additions, always jump straight to LCM values. LCM of 18 and 15 is 90. Efficiencies are 5 and 6. Together they do 11 per day for 3 days (33 units). Remaining 57 is done by B (speed 6) in 57/6 = 9.5 days. Total time = 3 + 9.5 = 12.5 days. Common Pitfall: A very common mistake is finding the time B takes for the *remaining* work (9(1/2) days) and selecting that as the answer. You must carefully read whether the question asks for the remaining time or the total time. Final Answer: Therefore, the correct answer is 12(1/2) days.

Question 17

A machine P can print one lakh books in 8 hours, machine Q can print the same number of books in 10 hours while machine R can print them in 12 hours. All the machines are started at 9 a.m. while machine P is closed at 11 a.m. and the remaining two m…

Accepted Answer

Concept & Total Work and Rate: This problem relies on assigning a "Total Work" unit value using the Least Common Multiple (LCM) of the given times to find individual machine printing rates, then tracking the work completed in distinct time phases. Step-by-Step Solution: • Given: P prints in 8 hrs, Q in 10 hrs, R in 12 hrs. They run from 9 a.m. to 11 a.m. (2 hours). Then P closes, and Q + R finish the rest. • Calculation: Let Total Work = LCM(8, 10, 12) = 120 units. • Rate of Machine P = 120 / 8 = 15 units/hr. • Rate of Machine Q = 120 / 10 = 12 units/hr. • Rate of Machine R = 120 / 12 = 10 units/hr. • For the first 2 hours (9 a.m. to 11 a.m.), P, Q, and R work together. • Combined rate of (P + Q + R) = 15 + 12 + 10 = 37 units/hr. • Work done in 2 hours = 37 × 2 = 74 units. • Remaining work = Total Work - Work Done = 120 - 74 = 46 units. • After 11 a.m., P is closed. Q and R continue working. • Combined rate of (Q + R) = 12 + 10 = 22 units/hr. • Time taken by Q and R to finish the remaining work = Remaining Work / (Q + R rate) = 46 / 22 = 23 / 11 hours. • 23 / 11 \approx 2.09 hours, which is approximately 2 hours and 5 minutes. • Finish time = 11:00 a.m. + 2 hours 5 minutes = 1:05 p.m. • The closest option is approximately 1 p.m. Exam Strategy & Shortcut: Mentally find the LCM (120) and rates (15, 12, 10). Sum the rates for the first phase: 37. Multiply by 2 hours = 74. Subtract from 120 to get 46. Sum the rates for the second phase: 22. Divide 46 by 22, which is slightly more than 2. Thus, the completion time is roughly 2 hours after 11 a.m., placing it closest to 1 p.m. Common Pitfall: A common mistake is forgetting to add the remaining time to the time the initial phase ended (11 a.m.), or struggling to convert fractional hours into standard time, which isn't strictly necessary here since "approximately" allows for selecting the nearest whole hour. Final Answer: Therefore, the correct answer is 1 p.m.

Question 18

X and Y can do a piece of work in 20 days and 12 days respectively. X started the work alone and then after 4 days Y joined him till the completion of the work. How long did the work last ?

Accepted Answer

Concept & Sequential Work: Track the amount of work completed in chronological phases. Subtract the initial work completed by the solo worker from the total work, then divide the remainder by the combined efficiency of all current workers to find the remaining time. Step-by-Step Solution: • Given: X can do the work in 20 days, Y in 12 days. X works alone for 4 days, then Y joins. • Calculation: Let Total Work = LCM(20, 12) = 60 units. • Efficiency of X = 60 / 20 = 3 units/day. • Efficiency of Y = 60 / 12 = 5 units/day. • X starts alone and works for 4 days. Work done by X = 3 × 4 = 12 units. • Remaining work = 60 - 12 = 48 units. • Y joins X, so they now work together. • Combined efficiency of (X + Y) = 3 + 5 = 8 units/day. • Time taken by (X + Y) to finish the remaining work = 48 / 8 = 6 days. • The question asks how long the work lasted in total. • Total time = Time X worked alone + Time X and Y worked together = 4 + 6 = 10 days. Exam Strategy & Shortcut: Use the LCM (60). X does 3 units/day, Y does 5 units/day. X does 12 units in 4 days. 48 units remain. Together they do 8 units/day. 48 / 8 = 6 days. Total time is 4 + 6 = 10 days. Common Pitfall: A frequent error is calculating the time for the second phase (6 days) and choosing that as the final answer. The question specifically asks "How long did the work last?", implying the total duration from start to finish. Final Answer: Therefore, the correct answer is 10 days.

Question 19

A completes $\frac{7}{10}$ of a work in 15 days. Then he completes the remaining work with the help of B in 4 days. The time required for A and B together to complete the entire work is (S.S.C., 2005)

Accepted Answer

Concept & Fractional Work Scaling: If you know the time taken by a group of workers to complete a specific fraction of a job, you can find the time it takes them to complete the entire job by setting up a simple proportion or multiplying the time by the reciprocal of the work fraction. Step-by-Step Solution: • Given: A completes 7/10 of a work in 15 days. A and B together complete the remaining work in 4 days. • Calculation: First, determine what fraction of the work remains after A's solo portion. • Remaining work = 1 - 7/10 = 3/10. • We are given that A and B working together can complete this 3/10 fraction of the work in exactly 4 days. • To find the time required for A and B together to complete 1 entire work (the whole job), we scale the time up. • Let T be the total time for A and B to do the whole work. • (3/10) × Total Work = 4 days • Total Time T = 4 × (10 / 3) = 40 / 3 days. • Converting the improper fraction to a mixed number: 40 / 3 = 13(1/3) days. Exam Strategy & Shortcut: Recognize that the information about A taking 15 days is a distractor and not needed to solve the problem. Focus only on the final phase: A and B do 3/10 of the work in 4 days. The time for the full work is simply 4 × (10/3) = 40/3 = 13(1/3) days. Common Pitfall: A common pitfall is getting bogged down trying to calculate A's individual efficiency from the first sentence, leading to unnecessary complex calculations with B's efficiency. Final Answer: Therefore, the correct answer is 13(1/3) days.

Question 20

A man and a boy can do a piece of work in 24 days. If the man works alone for the last 6 days, it is completed in 26 days. How long would the boy take to do it alone? (S.S.C., 2005)

Accepted Answer

Concept & Work Breakdown by Phases: When a total timeline is given alongside a solo finishing phase, subtract the solo days from the total days to find how long the workers collaborated. Use the fraction of work completed together to deduce the solo worker's speed from the remainder. Step-by-Step Solution: • Given: Man and Boy together take 24 days. The total work takes 26 days if the Man works alone for the last 6 days. • Calculation: Since the total job took 26 days and the Man worked alone for the last 6 days, the Man and Boy must have worked together for the initial period. • Time worked together = 26 - 6 = 20 days. • The Man and Boy's combined 1-day work is 1/24. • Work completed by them together in 20 days = 20 × (1/24) = 20/24 = 5/6. • Remaining work = 1 - 5/6 = 1/6. • This remaining 1/6 of the work was completed by the Man alone in 6 days. • So, the Man's 1-day work = (1/6) / 6 = 1/36. (This means the man alone takes 36 days). • We need the Boy's 1-day work. • Combined 1-day work = Man's 1-day work + Boy's 1-day work • 1/24 = 1/36 + Boy's 1-day work • Boy's 1-day work = 1/24 - 1/36 • Find the LCM of 24 and 36, which is 72. • Boy's 1-day work = (3 - 2) / 72 = 1/72. • Therefore, the boy alone would take 72 days. Exam Strategy & Shortcut: They worked together for 20 out of the required 24 days, meaning they finished 20/24 = 5/6 of the work. The man finished the remaining 1/6 in 6 days. Thus, the man does 1/6 work in 6 days, or 1/36 per day. Total efficiency is 1/24. Boy = 1/24 - 1/36 = 1/72. So, 72 days. Common Pitfall: A frequent error is assuming the man and boy worked together for 26 days minus 6 days of the man alone, but misinterpreting the timeline and calculating the boy's remaining workload backward. Always anchor the fractional work completed. Final Answer: Therefore, the correct answer is 72 days.

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