Question 1

A can do a job in 10 days. B is 60 percent more efficient than A. In how many days can B alone complete the same job?

Accepted Answer

Introduction / Context: Efficiency comparisons translate into rate multipliers. If B is 60 percent more efficient than A, B’s work rate is 1.6 times A’s rate. Time is the reciprocal of rate for one whole job. Given Data / Assumptions: A time = 10 days ⇒ A rate = 1/10 per day. B efficiency = 160 percent of A. Concept / Approach: Multiply A’s rate by 1.6 to get B’s rate, then take the reciprocal to obtain B’s time to finish one job. Step-by-Step Solution: B rate = 1.6 * (1/10) = 0.16 per dayB time = 1 / 0.16 = 6.25 days = 6 1/4 days Verification / Alternative check: If A does 1 unit in 10 days, in 6.25 days A would do 0.625 units. Since B is 1.6 times as fast, B’s output in 6.25 days is 1 unit, confirming the calculation. Why Other Options Are Wrong: 6 and 6 2/3 arise from rounding or misinterpreting 60 percent more as 60 percent of A; 8 days assumes B is slower than reality; 5 days overestimates B’s efficiency. Common Pitfalls: Confusing 60 percent more with 60 percent of; converting 0.25 to fractional days incorrectly; mixing time and rate comparisons directly. Final Answer: 6 1/4

Question 2

A, B, and C together earn Rs 150 per day. A and C together earn Rs 94, and B and C together earn Rs 76. What is C’s daily earning?

Accepted Answer

Introduction / Context: This is a system of linear equations in daily earnings. Summations for different pairs allow isolating each individual value by subtraction. Given Data / Assumptions: A + B + C = 150 A + C = 94 B + C = 76 Concept / Approach: Use subtraction to eliminate variables and solve for C directly, then cross check with any other pair equation. Step-by-Step Solution: From A + B + C and A + C: B = 150 − 94 = 56From B + C = 76 ⇒ 56 + C = 76 ⇒ C = 20 Verification / Alternative check: Then A = 94 − C = 74. Check with total: A + B + C = 74 + 56 + 20 = 150, consistent with the first statement. Why Other Options Are Wrong: Rs. 75, Rs. 56, Rs. 34, Rs. 30 do not satisfy all three equations simultaneously. Common Pitfalls: Solving for the wrong variable; arithmetic errors when subtracting; assuming equal shares without basis. Final Answer: Rs. 20

Question 3

A alone can finish a work in 10 days and B alone in 15 days. Working together, they finish the work. Out of a total wage of Rs 75, what is A’s share?

Accepted Answer

Introduction / Context: Wages are split in proportion to work done. If two workers complete a job together, each person’s share equals their rate divided by the total rate, times the total wage. Given Data / Assumptions: A time = 10 days ⇒ A rate = 1/10 per day. B time = 15 days ⇒ B rate = 1/15 per day. Total wage = Rs 75. Concept / Approach: Share of A = (A rate / (A rate + B rate)) * total wage. This is because the entire job output is proportional to rates, not times. Step-by-Step Solution: A rate = 1/10; B rate = 1/15A share fraction = (1/10) / (1/10 + 1/15) = (1/10) / (5/30) = 3/5A share = (3/5) * 75 = 45 Verification / Alternative check: Check B share: (2/5) * 75 = 30. The two shares add to Rs 75 as expected. Why Other Options Are Wrong: Rs. 30 is B’s share; Rs. 37.50, Rs. 50, Rs. 40 are inconsistent with proportional rates. Common Pitfalls: Splitting wages equally; using times instead of rates; arithmetic slips converting fractions. Final Answer: Rs. 45

Question 4

Factory A produces x cars per hour. Factory B produces y cars every 2 hours. How many cars do both factories together produce in 8 hours?

Accepted Answer

Introduction / Context: Translate textual rates into per hour rates and aggregate outputs across the specified time window. Then sum outputs from both factories. Given Data / Assumptions: A rate = x cars per hour. B rate = y cars every 2 hours = y/2 cars per hour. Time window = 8 hours. Concept / Approach: Total cars = (rate of A + rate of B) * time. Convert B to per hour first, then multiply by 8 and simplify algebraically. Step-by-Step Solution: A output in 8 h = 8xB output in 8 h = 8 * (y/2) = 4yTotal = 8x + 4y = 4(2x + y) Verification / Alternative check: Unit check: both terms have units cars. Factorization 8x + 4y = 4(2x + y) is algebraically correct. Why Other Options Are Wrong: 8(x + y) incorrectly treats B as y per hour; (8x + y)/2 and 16(x + y) are algebraic or rate conversion errors; 2(4x + y) equals 8x + 2y, which undercounts B by half. Common Pitfalls: Forgetting to convert B’s rate to per hour; distributing the 8 incorrectly; misreading parentheses in options. Final Answer: 4(2x + y)

Question 5

A is twice as efficient as B, and together they finish a job in 14 days. In how many days can A alone finish the job?

Accepted Answer

Introduction / Context: Relative efficiency problems can be handled by assigning a variable to one worker’s rate and expressing the other’s rate as a multiple. Sum the rates to meet the joint time. Given Data / Assumptions: A is twice as efficient as B ⇒ rate of A = 2b if rate of B = b. Together they take 14 days ⇒ combined rate = 1/14. Concept / Approach: Set B rate to b, then A rate is 2b. Add to get 3b and equate to 1/14 to solve for b, then compute A’s time as 1 divided by A’s rate. Step-by-Step Solution: 2b + b = 3b = 1/14b = 1/42A rate = 2b = 1/21 ⇒ A alone time = 21 days Verification / Alternative check: Check combined: 1/21 + 1/42 = 3/42 = 1/14, consistent with the given joint time. Why Other Options Are Wrong: 11, 28, 42, 14 do not satisfy the joint time when combined with B’s rate and the 2:1 efficiency relation. Common Pitfalls: Misreading twice as efficient; averaging times instead of rates; arithmetic slips with fractional rates. Final Answer: 21 days

Question 6

Fourteen workers can make 1400 toys in 5 days. After the first day, 14 more workers join. How many additional days are needed to finish the remaining toys?

Accepted Answer

Introduction / Context: Output-per-worker-per-day can be derived from an initial production plan. Once additional workers join, the daily output changes and we compute remaining time accordingly. Given Data / Assumptions: 14 workers produce 1400 toys in 5 days. After 1 day, 14 more workers join (total 28). Find extra days to finish the remaining toys. Concept / Approach: First compute per worker daily productivity p from the original plan. Subtract the first day’s production at 14p from the total to get the remainder. Divide the remainder by the new daily output 28p. Step-by-Step Solution: 14 * p * 5 = 1400 ⇒ p = 1400 / 70 = 20 toys per worker per dayDay 1 production = 14 * 20 = 280 toysRemaining = 1400 − 280 = 1120 toysNew daily output = 28 * 20 = 560 toysDays to finish = 1120 / 560 = 2 days Verification / Alternative check: Total elapsed time = 1 + 2 = 3 days which is less than the original 5 day plan, as expected when doubling the workforce after day 1. Why Other Options Are Wrong: 3, 4, 3.5, 1 days do not divide the remainder 1120 by 560 correctly and lead to under or over production. Common Pitfalls: Recomputing p incorrectly; forgetting to subtract the first day’s production; using 14 rather than 28 for the remaining days. Final Answer: 2 days

Question 7

Twenty four boys can complete a job in 12 days, while sixteen men can complete the same job in 9 days. In how many days can 12 boys and 12 men together complete the job?

Accepted Answer

Introduction / Context: This problem compares two worker types. We equate the total work from each scenario to relate one man’s productivity to one boy’s productivity, then compute the combined team rate. Given Data / Assumptions: 24 boys finish in 12 days. 16 men finish in 9 days. Find time for 12 boys and 12 men together. Concept / Approach: Let 1 boy per day produce b units, 1 man per day produce m units, and let total work be W. From each scenario, write W in terms of b and m to find m in terms of b. Then add rates for the mixed team and compute time = W / rate. Step-by-Step Solution: W = 24 * b * 12 = 288bW = 16 * m * 9 = 144mEquate: 288b = 144m ⇒ m = 2bMixed team daily rate = 12b + 12m = 12b + 24b = 36bTime = W / rate = 288b / 36b = 8 days Verification / Alternative check: Using man boy equivalence 1 man = 2 boys, the mixed team equals 12 + 24 = 36 boy equivalents, and 288 boy days / 36 equals 8 days. Why Other Options Are Wrong: 6, 7, 10, 9 assume the wrong boy to man equivalence or misdivide total boy days by the combined equivalent rate. Common Pitfalls: Adding times rather than rates; treating men and boys as identical; miscomputing the equivalence ratio from the two full team cases. Final Answer: 8 days

Question 8

Twenty four men can complete a job in 16 days, while thirty six women can complete the same job in 24 days. In how many days will 8 men and 6 women together complete the job?

Accepted Answer

Introduction / Context: Different categories of workers have different per person rates. We extract each category’s per person daily rate from the given completions and then add them for the mixed group. Given Data / Assumptions: 24 men finish in 16 days. 36 women finish in 24 days. Find completion time for 8 men and 6 women together. Concept / Approach: Compute 1 man per day rate and 1 woman per day rate from the team completions, then compute the mixed rate and take the reciprocal to get time. Step-by-Step Solution: Men team rate = 1/16 ⇒ per man rate r_m = (1/16) / 24 = 1/384Women team rate = 1/24 ⇒ per woman rate r_w = (1/24) / 36 = 1/864Mixed rate = 8 * (1/384) + 6 * (1/864) = 1/36Time = 1 / (1/36) = 36 days Verification / Alternative check: Convert women to man equivalents: 1 woman = 4/9 man, then 8 men + 6 women = 8 + 6*(4/9) = 8 + 8/3 = 32/3 men; time = (24 men * 16 days) / (32/3) ≈ 36 days, consistent. Why Other Options Are Wrong: 28, 32, 24, 30 correspond to incorrect rate conversions or algebraic slips while combining categories. Common Pitfalls: Assuming men and women have the same per person rate; adding days instead of rates; arithmetic errors with 384 and 864 denominators. Final Answer: 36 days

Question 9

A alone can complete a job in 12 days and B alone in 24 days. Working together, in how many days can they complete the job?

Accepted Answer

Introduction / Context: When two workers collaborate, the combined rate is the sum of individual rates. The completion time is the reciprocal of the combined rate for a unit job. Given Data / Assumptions: A: 12 days ⇒ 1/12 per day. B: 24 days ⇒ 1/24 per day. Concept / Approach: Sum the rates and invert to get time. Step-by-Step Solution: Combined rate = 1/12 + 1/24 = 1/8 per dayTime = 1 / (1/8) = 8 days Verification / Alternative check: In 8 days, A produces 8/12 = 2/3 and B produces 8/24 = 1/3. Total equals 1 job, confirming the result. Why Other Options Are Wrong: 6 and 4 underestimate the time; 10 and 12 overestimate; only 8 days matches the real sum of rates. Common Pitfalls: Averaging times instead of summing rates; arithmetic slips with fractions 1/12 and 1/24. Final Answer: 8 days

Question 10

A can finish a job in 10 days and B in 20 days. They start together, but A leaves 2 days before completion. In how many days is the whole job finished?

Accepted Answer

Introduction / Context: Staggered participation requires expressing total work as the sum of contributions over the exact durations each worker is present. Create an equation in terms of the unknown total time and solve. Given Data / Assumptions: A: 10 days ⇒ 1/10 per day. B: 20 days ⇒ 1/20 per day. A works until 2 days before the end; B works the entire duration. Concept / Approach: Let total time be T days. Then A works T − 2 days and B works T days. Set total work to 1 and solve for T. Step-by-Step Solution: (T − 2) * (1/10) + T * (1/20) = 1Multiply by 20: 2(T − 2) + T = 203T − 4 = 20 ⇒ 3T = 24 ⇒ T = 8 days Verification / Alternative check: Check contributions: A works 6 days at 1/10 = 0.6; B works 8 days at 1/20 = 0.4; total is 1 job. Why Other Options Are Wrong: 7 2/3, 7, 6, 9 days fail the linear equation or give totals less or more than the full job. Common Pitfalls: Letting A also work the last 2 days; mixing time and rate averages; arithmetic mistakes solving the linear equation. Final Answer: 8 days

Question 11

A and B can complete a work in 10 hours, B and C in 15 hours, and A and C in 12 hours. How long will B alone take to complete the work?

Accepted Answer

Introduction / Context: Pairwise completion information allows us to compute the sum of individual hourly rates and then isolate any one worker’s rate by subtraction. Given Data / Assumptions: A + B: 10 h ⇒ 1/10 per hour. B + C: 15 h ⇒ 1/15 per hour. A + C: 12 h ⇒ 1/12 per hour. Concept / Approach: Let a, b, c be hourly rates. Then a + b = 1/10, b + c = 1/15, a + c = 1/12. Sum to get 2(a + b + c), divide by 2 for the total rate, then subtract (a + c) to get b. Step-by-Step Solution: 2(a + b + c) = 1/10 + 1/15 + 1/12 = 1/4a + b + c = 1/8b = (a + b + c) − (a + c) = 1/8 − 1/12 = 1/24Time for B alone = 1 / (1/24) = 24 h Verification / Alternative check: Compute a and c similarly and verify that the pair sums equal the given rates. The arithmetic is consistent. Why Other Options Are Wrong: 12 h, 16 h, 20 h, 18 h do not match the extracted b when recombined into pair rates. Common Pitfalls: Adding hours instead of rates; forgetting to divide the pair sum by 2; sign mistakes while subtracting to isolate b. Final Answer: 24 h

Question 12

Work completion after partial progress by a faster coworker A can finish a job in 10 days, while B can finish the same job in 6 days. B works alone for the first 4 days. How long will A take to complete the remaining work, starting from that point?

Accepted Answer

Introduction / Context: Questions involving “work done before/after” test your ability to compute partial work using individual rates and then convert the remaining work into time for another worker. This problem features two workers with different efficiencies and a hand-off in the middle. Given Data / Assumptions: A completes the job in 10 days. B completes the job in 6 days. B works alone for the first 4 days. Total work is normalized to 1 unit. Concept / Approach: The rate approach: If a worker completes a job in T days, the daily rate is 1/T. Work done = rate * time. Remaining work = total work − completed work. Step-by-Step Solution: Rate of A = 1/10 per dayRate of B = 1/6 per dayWork done by B in 4 days = 4 * (1/6) = 2/3Remaining work after B = 1 − 2/3 = 1/3Time for A to do the remaining work = (1/3) / (1/10) = 10/3 days Verification / Alternative check: Convert to hours if desired: 10/3 days ≈ 3.333... days. Multiplying A’s rate 1/10 by 10/3 yields exactly 1/3 of the job, matching the remaining portion. Why Other Options Are Wrong: 11/3 days: Overestimates the needed time; A would do more than the remaining 1/3. 7/3 days: Underestimates; A would not finish the remaining part. 4 days: Assumes slower-than-necessary completion; A’s rate is sufficient to do it in less. None of these: Incorrect because 10/3 is available. Common Pitfalls: Mixing up who worked first, or adding rates/time incorrectly. Always compute remaining work correctly before dividing by the finisher’s rate. Final Answer: 10/3 days

Question 13

Missing workers from day one change the timeline A contractor plans to complete a job in 18 days with a certain number of men. However, 12 men are absent from the very first day and the remaining crew finishes the work in 30 days. How many men were …

Accepted Answer

Introduction / Context: This is a classic “planned strength vs. actual strength” time and work problem. We equate total work using rate * time with different crew sizes. Given Data / Assumptions: Planned duration = 18 days with M men. Actual: (M − 12) men finish in 30 days. Each man has the same constant productivity. Concept / Approach: Let a single man’s daily rate be r. Total work W can be written two ways and equated. Step-by-Step Solution: Planned work: W = M * r * 18Actual work: W = (M − 12) * r * 30Equate: M * 18 = (M − 12) * 3018M = 30M − 360 ⇒ 12M = 360 ⇒ M = 30 Verification / Alternative check: Plug back: Planned capacity per day = 30r; in 18 days gives 540r. Actual capacity per day = 18r; in 30 days gives 540r. Matches. Why Other Options Are Wrong: 40, 24, 18: Do not satisfy the equation 18M = 30(M − 12). None of these: Incorrect because 30 fits perfectly. Common Pitfalls: Forgetting to hold total work constant or mixing up which team size applies to which duration. Final Answer: 30

Question 14

Helper joins mid-way on a construction task A mason can build a tank in 12 hours. He works alone for 6 hours, then a boy helps and they finish the job in 5 more hours. How long would the boy take to complete the entire job alone?

Accepted Answer

Introduction / Context: When a helper joins partway, compute the initial portion done by the first worker, then infer the combined rate to deduce the helper’s individual rate. Given Data / Assumptions: Mason completes job in 12 h ⇒ rate = 1/12 per hour. Mason works alone for 6 h ⇒ completes 1/2 of the job. Remaining 1/2 is done by mason + boy in 5 h. Concept / Approach: Combined rate = remaining work / time. Boy’s rate = combined rate − mason’s rate. Time for boy alone = 1 / boy’s rate. Step-by-Step Solution: Work after 6 h alone = 6 * (1/12) = 1/2Remaining work = 1/2Combined rate (mason + boy) = (1/2) / 5 = 1/10 per hourMason’s rate = 1/12 ⇒ Boy’s rate = 1/10 − 1/12 = 1/60 per hourBoy alone time = 1 / (1/60) = 60 h Verification / Alternative check: In 5 hours together they do 5 * (1/10) = 1/2, which matches the remaining half. Why Other Options Are Wrong: 45 h, 30 h, 64 h do not match the inferred boy’s rate 1/60. Common Pitfalls: Assuming linear sharing without first computing combined rate from the second phase. Final Answer: 60 h

Question 15

Relative efficiencies from joint completion time A and B can complete a job together in 12 days. A is exactly 2 times as efficient as B. In how many days can B alone complete the job?

Accepted Answer

Introduction / Context: This problem leverages the ratio of efficiencies. If A is 2 times as efficient as B, A’s rate is double B’s rate. From the joint time we can extract each individual time. Given Data / Assumptions: A is 2x as efficient as B. Together they finish in 12 days. Total work normalized to 1. Concept / Approach: If B’s rate is b, A’s rate is 2b. Together rate = 3b = 1/12 ⇒ b = 1/36. Step-by-Step Solution: Let B’s rate = bA’s rate = 2bTogether: 2b + b = 3b = 1/12 ⇒ b = 1/36Therefore, B alone takes 1 / b = 36 days Verification / Alternative check: Check rates: A = 2/36 = 1/18; B = 1/36; Together = 1/18 + 1/36 = 3/36 = 1/12. Why Other Options Are Wrong: 24, 18, 12 are not consistent with the 3b = 1/12 relationship. Common Pitfalls: Confusing “2 times as efficient” with “2 days less” or miscomputing sum of rates. Final Answer: 36 days

Question 16

Comparing speeds to infer joint time A is three times as efficient as B and thus completes the job 30 days sooner than B. In how many days will A and B finish the job working together from start to finish?

Accepted Answer

Introduction / Context: The problem gives a relative efficiency (A is 3x B) and a difference in their individual completion times (A finishes 30 days sooner). From this, we can find individual times and then the joint time. Given Data / Assumptions: A is three times as efficient as B. A needs 30 fewer days than B to complete the job alone. Concept / Approach: If B alone takes t days, then A alone takes t/3 days (because A is 3x as fast). The difference t − t/3 = 30 allows solving for t. Then combine rates. Step-by-Step Solution: t − t/3 = 30 ⇒ (2/3)t = 30 ⇒ t = 45 days (B’s time)A’s time = 45/3 = 15 daysJoint rate = 1/15 + 1/45 = 4/60 = 1/15 + 1/45 = 4/45Joint time = 1 / (4/45) = 45/4 days = 11.25 days Verification / Alternative check: In 11.25 days, A contributes 11.25/15 = 0.75 of the job, B contributes 11.25/45 = 0.25, totaling 1 job. Why Other Options Are Wrong: 15 days is A-alone time, not joint. 45/5 (=9) and 12 days do not match the combined rate. Common Pitfalls: Equating “3 times as efficient” with “30 days less” without forming equations, or adding times instead of rates. Final Answer: 45/4 days

Question 17

Repairing a partially broken stem: infer Z’s daily percentage X can do 20% of the work per day, Y can do 25% of the work per day, and together X, Y, and Z can do 50% of the work per day. What percentage of the work per day is done by Z alone?

Accepted Answer

Introduction / Context: This question asks for Z’s solo percentage contribution when daily percentages for X and Y are known, and a combined daily percentage for X+Y+Z is given. The original stem had missing words; we apply recovery to make it solvable without altering the core intent. Given Data / Assumptions: X does 20% of the job per day ⇒ 0.20 per day. Y does 25% per day ⇒ 0.25 per day. Together X+Y+Z do 50% per day ⇒ 0.50 per day. Concept / Approach: Percentages per day add for independent workers. Thus, Z’s rate is (X+Y+Z) − (X+Y). Step-by-Step Solution: Rate(X) = 0.20Rate(Y) = 0.25Rate(X+Y+Z) = 0.50Therefore, Rate(Z) = 0.50 − (0.20 + 0.25) = 0.05 = 5% Verification / Alternative check: 0.20 + 0.25 + 0.05 = 0.50, which matches the combined daily percentage. Why Other Options Are Wrong: 10%, 15%, 20% overstate Z’s share given the fixed total of 50%. Common Pitfalls: Confusing percentage of work with fraction of the day; here percentages are per-day work fractions. Final Answer: 5%

Question 18

Men–women–children equivalence and a family crew Either 4 men or 6 women or 10 children can paint a house in 5 days. A couple (1 man + 1 woman) and their 5 sons (children) are hired. In how many days will they finish the job?

Accepted Answer

Introduction / Context: When different groups can complete a job in the same time, we can infer equivalent individual rates. Once we know the man, woman, and child rates, we sum the family’s daily rate and invert to get total time. Given Data / Assumptions: 4 men in 5 days ⇒ 1 job. 6 women in 5 days ⇒ 1 job. 10 children in 5 days ⇒ 1 job. Crew used: 1 man + 1 woman + 5 children. Concept / Approach: Let the job be 1 unit. Compute per-day per-person rates from the equivalences, then sum the family’s rates. Step-by-Step Solution: Rate(man) = 1 / (4 * 5) = 1/20 per dayRate(woman) = 1 / (6 * 5) = 1/30 per dayRate(child) = 1 / (10 * 5) = 1/50 per dayFamily rate = 1/20 + 1/30 + 5 * (1/50) = 1/12 + 1/10 = 11/60 per dayTime = 1 / (11/60) = 60/11 days ≈ 5.45 days Verification / Alternative check: Check partial: In 5 days, they would do 5 * 11/60 = 11/12 of the job, leaving 1/12; another 1/ (11/60) − 5 ≈ 0.45 day finishes it. Why Other Options Are Wrong: 56/11 ≈ 5.09, 11/2 = 5.5, and 6 are not the exact reciprocal of 11/60. Common Pitfalls: Forgetting that “4 men in 5 days” implies a single man-day rate of 1/20. Final Answer: 60/11 days

Question 19

Mixed grazing rates for goats and sheep Either 4 goats or 6 sheep can graze a field in 50 days. If 2 goats and 9 sheep graze together, in how many days will they finish the field?

Accepted Answer

Introduction / Context: This is a rates-combination problem using equivalent-team times. We find individual animal rates and sum them for the new team. Given Data / Assumptions: 4 goats in 50 days ⇒ 1 job. 6 sheep in 50 days ⇒ 1 job. Team: 2 goats + 9 sheep. Concept / Approach: Compute per-day per-animal rates from the team times, add them, and invert to get time. Step-by-Step Solution: Rate(goat) = 1 / (4 * 50) = 1/200 per dayRate(sheep) = 1 / (6 * 50) = 1/300 per dayTeam rate = 2*(1/200) + 9*(1/300) = 1/100 + 3/100 = 1/25 per dayTime = 1 / (1/25) = 25 days Verification / Alternative check: If they worked 25 days, they would do 25 * (1/25) = 1 job, confirming correctness. Why Other Options Are Wrong: 50, 75, 100 days correspond to incorrect combined rates. Common Pitfalls: Failing to convert team times into per-animal rates before recombining. Final Answer: 25 days

Question 20

Inferring efficiency ratio from a scaled comparison The work done by 8 men and 4 boys in 1 day is 7 times the work done by 1 man and 1 boy in 1 day. What is the ratio of the daily work of 1 man to that of 1 boy?

Accepted Answer

Introduction / Context: Scaled comparisons let us deduce relative efficiencies. Here, a large crew’s one-day output equals seven times a small crew’s one-day output, which yields a linear equation in the man and boy daily rates. Given Data / Assumptions: (8 men + 4 boys) in 1 day = 7 × (1 man + 1 boy) in 1 day. Let daily rate(man) = M and daily rate(boy) = B. Concept / Approach: Translate the sentence into 8M + 4B = 7(M + B). Solve for M in terms of B to get the ratio M:B. Step-by-Step Solution: 8M + 4B = 7M + 7B8M − 7M = 7B − 4B ⇒ M = 3BTherefore, M : B = 3 : 1 Verification / Alternative check: Substitute M = 3B: LHS = 8(3B) + 4B = 24B + 4B = 28B; RHS = 7(3B + B) = 7(4B) = 28B. Balanced. Why Other Options Are Wrong: 2 : 1, 1 : 1, 1 : 2 do not satisfy the equality after substitution. Common Pitfalls: Dropping coefficients or misplacing terms when moving 7(M + B) across the equality. Final Answer: 3 : 1

Time and Work Questions