Question 1

Inferring individual daily pay from combined payouts: A, B and C together get ₹ 5400 for a work done in 36 days. A and C together get ₹ 1880 for 20 days of the same work, while B and C together get ₹ 3040 for 40 days. What is C’s daily earning?

Accepted Answer

Introduction / Context:The problem provides total payments over different durations for various combinations of workers. The key is to convert each total to a per-day amount and then set up simultaneous equations to isolate each person’s daily earning. We are asked specifically for C’s daily pay. Given Data / Assumptions: (A+B+C) total = ₹ 5400 in 36 days ⇒ per day = 5400/36 = ₹ 150 (A+C) total = ₹ 1880 in 20 days ⇒ per day = 1880/20 = ₹ 94 (B+C) total = ₹ 3040 in 40 days ⇒ per day = 3040/40 = ₹ 76 Concept / Approach:Let a, b, c be daily earnings. Then a+b+c=150, a+c=94, b+c=76. Subtract to solve for each variable, focusing on c. Step-by-Step Solution:From a+b+c=150 and a+c=94 ⇒ b = 150 − 94 = 56From b+c=76 ⇒ 56 + c = 76 ⇒ c = 20 Verification / Alternative check:Recover a: a = 94 − c = 94 − 20 = 74. Check sums: a+b+c = 74 + 56 + 20 = 150 — consistent. Why Other Options Are Wrong:₹ 25, ₹ 30, ₹ 50 mismatch the linear system results; only ₹ 20 satisfies all three equations simultaneously. Common Pitfalls:Forgetting to convert totals to per-day amounts; arithmetic slips in subtraction; mixing up variables when substituting back. Final Answer:₹ 20

Question 2

Equivalent wages across categories: The combined daily wages of 3 men, 2 women, and 4 boys is ₹ 26. Given that wages of 3 men equal wages of 4 women, and wages of 2 women equal wages of 3 boys, find the total daily wages for a group of 4 men, 3 wome…

Accepted Answer

Introduction / Context:We must reconcile wage relationships between men, women, and boys to a common unit and then compute the total for a different combination. Such problems rely on expressing all categories in terms of one variable using the given equivalences. Given Data / Assumptions: 3 men = 4 women (in wages) 2 women = 3 boys (in wages) 3 men + 2 women + 4 boys = ₹ 26 (per day) Find: 4 men + 3 women + 2 boys Concept / Approach:Let the daily wage of a woman be w. Then man's daily wage m and boy's daily wage b can be expressed in terms of w using the given equalities. Substitute into the known total to find w, then compute the requested combination. Step-by-Step Solution:From 3m = 4w ⇒ m = 4w/3From 2w = 3b ⇒ b = 2w/3Total given: 3m + 2w + 4b = 3*(4w/3) + 2w + 4*(2w/3) = 4w + 2w + 8w/3 = 26w/326w/3 = 26 ⇒ w = 3Then m = 4, b = 2Desired: 4m + 3w + 2b = 4*4 + 3*3 + 2*2 = 16 + 9 + 4 = ₹ 29 Verification / Alternative check:Plug back: 3m + 2w + 4b = 12 + 6 + 8 = 26, consistent with the given total. Why Other Options Are Wrong:₹ 35, ₹ 65, ₹ 20 do not result from the derived unit wages m = 4, w = 3, b = 2. Common Pitfalls:Mistaking men–women equivalence as m = 3/4 w instead of m = 4/3 w; mixing wage and headcount proportions; arithmetic slips in fraction addition. Final Answer:₹ 29

Question 3

Numbers employed and wage ratios: Men, women, and children are employed in the proportion 3 : 2 : 1, and their daily wage rates are in the ratio 5 : 3 : 2 respectively. When 90 men are employed, the total daily wages paid to all (men + women + child…

Accepted Answer

Introduction / Context:This item combines headcount proportions with wage-rate ratios. Knowing the men:woman:child staffing proportion and their wage ratios allows us to set up the total daily wage equation and solve for the man’s daily wage explicitly. Given Data / Assumptions: Headcount ratio = 3 : 2 : 1 Wage-rate ratio (per day) = 5 : 3 : 2 Men employed = 90 Total daily wages for all categories = ₹ 10,350 Concept / Approach:If 3 units correspond to 90 men, the scaling factor is 30. Therefore, there are 60 women and 30 children. If the base wage unit is x, man = 5x, woman = 3x, child = 2x. Form the total daily payout and solve for x, then compute 5x. Step-by-Step Solution:Women = 60, Children = 30Total per day = 90*(5x) + 60*(3x) + 30*(2x) = 450x + 180x + 60x = 690x690x = 10,350 ⇒ x = 15Man’s daily wage = 5x = 5 * 15 = ₹ 75 Verification / Alternative check:Compute women and children wages: women = 60*45 = 2700, children = 30*30 = 900; men = 90*75 = 6750. Sum = 6750 + 2700 + 900 = 10,350. Why Other Options Are Wrong:₹ 45 is the woman’s wage at x = 15; ₹ 57.50 or ₹ 115 do not match any derived category at this x. Common Pitfalls:Confusing headcount and wage ratios; forgetting to scale the staffing counts using the 3 : 2 : 1 ratio. Final Answer:₹ 75

Question 4

Inferring relative productivity from completion times: 2 men and 1 woman can complete a work in 14 days, while 4 women and 2 men can do the same work in 8 days. If a man earns ₹ 90 per day under pay proportional to productivity, what is a woman’s da…

Accepted Answer

Introduction / Context:This question links work rates to wages. If wages are proportional to productivity, then the ratio of a man’s wage to a woman’s wage must match the ratio of their work rates. We use the two completion scenarios to deduce the man-to-woman productivity ratio. Given Data / Assumptions: (2m + 1w) completes in 14 days ⇒ rate = (2M + W) (2m + 4w) completes in 8 days ⇒ rate = (2M + 4W) Let total work = 1 unit; M and W denote man and woman daily work rates Man’s daily wage = ₹ 90; wages ∝ rates Concept / Approach:Write rates as 1/14 = 2M + W and 1/8 = 2M + 4W after normalizing by total work. Eliminate M to find the ratio M : W. Then use wage proportionality to scale from man’s wage ₹ 90 to woman’s wage. Step-by-Step Solution:2M + W = 1/142M + 4W = 1/8Subtract first from second: 3W = 1/8 − 1/14 = (7 − 4)/56 = 3/56 ⇒ W = 1/56Then 2M + W = 1/14 ⇒ 2M = 1/14 − 1/56 = (4 − 1)/56 = 3/56 ⇒ M = 3/112Ratio M : W = (3/112) : (1/56) = (3/112) : (2/112) = 3 : 2Wage ratio man : woman = 3 : 2; man = ₹ 90 ⇒ woman = (2/3)*90 = ₹ 60 Verification / Alternative check:Productivity proportionality is consistent with many wage problems: if you double the workers of one type, the team rate adjusts linearly, which aligns with our equations. Why Other Options Are Wrong:₹ 48 or ₹ 72 imply different productivity ratios; ₹ 135 exceeds the man’s wage, contradicting the 3 : 2 ratio. Common Pitfalls:Using days (inverse) instead of rates directly; miscomputing 1/8 − 1/14; assuming equal wages despite different productivities. Final Answer:₹ 60

Question 5

Alternate-day working and fair split: Total wages for completing a job is ₹ 1280. A alone can finish in 8 days, B alone in 12 days. If they work on alternate days starting with A and stop exactly when the work finishes, what is A’s share of the mone…

Accepted Answer

Introduction / Context:When workers alternate days, the fair split should still reflect actual work done. We compute the cumulative work over cycles until completion, including any partial last day, then use those fractions to divide the fixed ₹ 1280. The alternation begins with A. Given Data / Assumptions: A's rate = 1/8 job per day B's rate = 1/12 job per day Work alternates daily: A then B, and so on, until job completes Total wage = ₹ 1280 Concept / Approach:One 2-day cycle produces (1/8 + 1/12) = 5/24 of the job. After 4 full cycles (8 days), they complete 20/24 = 5/6. Day 9 (A's turn) adds 1/8 = 3/24, reaching 23/24. Only 1/24 remains, which B completes in half a day (since B does 1/12 per full day). Split money by total work each has completed. Step-by-Step Solution:A's work: 4 full days in cycles + day 9 = 5 days total ⇒ 5*(1/8) = 5/8B's work: 4 full days + partial last day = 4*(1/12) + 1/24 = 8/24 + 1/24 = 9/24 = 3/8Check: 5/8 + 3/8 = 1A's share = 5/8 * 1280 = ₹ 800 Verification / Alternative check:Cycle method is robust; alternatively integrate until cumulative sum hits 1, accounting for partial-day completion on the final day for B. Why Other Options Are Wrong:₹ 500, ₹ 600, ₹ 700 do not correspond to the exact 5/8 fraction of the total payment. Common Pitfalls:Dividing by number of days worked instead of work done; missing that B only works a fraction of the last day; rounding prematurely. Final Answer:₹ 800

Question 6

Determining men and women daily wages first: 4 men and 6 women earn ₹ 1600 in 5 days; 3 men and 7 women earn ₹ 1740 in 6 days. In how many days will 7 men and 6 women earn ₹ 3760?

Accepted Answer

Introduction / Context:We first extract the per-day earnings equations for men and women using the two scenarios. With those unit wages known, we can compute the daily earnings of the new team and then determine how many days are required to reach ₹ 3760. Given Data / Assumptions: (4m + 6w) for 5 days = ₹ 1600 ⇒ per day = 1600/5 = 320 (3m + 7w) for 6 days = ₹ 1740 ⇒ per day = 1740/6 = 290 Find days d for (7m + 6w) to earn ₹ 3760 Concept / Approach:Let M and W be daily wages of a man and a woman. Solve 4M + 6W = 320 and 3M + 7W = 290. Then compute per-day for (7M + 6W) and divide 3760 by that to get the time required. Step-by-Step Solution:4M + 6W = 3203M + 7W = 290Solving ⇒ M = 50, W = 20Daily for 7 men + 6 women = 7*50 + 6*20 = 350 + 120 = ₹ 470Days d = 3760 / 470 = 8 Verification / Alternative check:Back-substitute M, W into both original equations to verify correctness before using them for the final calculation — both equalities hold. Why Other Options Are Wrong:6, 10, 12 days correspond to daily totals not equal to ₹ 470; they do not reach ₹ 3760 exactly. Common Pitfalls:Arithmetic slips in solving the two-variable system; forgetting to convert totals to per-day values; mixing up rates and wages. Final Answer:8 days

Question 7

Two men with a helper (money division by work): The first man alone can finish a work in 7 days and the second man alone in 8 days. Together with a boy they finish it in 3 days. If the total amount is to be divided proportional to work done, which s…

Accepted Answer

Introduction / Context:When a lump sum is split by contribution, each person’s share equals their fraction of the total work. With two men of known solo times and a boy helping so the job finishes in 3 days, we derive each one’s share fraction and compare with the given dollar splits to identify the correct one. Given Data / Assumptions: Rate(man1) = 1/7 per day Rate(man2) = 1/8 per day Total rate with boy = 1/3 per day Division is proportional to work done in 3 days Concept / Approach:Boy’s rate = total rate − (man1 + man2) rate. Each share fraction equals 3 * (individual rate), since they work for 3 days, and these fractions sum to 1. Reduce to a clean integer ratio and see which option matches. Step-by-Step Solution:Rate(man1) + Rate(man2) = 1/7 + 1/8 = (8 + 7)/56 = 15/56Total rate = 1/3 = 56/168; (man1 + man2) = 45/168Rate(boy) = 56/168 − 45/168 = 11/168Share fractions over 3 days: man1 = 3*(1/7) = 3/7; man2 = 3*(1/8) = 3/8; boy = 3*(11/168) = 33/168Convert to integer ratio: multiply by 168 ⇒ 72 : 63 : 33 ⇒ simplify to 24 : 21 : 11Check option splits for this 24:21:11 ratio → $600:$525:$275 fits exactly Verification / Alternative check:Any total consistent with the 24:21:11 ratio yields the same proportional split; the provided option matches this ratio precisely. Why Other Options Are Wrong:Other splits reduce to different ratios (e.g., 6:5:3 or 12:11:5), not 24:21:11. Common Pitfalls:Forgetting to multiply rates by the common duration (3 days); arithmetic slips when reducing fractions; comparing shares by amounts instead of checking ratios. Final Answer:$600, $525, $275

Question 8

Two men and a boy complete a job: Two men undertake a piece of work for $600. Individually, they can finish it in 6 days and 8 days respectively. With the help of a boy, they complete the work in 3 days. What is the boy’s share (in dollars)?

Accepted Answer

Introduction / Context:We distribute a fixed amount ($600) according to each participant’s contribution to the work. Two men of known solo times and a boy together finish in 3 days, letting us compute the boy’s rate and thus his monetary share. Given Data / Assumptions: Rate(man1) = 1/6 per day Rate(man2) = 1/8 per day Total rate with boy = 1/3 per day Total amount to share = $600 Concept / Approach:Find the boy’s daily rate by subtracting the two men’s combined rate from the total. The share fraction for each person equals their rate divided by the total rate, or equivalently rate * 3 days since they all worked 3 days. Multiply the boy’s fraction by $600 to get his share. Step-by-Step Solution:Rate(man1) + Rate(man2) = 1/6 + 1/8 = (4 + 3)/24 = 7/24Total rate = 1/3 = 8/24Boy’s rate = 8/24 − 7/24 = 1/24Boy’s share fraction = (1/24) / (1/3) = (1/24) * 3 = 1/8Boy’s share = 1/8 * $600 = $75 Verification / Alternative check:Men’s fractions: (1/6)/(1/3) = 1/2 and (1/8)/(1/3) = 3/8. Together with 1/8 sum to 1; shares are $300, $225, and $75, totaling $600. Why Other Options Are Wrong:$300 and $225 belong to the two men; $100 would overpay the boy versus his 1/8 contribution. Common Pitfalls:Allocating by days rather than rates; forgetting to normalize by the total combined rate; arithmetic slips with fractions. Final Answer:$75

Question 9

Work and wages — man vs. boy sharing based on efficiency (5 days): A man and a boy together earn total wages of $800 for 5 days of joint work. The man's efficiency is three times the boy's efficiency. Assuming wages are shared in direct proportion t…

Accepted Answer

Introduction / Context:This problem from work-and-wages tests proportional sharing. When workers have different efficiencies, their wage shares are divided in the same ratio as their effective work contributions. The question asks for the boy's daily wage given the total pay for 5 days and a stated efficiency ratio. Given Data / Assumptions: Total wages for 5 days together = $800. Man's efficiency : Boy's efficiency = 3 : 1. Payment is proportional to work done (standard assumption in wage-sharing problems). We need the boy's daily wage in dollars per day. Concept / Approach:If the boy's work rate is b units/day, the man's is 3b units/day. Together per day they contribute 4b units. Over 5 days, total work = 5 * 4b = 20b units. Wages split in the ratio of their contributions: man gets 3/4 of the total and boy gets 1/4 of the total. Then divide the boy's 5-day share by 5 to obtain his daily wage. Step-by-Step Solution:Work rates: boy = b, man = 3b.Daily combined rate = b + 3b = 4b.Total wages for 5 days = $800.Share ratio = 3 : 1 ⇒ Boy's share = (1/4) * 800 = $200 for 5 days.Boy's daily wage = 200 / 5 = $40 per day. Verification / Alternative check:If the boy earns $40/day, in 5 days he gets $200 and the man gets $600. That respects the 3:1 split and sums to $800, consistent with the prompt. Why Other Options Are Wrong: $44 and $56 break the exact 3:1 sharing for a total of $800 in 5 days. $76 is far too high for the boy given he does only one quarter of the work. Common Pitfalls: Dividing by 5 too early (before applying the share) or mixing up man/boy ratios. Final Answer:$40

Question 10

Contract sharing — three workers A, B, C paid $575: A and C together are supposed to complete 19/23 of the job. Assuming payment is proportional to actual work share, how much should B be paid?

Accepted Answer

Introduction / Context:This is a direct work-sharing allocation. The total contract value is known. When the shares of the work completed by some parties are known, the remainder belongs to the other party, and payment follows the same proportion. Given Data / Assumptions: Total amount for the full work = $575. A + C together do 19/23 of the work. Hence B does the remaining 4/23 of the work. Payment is proportional to work completed. Concept / Approach:The person's pay = (their fraction of total work) * (total contract amount). Compute B's fraction and multiply by $575. Step-by-Step Solution:Work share of A + C = 19/23.Work share of B = 1 − 19/23 = 4/23.B's pay = (4/23) * 575.Compute 575 / 23 = 25, hence B's pay = 4 * 25 = $100. Verification / Alternative check:A + C together get (19/23)*575 = 19*25 = $475. B gets $100. The sum $475 + $100 = $575, confirming consistency. Why Other Options Are Wrong: $210 and $200 correspond to larger fractions than 4/23; they overpay B. $475 is the share for A + C, not for B. Common Pitfalls: Mistaking 19/23 as B's share or forgetting that all shares must sum to 1. Final Answer:$100

Question 11

A and B take a $4500 contract; A alone 8 days, B alone 12 days: With C's help they finish in 4 days. Assuming payment is proportional to work done, what is C's share of the money?

Accepted Answer

Introduction / Context:In joint work, individual daily work rates add. Once the combined completion time is known, each worker’s portion of the total work is rate * time. Payment is then allocated in that same proportion. Given Data / Assumptions: A alone finishes in 8 days ⇒ rate(A) = 1/8 work/day. B alone finishes in 12 days ⇒ rate(B) = 1/12 work/day. A + B + C finish in 4 days ⇒ total rate = 1/4 work/day. Total payment = $4500. Concept / Approach:Find C's daily rate by subtraction from the combined rate. Then compute C's share of total work in 4 days and multiply by the contract amount. Step-by-Step Solution:rate(A) = 1/8, rate(B) = 1/12.rate(A) + rate(B) = 1/8 + 1/12 = 3/24 + 2/24 = 5/24.rate(A+B+C) = 1/4 ⇒ rate(C) = 1/4 − 5/24 = (6 − 5)/24 = 1/24.C's share of work over 4 days = 4 * (1/24) = 1/6 of the job.C's share of money = (1/6) * 4500 = $750. Verification / Alternative check:A’s share of work in 4 days = 4*(1/8) = 1/2; B’s = 4*(1/12) = 1/3; C’s = 1/6. These add to 1, validating the rates. The payouts in the same ratio sum to $4500. Why Other Options Are Wrong: $2250 and $1500 are too large for C's small rate. $375 is half of the correct $750, inconsistent with rate(C) = 1/24. Common Pitfalls: Using 4 days as a divisor for money directly without computing each worker’s share of the job. Final Answer:$750

Question 12

A, B, C can finish alone in 8, 12, 16 days; total payment ₹1440: With D's help, they finish in 3 days. Assuming proportional sharing, how much does D get?

Accepted Answer

Introduction / Context:When several workers collaborate, total daily rate is the sum of individual rates. Once the group completion time is known, each worker’s contribution equals their rate multiplied by total days. Payment is then split by contribution fraction. Given Data / Assumptions: A alone: 8 days ⇒ rate(A) = 1/8. B alone: 12 days ⇒ rate(B) = 1/12. C alone: 16 days ⇒ rate(C) = 1/16. All four finish in 3 days ⇒ group rate = 1/3. Total payment = ₹1440. Concept / Approach:First find rate(A+B+C). Subtract from group rate to get rate(D). Multiply rate(D) by 3 days to get D’s share of total work, and then D’s share of money. Step-by-Step Solution:rate(A) + rate(B) + rate(C) = 1/8 + 1/12 + 1/16.Use denominator 48: 6/48 + 4/48 + 3/48 = 13/48.Group rate = 1/3 = 16/48 ⇒ rate(D) = 16/48 − 13/48 = 3/48 = 1/16.D’s share of work in 3 days = 3 * (1/16) = 3/16.D’s payment = (3/16) * 1440 = 270. Verification / Alternative check:A, B, C contribute 13/48 per day; D contributes 1/16 per day. Over 3 days, total work is (13/48 + 1/16) * 3 = (13/48 + 3/48) * 3 = (16/48) * 3 = 1, as required. Why Other Options Are Wrong: ₹265 and ₹300 do not match the exact 3/16 share of ₹1440. “None of these” is unnecessary since a precise value exists. Common Pitfalls: Miscalculating common denominators or forgetting to multiply by the total number of days. Final Answer:₹ 270

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