Question 1

Karim invests Rs. 30,000 for one year. What amount should Raunaq invest for one year so that the profit ratio (Karim : Raunaq) is 2 : 3?

Accepted Answer

Introduction / Context: With equal time periods, profit shares are simply proportional to invested capital. We are given the desired profit ratio and Karim’s capital, and must find Raunaq’s capital accordingly. Given Data / Assumptions: Karim: Rs. 30,000 for 1 year. Desired profit ratio Karim : Raunaq = 2 : 3. Time for both partners is equal, so ratio equals capitals. Concept / Approach: Set 30,000 : x = 2 : 3 and solve for x using cross-multiplication. Step-by-Step Solution: 30,000 / x = 2 / 3. Cross-multiply: 2x = 90,000 ⇒ x = 45,000. Therefore, Raunaq must invest Rs. 45,000. Verification / Alternative check: Capitals 30,000 and 45,000 form the ratio 2 : 3; profits will follow the same ratio since time is identical. Why Other Options Are Wrong: Rs. 20,000, Rs. 40,000, and Rs. 18,000 do not yield the 2 : 3 ratio with 30,000. Common Pitfalls: Using 3 : 2 instead of 2 : 3 or forgetting that equal time periods eliminate the need for time weighting. Final Answer: Rs. 45000

Question 2

A, B, and C are partners. Twice A’s capital equals thrice B’s capital; B’s capital is four times C’s capital. Out of a total profit of Rs. 5,940, what is C’s share?

Accepted Answer

Introduction / Context: We translate the relations among capitals into a simple ratio and then apply it to split the profit. No time factor is mentioned, so time is equal for all partners. Given Data / Assumptions: 2A = 3B ⇒ A = 3B/2. B = 4C. Total profit = Rs. 5,940. Concept / Approach: Express A and B in terms of C and form the ratio A : B : C. Then C’s share = (C’s part / total parts) * total profit. Step-by-Step Solution: B = 4C. A = 3B/2 = 3*(4C)/2 = 6C. So A : B : C = 6C : 4C : C = 6 : 4 : 1. Total parts = 6 + 4 + 1 = 11. C’s share = (1/11) * 5,940 = Rs. 540. Verification / Alternative check: Shares = Rs. 3,240 (A), Rs. 2,160 (B), Rs. 540 (C) sum to Rs. 5,940, matching the total. Why Other Options Are Wrong: Rs. 700 and Rs. 900 exceed the 1/11 part. Rs. 740 is not a clean 1/11 fraction of 5,940. Common Pitfalls: Misreading “twice equals thrice” and solving as A = 2B/3 instead of A = 3B/2, or forgetting to reduce to the simplest A : B : C ratio. Final Answer: Rs. 540

Question 3

A’s capital equals twice B’s, and B’s capital equals three times C’s. What is the ratio of capitals A : B : C?

Accepted Answer

Introduction / Context: Converting nested relations into a single three-term ratio is a standard step in partnership and ratio problems. We align all terms to a common base (here, C) to obtain A : B : C directly. Given Data / Assumptions: A = 2B. B = 3C. Concept / Approach: Substitute B = 3C into A = 2B to express all three in terms of C, then read off the ratio. Step-by-Step Solution: B = 3C. A = 2B = 2 * 3C = 6C. Therefore, A : B : C = 6C : 3C : C = 6 : 3 : 1. Verification / Alternative check: The ratio satisfies the original statements: A = 2B (6 : 3) and B = 3C (3 : 1). Why Other Options Are Wrong: 2 : 1 : 3 scrambles the order. 1 : 2 : 6 is inverted. 1 : 3 : 6 swaps positions and does not meet A = 2B. Common Pitfalls: Mixing the directions (e.g., thinking B = 2A) or failing to keep a consistent base when forming the combined ratio. Final Answer: 6 : 3 : 1

Question 4

Three partners invest Rs. 26,000, Rs. 34,000, and Rs. 10,000. If the total profit is Rs. 3,500, what is B’s share of profit?

Accepted Answer

Introduction / Context: With equal time, profits divide in proportion to capitals invested. We calculate B’s fraction of the total capital and apply it to the total profit to get B’s share. Given Data / Assumptions: Capitals: A = 26,000; B = 34,000; C = 10,000. Total profit = Rs. 3,500. Time period identical for all partners. Concept / Approach: B’s share = (B’s capital / total capital) * total profit. Step-by-Step Solution: Total capital = 26,000 + 34,000 + 10,000 = 70,000. B’s fraction = 34,000 / 70,000 = 17/35. B’s profit = 3,500 * (17/35) = 100 * 17 = Rs. 1,700. Verification / Alternative check: Remaining profit = 3,500 − 1,700 = 1,800 to be split between A and C in 26 : 10; this checks out proportionally. Why Other Options Are Wrong: Rs. 1,300 and Rs. 1,500 do not match 17/35 of 3,500. Rs. 500 is far too small. Common Pitfalls: Using percentages incorrectly or forgetting to sum all capitals before computing the fraction. Final Answer: Rs. 1700

Question 5

A, B, and C contract a job for Rs. 550. Together, A and B are expected to complete 7/11 of the work. What is C’s monetary share?

Accepted Answer

Introduction / Context: Payment is distributed in proportion to the work each party is supposed to accomplish. If A and B together handle a known fraction, the remainder belongs to C, and so does the corresponding money share. Given Data / Assumptions: Total contract value = Rs. 550. A + B are to perform 7/11 of the work. C’s share corresponds to the remaining 4/11 of the work. Concept / Approach: Money share ∝ work share. So C’s money = (4/11) * 550 because 1 − 7/11 = 4/11. Step-by-Step Solution: C’s fraction = 1 − 7/11 = 4/11. C’s share = (4/11) * 550 = 4 * 50 = Rs. 200 (since 550/11 = 50). Verification / Alternative check: A + B together get (7/11) * 550 = Rs. 350. Rs. 350 + Rs. 200 = Rs. 550, matching the total payment. Why Other Options Are Wrong: Rs. 300 and Rs. 400 do not equal 4/11 of 550. “1831/3” is not a valid amount for this exact fraction and total. Common Pitfalls: Confusing the fractional remainder or miscomputing 550/11. Always subtract the completed fraction from 1 to get the remainder for C. Final Answer: Rs. 200

Question 6

Subodh invests Rs. 45,000. After 4 months, Nepal joins with Rs. 30,000. At year-end, the total profit is Rs. 13,000. What is the difference between their profit shares?

Accepted Answer

Introduction / Context: Time-weighted capital determines shares when partners join later. We compute each partner’s capital * months, obtain the ratio, split the profit, and then find the difference in shares. Given Data / Assumptions: Subodh: Rs. 45,000 invested for 12 months. Nepal: Rs. 30,000 invested after 4 months ⇒ for 8 months. Total profit = Rs. 13,000. Concept / Approach: Profit ratio = (capital * time). Then difference = (Subodh’s share − Nepal’s share) after distributing the total profit according to their parts. Step-by-Step Solution: Subodh weight = 45,000 * 12 = 540,000. Nepal weight = 30,000 * 8 = 240,000. Ratio = 540 : 240 = 9 : 4. Total parts = 13; Subodh’s share = (9/13) * 13,000 = Rs. 9,000. Nepal’s share = (4/13) * 13,000 = Rs. 4,000. Difference = 9,000 − 4,000 = Rs. 5,000. Verification / Alternative check: Shares 9,000 and 4,000 add to 13,000 and reflect the 9 : 4 ratio accurately. Why Other Options Are Wrong: Rs. 7,000, Rs. 3,000, and Rs. 9,000 do not match the computed Rs. 5,000 difference; hence “None of these” is correct. Common Pitfalls: Using 4 months instead of 8 months for Nepal, or forgetting to multiply capital by time in months before forming the ratio. Final Answer: None of these

Question 7

Rakesh invests Rs. 70,000. After 8 months, Vinod joins with Rs. 1,80,000. In what ratio should the year-end profit be divided?

Accepted Answer

Introduction / Context: When partners invest at different times, compute time-weighted capitals to obtain the profit-sharing ratio. Here, one partner joins late with a larger capital for fewer months. Given Data / Assumptions: Rakesh: Rs. 70,000 for 12 months. Vinod: Rs. 1,80,000 for 4 months (joined after 8 months). Profits divide in proportion to capital * time. Concept / Approach: Compute each partner’s product capital * months, then reduce to the simplest ratio for profit division. Step-by-Step Solution: Rakesh weight = 70,000 * 12 = 840,000. Vinod weight = 1,80,000 * 4 = 720,000. Ratio Rakesh : Vinod = 840,000 : 720,000 = 7 : 6. Verification / Alternative check: Divide both by 120,000 to get 7 : 6. No listed option equals 7 : 6, confirming “None of these.” Why Other Options Are Wrong: 8 : 7, 6 : 5, or 7 : 5 disagree with the precise time-weighted calculation. Common Pitfalls: Using 8 months for Vinod instead of 4, or comparing capital values without adjusting for time. Always multiply capital by the number of months invested. Final Answer: None of these

Question 8

A, B, and C can complete a job alone in 20, 25, and 30 days respectively. If they jointly earn Rs. 2,220 for the whole job, by how much does A’s share exceed B’s share?

Accepted Answer

Introduction / Context: When wages are shared for a job completed together, each person’s share is proportional to their work rate (i.e., 1/time taken alone). We compute shares and then find the difference between A and B. Given Data / Assumptions: A alone: 20 days; B alone: 25 days; C alone: 30 days. Total payment = Rs. 2,220 for the whole job. Work shares proportional to rates 1/20, 1/25, 1/30. Concept / Approach: Convert to a common base (e.g., LCM 300). Effective daily works: A = 15, B = 12, C = 10 units. Then divide money in the ratio 15 : 12 : 10 and compare A and B. Step-by-Step Solution: Work rate ratio = 1/20 : 1/25 : 1/30 = 15 : 12 : 10. Sum parts = 37. A’s share = (15/37) * 2,220; B’s share = (12/37) * 2,220. Difference = (3/37) * 2,220 = 180 (since 2,220/37 = 60). Verification / Alternative check: Individual shares: A = 900, B = 720, C = 600; they add to 2,220, and A − B = 180. Why Other Options Are Wrong: Rs. 120, Rs. 300, and Rs. 600 do not match the exact difference from the computed shares. Common Pitfalls: Dividing by times instead of using rates or missing simplification using the LCM for clean integers. Final Answer: Rs. 180

Question 9

Alok invests Rs. 75,000. After 3 months, Chandan joins with Rs. 60,000. If the annual profit is Rs. 16,000, what is Chandan’s share?

Accepted Answer

Introduction / Context: Profit shares are proportional to capital * time. A later-joining partner contributes for fewer months, reducing his weighted share even with a sizable capital. Given Data / Assumptions: Alok: Rs. 75,000 for 12 months. Chandan: Rs. 60,000 for 9 months (joined after 3 months). Total profit = Rs. 16,000. Concept / Approach: Compute weights: 75,000 * 12 and 60,000 * 9. Reduce the ratio, then split Rs. 16,000 accordingly to find Chandan’s share. Step-by-Step Solution: Alok weight = 900,000. Chandan weight = 540,000. Ratio = 900 : 540 = 5 : 3. Total parts = 8; Chandan’s share = (3/8) * 16,000 = Rs. 6,000. Verification / Alternative check: Alok’s share = (5/8) * 16,000 = Rs. 10,000. Sum = 16,000, consistent with the ratio 5 : 3. Why Other Options Are Wrong: Rs. 10,000 is Alok’s share. Rs. 8,000 and Rs. 4,500 do not match the 5 : 3 ratio split. Common Pitfalls: Using raw capitals without time weighting or miscounting the months for Chandan’s investment. Final Answer: Rs. 6000

Question 10

Gopal invests Rs. 3,000. After 6 months, Dinesh joins with Rs. 2,000. If the total profit at year-end is Rs. 2,600, what is Dinesh’s share?

Accepted Answer

Introduction / Context: A partner joining halfway contributes for fewer months. Profit shares must reflect capital * time. We compute each partner’s weighted share and then apply it to the total profit. Given Data / Assumptions: Gopal: Rs. 3,000 for 12 months. Dinesh: Rs. 2,000 for 6 months. Total profit = Rs. 2,600. Concept / Approach: Find the ratio (3,000 * 12) : (2,000 * 6) and then allocate Rs. 2,600 accordingly to obtain Dinesh’s share. Step-by-Step Solution: Gopal weight = 36,000. Dinesh weight = 12,000. Ratio = 3 : 1. Dinesh’s share = (1/4) * 2,600 = Rs. 650. Verification / Alternative check: Gopal’s share = 1,950; Dinesh’s = 650; sum to 2,600 and respect the 3 : 1 ratio. Why Other Options Are Wrong: Rs. 1,733 and Rs. 1,950 are not Dinesh’s share; Rs. 866 is not a clean quarter of 2,600. Common Pitfalls: Forgetting to weight by time or miscounting months. Always multiply capital by months invested before forming the ratio. Final Answer: Rs. 650

Question 11

A, B, and C start a business. A invests three times B’s investment, and B invests two-thirds of C’s investment. What is the ratio of their capitals A : B : C?

Accepted Answer

Introduction / Context: Translating linked statements into a single three-term ratio is common in partnership problems. We use B as the base to express A and C, then simplify the ratio. Given Data / Assumptions: A = 3 * B. B = (2/3) * C ⇒ C = (3/2) * B. Concept / Approach: Write A : B : C in terms of B, then clear fractions by multiplying all terms with a common multiple to express the ratio in integers. Step-by-Step Solution: A : B : C = 3B : B : 1.5B. Multiply by 2 to remove the decimal: 6B : 2B : 3B. Thus, A : B : C = 6 : 2 : 3. Verification / Alternative check: The ratio is consistent with the statements: A = 3B (6 : 2) and B = (2/3)C (2 : 3). Why Other Options Are Wrong: 3 : 9 : 2 and 6 : 10 : 15 do not follow from the given relationships. 5 : 3 : 2 is unrelated. Common Pitfalls: Inverting B and C or mishandling the two-thirds relationship. Carefully convert B = 2/3 of C into C = 3/2 of B first. Final Answer: 6 : 2 : 3

Question 12

A, B, and C share a sum in the ratio 2 : 3 : 7. If A and B together receive Rs. 1,500 less than C, what is A’s share?

Accepted Answer

Introduction / Context: This is a ratio allocation with an extra condition comparing a combined share to another. We set the common multiplier and solve using the given difference to find A’s share. Given Data / Assumptions: A : B : C = 2 : 3 : 7. (A + B) is Rs. 1,500 less than C. Find A’s amount. Concept / Approach: Let shares be 2k, 3k, 7k. Then (2k + 3k) = 7k − 1,500 ⇒ 5k = 7k − 1,500. Solve for k and then compute A = 2k. Step-by-Step Solution: 5k = 7k − 1,500 ⇒ 2k = 1,500 ⇒ k = 750. A’s share = 2k = 1,500. Verification / Alternative check: B = 3k = 2,250; C = 7k = 5,250. Then A + B = 3,750 which is Rs. 1,500 less than 5,250, matching the condition. Why Other Options Are Wrong: Rs. 1,000 and Rs. 2,000 do not satisfy the difference condition when ratios are respected. “Date inadequate” is incorrect; data is sufficient. Common Pitfalls: Misinterpreting “less than” or mixing the roles of A + B and C. Always express all in a single variable k first. Final Answer: Rs. 1500

Question 13

Three hikers A, B, and C start with Rs. 50 each and agree to share expenses equally. At the end, A has Rs. 20, B has Rs. 30, and C has Rs. 40 left. How should they settle their accounts?

Accepted Answer

Introduction / Context: This is a settlement-of-expenses problem. Everyone begins equally and agrees to share equally, so each person’s fair share of expenses is total spent divided equally. Transfers settle over- or under-payments. Given Data / Assumptions: Each started with Rs. 50. End balances: A = Rs. 20, B = Rs. 30, C = Rs. 40. They share expenses equally. Concept / Approach: Amount spent by each = 50 − ending balance. Compute total spent, divide by 3 for each person’s fair share, then balance by transfers from under-spenders to over-spenders. Step-by-Step Solution: A spent = 50 − 20 = 30. B spent = 50 − 30 = 20. C spent = 50 − 40 = 10. Total spent = 30 + 20 + 10 = 60 ⇒ fair share per person = 60 / 3 = 20. A overpaid by 10; B paid exactly; C underpaid by 10. Therefore, C must pay Rs. 10 to A. Verification / Alternative check: After transfer, each effectively bears Rs. 20 of expense. Balances align with equal sharing. Why Other Options Are Wrong: Payments from A to B or C would worsen the disparity. “Can’t be settled” is incorrect; a single Rs. 10 transfer balances accounts. Common Pitfalls: Forgetting to convert ending balances to amounts spent, or misallocating who overpaid vs. underpaid. Final Answer: C must pay Rs. 10 to A

Question 14

Ravi and Kavi invest Rs. 8,000 and Rs. 72,000 respectively in a business. What is the ratio of their profits at the end of one year?

Accepted Answer

Introduction / Context: With equal time periods, profit ratio equals capital ratio. We simply reduce 8,000 : 72,000 to the simplest form to find the profit ratio between Ravi and Kavi. Given Data / Assumptions: Ravi invests Rs. 8,000. Kavi invests Rs. 72,000. Time period is the same for both partners. Concept / Approach: Profit ratio = capital ratio when time is equal. Reduce 8,000 : 72,000 by their greatest common divisor. Step-by-Step Solution: 8,000 : 72,000 = 8 : 72 = 1 : 9. Thus, Ravi : Kavi = 1 : 9. Verification / Alternative check: If total profit were P, Ravi gets P/10 and Kavi gets 9P/10, maintaining the 1 : 9 ratio. Why Other Options Are Wrong: 2 : 9, 5 : 9, and 7 : 9 do not reduce from 8,000 : 72,000. Common Pitfalls: Forgetting to reduce the ratio completely or mistakenly adding capitals instead of comparing them. Final Answer: 1 : 9

Question 15

P starts a business with Rs. 3,00,000. Q joins after 3 months with Rs. 12,00,000. What is the ratio of their profits at the end of one year?

Accepted Answer

Introduction / Context: For unequal joining times, profits are proportional to capital * time. Q joins later but with a larger capital; we compute time-weighted contributions for both partners to find the final ratio. Given Data / Assumptions: P: Rs. 3,00,000 for 12 months. Q: Rs. 12,00,000 for 9 months (joined after 3 months). Profits ∝ capital * time. Concept / Approach: Multiply each capital by months invested to get weights, then reduce to simplest ratio P : Q. Step-by-Step Solution: P weight = 3,00,000 * 12 = 36,00,000. Q weight = 12,00,000 * 9 = 1,08,00,000. Ratio P : Q = 36 : 108 = 1 : 3. Verification / Alternative check: Dividing by 36 confirms 1 : 3. Any total profit will split in the proportion 1 part to P and 3 parts to Q. Why Other Options Are Wrong: 2 : 5, 3 : 5, and 5 : 1 do not match the calculated 1 : 3 weight ratio. Common Pitfalls: Ignoring the late entry of Q and using only capital ratio; time must be included in the weighting. Final Answer: 1 : 3

Question 16

Ram, Shyam, and Kamal invest in the ratio 3 : 4 : 7. If the annual profit is Rs. 21,000, what is Kamal’s share?

Accepted Answer

Introduction / Context: Profit shares in partnership with equal time are directly proportional to capitals. With the ratio given, we compute Kamal’s fraction of the total and multiply by the total profit to obtain his share. Given Data / Assumptions: Capital ratio = 3 : 4 : 7 (Ram : Shyam : Kamal). Total profit = Rs. 21,000. Time period is the same for all. Concept / Approach: Kamal’s fraction = 7 / (3 + 4 + 7) = 7 / 14 = 1/2. Kamal’s share = 1/2 of the total profit. Step-by-Step Solution: Sum of ratio parts = 3 + 4 + 7 = 14. Kamal’s fraction = 7 / 14 = 1/2. Kamal’s share = (1/2) * 21,000 = Rs. 10,500. Verification / Alternative check: Remaining Rs. 10,500 splits between Ram and Shyam in 3 : 4 (4,500 and 6,000), and totals reconcile to Rs. 21,000. Why Other Options Are Wrong: Rs. 12,500 and Rs. 15,000 exceed Kamal’s exact half; Rs. 10,000 is too low. Common Pitfalls: Not summing all ratio parts before computing the fraction, or mixing up which partner corresponds to 7 parts. Final Answer: ₹ 10500

Question 17

Rakesh, Dinesh, and Mahesh start a partnership business by investing ₹5000, ₹8000, and ₹12000 respectively. At the end of one year the total profit is ₹12500. Based strictly on capital–time proportionality (equal time), what is Dinesh’s share of the…

Accepted Answer

Introduction / Context: This problem tests basic partnership theory. When partners invest for the same duration, the profit is divided in proportion to their invested capitals. No salary or interest is mentioned, so only capital matters. Given Data / Assumptions: Investments: Rakesh = ₹5000, Dinesh = ₹8000, Mahesh = ₹12000. Time = 1 year for all (equal). Total profit = ₹12500. No other adjustments such as interest on capital or drawings. Concept / Approach: With equal time, profit share ∝ capital. Compute the investment ratio, convert to parts, and multiply Dinesh’s fraction by total profit. Step-by-Step Solution: Capital ratio = 5000 : 8000 : 12000 = 5 : 8 : 12Sum of parts = 5 + 8 + 12 = 25Dinesh’s fraction = 8/25Dinesh’s share = 12500 * (8/25) = 12500 * 0.32 = ₹4000 Verification / Alternative check: Rakesh’s share = 12500*(5/25)=₹2500; Mahesh’s share = 12500*(12/25)=₹6000; total = 2500+4000+6000 = ₹12500, consistent. Why Other Options Are Wrong: ₹4500, ₹6000, and ₹7500 correspond to different fractions (9/25, 12/25, 15/25) and do not match the 8-part share. ₹3200 implies 8/31.25 parts, which is not the given ratio. Common Pitfalls: Dividing profit equally or taking simple arithmetic mean of capitals. Always use proportional allocation to capital when times are equal. Final Answer: ₹4000

Question 18

Reena, Reema, and Meera rent a car and use it for 14 h, 16 h, and 22 h respectively. If the total rent for the car is ₹2080 and cost is shared in proportion to hours used, how much must Meera pay?

Accepted Answer

Introduction / Context: Shared rentals are commonly split in proportion to usage. Here, the usage metric is hours the car was used by each person. Each person pays total rent multiplied by their usage fraction. Given Data / Assumptions: Hours: Reena = 14, Reema = 16, Meera = 22. Total rent = ₹2080. Cost split ∝ hours used (uniform hourly rate). Concept / Approach: Compute total hours and then Meera’s fraction = Meera’s hours / total hours. Multiply this fraction by the total rent to get her payment. Step-by-Step Solution: Total hours = 14 + 16 + 22 = 52Meera’s fraction = 22 / 52Meera’s share = 2080 * (22 / 52)Simplify: 22/52 = 11/26; 2080/26 = 80; 80*11 = ₹880 Verification / Alternative check: Rate per hour = 2080 / 52 = ₹40. Meera’s cost = 22 * 40 = ₹880, same result. Why Other Options Are Wrong: ₹1280, ₹1040, and ₹960 imply wrong hour totals or non-proportional splitting. ₹820 does not match any sensible round calculation at ₹40 per hour. Common Pitfalls: Splitting rent equally or proportionate to number of people instead of actual hours. Always use the agreed metric (hours). Final Answer: ₹880

Question 19

Pinku, Rinku, and Tinku divide ₹4200 in the ratio 7 : 8 : 6. If ₹200 is added to each person’s share, what is the new ratio of their amounts?

Accepted Answer

Introduction / Context: This question mixes proportional division and then a fixed increment to each share. First compute actual shares using the original ratio, then add the constant and reduce to the simplest ratio. Given Data / Assumptions: Total = ₹4200; ratio = 7 : 8 : 6. Constant addition = ₹200 to each share. Concept / Approach: Convert the ratio into actual rupee amounts by dividing the total proportionally. Then apply the same increment to all three and finally reduce the triplet to a simplest ratio by dividing by any common factor. Step-by-Step Solution: Sum of parts = 7 + 8 + 6 = 21Value of 1 part = 4200 / 21 = ₹200Original shares: Pinku = 7*200 = ₹1400; Rinku = 8*200 = ₹1600; Tinku = 6*200 = ₹1200After adding ₹200 each: 1600, 1800, 1400New ratio = 1600 : 1800 : 1400 = 16 : 18 : 14 = 8 : 9 : 7 Verification / Alternative check: The common factor 200 divides all three augmented shares, confirming 8 : 9 : 7. Why Other Options Are Wrong: 8 : 9 : 5 and 7 : 9 : 5 do not reflect the equal increment effect. 7 : 8 : 6 is the original, not the updated ratio. Common Pitfalls: Adding ₹200 to the ratio numbers directly. Ratios must be converted to actual values before adding constants. Final Answer: 8 : 9 : 7

Question 20

P, Q, and R start a business. P invests three times as much as Q, and Q invests two-thirds as much as R. Find the ratio of their capitals P : Q : R.

Accepted Answer

Introduction / Context: Partnership setups often give chained relationships between capitals. Translate the verbal conditions into algebra and then reduce to a clean integer ratio. Given Data / Assumptions: P = 3 × Q Q = (2/3) × R All capitals are positive. Concept / Approach: Express P and Q in terms of R, then form P : Q : R and scale to remove fractions. Multiplying all terms by the least common multiple of denominators yields an integer ratio. Step-by-Step Solution: Given Q = (2/3)RThen P = 3Q = 3*(2/3)R = 2RP : Q : R = 2R : (2/3)R : RCancel R and clear denominator by ×3 ⇒ 6 : 2 : 3 Verification / Alternative check: Check back: If R = 3, then Q = 2, P = 6; indeed P is thrice Q, and Q is two-thirds of R. Why Other Options Are Wrong: Other orders or numbers violate either “P is 3 times Q” or “Q is 2/3 of R.” Common Pitfalls: Confusing “Q is 2/3 of R” with “R is 2/3 of Q.” Read the direction carefully. Final Answer: 6 : 2 : 3

Partnership Questions