Question 1

A, B and C invest in the ratio 7⁄2 : 4⁄3 : 6⁄5. After 4 months, A increases his investment by 50%. If the total profit after one year is Rs. 21,600, what is B's share?

Accepted Answer

Problem restatementProfits are proportional to (capital × time). A invests in two phases: first 4 months at his original capital; next 8 months at 1.5 times that capital. B and C keep their capitals constant for 12 months. Given data Capital ratios: A : B : C = 7⁄2 : 4⁄3 : 6⁄5 Time: total 12 months; A increases capital after 4 months by 50% Total profit = Rs. 21,600 Concept/ApproachConvert fractional ratios to a common integer base and compute time-weighted contributions. Step-by-step calculationChoose base so that A = 105, B = 40, C = 36 (LCM of denominators 2,3,5 → 30; so 7⁄2=105⁄30, 4⁄3=40⁄30, 6⁄5=36⁄30)A's contribution = 4×A + 8×(1.5A) = 4A + 12A = 16A = 16×105 = 1680B's contribution = 12×B = 12×40 = 480C's contribution = 12×C = 12×36 = 432Total = 1680 + 480 + 432 = 2592B's profit share fraction = 480 ÷ 2592 = 5 ÷ 27B's profit = 21,600 × (5÷27) = Rs. 4,000 Verification/AlternativeFraction 5⁄27 ≈ 0.185185; 0.185185 × 21,600 = 4,000. Common pitfallsMultiplying A's capital by 1.5 for the full 12 months (the increase starts after 4 months only). Final AnswerRs. 4,000

Question 2

A starts a business with Rs. 3,500. After 5 months, B joins as a partner. After one year from the start, profits are shared in the ratio 2 : 3 (A : B). What is B's capital contribution?

Accepted Answer

Problem restatementProfit shares reflect capital × time. A invests for 12 months, B for the last 7 months. The final ratio A : B = 2 : 3. Find B's capital. Given data A's capital = 3,500 for 12 months ⇒ 3,500 × 12 B joins after 5 months ⇒ time = 7 months with capital = x Profit ratio = 2 : 3 (A : B) Concept/ApproachSet the ratio of capital-months equal to 2 : 3 and solve for x. Step-by-step calculationA : B = (3,500 × 12) : (x × 7) = 2 : 342,000 : 7x = 2 : 3 ⇒ 42,000 × 3 = 7x × 2126,000 = 14x ⇒ x = 9,000 Verification/AlternativeA's share units = 42,000; B's units with x = 9,000 → 63,000, giving 42,000 : 63,000 = 2 : 3. Common pitfallsTaking B's time as 12 months (he joined 5 months late). Final AnswerRs. 9,000

Question 3

A and B invest Rs. 20,000 and Rs. 15,000 respectively. After 6 months, C joins with Rs. 20,000. If the total profit at the end of 2 years from start is Rs. 25,000, what is B's share?

Accepted Answer

Problem restatementCompute profit shares via capital × time. A and B invest from the start; C enters after 6 months. The business runs for 24 months from the start. Given data A = 20,000 for 24 months ⇒ 480,000 units B = 15,000 for 24 months ⇒ 360,000 units C = 20,000 for 18 months ⇒ 360,000 units Total profit = Rs. 25,000 Concept/ApproachProfit share = (partner's units) ÷ (total units) × total profit. Step-by-step calculationTotal units = 480,000 + 360,000 + 360,000 = 1,200,000B's fraction = 360,000 ÷ 1,200,000 = 0.30B's profit = 0.30 × 25,000 = Rs. 7,500 Verification/AlternativeShares are proportional: A : B : C = 480 : 360 : 360 = 4 : 3 : 3; total parts = 10; B gets 3⁄10 of 25,000 = 7,500. Common pitfallsUsing 24 months for C as well (C joined after 6 months; only 18 months counted). Final AnswerRs. 7,500

Question 4

Aman invests ₹70,000 from the start. Rakhi joins after 6 months with ₹1,05,000, and Sagar joins 6 months later with ₹1,40,000. If the profit is distributed 3 years after Aman started, in what ratio should Aman : Rakhi : Sagar share the profit?

Accepted Answer

Problem restatementProfits in partnerships are divided in proportion to capital × time. We are given different joining times over a total horizon of 3 years (36 months). Given data Aman: ₹70,000 invested for 36 months. Rakhi: ₹1,05,000 invested for 30 months (joins after 6 months). Sagar: ₹1,40,000 invested for 24 months (joins after another 6 months). Concept/ApproachProfit shares are proportional to the product of amount invested and duration invested. Step-by-step calculation Aman's units = 70,000 × 36 = 25,20,000 Rakhi's units = 1,05,000 × 30 = 31,50,000 Sagar's units = 1,40,000 × 24 = 33,60,000 Divide all by 10,000 → 252 : 315 : 336 Simplify by 21 → 12 : 15 : 16 Verification/AlternativeCheck months: 36, 30, 24 correctly reflect 0, 6, 12-month join points across 36-month horizon. Common pitfalls Using only capital ratios and ignoring time weighting. Using 12 and 18 months for later partners (misreading the 6-month gaps). Final Answer12 : 15 : 16

Question 5

A and B invest in the ratio 3 : 5. After 6 months, C joins with the same amount as B. At the end of one year, in what ratio should A : B : C share the profit?

Accepted Answer

Problem restatementDetermine the profit-sharing ratio when partners invest different amounts and join at different times within a 1-year (12-month) period. Given data Capital ratio initially: A : B = 3x : 5x for 12 months. C joins after 6 months with capital equal to B, i.e., 5x for 6 months. Concept/ApproachProfit share ∝ (capital × time). Step-by-step calculation A units = 3x × 12 = 36x B units = 5x × 12 = 60x C units = 5x × 6 = 30x Ratio = 36 : 60 : 30 = 6 : 10 : 5 Common pitfalls Forgetting that C's capital works only for 6 months. Final Answer6 : 10 : 5

Question 6

Arun, Kamal, and Vinay invest ₹8,000, ₹4,000, and ₹8,000 respectively. Arun withdraws after 6 months. If the total profit after 8 months is ₹4,005, what is Kamal's share?

Accepted Answer

Problem restatementCompute each partner's time-weighted capital and then allocate the profit accordingly. Arun leaves at month 6; business ends at month 8. Given data Arun: ₹8,000 for 6 months. Kamal: ₹4,000 for 8 months. Vinay: ₹8,000 for 8 months. Total profit = ₹4,005. Concept/ApproachShares ∝ capital × time (money-months). Step-by-step calculation Arun units = 8,000 × 6 = 48,000 Kamal units = 4,000 × 8 = 32,000 Vinay units = 8,000 × 8 = 64,000 Ratio = 48,000 : 32,000 : 64,000 = 3 : 2 : 4 Sum of parts = 3 + 2 + 4 = 9 One part = 4,005 ÷ 9 = 445 Kamal's share = 2 × 445 = ₹890 VerificationArun's share = 3 × 445 = ₹1,335; Vinay's = 4 × 445 = ₹1,780; total = 1,335 + 890 + 1,780 = ₹4,005. Common pitfalls Dividing profit by the raw capital ratio without time-weighting. Final Answer₹890

Question 7

A and B invest in the ratio 3 : 2. If 5% of the total profit goes to charity and A's share of the remainder is ₹855, what is the total profit?

Accepted Answer

Problem restatementA fixed 5% of the total profit P is donated. The remaining 95% is divided between A and B in the ratio 3 : 2, with A's portion equal to ₹855. Find P. Given data Charity = 5% of P → distributable = 95% of P. A's share from distributable = (3/5) × 0.95P = ₹855. Concept/ApproachSet up an equation for A's share and solve for P. Step-by-step calculation (3/5) × 0.95P = 0.57P = 855 P = 855 ÷ 0.57 = ₹1,500 VerificationDistributable = 0.95 × 1,500 = ₹1,425. A's share = 3/5 × 1,425 = ₹855 (matches). Common pitfalls Applying 5% charity after taking A's share instead of before. Final Answer₹1,500

Question 8

A, B, and C rent a pasture. A puts 10 oxen for 7 months, B puts 12 oxen for 5 months, and C puts 15 oxen for 3 months. If the total rent is ₹175, how much should C pay?

Accepted Answer

Problem restatementRent is shared in proportion to oxen × months. Given data A: 10 × 7 = 70 ox-months. B: 12 × 5 = 60 ox-months. C: 15 × 3 = 45 ox-months. Total rent = ₹175. Concept/ApproachShare ∝ ox-months. Step-by-step calculation Total ox-months = 70 + 60 + 45 = 175 C's fraction = 45/175 C's rent = 45/175 × ₹175 = ₹45 Final Answer₹45

Question 9

A begins a business with ₹85,000. Later, B joins with ₹42,500. If year-end profits are shared in the ratio 3 : 1 (A : B), for how many months did B's capital remain invested?

Accepted Answer

Problem restatementFind B's joining duration given capitals and the final profit ratio over a 12-month year. Given data A: ₹85,000 for 12 months. B: ₹42,500 for t months. Profit ratio A : B = 3 : 1. Concept/ApproachProfit share ∝ capital × time. Step-by-step calculation 85,000 × 12 : 42,500 × t = 3 : 1 10,20,000 : 42,500t = 3 : 1 10,20,000 ÷ (42,500t) = 3 ⇒ 10,20,000 = 1,27,500t t = 10,20,000 ÷ 1,27,500 = 8 months Final Answer8 months

Question 10

A, B, C subscribe ₹50,000 in total. A invests ₹4,000 more than B, and B invests ₹5,000 more than C. If the total profit is ₹35,000, how much does A receive?

Accepted Answer

Problem restatementFind each person's capital from the given differences and the total capital, then allocate profit proportional to capital (assuming equal time). Given data Let C = x ⇒ B = x + 5,000 ⇒ A = x + 9,000. A + B + C = 50,000. Total profit = ₹35,000. Step-by-step calculation x + (x + 5,000) + (x + 9,000) = 50,000 3x + 14,000 = 50,000 ⇒ x = 12,000 C = 12,000, B = 17,000, A = 21,000 A's share = (21,000/50,000) × 35,000 = ₹14,700 VerificationShares sum correctly if computed similarly for B and C. Final Answer₹14,700

Question 11

Three partners share profit in the ratio 5 : 7 : 8 for working periods of 14, 8, and 7 months respectively. What was the ratio of their investments?

Accepted Answer

Problem restatementGiven profit shares and the months each partner remained invested, back out the ratio of investments. Concept/ApproachProfit ∝ (investment × time). If I1, I2, I3 are investments and t1, t2, t3 their durations, then: I1t1 : I2t2 : I3t3 = 5 : 7 : 8 Thus, I1 : I2 : I3 = (5/14) : (7/8) : (8/7) Step-by-step calculation Multiply each by LCM(14, 8, 7) = 56: (5/14)×56 = 20 (7/8)×56 = 49 (8/7)×56 = 64 Investment ratio = 20 : 49 : 64 Common pitfalls Inverting time factors (dividing instead of multiplying by months). Final Answer20 : 49 : 64

Question 12

A invests ₹6,500 for 6 months; B invests ₹8,400 for 5 months; C invests ₹10,000 for 3 months. A, as the working partner, receives 5% of the profit as remuneration. If total profit is ₹7,400, what is B's share?

Accepted Answer

Problem restatementFirst allocate 5% of the profit to the working partner (A). Distribute the remaining profit in the ratio of capital × time among A, B, C. Compute B's amount. Given data Total profit = ₹7,400. Working partner (A) gets 5% = ₹370. Distributable remainder = ₹7,400 − ₹370 = ₹7,030. Capital-months: A: 6,500 × 6 = 39,000 B: 8,400 × 5 = 42,000 C: 10,000 × 3 = 30,000 Concept/ApproachShares of the remainder are proportional to 39,000 : 42,000 : 30,000 = 13 : 14 : 10. Step-by-step calculation Total parts = 13 + 14 + 10 = 37 Each part = 7,030 ÷ 37 = 190 B's share = 14 × 190 = ₹2,660 VerificationA's and C's shares of the remainder are 13×190 = ₹2,470 and 10×190 = ₹1,900 respectively. Adding A's remuneration: A gets ₹2,470 + ₹370 = ₹2,840; totals sum to ₹7,400. Final Answer₹2,660

Question 13

A and B invest in the ratio 4 : 5. After 3 months, A withdraws 1/4 of his capital and B withdraws 1/5 of his capital. If the total profit at the end of 10 months is ₹760, what is A's share?

Accepted Answer

Problem restatementCapitals change once during the year. Use time-weighted capitals to split the profit. Given data Initial capitals: A = 4k, B = 5k. After 3 months: A withdraws 1/4 ⇒ new A = 3k; B withdraws 1/5 ⇒ new B = 4k. Total period = 10 months. Concept/ApproachCompute capital-months before and after the change. Step-by-step calculation First 3 months: A = 4k × 3 = 12k; B = 5k × 3 = 15k Next 7 months: A = 3k × 7 = 21k; B = 4k × 7 = 28k Totals: A = 12k + 21k = 33k; B = 15k + 28k = 43k Profit ratio A : B = 33 : 43 A's share = ₹760 × (33/(33+43)) = 760 × (33/76) = ₹330 Verification760 ÷ 76 = 10; 10 × 33 = ₹330 (A), remainder ₹430 (B). 330 : 430 = 33 : 43. Common pitfalls Splitting by initial 4 : 5 ratio without time adjustments after withdrawals. Final Answer₹330

Question 14

Simran invests Rs. 50,000 to start a software business. After 6 months, Nanda joins with Rs. 80,000. After 3 years, the profit is Rs. 24,500. What is Simran's share of the profit?

Accepted Answer

Problem restatementProfits in a partnership are divided in proportion to (capital × time). Simran invests first; Nanda joins after 6 months. Compute time-weighted capitals and apportion the total profit. Given data Simran's capital = Rs. 50,000 for 3 years. Nanda's capital = Rs. 80,000 for 2.5 years (joins after 6 months). Total profit = Rs. 24,500. Concept/ApproachUse capital–month (or capital–year) products to get their profit-sharing ratio, then apply the ratio to the total profit. Step-by-Step calculationSimran's units = 50,000 × 3 = 150,000Nanda's units = 80,000 × 2.5 = 200,000Ratio (Simran : Nanda) = 150,000 : 200,000 = 3 : 4Total parts = 3 + 4 = 7Simran's share = (3/7) × 24,500 = 10,500 Verification/AlternativeIf Nanda had also invested for 3 years, the ratio would be 5:8; reducing Nanda's time to 2.5 years scales her part to 4 and yields 3:4 overall—consistent. Common pitfallsUsing plain capital ratio (50,000:80,000) without accounting for different investment durations. Final AnswerRs. 10,500

Question 15

Partnership with time-weighted capitals: Pramod starts a business with ₹ 40000. After 4 months, Vikas joins with ₹ 60000. At the end of 12 months, total profit is ₹ 16000. Find Vikas’s share in the profit.

Accepted Answer

Introduction / Context:In partnership problems, profit is divided in the ratio of capital × time (money product). When partners join at different times, weight each investment by its active duration to get fair shares. Given Data / Assumptions: Pramod: ₹ 40000 for 12 months. Vikas: ₹ 60000 for 8 months (since he joined after 4 months). Total profit = ₹ 16000. Concept / Approach:Compute money products and use them as share weights. Ratio = (40000*12) : (60000*8). Divide profit in this ratio to obtain Vikas’s share. Step-by-Step Solution:Pramod weight = 40000 * 12 = 480000.Vikas weight = 60000 * 8 = 480000.Weights ratio = 1 : 1. Total profit split equally.Vikas’s share = 16000 / 2 = ₹ 8000. Verification / Alternative check:If profit per weight unit is the same, equal weights imply equal profits. The computation confirms symmetry in money-time product. Why Other Options Are Wrong: ₹ 4000 and ₹ 10000 require asymmetric weights, which are not present. ₹ 12000 contradicts equal weightage. Common Pitfalls: Ignoring time and dividing profit only by capitals. Assuming equal months for both partners despite the staggered start. Final Answer:Rs. 8000

Question 16

Inferring investment duration from profit ratio: A and B invest in the ratio 12 : 11. Their annual profits are in the ratio 4 : 1. If A invested for 11 months, for how many months did B invest?

Accepted Answer

Introduction / Context:Profit in partnerships is proportional to (capital × time). Knowing the capital ratio and profit ratio, we can infer one partner’s investment duration from the other’s time period. Given Data / Assumptions: Capital ratio A : B = 12 : 11. Profit ratio A : B = 4 : 1. A’s time = 11 months; B’s time = t months (unknown). Concept / Approach:Set up the equality: (12 × 11) : (11 × t) = 4 : 1. Solve for t by cross-multiplying. This directly uses the money-time product model of profit sharing. Step-by-Step Solution:(12 × 11) / (11 × t) = 4 / 1.132 / (11t) = 4 ⇒ 132 = 44t ⇒ t = 3 months. Verification / Alternative check:Check weights: A weight = 12 × 11 = 132; B weight = 11 × 3 = 33; ratio 132 : 33 = 4 : 1, matching the profit ratio. Why Other Options Are Wrong: 6 months gives ratio 132 : 66 = 2 : 1, not 4 : 1. 32/3 and 4 months also fail to match the required ratio. Common Pitfalls: Cancelling the 11s prematurely without maintaining the ratio relationship properly. Final Answer:3 months

Question 17

Partnership profits with unequal capital and time: A invests thrice as much as B and keeps the money for twice B’s period. If B’s profit share is Rs. 4,000, what is the total profit of the firm?

Accepted Answer

Introduction / Context: In partnership questions, profit sharing is proportional to capital * time. Here A invests more money and for a longer period than B. We are given B’s profit and must compute the total profit of the business. Given Data / Assumptions: A’s capital is 3 times B’s capital. A’s time in business is 2 times B’s time. B’s profit share = Rs. 4,000. Profits are divided in the ratio of (capital * time). Concept / Approach: If B’s capital is 1 unit for 1 unit of time, B’s weight = 1 * 1 = 1. A’s weight = 3 * 2 = 6. Therefore, A : B = 6 : 1. Total parts = 7. Each part equals B’s profit share since B has 1 part. Step-by-Step Solution: Profit ratio A : B = (3 * 2) : (1 * 1) = 6 : 1. Total parts = 6 + 1 = 7. 1 part = Rs. 4,000 (B’s share). Total profit = 7 * 4,000 = Rs. 28,000. Verification / Alternative check: A’s share = 6 * 4,000 = Rs. 24,000; B’s share = Rs. 4,000. Sum = Rs. 28,000, consistent with the ratio 6 : 1. Why Other Options Are Wrong: Rs. 24,000 and Rs. 20,000 underestimate total parts. Rs. 16,000 contradicts the given B’s share of Rs. 4,000. Common Pitfalls: Ignoring time in the ratio or adding the multipliers incorrectly. Always compute capital * time for each partner before forming the ratio. Final Answer: Rs. 28,000

Question 18

Three partners invest Rs. 27,000, Rs. 81,000, and Rs. 72,000 respectively. If Ram’s profit share is Rs. 36,000 after one year, what is the total profit?

Accepted Answer

Introduction / Context: When all partners invest for the same time, profits are divided in the ratio of their capitals. Given Ram’s share and his capital, we back-calculate the total profit. Given Data / Assumptions: Dilip = Rs. 27,000; Ram = Rs. 81,000; Amar = Rs. 72,000. Time = 1 year for all. Ram’s profit = Rs. 36,000. Concept / Approach: Ram’s fraction of profit = Ram’s capital / total capital. Then total profit = Ram’s share / Ram’s fraction. Step-by-Step Solution: Total capital = 27,000 + 81,000 + 72,000 = 180,000. Ram’s fraction = 81,000 / 180,000 = 9 / 20 = 0.45. Total profit = 36,000 / 0.45 = Rs. 80,000. Verification / Alternative check: Check shares with total Rs. 80,000: Ram gets 0.45 * 80,000 = Rs. 36,000, matching the given. Others would get Rs. 12,000 and Rs. 32,000 respectively, summing to Rs. 80,000. Why Other Options Are Wrong: Rs. 108,000 and Rs. 16,000 do not satisfy Ram’s 45% share equalling Rs. 36,000. “None of these” is incorrect because Rs. 80,000 is valid. Common Pitfalls: Dividing by the wrong total or miscomputing the fractional share. Ensure the ratio uses the sum of all capitals for the same time period. Final Answer: Rs. 80,000

Question 19

Rs. 700 is to be divided among A, B, and C such that A gets half of B’s share and B gets half of C’s share. What is C’s share?

Accepted Answer

Introduction / Context: This division problem relates the three shares by simple fractions. By expressing all shares in terms of C, the total can be solved quickly. Given Data / Assumptions: A = 1/2 of B. B = 1/2 of C. Total A + B + C = Rs. 700. Concept / Approach: Express B and A in terms of C to compute the sum: C + C/2 + C/4. Then equate to 700 and solve for C. Step-by-Step Solution: B = C/2; A = B/2 = C/4. Sum = C + C/2 + C/4 = (4C + 2C + C)/4 = 7C/4. 7C/4 = 700 ⇒ 7C = 2800 ⇒ C = 400. Verification / Alternative check: Then B = 200 and A = 100. Total = 100 + 200 + 400 = 700, consistent with the condition A = 1/2 B and B = 1/2 C. Why Other Options Are Wrong: Rs. 200 or Rs. 300 do not satisfy the fractional relationships. Rs. 600 would overshoot the total when combined with A and B. Common Pitfalls: Confusing “half as much as” with “twice as much as”, or not expressing all terms consistently in terms of a single variable. Final Answer: Rs. 400

Question 20

Jayant invests Rs. 6,000 for 12 months. After 6 months, Madhu joins with Rs. 4,000 for the last 6 months. If the total profit is Rs. 5,200, what is Madhu’s share?

Accepted Answer

Introduction / Context: When partners join at different times, use capital * time to determine profit shares. Here we compute Jayant’s and Madhu’s time-weighted capitals and then divide the profit accordingly. Given Data / Assumptions: Jayant: Rs. 6,000 for 12 months. Madhu: Rs. 4,000 for 6 months. Total profit = Rs. 5,200. Concept / Approach: Profit share ratio = (6000 * 12) : (4000 * 6). Convert to simplest ratio and allocate the total profit by parts. Step-by-Step Solution: Jayant weight = 6,000 * 12 = 72,000. Madhu weight = 4,000 * 6 = 24,000. Ratio = 72,000 : 24,000 = 3 : 1. Total parts = 3 + 1 = 4; each part = 5,200 / 4 = 1,300. Madhu’s share = 1,300. Verification / Alternative check: Jayant’s share = 3 * 1,300 = 3,900; Madhu’s share = 1,300. Sum = 5,200, confirming the distribution. Why Other Options Are Wrong: 2,080, 1,800, and 2,600 are inconsistent with the 3 : 1 time-weighted ratio. Common Pitfalls: Forgetting to multiply by months invested or mistakenly using raw capitals without time adjustment. Final Answer: Rs. 1300

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