Question 1

Two businessmen A and B invest in a business in the ratio 5 : 8. They decide that 30% of the total profit will be reinvested into the business and only the remaining 70% will be distributed between them in the same ratio. If A's share of the distrib…

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Introduction / Context:This problem is based on a simple partnership scenario where two partners share profit in a fixed ratio but also choose to reinvest a part of the profit back into the business. The key idea is distinguishing between total profit and the portion that is actually distributed to the partners. Understanding this distinction and handling percentage and ratio calculations correctly is essential for solving such questions commonly asked in quantitative aptitude exams. Given Data / Assumptions:- Two partners A and B invest in the ratio 5 : 8.- Let total profit be P rupees.- 30% of the total profit is reinvested in the business.- Therefore, 70% of the total profit is distributed among A and B.- The distributed portion is shared in the ratio 5 : 8.- A's share from the distributed profit is Rs. 87,500.- We are required to find the total profit P. Concept / Approach:The approach involves two layers of proportional reasoning. First, only 70% of the profit is available for distribution. Second, that 70% is divided between A and B according to their capital ratio of 5 : 8. The share received by A is therefore a fixed fraction of the total profit. By expressing A's share in terms of P and equating it to Rs. 87,500, we can solve a simple linear equation to obtain the value of P. Step-by-Step Solution:Step 1: Let total profit be P.Step 2: Reinvestment is 30% of P, so distributed profit = 70% of P.Step 3: Therefore, distributed profit = 0.70 * P.Step 4: This distributed amount is shared in the ratio 5 : 8, so total ratio parts = 5 + 8 = 13.Step 5: A's share = (5 / 13) of the distributed profit.Step 6: Hence, A's share = (5 / 13) * 0.70 * P.Step 7: Note that 0.70 = 7 / 10, so A's share = (5 / 13) * (7 / 10) * P = (35 / 130) * P = (7 / 26) * P.Step 8: We are given A's share = Rs. 87,500, so (7 / 26) * P = 87,500.Step 9: Solve for P: P = 87,500 * (26 / 7).Step 10: 87,500 * 26 = 2,275,000 and dividing by 7 gives P = 325,000.Therefore, the total profit is Rs. 3,25,000. Verification / Alternative check:Take P = 325,000.Distributed profit = 70% of 325,000 = 0.70 * 325,000 = 227,500.A's share = (5 / 13) * 227,500.Compute: 227,500 / 13 = 17,500 and 17,500 * 5 = 87,500.This matches the given information, so the result is consistent. Why Other Options Are Wrong:- 227000: This is close to the distributed profit, not the total profit, and ignores the reinvested portion.- 250000: This would give a smaller distributed amount and therefore a smaller share for A than 87,500.- 375000: This would produce an A share larger than 87,500 if calculated correctly, so it does not fit the data. Common Pitfalls:- Treating Rs. 87,500 as A's share out of the entire profit rather than from the 70% portion.- Using the ratio 5 : 8 directly on the total profit P instead of on the distributed profit only.- Mistakes in handling percentages, especially not converting 30% reinvestment into 70% distribution correctly.- Arithmetic errors when solving (7 / 26) * P = 87,500. Final Answer:The total profit earned by the business was Rs. 3,25,000.

Question 2

A and B start a partnership business with initial investments in the ratio 5 : 6. After 6 months, C joins them by investing an amount equal to two-thirds of B's capital. At the end of the year, C's share of the profit is Rs. 21,600. What is the tota…

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Introduction / Context:This partnership question deals with different joining times and different capital amounts. It focuses on the concept that profit in a partnership is divided in proportion to the product of capital and time, often called money-months or money-time units. Here, A and B start the business together, while C joins later with a specific fraction of B's capital. Based on C's known share of profit, we must work backwards to determine the total profit. Such questions are common in aptitude tests and help reinforce the idea of weighted contributions over time. Given Data / Assumptions:- A and B invest in the ratio 5 : 6.- Let B's capital be 6x; then A's capital is 5x.- C joins after 6 months with capital equal to two-thirds of B's capital.- Thus, C's capital is (2/3) * 6x = 4x.- Total business duration is 12 months from the start.- A and B remain invested for 12 months, C for 6 months.- C's share of profit after one year is Rs. 21,600.- We must find the total profit of the business. Concept / Approach:The key concept is that profit is shared in the ratio of capital multiplied by the time period for which it is invested. This forms a proportionality ratio. We compute money-time units for each partner: capital * months. These units give us the ratio in which total profit is divided among A, B and C. Once we know the ratio, we can treat C's share as a fraction of the whole and solve for total profit using a simple proportion. Step-by-Step Solution:Step 1: Let B's capital be 6x; A's capital is 5x.Step 2: C joins after 6 months with capital (2/3) of B's capital, so C's capital = 4x.Step 3: Time invested:- A: 12 months.- B: 12 months.- C: 6 months (joins after 6 months in a 12 month year).Step 4: Money-time units:- A: 5x * 12 = 60x.- B: 6x * 12 = 72x.- C: 4x * 6 = 24x.Step 5: Ratio of profit shares = 60x : 72x : 24x = 5 : 6 : 2 (dividing by 12x).Step 6: Total parts = 5 + 6 + 2 = 13 parts.Step 7: C gets 2 parts out of 13, and this is given as Rs. 21,600.Step 8: So 2 / 13 of total profit = 21,600.Step 9: Total profit = 21,600 * (13 / 2) = 21,600 * 6.5 = 140,400. Verification / Alternative check:Compute 1 part of profit: 140,400 / 13 = 10,800.A's share = 5 * 10,800 = 54,000.B's share = 6 * 10,800 = 64,800.C's share = 2 * 10,800 = 21,600.Sum = 54,000 + 64,800 + 21,600 = 140,400, which matches the calculated total profit, confirming the solution. Why Other Options Are Wrong:- 46800: Too small; it ignores the ratio 5 : 6 : 2 and would make C's share unrealistically large as a fraction of total profit.- 56160: Also too small; it does not produce C's share of 21,600 when using the ratio 5 : 6 : 2.- 70200: Exactly half of the correct answer, corresponding to misinterpreting C's share as 4 out of 13 instead of 2 out of 13. Common Pitfalls:- Forgetting that C joins after 6 months and incorrectly using 12 months instead of 6 for C's time factor.- Using only the capital ratio 5 : 6 : 4 without incorporating the different time periods.- Miscalculating the ratio by neglecting to reduce 60 : 72 : 24 to 5 : 6 : 2.- Doing percentage-based thinking instead of working directly with proportional parts. Final Answer:The total profit of the business at the end of the year is Rs. 1,40,400.

Question 3

A starts a taxi service by investing Rs. 25 lakhs. After 3 months, B joins the business by investing Rs. 40 lakhs, and 4 months after B joins (that is, after a total of 7 months from the start), C also joins by investing Rs. 50 lakhs. One year after…

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Introduction / Context:This is a classic partnership problem with unequal joining times and unequal capital contributions. The partners A, B and C invest different amounts and join the business at different times. Because profit is proportional to both the amount of capital invested and the duration of the investment, we must compute effective contributions in money-time units. The question asks for C's share of the total profit earned over one year from the start of the business. Given Data / Assumptions:- A invests Rs. 25 lakhs from the beginning and remains for 12 months.- B joins after 3 months with Rs. 40 lakhs and remains until the end of the year.- C joins 4 months after B, which is 7 months after A started, with Rs. 50 lakhs and remains until the end of the year.- Total time considered is 12 months from A's start.- Total profit at the end of the year is Rs. 2,73,000.- We are required to find C's share of this profit. Concept / Approach:The key concept is capital-time weighting. Profit share is proportional to capital * time (for example, rupees multiplied by months). For each partner, we compute the product of their capital and the number of months that capital remained invested. The ratio of these products across all partners gives the ratio in which profit is divided. Then, we multiply the total profit by C's fraction of the total ratio to find C's share. Step-by-Step Solution:Step 1: Determine the time each partner's capital is invested.- A: invests for the full 12 months.- B: joins after 3 months, so invests for 12 - 3 = 9 months.- C: joins 4 months after B, i.e., after 7 months from start, so invests for 12 - 7 = 5 months.Step 2: Compute money-time units (capital * months).- A: 25 lakhs * 12 = 300 lakh-months.- B: 40 lakhs * 9 = 360 lakh-months.- C: 50 lakhs * 5 = 250 lakh-months.Step 3: Profit sharing ratio = 300 : 360 : 250.Step 4: Simplify the ratio by dividing by 10: 30 : 36 : 25.Step 5: Total parts = 30 + 36 + 25 = 91.Step 6: C's share fraction = 25 / 91.Step 7: Total profit = Rs. 2,73,000. So C's profit share = (25 / 91) * 2,73,000.Step 8: Compute (25 / 91) * 2,73,000 = 2,73,000 * 25 / 91.Step 9: 2,73,000 / 91 = 3,000. Thus C's share = 3,000 * 25 = Rs. 75,000. Verification / Alternative check:Compute A and B shares to ensure the total adds up.Each part of profit = 2,73,000 / 91 = 3,000.A's share = 30 * 3,000 = Rs. 90,000.B's share = 36 * 3,000 = Rs. 1,08,000.C's share = 25 * 3,000 = Rs. 75,000.Total = 90,000 + 1,08,000 + 75,000 = 2,73,000, which matches the given profit, confirming the answer. Why Other Options Are Wrong:- 100000: Larger than C's actual share; would require C to have a bigger ratio than 25/91.- 125000: Implies C gets nearly half the total profit, inconsistent with the calculated ratio.- 150000: Equal to more than half of the total profit, which is impossible given three partners with substantial contributions. Common Pitfalls:- Ignoring the different joining times and simply using the capital ratio 25 : 40 : 50.- Mixing up months and years, or miscalculating the duration each partner stays in the business.- Not simplifying the ratio correctly and making arithmetic mistakes while finding C's share.- Applying percentage ideas instead of the precise capital-time multiplication method, which is the correct approach for partnership timing questions. Final Answer:C's share of the profit is Rs. 75,000.

Question 4

A, B and C invest in a business in the ratio 4 : 5 : 7. C is a sleeping partner, so his share of profit will be only half of what it would have been if he were a working partner. The firm earns a total profit of Rs. 36,000, out of which 25% is reinv…

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Introduction / Context:This partnership problem involves three partners where one partner, C, is a sleeping partner and therefore receives a reduced share of the profit compared to what his capital ratio alone would entitle him to. The business also reinvests a portion of the profit, meaning that only the remaining part is actually distributed. The task is to calculate B's share of the distributed profit. This type of question strengthens understanding of effective profit-sharing ratios when modifications like sleeping partners and reinvestment occur. Given Data / Assumptions:- Capital investment ratio of A : B : C is 4 : 5 : 7.- C is a sleeping partner.- C's profit share is half of what it would have been if he were a working partner.- Total profit = Rs. 36,000.- 25% of the profit is reinvested in the business.- Thus, 75% of the total profit is distributed among the partners.- We must find B's share in the distributed profit. Concept / Approach:The solution requires adjusting the profit-sharing ratio to account for the reduced share of the sleeping partner. Initially, the ratio 4 : 5 : 7 represents the capital-based shares if everyone were working. For C, we cut his share to half by multiplying his ratio part by 1/2. This yields an effective profit-sharing ratio. We then apply this adjusted ratio to the amount actually distributed (75% of Rs. 36,000) to obtain each partner's share, and particularly B's share. Step-by-Step Solution:Step 1: Base profit-sharing ratio from capital is A : B : C = 4 : 5 : 7.Step 2: Since C is a sleeping partner, his share is half of his normal share.Step 3: C's effective part = 7 * (1/2) = 3.5.Step 4: Effective profit-sharing ratio becomes A : B : C = 4 : 5 : 3.5.Step 5: Sum of these effective parts = 4 + 5 + 3.5 = 12.5.Step 6: Total profit is Rs. 36,000. Only 75% is distributed because 25% is reinvested.Step 7: Distributed profit = 0.75 * 36,000 = Rs. 27,000.Step 8: B's share = (B's effective part / total effective parts) * distributed profit.Step 9: B's share = (5 / 12.5) * 27,000.Step 10: 5 / 12.5 = 0.4, so B's share = 0.4 * 27,000 = Rs. 10,800. Verification / Alternative check:Consider each unit of the effective ratio as representing an equal amount of profit.Total effective parts = 12.5, distributed profit = 27,000.Value per part = 27,000 / 12.5 = 2,160.A's share = 4 * 2,160 = 8,640.B's share = 5 * 2,160 = 10,800.C's share = 3.5 * 2,160 = 7,560.Total = 8,640 + 10,800 + 7,560 = 27,000, which matches the distributed profit, confirming our calculation. Why Other Options Are Wrong:- 7560: This corresponds approximately to C's effective share, not B's share.- 8640: This matches A's share rather than B's share based on the effective ratio.- 9200: This value does not correspond to any correct part multiple from the effective ratio and arises from incorrect percentage or ratio manipulation. Common Pitfalls:- Forgetting to reduce C's share by half and incorrectly using the original ratio 4 : 5 : 7.- Applying the ratio directly to the total profit of Rs. 36,000 instead of to the distributed portion of Rs. 27,000.- Miscalculating 25% and 75%, or using 25% as the distributed amount instead of the reinvested amount.- Treating 3.5 as 3 or 4 when simplifying the ratio, which distorts the final shares. Final Answer:B receives a profit share of Rs. 10,800 from the distributed amount.

Question 5

S, T and U started a business by investing Rs. 4,200, Rs. 6,300 and Rs. 8,400 respectively. S withdrew from the business after 3 months, T withdrew after 6 months, and U remained invested for the full one year. At the end of the year, the total prof…

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Introduction / Context:This question examines profit sharing when partners invest for different lengths of time. Although each person invests a certain capital amount, they do not keep their money in the business for the same duration. Therefore, the share of profit must be based on both capital and time. This approach is crucial in partnership problems and is frequently tested in banking, SSC and other competitive exams. Given Data / Assumptions:- Initial investments are S = Rs. 4,200, T = Rs. 6,300, U = Rs. 8,400.- S withdraws after 3 months, so S is invested for 3 months.- T withdraws after 6 months, so T is invested for 6 months.- U remains invested for the full 12 months (1 year).- Total profit at the end of the year is Rs. 6,000.- We need to find U's share of the profit. Concept / Approach:The central concept is that profit is proportional to capital multiplied by time of investment. These products are often referred to as money-time units, such as rupee-months. We calculate these units for each partner, form a ratio, and then divide the total profit according to that ratio. Finally, U's fraction of the total is applied to Rs. 6,000 to find U's share. Step-by-Step Solution:Step 1: Determine the duration of investment for each partner.- S: 3 months.- T: 6 months.- U: 12 months.Step 2: Compute money-time units (capital * time).- S: 4,200 * 3 = 12,600.- T: 6,300 * 6 = 37,800.- U: 8,400 * 12 = 100,800.Step 3: Form the ratio of profit shares = 12,600 : 37,800 : 100,800.Step 4: Simplify the ratio by dividing all three terms by 4,200.- S: 12,600 / 4,200 = 3.- T: 37,800 / 4,200 = 9.- U: 100,800 / 4,200 = 24.So the ratio is S : T : U = 3 : 9 : 24.Step 5: Sum of ratio parts = 3 + 9 + 24 = 36.Step 6: U's share fraction = 24 / 36 = 2 / 3.Step 7: Total profit = Rs. 6,000. So U's share = (2 / 3) * 6,000 = Rs. 4,000. Verification / Alternative check:Compute each partner's share directly.Each part of the ratio corresponds to 6,000 / 36 = Rs. 166.67 approximately.S's share = 3 * 166.67 ≈ 500.T's share = 9 * 166.67 ≈ 1,500.U's share = 24 * 166.67 ≈ 4,000.The approximate sum is 500 + 1,500 + 4,000 = 6,000, consistent with total profit. Exact fractional calculations give exactly Rs. 4,000 for U. Why Other Options Are Wrong:- 4200: This exceeds the total profit share that U can reasonably get given the ratio and total of 6,000.- 3200: This would correspond to a smaller fraction than 2/3 of the total profit and does not match the money-time ratio.- 3600: This is 60% of the profit, but according to the ratio, U must receive 2/3, that is about 66.67% of the profit. Common Pitfalls:- Ignoring time and using only the capital amounts 4,200 : 6,300 : 8,400 which would be incorrect.- Miscalculating the duration, for example assuming S and T also stay for 12 months.- Arithmetic mistakes when simplifying the ratio or multiplying by the total profit.- Forgetting to express U's share as a fraction of the total ratio before applying it to the profit. Final Answer:U's share of the profit is Rs. 4,000.

Question 6

A profit of Rs. 1,35,200 has to be divided among three partners Aashay, Bhairav and Chetan in the ratio 4 : 3 : 6. How much money, in rupees, does Chetan receive from this profit?

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Introduction / Context:This is a straightforward ratio-based profit-sharing question. The total profit is known and the ratio in which it must be divided among three partners is given. Students must apply the concept of proportional division to determine the exact amount each partner receives. Here, we are specifically asked for Chetan's share. Such questions are common in basic arithmetic and partnership topics and help reinforce the idea of distributing a total amount according to a given ratio. Given Data / Assumptions:- Total profit to be shared = Rs. 1,35,200.- Profit-sharing ratio among Aashay, Bhairav and Chetan is 4 : 3 : 6.- We need to find Chetan's share of the profit.- No time factor or extra conditions such as working or sleeping partners are mentioned. Concept / Approach:The core concept is division of a total quantity in a given ratio. If a total S is to be divided in the ratio p : q : r, then the sum of the parts is p + q + r. Each person's share is the fraction of the total given by their ratio part divided by the total of the parts. Here, we compute the total of the ratio 4 + 3 + 6, then calculate Chetan's fractional share as 6 divided by this sum, and apply it to the profit of Rs. 1,35,200. Step-by-Step Solution:Step 1: Ratio of shares is Aashay : Bhairav : Chetan = 4 : 3 : 6.Step 2: Sum of the ratio parts = 4 + 3 + 6 = 13.Step 3: Chetan's fraction of the total profit is 6 / 13.Step 4: Total profit = Rs. 1,35,200.Step 5: Chetan's share = (6 / 13) * 1,35,200.Step 6: First compute 1,35,200 / 13.- 13 * 10,000 = 1,30,000, leaving 5,200.- 13 * 400 = 5,200.So 1,35,200 / 13 = 10,400.Step 7: Now multiply by 6: Chetan's share = 10,400 * 6 = 62,400.Therefore, Chetan gets Rs. 62,400. Verification / Alternative check:We can verify by calculating all three shares.Each part value = 1,35,200 / 13 = 10,400.Aashay's share = 4 * 10,400 = 41,600.Bhairav's share = 3 * 10,400 = 31,200.Chetan's share = 6 * 10,400 = 62,400.Check the sum: 41,600 + 31,200 + 62,400 = 1,35,200, which matches the total profit. Hence, the distribution is correct. Why Other Options Are Wrong:- 31200: This corresponds to Bhairav's share (3 parts), not Chetan's share.- 41600: This is Aashay's share (4 parts), not Chetan's share.- 93600: This would require Chetan to have 9 parts out of 13, which he does not; it is more than his entitled share based on the ratio. Common Pitfalls:- Adding the ratio numbers incorrectly or forgetting to include one of the partners when summing.- Failing to divide the total profit by the sum of the ratio parts before assigning individual shares.- Mixing up whose ratio part corresponds to which partner while matching the options.- Performing percentage calculations instead of direct ratio-based division, which is more straightforward here. Final Answer:Chetan receives Rs. 62,400 from the profit.

Question 7

Three friends, 'X', 'Y', and 'Z', decided to start a business together. They made investments of ₹6,000, ₹8,000, and ₹5,000, respectively. The time for which 'X', 'Y', and 'Z' kept their money invested is 't' months, (t - 3) months, and (t + 3) mont…

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Concept & Partnership Ratios: In partnership problems, profit is distributed based on the "Investment × Time" ratio. Additionally, when a working partner receives a salary, it must be deducted from the total profit before distributing the remainder according to the ratio. Profit Ratio = (I_1 × T_1) : (I_2 × T_2) : (I_3 × T_3) Step-by-Step Solution: 1. Find Profit Sharing Ratios: Investments: I_X = 6000, I_Y = 8000, I_Z = 5000 Time: T_X = t, T_Y = t - 3, T_Z = t + 3 Ratio of Profit Share (X : Y : Z) = 6000t : 8000(t - 3) : 5000(t + 3) = 6t : 8(t - 3) : 5(t + 3) Total units of ratio = 6t + 8t - 24 + 5t + 15 = 19t - 9 2. Calculate Distributable Profit: X receives a salary of ₹3000 per month for t months = 3000t. Remaining profit to distribute = 81000 - 3000t. 3. Formulate the Equation: Total amount X received = Salary + Profit Share = 3000t + (6t/19t - 9) × (81000 - 3000t) Total amount Z received = Profit Share = (5(t + 3)/19t - 9) × (81000 - 3000t) Given: X's total - Z's total = 25000 3000t + (6t - 5(t + 3)/19t - 9) × 3000(27 - t) = 25000 3t + (t - 15/19t - 9) × 3(27 - t) = 25 4. Solve for t: 3t(19t - 9) + 3(27t - t^2 - 405 + 15t) = 25(19t - 9) 57t^2 - 27t + 3(-t^2 + 42t - 405) = 475t - 225 57t^2 - 27t - 3t^2 + 126t - 1215 = 475t - 225 54t^2 + 99t - 1215 = 475t - 225 54t^2 - 376t - 990 = 0 Dividing by 2 gives: 27t^2 - 188t - 495 = 0 Using the quadratic formula: t = \frac{188 \pm \sqrt{35344 + 53460}}{54} = \frac{188 \pm \sqrt{88804}}{54} = (188 \pm 298/54) t = (486/54) = 9 (Ignoring the negative root) 5. Calculate Y's Share: With t = 9, total remaining profit = 81000 - 3000(9) = 54000. Ratio X : Y : Z = 6(9) : 8(6) : 5(12) = 54 : 48 : 60 = 9 : 8 : 10. Y's share = (8/9 + 8 + 10) × 54000 = (8/27) × 54000 = 8 × 2000 = 16000. Exam Strategy & Shortcut: To avoid large numbers when solving quadratics in partnership questions, divide equations by common factors early (like dividing the whole equation by 1000). Also, if you suspect an integer root for time, you can plug in values like 6, 9, 12 into 3t + \dots = 25 to save time over expanding the full quadratic. Common Pitfall: Forgetting to subtract the manager's salary from the gross profit before splitting the remaining funds into the investment ratios is the most common reason answers go off-track. Final Answer: Therefore, the correct answer is ₹16,000.

Question 8

Prakash, Sachin and Anil started a business jointly investing ₹ $11$ lakh, ₹ $16.5$ lakh and ₹ $8.25$ lakh respectively. The profit earned by them in the business at the end of $3$ years was ₹ $19.5$ lakh. What will be $50\%$ of Anil's share in the …

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Concept & Equal Time Partnership: When investments are made for the same duration, the profit is shared in the ratio of the initial capital investments. Profit Ratio = Investment_1 : Investment_2 : Investment_3 Step-by-Step Solution: • Given: • Prakash's investment = ₹ 11 lakh • Sachin's investment = ₹ 16.5 lakh • Anil's investment = ₹ 8.25 lakh • Total Profit = ₹ 19.5 lakh • Calculation/Deduction: 1. Find the ratio of their investments: Prakash : Sachin : Anil = 11 : 16.5 : 8.25 2. Multiply by 100 to remove the decimals: Ratio = 1100 : 1650 : 825 3. Simplify the ratio by dividing by their highest common factor (275): Ratio = 4 : 6 : 3 4. Sum the parts of the ratio to find the total denominator: Total parts = 4 + 6 + 3 = 13 5. Calculate Anil's total share in the profit (he represents 3 parts): Anil's Share = 19.5 × ((3/13)) Anil's Share = 1.5 × 3 = 4.5 lakh 6. The question asks for 50% of Anil's share: 50% of 4.5 lakh = (4.5/2) = 2.25 lakh Exam Strategy & Shortcut: When simplifying 1100 : 1650 : 825, notice that 825 × 2 = 1650, so the ratio between the last two is 2 : 1 (or 6:3). Knowing 11 × 75 = 825, you can divide by common factors like 25 and 11 rapidly. Crucially, don't forget the final step: the question specifically asks for 50% of Anil's share, not his full share. Common Pitfall: A major trap is calculating Anil's full share (4.5 lakh) and immediately selecting option (d). Always re-read the last sentence of the question. Final Answer: Therefore, the correct answer is ₹ 2.25 lakh.

Question 9

Gautam started a business with a sum of ₹ 60000. Jatin joined him 8 months later with a sum of ₹ 35000. At what respective ratio will the two share the profit after two years?

Accepted Answer

Concept & Active Investment Duration: For partnerships spanning multiple years, carefully convert the total duration into months. Calculate the active duration for late joiners by subtracting their joining delay from the total duration. Profit Ratio = (C_1 × T_1) : (C_2 × T_2) Step-by-Step Solution: • Given: • Total duration = 2 years = 24 months. • Gautam's capital = ₹ 60000 (invested for the full 24 months). • Jatin's capital = ₹ 35000 (joined 8 months later). • Calculation/Deduction: 1. Determine Jatin's active time: Jatin's time = 24 - 8 = 16 months 2. Set up the profit ratio based on Capital × Time: Gautam : Jatin = (60000 × 24) : (35000 × 16) 3. Simplify by dividing by 1000 (drop three zeros): Ratio = (60 × 24) : (35 × 16) 4. Factor and simplify further. Divide by 8: Ratio = (60 × 3) : (35 × 2) Ratio = 180 : 70 5. Divide by 10: Ratio = 18 : 7 Exam Strategy & Shortcut: To quickly simplify (60 × 24) : (35 × 16), look for common denominators across both sides of the ratio. 24 and 16 are divisible by 8 (leaving 3 and 2). 60 and 35 are divisible by 5 (leaving 12 and 7). So the ratio simplifies directly to (12 × 3) : (7 × 2) \implies 36 : 14 \implies 18 : 7. Common Pitfall: Students often miss that the total duration is "two years" and mistakenly calculate using a 12-month base, which completely skews the time ratio. Always verify the total timeline. Final Answer: Therefore, the correct answer is 18 : 7.

Question 10

Dilip, Ram and Avtar started a shop by investing ₹ 2700, ₹ 8100 and ₹ 7200 respectively. At the end of one year, the profit earned was distributed. If Ram's share was ₹ 3600, what was their total profit?

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Concept & Reverse Profit Calculation: When you are given one partner's profit share and the investment ratio, you can find the value of a single "ratio part" and use it to determine the total profit. Total Profit = Value of One Part × Total Parts Step-by-Step Solution: • Given: • Investments: Dilip = ₹ 2700, Ram = ₹ 8100, Avtar = ₹ 7200. • Time period is identical for all (1 year). • Ram's profit share = ₹ 3600. • Calculation/Deduction: 1. Find the profit-sharing ratio (which equals the investment ratio): Dilip : Ram : Avtar = 2700 : 8100 : 7200 2. Simplify by dividing by 100: Ratio = 27 : 81 : 72 3. Divide by their highest common factor (9): Ratio = 3 : 9 : 8 4. Calculate total parts in the ratio: Total Parts = 3 + 9 + 8 = 20 5. Equate Ram's ratio parts to his actual profit share: 9 parts = 3600 1 part = (3600/9) = 400 6. Calculate total profit using the total parts: Total Profit = 20 parts × 400 = 8000 Exam Strategy & Shortcut: Simplify the ratio quickly: 27:81:72 \implies 3:9:8. You know Ram represents 9 parts. Mentally map 9 \rightarrow 3600, which means a multiplier of 400. The total is 20 parts. 20 × 400 = 8000. The whole calculation can be done mentally in under 15 seconds. Common Pitfall: A mistake is trying to set up a complex algebraic equation (e.g., 3x + 9x + 8x = Y) and getting lost in variables. The unitary method (finding the value of 1 part) is much faster and less prone to error. Final Answer: Therefore, the correct answer is ₹ 8000.

Question 11

A, B and C enter into a partnership. A invests some money at the beginning, B invests double the amount after six months and C invests thrice the amount after eight months. If the annual profit be ₹ 27000; C's share (in ₹) is

Accepted Answer

Concept & Proportional Investment: When investments are stated in relative terms (double, thrice), express them algebraically using a base variable (e.g., let initial amount be x). The equivalent capital is then found by multiplying by the active months. Step-by-Step Solution: • Given: • Annual profit duration = 12 months • Total Profit = ₹ 27000 • Define Investments and Time: • Let A's initial investment be x. A invested for 12 months. • B invests double the amount = 2x. He joins after 6 months, so his time = (12 - 6) = 6 months. • C invests thrice the amount = 3x. He joins after 8 months, so his time = (12 - 8) = 4 months. • Calculate Profit Ratio: • Ratio = (A's capital) : (B's capital) : (C's capital) • Ratio = (x × 12) : (2x × 6) : (3x × 4) • Ratio = 12x : 12x : 12x • Ratio = 1 : 1 : 1 • Calculate C's Share: • The profit is distributed equally among all three partners because their equivalent capitals are identical. • Sum of ratio parts = 1 + 1 + 1 = 3 • C's Share = ((1/3)) × 27000 = ₹ 9000 Exam Strategy & Shortcut: Notice the symmetry immediately: 1 (base) × 12 (full time) 2 (double) × 6 (half time) 3 (triple) × 4 (one-third time) All products equal 12. Thus, the profit is split equally into three parts. 27000 / 3 = 9000. Common Pitfall: Assuming that because someone invested more money later (like C investing thrice the amount), they automatically get a larger share of the profit. Time is just as critical as the investment amount! Final Answer: Therefore, the correct answer is ₹ 9000.

Question 12

A and B entered into a partnership investing ₹ 16000 and ₹ 12000 respectively. After 3 months, A withdrew ₹ 5000 while B invested ₹ 5000 more. After 3 more months, C joins the business with a capital of ₹ 21000. The share of B exceeds that of C, out…

Accepted Answer

Concept & Profit Distribution: In a partnership, if investments are made for different time periods, the profit is distributed in the ratio of the product of capital and time for each partner. Profit Ratio = (C_1 × T_1) : (C_2 × T_2) : (C_3 × T_3) Step-by-Step Solution: • A's Investment: Invests ₹ 16000 for 3 months. Then withdraws ₹ 5000, leaving ₹ 11000 for the remaining 9 months. • A's equivalent capital = (16000 × 3) + (11000 × 9) = 48000 + 99000 = 147000 • B's Investment: Invests ₹ 12000 for 3 months. Then invests ₹ 5000 more, making it ₹ 17000 for the remaining 9 months. • B's equivalent capital = (12000 × 3) + (17000 × 9) = 36000 + 153000 = 189000 • C's Investment: Joins after 6 months (3 months + 3 more months) with ₹ 21000. So, C's money is invested for 12 - 6 = 6 months. • C's equivalent capital = 21000 × 6 = 126000 • Profit Ratio: • A : B : C = 147000 : 189000 : 126000 • Dividing by 1000: 147 : 189 : 126 • Dividing by 21: 7 : 9 : 6 • Total Profit: ₹ 26400 • Total ratio parts = 7 + 9 + 6 = 22 parts • Difference between B's and C's shares is (9 - 6) = 3 parts. • Difference = (3/22) × 26400 = 3 × 1200 = 3600 Exam Strategy & Shortcut: Instead of calculating B's and C's shares individually and then subtracting, directly calculate the difference in their ratio parts (9 - 6 = 3 parts) and find the value of those 3 parts relative to the total profit. Common Pitfall: Miscalculating the number of months each partner's money is invested. Carefully read phrases like "After 3 more months" to determine the total elapsed time before a new event occurs. Final Answer: Therefore, the correct answer is ₹ 3600.

Question 13

Anu is a working partner and Bimla is a sleeping partner in a business. Anu puts in ₹ 5000 and Bimla puts in ₹ 6000. Anu receives 12.5% of the profit for managing the business and the rest is divided in proportion to their capitals. What does each g…

Accepted Answer

Concept & Working Partner Commission: When a partnership includes a working partner, their management fee or commission is deducted from the total profit first. The remaining profit is then distributed among all partners (including the working partner) based on their capital investment ratio. Step-by-Step Solution: • Capital Ratio: • Anu : Bimla = 5000 : 6000 = 5 : 6 • Anu's Management Fee: • Anu receives 12.5% of the total profit. • 12.5% = (1/8) • Fee = (1/8) × 880 = ₹ 110 • Remaining Profit: • Total Profit - Management Fee = 880 - 110 = ₹ 770 • Distributing the Remainder: • The remaining ₹ 770 is split in the ratio 5 : 6. • Total ratio parts = 5 + 6 = 11 • Value of 1 part = (770/11) = ₹ 70 • Anu's investment share = 5 × 70 = ₹ 350 • Bimla's investment share = 6 × 70 = ₹ 420 • Total Individual Earnings: • Anu's Total = Management Fee + Investment Share = 110 + 350 = ₹ 460 • Bimla's Total = Investment Share = ₹ 420 Exam Strategy & Shortcut: Memorize fraction equivalents for common percentages to speed up calculations. Knowing that 12.5% = (1/8) allows you to instantly calculate Anu's fee as (880/8) = 110 mentally. Common Pitfall: Forgetting to add the working partner's management fee back to their investment share at the end. Anu gets *both* the ₹ 110 and the ₹ 350. Final Answer: Therefore, the correct answer is ₹ 460, ₹ 420.

Question 14

If 4 (A's capital) = 6 (B's capital) = 10 (C's capital), then out of a profit of ₹ 4650, C will receive

Accepted Answer

Concept & Equated Multiples: When investments are given as equated multiples (e.g., xA = yB = zC), find the ratio of their investments by dividing the entire equation by the Least Common Multiple (LCM) of the coefficients x, y, and z. Step-by-Step Solution: • Given Equality: • 4A = 6B = 10C • Finding the Ratio: • Find the LCM of the coefficients 4, 6, and 10. • LCM(4, 6, 10) = 60. • Divide the entire equation by 60: (4A/60) = (6B/60) = (10C/60) (A/15) = (B/10) = (C/6) • This gives the direct investment (and thus profit) ratio: A : B : C = 15 : 10 : 6 • Distributing the Profit: • Total ratio parts = 15 + 10 + 6 = 31 parts. • Total profit = ₹ 4650. • Value of 1 part = (4650/31) = ₹ 150. • C's share corresponds to 6 parts. • C's share = 6 × 150 = ₹ 900. Exam Strategy & Shortcut: Instead of LCM, you can use the "cover-up" method. To find A's ratio part, cover 4 and multiply the others (6 × 10 = 60). For B, cover 6 (4 × 10 = 40). For C, cover 10 (4 × 6 = 24). The ratio is 60 : 40 : 24. Divide by 4 to simplify to 15 : 10 : 6. Common Pitfall: Assuming the ratio is directly 4 : 6 : 10. If 4A = 10C, A must be larger than C for the equality to hold, meaning the ratio is inverted relative to the coefficients. Final Answer: Therefore, the correct answer is ₹ 900.

Question 15

A, B and C enter into a partnership by making investments in the ratio 3 : 5 : 7. After a year, C invests another ₹ 337600 while A withdraws ₹ 45600. The ratio of investments then changes to 24 : 59 : 167. How much did A invest initially?

Accepted Answer

Concept & Equating Ratios: When an initial ratio is given, assign variables (3x, 5x, 7x). Apply the stated absolute changes (additions/withdrawals) to form expressions for the new amounts. Equate the ratio of these new expressions to the given final ratio to solve for the variable. Step-by-Step Solution: • Let the initial investments of A, B, and C be 3x, 5x, and 7x respectively. • Changes after one year: • A's new investment = 3x - 45600 • B's new investment = 5x (no change mentioned) • C's new investment = 7x + 337600 • New Ratio: • The new investments are in the ratio 24 : 59 : 167. • (3x - 45600) : 5x : (7x + 337600) = 24 : 59 : 167 • Solving for x: • We can use any two parts of the ratio to form an equation. Using A and B is easiest because B's term is simple (5x). • (3x - 45600/5x) = (24/59) • Cross-multiply: • 59(3x - 45600) = 24(5x) • 177x - 2690400 = 120x • 177x - 120x = 2690400 • 57x = 2690400 • x = (2690400/57) = 47200 • Calculate A's Initial Investment: • A's initial investment = 3x • 3 × 47200 = 141600 Exam Strategy & Shortcut: Always choose the terms that create the simplest equation. Pairing A and B ((3x - \dots/5x)) avoids multiplying large numbers like 167 or 337600 until absolutely necessary. Common Pitfall: Using the C term to solve for x makes the arithmetic much more prone to errors due to large multiplication steps. Final Answer: Therefore, the correct answer is ₹ 141600.

Question 16

A, B and C enter into a partnership. A contributes one-third of the capital while B contributes as much as A and C together contribute. If the profit at the end of the year amounts to ₹ 900, what would C receive?

Accepted Answer

Concept & Formula: The profit in a partnership is distributed in proportion to the total capital invested and the time period of investment. When the time period is identical, profit is distributed based on the ratio of investments. Profit Ratio = Capital Ratio Step-by-Step Solution: 1. Let the total capital be 1. 2. Capital contributed by A = (1/3). 3. We are given that B contributes as much as A and C together: B = A + C. 4. Total capital is A + B + C = 1. 5. Substituting B with A + C, we get: A + (A + C) + C = 1, which simplifies to 2(A + C) = 1, so A + C = (1/2). 6. Since A's share is (1/3), C's share is: C = (1/2) - (1/3) = (3/6) - (2/6) = (1/6). 7. Now, we find B's share: B = A + C = (1/2). 8. The ratio of their investments (A : B : C) is (1/3) : (1/2) : (1/6). Multiplying by 6 to clear denominators, we get 2 : 3 : 1. 9. C's share of the profit = (1/6) of total profit = (1/6) × 900 = 150. Exam Strategy & Shortcut: Instead of finding B, recognize that if B = A + C, then B provides half the capital (since A + B + C = 1 implies B + B = 1 \Rightarrow B = (1/2)). Since A is (1/3), C must make up the difference to half: (1/2) - (1/3) = (1/6). The ratio is direct. Common Pitfall: Misinterpreting "B contributes as much as A and C together" as B taking the majority of the profit without solving for C's specific fraction first. Final Answer: Therefore, the correct answer is ₹ 150.

Question 17

A and B started a business jointly. A's investment was thrice the investment of B and the period of his investment was two times the period of investment of B. If B received ₹ 4000 as profit, then their total profit is:

Accepted Answer

Concept & Formula: The profit share is directly proportional to the product of capital invested and time period of investment. Ratio of Profit = (Capital_A × Time_A) : (Capital_B × Time_B) Step-by-Step Solution: 1. Let B's investment be x and the period of B's investment be y. 2. Then, A's investment is 3x and the period of A's investment is 2y. 3. Find the ratio of their profits: A : B = (3x × 2y) : (x × y) = 6xy : xy = 6 : 1. 4. This means out of every 7 parts of profit, A gets 6 parts and B gets 1 part. 5. We are given that B's profit share (1 part) is ₹ 4000. 6. The total profit consists of 7 parts (6 parts for A + 1 part for B). 7. Total profit = 7 × 4000 = 28000. Exam Strategy & Shortcut: Multiply the ratios directly: Investment ratio is 3:1. Time ratio is 2:1. Profit ratio = 3 × 2 : 1 × 1 = 6:1. Total profit is 6+1=7 times B's profit. 7 × 4000 = 28000. Common Pitfall: Finding A's profit (24000) and incorrectly choosing that as the final answer instead of summing A and B for the *total* profit. Final Answer: Therefore, the correct answer is ₹ 28000.

Question 18

Swati and Rajni enter into a partnership with their capitals in the ratio 5 : 6. At the end of 7 months Swati withdraws her capital. If they receive the profit in the ratio of 5 : 9, find how long was Rajni's capital used.

Accepted Answer

Concept & Profit Ratio Formula: The ratio of profits is equal to the ratio of the products of capital and the time it was invested. (Profit_A/Profit_B) = (Capital_A × Time_A/Capital_B × Time_B) Step-by-Step Solution: 1. The ratio of initial capitals is C_{Swati} : C_{Rajni} = 5 : 6. Let Swati's capital be 5 units and Rajni's capital be 6 units. 2. Swati invested for 7 months, so Time_{Swati} = 7. 3. Let Rajni's money be used for t months, so Time_{Rajni} = t. 4. The profit ratio is given as Profit_{Swati} : Profit_{Rajni} = 5 : 9. 5. Substituting the values into the formula: (5 × 7/6 × t) = (5/9) 6. Simplify by cancelling the 5 in the numerators: (7/6t) = (1/9) 7. Cross-multiply: 6t = 63 8. t = (63/6) = 10.5 months. 9. Since 10.5 months is not among the given numerical options (10, 12, 14), the answer is "None of these". Exam Strategy & Shortcut: Write down the relationship directly: (5 × 7) / (6 × t) = 5 / 9. Mentally drop the 5s. You have 7 / 6t = 1 / 9 \Rightarrow 6t = 63 \Rightarrow t = 10.5. It takes less than 15 seconds to realize the answer isn't an integer, pointing straight to "None of these". Common Pitfall: Guessing "12 months" or "14 months" simply because 10.5 is a fraction, assuming a calculation error was made, instead of confidently trusting the math and selecting "None of these". Final Answer: Therefore, the correct answer is None of these.

Question 19

A and B started a business by investing ₹ 2400 and ₹ 3600 respectively. At the end of 4th month from the start of the business, C joined with ₹ 'X'. After 8 months from the start of the business. B withdrew ₹600. If C's share is ₹ 8000 in the annual…

Accepted Answer

Concept & Reverse Profit Sharing: When the final profit share and total profit are known but an investment is unknown, set up an equation equating the ratio of the unknown equivalent capital to the known equivalent capital with the ratio of their respective profits. \frac{C's Capital}{(A+B)'s Capital} = \frac{C's Profit}{(A+B)'s Profit} Step-by-Step Solution: • Given: • Business duration is 12 months (annual profit). • A's investment = ₹2400 for 12 months. • B's investment = ₹3600 for 8 months, then withdraws ₹600, meaning (3600 - 600) = ₹3000 for the remaining 4 months. • C's investment = ₹X for 8 months (joined after 4 months). • Total profit = ₹22,500; C's share = ₹8,000. • Calculation: • A's equivalent capital = 2400 × 12 = 28800 • B's equivalent capital = (3600 × 8) + (3000 × 4) = 28800 + 12000 = 40800 • C's equivalent capital = 8X • Sum of A and B's equivalent capitals = 28800 + 40800 = 69600 • Sum of A and B's profit = 22500 - 8000 = 14500 • Equating the ratios: (8X/69600) = (8000/14500) • Simplify the profit ratio: (8000/14500) = (80/145) = (16/29) • (8X/69600) = (16/29) • 8X = 69600 × ((16/29)) • 69600 / 29 = 2400 • 8X = 2400 × 16 = 38400 • X = (38400/8) = 4800 Exam Strategy & Shortcut: Instead of finding C's proportion over the *total* sum (which requires adding 8X to the denominator), compare C directly against the sum of A and B. This keeps the unknown variable X isolated in the numerator, solving the equation faster. Common Pitfall: A frequent error is calculating C's investment time as 4 months instead of 8 months. "At the end of 4th month... C joined" means C was active for the remaining 12 - 4 = 8 months. Final Answer: Therefore, the correct answer is ₹ 4800.

Question 20

Directions: Each of these questions is followed by three statements. You have to study the question and all the three statements given to decide whether any information provided in the statement(s) is redundant and can be dispensed with while answer…

Accepted Answer

Concept & Ratio of Equivalent Capitals: To calculate a partner's share in the total profit, we need the ratio of their "equivalent capital" (which is Investment × Time). For Neeta's share, we specifically need Neeta's equivalent capital relative to the total equivalent capital of all partners combined. Step-by-Step Solution: • From the main question: Total profit = ₹ 50,000. Total time = 2 years (24 months). • From Statement I: Neeta's investment is ₹ 85,000. She invested for the full 24 months. • Neeta's equivalent capital = 85000 × 24. • From Statement III: Seeta and Geeta's combined investment is ₹ 2,50,000 (2.5 lakh). • From Statement II: Seeta and Geeta joined after 6 months, meaning their investment time is 24 - 6 = 18 months. • Seeta and Geeta's combined equivalent capital = 250000 × 18. • Combining all three statements gives us the total equivalent capital (85000 × 24 + 250000 × 18) and Neeta's individual part, allowing us to find her exact share. • Checking for Redundancy: • Discarding I: We lose Neeta's investment amount. • Discarding II: We lose the time duration (18 months) for Seeta and Geeta. • Discarding III: We lose the combined investment amount of Seeta and Geeta. • Therefore, no single statement can be completely dispensed with. Exam Strategy & Shortcut: In Data Sufficiency, do not waste time fully calculating the final numerical answer. Once you confirm you have the necessary variables (Neeta's Inv × Time and Total Inv × Time), stop and evaluate if any single statement can be dropped. If dropping a statement removes a crucial variable (like time), it is fully required. Common Pitfall: A common mistake is realizing that the 3 : 5 ratio in Statement II is not needed to find *Neeta's* share, and thus incorrectly assuming Statement II is completely redundant. You must read the *entire* statement to ensure no other critical data (like the 6-month delay) is lost by dispensing with it. Final Answer: Therefore, the correct answer is Information in all the three statements is required to answer the question.

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