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

Rahul, Arun and Sumit started a business. Rahul invested $\frac{1}{2}$ part, Arun $\frac{1}{3}$ part and rest of the capital was invested by Sumit. The ratio of their profits will be

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Concept & Partnership Ratio: When investments are made for the exact same time period, the profit sharing ratio is exactly equal to the ratio of their invested capitals. Profit Ratio = Investment Ratio Step-by-Step Solution: • Given: • Rahul's investment = (1/2) of total • Arun's investment = (1/3) of total • Calculation/Deduction: 1. Find Sumit's investment by subtracting Rahul's and Arun's shares from the whole (1): Sumit's share = 1 - ( (1/2) + (1/3) ) Sumit's share = 1 - ( (3 + 2/6) ) = 1 - (5/6) = (1/6) 2. Set up the ratio of investments for Rahul : Arun : Sumit: Ratio = (1/2) : (1/3) : (1/6) 3. Multiply the entire ratio by the Least Common Multiple (LCM) of the denominators (which is 6) to convert it into whole numbers: Ratio = ((1/2) × 6) : ((1/3) × 6) : ((1/6) × 6) Ratio = 3 : 2 : 1 Exam Strategy & Shortcut: To quickly clear fractions in a ratio like (1/2) : (1/3) : (1/6), simply multiply by the LCM of 2, 3, and 6. The LCM is 6, immediately yielding 3 : 2 : 1. Common Pitfall: A frequent mistake is to leave the ratio as fractions and incorrectly map the numerators without establishing a common denominator. Always normalize the ratio using the LCM. Final Answer: Therefore, the correct answer is 3 : 2 : 1.

Question 9

Anand and Deepak started a business investing ₹ 22500 and ₹ 35000 respectively. Out of a total profit of ₹ 13800, Deepak's share is

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Concept & Profit Distribution: Profits are distributed in proportion to the capital invested when the duration of investment is identical. Share of A = Total Profit × ( \frac{Ratio of A}{Total Ratio Parts} ) Step-by-Step Solution: • Given: • Anand's investment = ₹ 22500 • Deepak's investment = ₹ 35000 • Total profit = ₹ 13800 • Calculation/Deduction: 1. Determine the profit-sharing ratio based on investments: Anand : Deepak = 22500 : 35000 2. Simplify the ratio by removing two zeros: Ratio = 225 : 350 3. Divide both numbers by their highest common factor (25): Ratio = 9 : 14 4. Calculate the total number of ratio parts: Total Parts = 9 + 14 = 23 5. Calculate Deepak's share of the total profit: Deepak's Share = 13800 × ((14/23)) Deepak's Share = 600 × 14 = 8400 Exam Strategy & Shortcut: Quickly simplify the ratio 22500 : 35000 \rightarrow 225 : 350 \rightarrow 9 : 14. Knowing the total parts equal 23, you can check if 138 is divisible by 23 (23 × 6 = 138). The multiplier is 600. Multiply Deepak's ratio part (14) by 600 to get 8400. Common Pitfall: A common error is calculating Anand's share instead of Deepak's due to misreading the final question line. Always double-check whose share is requested before finalizing the calculation. Final Answer: Therefore, the correct answer is ₹ 8400.

Question 10

P, Q and R invested ₹ 45000, ₹ 70000 and ₹ 90000 respectively to start a business. At the end of 2 years, they earned a profit of ₹ 164000. What will be Q's share in the profit?

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Concept & Equal Time Partnership: In a partnership where all members invest their capital for an identical duration, the profit is distributed precisely in the ratio of their initial investments. Profit Ratio = Investment_1 : Investment_2 : Investment_3 Step-by-Step Solution: • Given: • Investments: P = ₹ 45000, Q = ₹ 70000, R = ₹ 90000 • Total Profit = ₹ 164000 • Time = 2 years (Constant for all, thus negligible for ratio) • Calculation/Deduction: 1. Find the ratio of their investments: P : Q : R = 45000 : 70000 : 90000 2. Simplify by canceling out three zeros from each term: P : Q : R = 45 : 70 : 90 3. Further simplify by dividing all terms by their highest common factor, 5: P : Q : R = 9 : 14 : 18 4. Sum the parts of the ratio to find the total denominator: Total parts = 9 + 14 + 18 = 41 5. Calculate Q's share (which is 14 parts out of 41): Q's Share = 164000 × ((14/41)) 6. Simplify and solve: (164000/41) = 4000 Q's Share = 4000 × 14 = 56000 Exam Strategy & Shortcut: Recognize that 41 × 4 = 164. Once you get the total parts as 41, the division step becomes mental math: 164000 / 41 = 4000. Q represents 14 units, so 14 × 4000 = 56000. Common Pitfall: A time-consuming mistake is multiplying each investment by 2 (for the years) before finding the ratio. Since the time is the same for all investors, it will eventually cancel out. Save time by ignoring uniform time periods immediately. Final Answer: Therefore, the correct answer is ₹ 56000.

Question 11

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 12

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 13

Simran started a software business by investing ₹ 50000. After six months, Nanda joined her with a capital of ₹ 80000. After 3 years, they earned a profit of ₹ 24500. What was Simran's share in the profit?

Accepted Answer

Concept & Multi-Year Profit Distribution: Calculate the profit-sharing ratio by multiplying each partner's capital by the number of months it was invested. When the result is not among the explicit numerical options, trust your calculation over guessing. Share of A = Total Profit × ( \frac{Ratio Part of A}{Total Ratio Parts} ) Step-by-Step Solution: • Given: • Total duration = 3 years = 36 months. • Simran's capital = ₹ 50000 (active for 36 months). • Nanda's capital = ₹ 80000 (joined 6 months later). • Total Profit = ₹ 24500. • Calculation/Deduction: 1. Determine Nanda's active time: Nanda's time = 36 - 6 = 30 months 2. Establish the ratio of their investments: Simran : Nanda = (50000 × 36) : (80000 × 30) 3. Simplify the ratio by removing four zeros from the capitals: Ratio = (5 × 36) : (8 × 30) 4. Simplify further (divide both by 6): Ratio = (5 × 6) : (8 × 5) Ratio = 30 : 40 = 3 : 4 5. Total parts = 3 + 4 = 7. 6. Calculate Simran's share (3 parts): Simran's Share = 24500 × ((3/7)) Simran's Share = 3500 × 3 = 10500 7. Compare with options. ₹ 10500 is not listed among (a), (b), (c), or (d). Exam Strategy & Shortcut: Simplify (5 × 36) : (8 × 30) quickly by noting 36/30 = 6/5. Thus the ratio is 5 × 6 : 8 × 5 \implies 30:40 \implies 3:4. Knowing the total parts is 7, 24500/7 = 3500. Simran's share is 3500 × 3 = 10500. Since 10500 is missing, immediately mark "None of these" without second-guessing. Common Pitfall: A common mistake is forcing an answer. Students might arrive at 10500, see 10250 in the options, assume they made a small arithmetic error, and mark the wrong option instead of confidently selecting "None of these". Final Answer: Therefore, the correct answer is None of these.

Question 14

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?

Accepted Answer

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 15

Aman started a business investing ₹ 70000. Rakhi joined him after six months with an amount of ₹ 105000 and Sagar joined them with ₹ 1.4 lakhs after another six months. The amount of profit earned should be distributed in what ratio among Aman, Rakh…

Accepted Answer

Concept & Formula: The core concept in partnership problems is that the profit is divided in the ratio of the product of the investment amount and the time period for which the investment was active. Ratio of Profit = (Investment_1 × Time_1) : (Investment_2 × Time_2) : \dots Step-by-Step Solution: • Given: • Aman's investment = ₹ 70000 • Rakhi's investment = ₹ 105000 • Sagar's investment = ₹ 1.4 lakhs = ₹ 140000 • Total duration of business = 3 years = 36 months • Determine Time Periods: • Aman invested for the full duration: 36 months • Rakhi joined after 6 months, so her money was invested for 36 - 6 = 30 months • Sagar joined after another 6 months (12 months from start), so his money was invested for 36 - 12 = 24 months • Calculate Profit Ratio: • Ratio = (70000 × 36) : (105000 × 30) : (140000 × 24) • Simplify by cancelling out common zeros (divide by 1000): • Ratio = (70 × 36) : (105 × 30) : (140 × 24) • Divide all terms by 35: • Ratio = (2 × 36) : (3 × 30) : (4 × 24) • Ratio = 72 : 90 : 96 • Divide all terms by 6: • Ratio = 12 : 15 : 16 Exam Strategy & Shortcut: Instead of multiplying large numbers, cancel out the common factors (like thousands and 35) from the investments *before* multiplying by the time periods. This significantly speeds up calculation. Common Pitfall: Forgetting to convert years to months or miscalculating the time Sagar's money was invested ("another six months" means 6+6=12 months late, so 36-12=24 months active). Final Answer: Therefore, the correct answer is 12 : 15 : 16.

Question 16

Sonia started a business investing ₹ 60000. After 6 months Vivek joined him with an amount of ₹ 140000. After 1 year Kirti also joined them with ₹ 120000. After 2 years the business yielded a total profit of ₹ 450000. What is the share of Vivek in t…

Accepted Answer

Concept & Ratio of Equivalent Capitals: Profits are distributed in proportion to the equivalent capital of each partner, which is the product of their investment and the time period it remained in the business. Equivalent Capital = Investment × Time Step-by-Step Solution: • Given: • Total duration = 2 years = 24 months • Sonia's investment = ₹ 60000 for 24 months • Vivek's investment = ₹ 140000 for (24 - 6) = 18 months • Kirti's investment = ₹ 120000 for (24 - 12) = 12 months • Total Profit = ₹ 450000 • Calculate Profit Ratio: • Ratio = (60000 × 24) : (140000 × 18) : (120000 × 12) • Cancel out 10000 from each: • Ratio = (6 × 24) : (14 × 18) : (12 × 12) • Ratio = 144 : 252 : 144 • Divide by 12: • Ratio = 12 : 21 : 12 • Divide by 3: • Ratio = 4 : 7 : 4 • Calculate Vivek's Share: • Sum of ratio parts = 4 + 7 + 4 = 15 • Vivek's Share = ((7/15)) × 450000 • Vivek's Share = 7 × 30000 = ₹ 210000 Exam Strategy & Shortcut: Quickly simplify the investment numbers by removing the common four zeros at the very beginning (6 : 14 : 12). Then simplify the time periods if they share a common factor (24, 18, 12 share a factor of 6). Investments: 6 : 14 : 12 -> 3 : 7 : 6 Times: 4 : 3 : 2 (divided by 6) Ratio = (3 × 4) : (7 × 3) : (6 × 2) = 12 : 21 : 12 = 4 : 7 : 4. Much faster! Common Pitfall: Misreading "After 1 year" for Kirti as meaning she invested for 1 year from that point. "After 1 year" means 12 months after the start, so out of the 24 months, her money was only active for the remaining 12 months. Final Answer: Therefore, the correct answer is ₹ 210000.

Question 17

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 18

A, B and C enter into a partnership. A initially invests ₹ 25 lakhs and adds another ₹ 10 lakhs after one year. B initially invests ₹ 35 lakhs and withdraws ₹ 10 lakhs after 2 years and C invests ₹ 30 lakhs. In what ratio should the profits be divid…

Accepted Answer

Concept & Multi-Phase Investments: To find the profit ratio, break down each partner's total time into distinct periods where their capital remains constant. Calculate the product of capital and time for each period, and sum them up. Equivalent Capital = (C_1 × T_1) + (C_2 × T_2) + \dots Step-by-Step Solution: • Given: • Total business duration = 3 years • Calculate Equivalent Capital (in lakhs): • A: Invests 25 for 1 year, then adds 10 (making it 35) for the remaining 2 years. • A = (25 × 1) + (35 × 2) = 25 + 70 = 95 • B: Invests 35 for 2 years, then withdraws 10 (leaving 25) for the remaining 1 year. • B = (35 × 2) + (25 × 1) = 70 + 25 = 95 • C: Invests 30 for the full 3 years without changes. • C = (30 × 3) = 90 • Calculate Profit Ratio: • Ratio = 95 : 95 : 90 • Divide all terms by 5: • Ratio = 19 : 19 : 18 • Match Options: • Option (c) is 20 : 19 : 18, which is incorrect. • Since 19 : 19 : 18 is not listed among (a), (b), or (c), the answer is "None of these". Exam Strategy & Shortcut: Always keep the units in "lakhs" to avoid writing unnecessary zeros. Since all time intervals are exact years, use years directly instead of converting to months. This keeps calculations small and manageable. Common Pitfall: Seeing a visually similar option like (c) 20 : 19 : 18 and mistakenly selecting it under time pressure. Always trust your final derived ratio. Final Answer: Therefore, the correct answer is None of these.

Question 19

Shekhar started a business investing ₹ 25000 in 2009. In 2010, he invested an additional amount of ₹ 10000 and Rajeev joined him with an amount of ₹ 35000. In 2011, Shekhar invested another additional amount of ₹ 10000 and Jatin joined them with an …

Accepted Answer

Concept & Annual Capital Adjustments: Treat each year as a separate block for partners whose investments change annually. The total equivalent capital is the sum of the capital deployed during each consecutive year. Step-by-Step Solution: • Given: • Total duration = 3 years (2009, 2010, 2011) • Total Profit = ₹ 150000 • Calculate Equivalent Capital: • Shekhar: • 2009 (1st year): ₹ 25000 × 1 year • 2010 (2nd year): (25000 + 10000) = ₹ 35000 × 1 year • 2011 (3rd year): (35000 + 10000) = ₹ 45000 × 1 year • Shekhar Total = 25000 + 35000 + 45000 = 105000 • Rajeev: • Joins in 2010. Remains for 2 years (2010, 2011). • Rajeev Total = 35000 × 2 = 70000 • Jatin: • Joins in 2011. Remains for 1 year (2011). • Jatin Total = 35000 × 1 = 35000 • Calculate Profit Ratio: • Ratio = 105000 : 70000 : 35000 • Divide all terms by 35000: • Ratio = 3 : 2 : 1 • Calculate Rajeev's Share: • Sum of ratio parts = 3 + 2 + 1 = 6 • Rajeev's Share = ((2/6)) × 150000 • Rajeev's Share = ((1/3)) × 150000 = ₹ 50000 Exam Strategy & Shortcut: Observe the numbers: 25k, 35k, 45k. The base multiples are evident early on. Shekhar = 25 + 35 + 45 = 105 Rajeev = 35 × 2 = 70 Jatin = 35 × 1 = 35 The numbers 105, 70, and 35 are all multiples of 35. This immediate simplification gives a 3:2:1 ratio, which makes calculating shares of 150,000 trivial (divide by 6). Common Pitfall: Not adding Shekhar's additional investments cumulatively. He added 10k in 2010 (now 35k) and *another* 10k in 2011. Some might mistakenly calculate 2011 as just 25k + 10k = 35k again. Final Answer: Therefore, the correct answer is ₹ 50000.

Question 20

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.

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