Question 1

A, B and C start a business where their initial capital investments are in the ratio 2 : 3 : 4. At the end of 6 months, A increases his capital so that his total capital becomes equal to C's initial capital. If B's annual share of profit is Rs. 3,00…

Accepted Answer

Introduction / Context: In this partnership question, three partners A, B and C invest in a ratio but one partner increases his capital after 6 months. Because the capital of one partner changes midway, we must compute effective investments taking time into account. We are told B's annual profit share and must determine the total profit of the business. Given Data / Assumptions: Initial capital ratio of A : B : C = 2 : 3 : 4. Business duration is 1 year (12 months). For the first 6 months, all partners invest in the original ratio 2 : 3 : 4. At the end of 6 months, A increases his capital so that his new capital equals C's initial capital. Thus, during the last 6 months A's capital becomes equal to 4 parts, the same as C's initial capital. B keeps his capital unchanged throughout the year. B's share of the annual profit is Rs. 3,000. Profit is shared in proportion to capital * time. Concept / Approach: We treat the year in two halves of 6 months each. For each half, we record the capital ratio of A, B and C and compute the effective investment by multiplying by 6 months. After summing each partner's effective investment, we obtain the overall ratio for profit sharing. Using B's share, we can then scale the ratio to find the total profit. Step-by-Step Solution: Step 1: Let the unit capital for the ratio be k. Initially: A = 2k, B = 3k, C = 4k. Step 2: For the first 6 months, effective investments are: A = 2k * 6 = 12k, B = 3k * 6 = 18k, C = 4k * 6 = 24k. Step 3: After 6 months, A increases his capital so that it becomes equal to C's initial 4k. So for the last 6 months: A = 4k, B = 3k, C = 4k. Step 4: Effective investments for the second 6 months are: A = 4k * 6 = 24k, B = 3k * 6 = 18k, C = 4k * 6 = 24k. Step 5: Total effective investments across the year: A = 12k + 24k = 36k, B = 18k + 18k = 36k, C = 24k + 24k = 48k. Step 6: Divide by 12k to simplify the ratio: A : B : C = 3 : 3 : 4. Step 7: Total number of parts = 3 + 3 + 4 = 10. B's share is 3 parts which correspond to Rs. 3,000. Step 8: Value of 1 part = 3,000 / 3 = Rs. 1,000. Step 9: Total profit = 10 * 1,000 = Rs. 10,000. Verification / Alternative check: We can verify by computing A's and C's shares. A gets 3 parts = Rs. 3,000 and C gets 4 parts = Rs. 4,000. Adding A's, B's and C's shares gives 3,000 + 3,000 + 4,000 = 10,000, which matches the total profit we calculated. The ratio of profits 3 : 3 : 4 is consistent with the ratio of effective investments 36k : 36k : 48k. Why Other Options Are Wrong: If the total profit were Rs. 8,640, Rs. 9,850 or Rs. 11,220, B's share calculated as 3/10 of that amount would not equal Rs. 3,000. For example, 3/10 of 8,640 is 2,592, not 3,000. Therefore those options do not satisfy the condition that B's annual share is exactly Rs. 3,000. Common Pitfalls: A common error is to use only the initial ratio 2 : 3 : 4 for the whole year, ignoring the increase in A's capital after 6 months. Another mistake is to add capital ratios directly across periods instead of computing capital * time. Always break the timeline into segments whenever there is a change in capital and then sum effective investments to build the final ratio. Final Answer: The total profit earned by the business is Rs. 10,000.

Question 2

A starts a business with Rs. 3,500. After 5 months, B joins A as a partner. At the end of one year, the profit is divided between A and B in the ratio 2 : 3. What is the amount of capital invested by B?

Accepted Answer

Introduction / Context: This is a partnership question with one partner starting the business and a second partner joining later. The final profits are divided in a known ratio. We must use the relation between profit share and capital * time to determine the capital B invested when he joined after 5 months. Given Data / Assumptions: A invests Rs. 3,500 at the start of the business. After 5 months, B joins the business as a partner. The total duration of the business is 1 year, that is, 12 months. A is invested for 12 months. B is invested for 12 - 5 = 7 months. The profit at the end of the year is divided in the ratio A : B = 2 : 3. Profit is proportional to capital * time. We must find B's capital. Concept / Approach: We compute the effective investments in terms of capital * time. Since the profit ratio equals the ratio of effective investments, we set up a proportion between A's and B's effective investments and the given ratio 2 : 3. Solving this proportion gives us the value of B's capital. Step-by-Step Solution: Step 1: Effective investment of A = capital * time = 3,500 * 12 = 42,000 rupee months. Step 2: Let B's capital be x rupees. B invests for 7 months, so effective investment of B = x * 7 = 7x rupee months. Step 3: The ratio of profits A : B is given as 2 : 3, so the ratio of effective investments must also be 2 : 3. Step 4: Therefore, 42,000 : 7x = 2 : 3. Step 5: Write this as a proportion: 42,000 / 7x = 2 / 3. Step 6: Cross multiply: 42,000 * 3 = 2 * 7x. Step 7: This gives 1,26,000 = 14x, so x = 1,26,000 / 14. Step 8: Compute x: x = 9,000. Thus B's capital is Rs. 9,000. Verification / Alternative check: Using A's effective investment 42,000 and B's effective investment 7 * 9,000 = 63,000, the ratio becomes 42,000 : 63,000. Dividing by 21,000 gives 2 : 3, which matches the given profit ratio. This confirms that the computed capital of B is consistent with the partnership conditions and the final profit sharing. Why Other Options Are Wrong: If B had invested Rs. 7,500, 8,000 or 8,500, the effective investments would be 52,500, 56,000 or 59,500 respectively. None of these yield a ratio of 42,000 : 7x equal to 2 : 3. Instead, the resulting ratios would be different and would not match the given profit division ratio. Common Pitfalls: A common mistake is to divide the capitals directly in the ratio 2 : 3 and ignore the fact that B joined late and therefore invested for less time. Another error is miscalculating the number of months B's capital was actually in the business, or setting up the proportion incorrectly. Always compute capital * time for each partner and form the ratio carefully. Final Answer: B's capital contribution is Rs. 9,000.

Question 3

P and Q invest in a business with capitals of Rs. 85,000 and Rs. 15,000 respectively. If the profit earned after 2 years is divided in proportion to their investments, in what ratio will the profit be shared between P and Q?

Accepted Answer

Introduction / Context: This is a basic partnership question where two partners invest fixed capitals for the same period of time. The profit at the end of the period is shared in proportion to their capital contributions. We must find the ratio in which P and Q will share the profit, given their capital investments. Given Data / Assumptions: P invests Rs. 85,000. Q invests Rs. 15,000. The business runs for 2 years, but both partners remain invested for the entire duration. Since their time of investment is the same, profit is shared directly in the ratio of their capitals. We must find the ratio of P's profit share to Q's profit share. Concept / Approach: When the time period is equal for all partners, profit sharing depends solely on capitals. Therefore, the profit ratio is simply the ratio of the capital amounts. We write 85,000 : 15,000 and simplify by dividing both numbers by their greatest common divisor to get the simplest integer ratio. Step-by-Step Solution: Step 1: Write the capital ratio: P : Q = 85,000 : 15,000. Step 2: Find a common factor. Both numbers are divisible by 5,000. Step 3: Divide each by 5,000: 85,000 / 5,000 = 17, 15,000 / 5,000 = 3. Step 4: Therefore the simplified ratio of profits P : Q = 17 : 3. Step 5: This ratio will be used to divide any profit amount earned by the business after 2 years. Verification / Alternative check: To verify, imagine a profit of Rs. 20,000. In the ratio 17 : 3, the total parts are 20. Each part is 20,000 / 20 = 1,000. P's share would then be 17 * 1,000 = 17,000 and Q's share would be 3 * 1,000 = 3,000. The division is consistent with the capital amounts, as P has invested much more and hence gets a much larger portion of the profit. Why Other Options Are Wrong: Ratios such as 17 : 23, 17 : 33 and 3 : 4 do not match the capital ratio 85,000 : 15,000. For example, 3 : 4 would suggest Q invests more than P, which is not true. Similarly, 17 : 23 and 17 : 33 would not reduce to 85,000 : 15,000 when scaled. Therefore these options are inconsistent with the given capitals and cannot be correct. Common Pitfalls: Students may attempt to include the time period even though it is the same for both partners, which is unnecessary. Another mistake is to simplify the ratio incorrectly or to divide by different factors. Always divide both capital amounts by the same largest possible number to obtain the simplest ratio. Final Answer: The profit will be shared between P and Q in the ratio 17 : 3.

Question 4

A and B start a business with their investments in the ratio 4 : 6. After 6 months, B withdraws his entire investment and C joins with twice the amount invested by B. If the total profit at the end of the year is Rs. 3,315, what is the share of C in…

Accepted Answer

Introduction / Context: This partnership question involves three partners at different stages: A and B start the business, B withdraws after 6 months and C joins with a different capital. The profit is calculated at the end of one year. We must determine how much of the total profit of Rs. 3,315 goes to C, using the capital * time concept to compute each partner's effective investment. Given Data / Assumptions: Initial investment ratio of A : B = 4 : 6. Let the common unit be k so A = 4k and B = 6k. B withdraws his capital completely after 6 months. After 6 months, C joins with twice B's investment, so C invests 2 * 6k = 12k. The business runs for a total of 12 months. A remains invested for all 12 months. B invests for the first 6 months only. C invests for the last 6 months only. Total profit at the end of the year is Rs. 3,315. Profit is shared in proportion to capital * time. Concept / Approach: Because capital contributions change over time, we compute the effective investment for each partner as capital multiplied by the number of months the capital is in the business. These effective investments define the ratio for profit sharing. Using that ratio, we can compute C's share by applying the ratio fraction to the total profit. Step-by-Step Solution: Step 1: Effective investment of A: A invests 4k for 12 months. Effective investment of A = 4k * 12 = 48k. Step 2: Effective investment of B: B invests 6k for 6 months. Effective investment of B = 6k * 6 = 36k. Step 3: Effective investment of C: C invests 12k (twice B's capital) for the last 6 months. Effective investment of C = 12k * 6 = 72k. Step 4: Total effective investments in terms of k: A : B : C = 48k : 36k : 72k. Step 5: Simplify by dividing by 12k: A : B : C = 4 : 3 : 6. Step 6: Total ratio parts = 4 + 3 + 6 = 13. Step 7: Each part of profit = 3,315 / 13 = 255. Step 8: C's share = 6 parts = 6 * 255 = Rs. 1,530. Verification / Alternative check: We can verify by also computing A's and B's shares. A gets 4 * 255 = 1,020 and B gets 3 * 255 = 765. Adding all three shares gives 1,020 + 765 + 1,530 = 3,315, which equals the total profit. The shares of A, B and C are also proportional to 4 : 3 : 6, confirming that the effective investment method is correctly applied. Why Other Options Are Wrong: The other options 450, 1,020 and 765 correspond to 1 part, 4 parts and 3 parts respectively in the 4 : 3 : 6 ratio. They represent possible shares of one of the other partners, not of C, whose share must clearly be the largest because C's effective investment 72k is greater than those of A and B. Hence these options cannot represent C's correct share. Common Pitfalls: Learners often forget that B withdraws completely after 6 months and mistakenly treat B as investing for the full year. Another common error is to assume that C invests for 6 months but with the same capital as B, overlooking the phrase "twice the amount of B". Carefully tracking both the amount and the time of each partner's investment is essential for such questions. Final Answer: C's share of the profit is Rs. 1,530.

Question 5

A, B and C hire a car for a total charge of Rs. 520 and use it for 7, 8 and 11 hours respectively. If the hire charges are to be shared in proportion to the hours of usage, how much should B pay?

Accepted Answer

Introduction / Context: This is a proportional sharing question related to hire charges. Three people A, B and C jointly hire a car and use it for different durations. The total hire charge must be divided among them in proportion to the number of hours each person uses the car. We must calculate the amount B should pay out of the total charge of Rs. 520. Given Data / Assumptions: A uses the car for 7 hours. B uses the car for 8 hours. C uses the car for 11 hours. Total hire charge for the car is Rs. 520. Charges are shared in proportion to the hours of usage. Each person's share = total charge * (individual hours / total hours). Concept / Approach: The underlying concept is direct proportion. Since the same car is used and the charge rate per hour is assumed constant, the cost borne by each person should be directly proportional to the time that person used the car. We first find the total hours of usage and then apply the proportional formula to find B's share of the cost. Step-by-Step Solution: Step 1: Add up the total hours of use: Total hours = 7 + 8 + 11 = 26 hours. Step 2: B uses the car for 8 hours. Step 3: B's fraction of the total usage = 8 / 26. Step 4: B's share of the hire charge = total charge * usage fraction = 520 * (8 / 26). Step 5: Simplify the fraction: First, 520 / 26 = 20. Step 6: Then B's share = 20 * 8 = Rs. 160. Step 7: So B should pay Rs. 160 as his part of the hire charges. Verification / Alternative check: We can compute the individual shares of A and C to verify. A's share = 520 * (7 / 26) = 20 * 7 = Rs. 140. C's share = 520 * (11 / 26) = 20 * 11 = Rs. 220. Adding all shares gives 140 + 160 + 220 = 520, which matches the total hire charge. This confirms that the proportional allocation is correct. Why Other Options Are Wrong: If B paid Rs. 150, Rs. 170 or Rs. 180, his share would no longer be exactly proportional to his 8 hours of usage. The implied rate per hour would be inconsistent with the rates for A and C. Only Rs. 160 maintains the same per hour rate for all three users and ensures that the sum of individual contributions equals the total charge. Common Pitfalls: A common mistake is to divide the total charge equally among all three people without considering different usage times. Another mistake is miscalculating the total hours or simplifying the fraction incorrectly. Always add all the hours first, find each person's fraction and then multiply that fraction by the total cost. Final Answer: The hire charges paid by B should be Rs. 160.

Question 6

A certain amount of money is divided among three partners P, Q and R in the ratio 3 : 7 : 6 respectively. If the difference between the share of P and the share of Q is Rs. 2700, what is the share of R in the total amount?

Accepted Answer

Introduction / Context: This aptitude question is based on the concept of dividing an amount of money in a given ratio and then using the difference between two shares to find the unknown share of a third person. Such questions are common in competitive exams under the topic of partnership and ratio proportion. Given Data / Assumptions: The amount is divided among P, Q and R in the ratio 3 : 7 : 6. The difference between the share of P and the share of Q is Rs. 2700. We need to find the share of R. The total amount is not directly required for the calculation. Concept / Approach: When an amount is divided in a ratio, each share is equal to ratio part multiplied by a common multiplying factor. The difference between any two shares is equal to the difference in their ratio parts multiplied by this same factor. By using the given difference, we can first find the factor and then use it to calculate the required share of R. Step-by-Step Solution: Step 1: Let the common multiplying factor be k. Then the shares are: P = 3k, Q = 7k, R = 6k. Step 2: The difference between the share of Q and the share of P is given as Rs. 2700. So, 7k - 3k = 4k. Step 3: Set this equal to the given difference: 4k = 2700. Step 4: Find k by dividing 2700 by 4. k = 2700 / 4 = 675. Step 5: Now calculate the share of R which is 6k. R = 6 * 675 = 4050. Step 6: Therefore the share of R in the total amount is Rs. 4050. Verification / Alternative check: We can quickly verify the logic by calculating the actual shares of P and Q. P gets 3 * 675 = Rs. 2025. Q gets 7 * 675 = Rs. 4725. The difference between these two amounts is 4725 - 2025 = Rs. 2700, which matches the given difference. Since R is 6 * 675 = Rs. 4050, our calculation is consistent with the ratio and the condition in the question, so the answer is reliable. Why Other Options Are Wrong: Rs. 2121 is not a multiple of 675, so it cannot match the ratio 3 : 7 : 6. Rs. 5040 would correspond to a factor of 840 which does not satisfy the difference condition of Rs. 2700 between P and Q. Rs. 3550 is neither a multiple of 675 nor consistent with the given ratio, so it cannot be correct. Common Pitfalls: One common mistake is to assume that the difference of 2700 is between the shares of Q and R rather than between P and Q. Another error is to try to find the total amount first, which is unnecessary and can waste time. Some learners also forget that the same common factor k must be used for all three shares once the ratio is given. Working directly with the ratio difference is simpler and less error prone. Final Answer: The share of R in the total amount is Rs. 4050.

Question 7

A sum of Rs. 4300 is distributed among three persons M, N and O. M receives 25% more than N and N receives 25% less than O. What is the share of O in the total amount?

Accepted Answer

Introduction / Context: This question tests understanding of percentage relationships between different shares in a partnership type distribution. The key idea is to convert the percentage statements into algebraic expressions, form equations and then solve for the required share, in this case the share of O from a fixed total amount of Rs. 4300. Given Data / Assumptions: Total amount distributed among M, N and O is Rs. 4300. M receives 25 percent more than N. N receives 25 percent less than O. We must find the amount received by O. All amounts are assumed to be positive real numbers. Concept / Approach: Percent increase and percent decrease can be converted into multiplicative factors. If someone receives 25 percent more than another, the multiplier is 1.25. If someone receives 25 percent less than another, the multiplier is 0.75. By letting one person's share be a variable, we can express all other shares in terms of that variable and then use the total sum to find exact numerical values. Step-by-Step Solution: Step 1: Let the share of N be N rupees. Step 2: M receives 25 percent more than N, so M = 1.25 * N = 5N / 4. Step 3: N receives 25 percent less than O, so N = 0.75 * O = 3O / 4. Step 4: From N = 3O / 4, express O in terms of N: O = 4N / 3. Step 5: Total amount is M + N + O = 4300. Step 6: Substitute M and O in terms of N: 5N / 4 + N + 4N / 3 = 4300. Step 7: Convert to a common denominator of 12: (15N + 12N + 16N) / 12 = 4300. Step 8: This gives 43N / 12 = 4300, so 43N = 4300 * 12 = 51600. Step 9: Therefore N = 51600 / 43 = 1200. Step 10: Now find O using O = 4N / 3 = 4 * 1200 / 3 = 1600. Verification / Alternative check: Using N = Rs. 1200 and O = Rs. 1600, M is 25 percent more than N. So M = 1.25 * 1200 = 1500. Check the total: M + N + O = 1500 + 1200 + 1600 = 4300, which matches the given sum. Also, N is 25 percent less than O because 1200 is 75 percent of 1600. This confirms that the calculations follow the percentage relationships correctly. Why Other Options Are Wrong: Rs. 1564 and Rs. 1632 do not give integer values for the other shares that add up to 4300 under the given percentage relationships. Rs. 1714 also fails to satisfy both the 25 percent more and 25 percent less conditions when back calculated. Only Rs. 1600 keeps all three values consistent with the data and the total sum. Common Pitfalls: Learners sometimes confuse which person receives more or less and may invert the percentage multipliers. Another mistake is to mix up 25 percent less with dividing by 1.25 instead of multiplying by 0.75. It is safer to work with clear algebraic expressions and a common denominator when adding fractional expressions, as done here with denominator 12. Final Answer: The share of O in the total amount is Rs. 1600.

Question 8

Three partners share the profit of a business in the ratio 4 : 6 : 8. They remain in the business for 12 months, 8 months and 6 months respectively. What was the ratio of their original investments?

Accepted Answer

Introduction / Context: This question checks your understanding of how partners share profit in a business when their capitals are invested for different periods of time. The key idea is that profit sharing depends on the product of capital and time, not only on the capital or the time separately. Given Data / Assumptions: The profit sharing ratio of three partners is 4 : 6 : 8. The time periods of investment are 12 months, 8 months and 6 months respectively. We have to find the ratio of their original investments. The total profit amount is not required for the solution. Concept / Approach: In partnership problems, each partner's share in profit is proportional to capital * time. If we know the final profit ratio and the time for each partner, we can reverse the process and find the investment ratio by dividing the profit ratio by the corresponding time periods. This effectively cancels out the effect of time and leaves only the ratio of capitals. Step-by-Step Solution: Step 1: Let the original investments of the three partners be I1, I2 and I3. Step 2: Their time periods are T1 = 12 months, T2 = 8 months and T3 = 6 months. Step 3: Since profit is proportional to capital * time, we have I1 * 12 : I2 * 8 : I3 * 6 = 4 : 6 : 8. Step 4: To find the investment ratio, divide each profit ratio term by the corresponding time. Step 5: So, I1 : I2 : I3 = 4 / 12 : 6 / 8 : 8 / 6. Step 6: Simplify each fraction: 4 / 12 = 1 / 3, 6 / 8 = 3 / 4, and 8 / 6 = 4 / 3. Step 7: Now express 1 / 3 : 3 / 4 : 4 / 3 in whole number form by taking the common denominator 12. Step 8: Multiply each term by 12: (1 / 3)*12 = 4, (3 / 4)*12 = 9, (4 / 3)*12 = 16. Step 9: Therefore, the ratio of their investments is 4 : 9 : 16. Verification / Alternative check: We can verify by checking that using 4, 9 and 16 as investment amounts with the given times gives back the profit ratio 4 : 6 : 8. Capital time products are 4 * 12 = 48, 9 * 8 = 72 and 16 * 6 = 96. The ratio 48 : 72 : 96 simplifies by dividing all terms by 12 to 4 : 6 : 8, which matches the profit ratio from the question. This confirms that the investment ratio 4 : 9 : 16 is correct. Why Other Options Are Wrong: 4 : 6 : 8 simply repeats the profit ratio and ignores the unequal time periods, so it cannot be the investment ratio. 4 : 7 : 8 and 4 : 8 : 16 do not produce the required profit ratio when multiplied by the respective time periods. Only 4 : 9 : 16 gives the correct final profit ratio of 4 : 6 : 8. Common Pitfalls: A frequent mistake is to equate the investment ratio directly to the profit ratio without considering the different times for which the capitals were invested. Another error is to forget to use a common multiple when converting fractional ratios to whole numbers, which can lead to incorrect simplification. Always remember that in partnership questions, capital time products determine the true ratio of profit sharing. Final Answer: The ratio of their investments is 4 : 9 : 16.

Question 9

Sandeep received Rs. 6000 as his share out of a total profit of Rs. 9000 that he and Saketh earned at the end of one year. If Sandeep invested Rs. 20,000 for 6 months and Saketh kept his investment for the whole year, what was the amount invested by…

Accepted Answer

Introduction / Context: This partnership problem asks you to work backwards from known profit shares and time periods to find an unknown capital investment. It connects ratio of profit, time of investment and the amount of capital each partner contributes to the business. Given Data / Assumptions: Total profit at the end of the year is Rs. 9000. Sandeep receives Rs. 6000 from this profit. Therefore Saketh receives Rs. 3000. Sandeep invested Rs. 20,000 for 6 months. Saketh invested some amount for the full 12 months. We must find the amount invested by Saketh. Concept / Approach: In partnerships, profit sharing is proportional to the product of capital and time. If one partner invests capital A1 for T1 months and another invests capital A2 for T2 months, then their profit shares are in the ratio A1 * T1 : A2 * T2. Here we know the profit ratio and one capital value with time, so we can find the unknown capital by forming a proportion and solving for the missing quantity. Step-by-Step Solution: Step 1: Determine the ratio of profit shares. Sandeep's share = Rs. 6000 and Saketh's share = Rs. 3000. Step 2: Profit ratio Sandeep : Saketh = 6000 : 3000 = 2 : 1. Step 3: Compute Sandeep's capital time product: 20,000 * 6 = 1,20,000. Step 4: Let Saketh's investment be x rupees for the full 12 months. Step 5: His capital time product is x * 12 = 12x. Step 6: Set up the proportion using capital time products: 1,20,000 : 12x = 2 : 1. Step 7: Cross multiply: 1,20,000 * 1 = 2 * 12x, which gives 1,20,000 = 24x. Step 8: Solve for x: x = 1,20,000 / 24 = 5000. Step 9: Therefore, Saketh invested Rs. 5000 for the whole year. Verification / Alternative check: If Saketh invested Rs. 5000 for 12 months, his capital time product is 5000 * 12 = 60,000. Sandeep's capital time product is 20,000 * 6 = 1,20,000. The ratio of their capital time products is 1,20,000 : 60,000 = 2 : 1, which matches the ratio of their profit shares 6000 : 3000. This confirms that the investment of Rs. 5000 by Saketh is correct. Why Other Options Are Wrong: Rs. 6200, Rs. 7600 and Rs. 8100 would produce capital time ratios that do not simplify to 2 : 1 when compared with Sandeep's product of 1,20,000. Therefore, none of these alternatives are consistent with the given profit distribution and time periods. Common Pitfalls: A common mistake is to compare only the capital amounts and ignore the time for which each amount is invested. Another error is to assume that larger capital always gets larger profit without checking the proportional relationship with time. Always multiply capital by time first to get a fair comparison in partnership questions. Final Answer: The amount invested by Saketh is Rs. 5000.

Question 10

In a business, A and C invested amounts in the ratio 2 : 1, while the ratio between the amounts invested by A and B was 3 : 2. If the total profit from the business is Rs. 1,57,300, how much amount does B receive as his share of the profit?

Accepted Answer

Introduction / Context: This is a partnership question where the ratios between investments are given indirectly. Instead of direct amounts, we know how A relates to B and C. The goal is to determine each partner's relative capital contribution and then use the total profit to compute the exact amount received by B. Given Data / Assumptions: The ratio of investments A : C is 2 : 1. The ratio of investments A : B is 3 : 2. Total profit from the business is Rs. 1,57,300. We need to find B's share of this profit. All partners remain in the business for the same period, so only capital ratios matter. Concept / Approach: When multiple ratios involving the same person are given, we can combine them to get a single three person ratio. The standard method is to express the common person's investment in a way that matches in both ratios and then deduce the other amounts. Once we know A : B : C, profit is distributed in that same ratio, because time of investment is equal for all partners. Step-by-Step Solution: Step 1: Let investments be A, B and C. Step 2: From A : C = 2 : 1, we can write A = 2k and C = k for some k. Step 3: From A : B = 3 : 2, we can write A = 3m and B = 2m for some m. Step 4: Since both expressions represent A, set 2k = 3m. Step 5: Choose convenient values that satisfy this, for example k = 3 and m = 2, giving A = 6. Step 6: Then B = 2m = 4 and C = k = 3. So A : B : C = 6 : 4 : 3. Step 7: Sum of ratio parts = 6 + 4 + 3 = 13. Step 8: B's fractional share of profit = 4 / 13. Step 9: Total profit is Rs. 1,57,300, so B's share = (4 / 13) * 1,57,300. Step 10: Compute this: 1,57,300 / 13 = 12,100 and 12,100 * 4 = 48,400. Verification / Alternative check: If B receives Rs. 48,400, then each ratio part is worth 1,57,300 / 13 = Rs. 12,100. A receives 6 parts = 6 * 12,100 = Rs. 72,600. C receives 3 parts = 3 * 12,100 = Rs. 36,300. Adding all three shares gives 72,600 + 48,400 + 36,300 = Rs. 1,57,300, which matches the total profit stated in the question. The ratio between their shares is 72,600 : 48,400 : 36,300 which simplifies to 6 : 4 : 3, consistent with our derived investment ratio. Why Other Options Are Wrong: Rs. 54,200, Rs. 64,000 and Rs. 74,000 do not correspond to the correct fraction 4 / 13 of the total profit. Using any of these values breaks the total profit sum or the A : B : C ratio, so they cannot be correct. Common Pitfalls: A frequent error is to try to combine the ratios by simple addition or subtraction instead of matching the common term. Another common mistake is to forget to include the third partner when distributing profit, which leads to wrong denominators for the fractions. Always derive a clean three term ratio first and then apply it to the total profit. Final Answer: The amount received by B as his share of profit is Rs. 48,400.

Question 11

A total amount of Rs. 700 is divided among A, B and C such that A receives half as much as B, and B receives half as much as C. What is the share of C?

Accepted Answer

Introduction / Context: This question checks your understanding of chained ratio relationships where each person's share is defined relative to another person. Instead of a direct ratio, we are given comparative statements, and we must convert them into an algebraic form to determine the individual shares from the total amount. Given Data / Assumptions: Total amount shared by A, B and C is Rs. 700. A receives half as much as B, so A is smaller than B. B receives half as much as C, so B is smaller than C. We must calculate C's share. Concept / Approach: When one quantity is half of another, we can express both using a single variable. Here, the relations A = B / 2 and B = C / 2 allow us to express all three shares in terms of C. Once every share is written in terms of C, we can use the total sum equation A + B + C = 700 to solve for C directly. Step-by-Step Solution: Step 1: Let the share of C be C rupees. Step 2: B receives half as much as C, so B = C / 2. Step 3: A receives half as much as B, so A = B / 2 = (C / 2) / 2 = C / 4. Step 4: Total amount is A + B + C = 700. Step 5: Substitute A and B in terms of C: C / 4 + C / 2 + C = 700. Step 6: Convert the fractions to a common denominator of 4: C / 4 + 2C / 4 + 4C / 4 = 700. Step 7: Combine numerators: (1C + 2C + 4C) / 4 = 7C / 4 = 700. Step 8: Multiply both sides by 4: 7C = 700 * 4 = 2800. Step 9: Divide by 7: C = 2800 / 7 = 400. Step 10: Therefore, C's share is Rs. 400. Verification / Alternative check: With C = Rs. 400, B is half of C, so B = 400 / 2 = 200. A is half of B, so A = 200 / 2 = 100. Check the total: A + B + C = 100 + 200 + 400 = 700, which matches the given total amount. Both comparison conditions are satisfied, so the value for C is consistent and correct. Why Other Options Are Wrong: Rs. 200 and Rs. 300 do not satisfy the chain of half relationships when we back compute A and B, and they do not sum to Rs. 700 with the other derived shares. Rs. 500 gives wrong totals and breaks the condition that B is half of C and A is half of B. Only Rs. 400 leads to a valid distribution of 100, 200 and 400 adding up to 700. Common Pitfalls: Students often reverse the relationships and treat C as half of B or B as half of A, which leads to incorrect equations. Another pitfall is to introduce unnecessary extra variables instead of expressing everything in terms of a single variable. Carefully reading the phrases "half as much as" and then converting them to clear equations helps to avoid confusion. Final Answer: The share of C is Rs. 400.

Question 12

A and B invest Rs. 1,16,000 and Rs. 1,44,000 respectively in a business. B keeps his capital invested for 6 months, while A invests for the full year. At the end of the year, B's share of profit is Rs. 9000. What is the total profit earned by the bu…

Accepted Answer

Introduction / Context: This partnership problem involves different capitals and different time periods of investment. The question gives the profit share of one partner and asks for the total profit. To solve this, we use the idea that profit shares are proportional to the products of capital and time for each partner. Given Data / Assumptions: A invests Rs. 1,16,000 for 12 months. B invests Rs. 1,44,000 for 6 months. B's share of profit at the end of the year is Rs. 9000. We need to find the total profit earned in the business. Concept / Approach: In partnership questions, each partner's share is proportional to capital * time. That means we first calculate the effective capital time products and then express them as a ratio. Knowing one partner's actual rupee share along with the ratio enables us to scale up and find the total profit. This avoids any need to know the exact total capital. Step-by-Step Solution: Step 1: Calculate A's capital time product: 1,16,000 * 12 = 13,92,000. Step 2: Calculate B's capital time product: 1,44,000 * 6 = 8,64,000. Step 3: The ratio of A's share to B's share is 13,92,000 : 8,64,000. Step 4: Simplify this ratio by dividing both terms by 48,000, giving 29 : 18. Step 5: So the profit ratio A : B is 29 : 18. Step 6: Total number of parts in the ratio is 29 + 18 = 47. Step 7: B gets 18 parts out of 47, and we know these 18 parts are worth Rs. 9000. Step 8: Value of one part = 9000 / 18 = 500. Step 9: Total profit = 47 * 500 = Rs. 23,500. Verification / Alternative check: If the total profit is Rs. 23,500 and one part is Rs. 500, then A's share is 29 parts = 29 * 500 = Rs. 14,500. B's share is 18 parts = 18 * 500 = Rs. 9000, which matches the given value. Adding the two shares, 14,500 + 9000 = 23,500, confirms that the total profit is consistent with the derived ratio and the information provided in the question. Why Other Options Are Wrong: Rs. 19,780, Rs. 21,700 and Rs. 20,050 do not yield a per part value that gives B's share exactly equal to Rs. 9000 while maintaining the ratio 29 : 18. When tested, they either give non integer values per part or give a B share different from Rs. 9000, so they must be rejected. Common Pitfalls: A usual mistake is to ignore the time periods and compare only capitals, which gives a wrong ratio. Another error is to miscalculate the simplification of the ratio or to forget to add the ratio parts before scaling to the total profit. Always compute capital time products first, then simplify the ratio carefully and finally convert ratio parts into money values. Final Answer: The total profit earned by the business is Rs. 23,500.

Question 13

A and B begin a business with investments of Rs. 3000 and Rs. 4000 respectively. After 8 months, A withdraws Rs. 1000 from his capital and B increases his capital by Rs. 1000. At the end of one year, their total profit is Rs. 630. What is A's share …

Accepted Answer

Introduction / Context: This question involves a partnership where both partners change their investments during the year. The concept tested is the calculation of effective capital time contributions for each partner and then using that to divide a known total profit between them. Given Data / Assumptions: A initially invests Rs. 3000. B initially invests Rs. 4000. After 8 months, A withdraws Rs. 1000, so his capital becomes Rs. 2000. After 8 months, B adds Rs. 1000, so his capital becomes Rs. 5000. Total business duration is 12 months. Total profit at the end of the year is Rs. 630. We must find A's share in this profit. Concept / Approach: When capital changes during the partnership, we split the time into segments where the capital remains constant and calculate capital time products for each segment. We then add these products for each partner to get their effective investments. The ratio of these effective investments gives the ratio of profit shares, which is applied to the total profit to find individual amounts. Step-by-Step Solution: Step 1: For A, from month 0 to month 8, capital is Rs. 3000. Step 2: Capital time for this period is 3000 * 8 = 24,000. Step 3: From month 8 to month 12, A's capital is Rs. 2000. Step 4: Capital time for this period is 2000 * 4 = 8000. Step 5: Total effective capital time for A is 24,000 + 8000 = 32,000. Step 6: For B, from month 0 to month 8, capital is Rs. 4000. Step 7: Capital time for this period is 4000 * 8 = 32,000. Step 8: From month 8 to month 12, B's capital is Rs. 5000. Step 9: Capital time for this period is 5000 * 4 = 20,000. Step 10: Total effective capital time for B is 32,000 + 20,000 = 52,000. Step 11: Ratio of A : B in capital time is 32,000 : 52,000 = 8 : 13. Step 12: Sum of ratio parts = 8 + 13 = 21. Step 13: A's share of profit = (8 / 21) * 630 = 240. Verification / Alternative check: If A gets Rs. 240, then B gets the remaining profit 630 - 240 = Rs. 390. Check the ratio: 240 : 390 simplifies by dividing by 30 to 8 : 13, which matches the capital time ratio 32,000 : 52,000. Therefore the distribution of profit is consistent with the investments and time periods, confirming that A's share is correct. Why Other Options Are Wrong: Rs. 75, Rs. 125 and Rs. 354 do not maintain the required 8 : 13 ratio when compared to B's share. Using any of these values leads either to a different ratio or to a total different from Rs. 630, so they cannot represent A's correct share of profit. Common Pitfalls: It is easy to forget to split the year into two parts and instead use only the original capitals for the whole 12 months. Another frequent error is to change the capital amounts but not adjust the months correctly when forming the capital time products. Always handle each time segment separately and then combine them carefully for each partner. Final Answer: A's share of the profit is Rs. 240.

Question 14

Rs. 1200 is divided among P, Q and R. P gets half of the total amount received by Q and R together, and Q gets one third of the total amount received by P and R together. What is the amount received by R?

Accepted Answer

Introduction / Context: This partnership style question uses relational statements between the shares of three people instead of simple ratios. The aim is to convert these relationships into equations, use the total amount, and solve systematically for the share of R. Given Data / Assumptions: Total amount shared among P, Q and R is Rs. 1200. P gets half of the combined amount received by Q and R. Q gets one third of the combined amount received by P and R. We must find R's share. Concept / Approach: When we have conditions like one person getting a fraction of the sum of the other two, we can represent each person's share with variables and set up equations. Solving a system of three equations gives exact values for P, Q and R. The key idea is to keep the expressions simple and combine them carefully, using substitution or simultaneous solving. Step-by-Step Solution: Step 1: Let the shares of P, Q and R be P, Q and R respectively in rupees. Step 2: Total amount condition: P + Q + R = 1200. Step 3: P gets half of the total received by Q and R, so P = 1/2 * (Q + R). Step 4: Q gets one third of the total received by P and R, so Q = 1/3 * (P + R). Step 5: Substitute P from Step 3 into the equation in Step 4 to express Q in terms of Q and R. However, it is more efficient to solve these three equations together. Step 6: From P = 1/2 * (Q + R), we can write 2P = Q + R. Step 7: From Q = 1/3 * (P + R), we can write 3Q = P + R. Step 8: Now use P + Q + R = 1200 along with 2P = Q + R and 3Q = P + R to solve. Solving this system gives P = 400, Q = 300 and R = 500. Step 9: Therefore, the amount received by R is Rs. 500. Verification / Alternative check: Check P's condition: Q + R = 300 + 500 = 800. Half of this is 800 / 2 = 400, which matches P's share. Check Q's condition: P + R = 400 + 500 = 900. One third of this is 900 / 3 = 300, which equals Q's share. The total P + Q + R = 400 + 300 + 500 = 1200, which matches the given total. All conditions are satisfied, so the values are correct. Why Other Options Are Wrong: Rs. 1100 and Rs. 1200 cannot be R's share because they are larger than or equal to the total or would leave negative or zero amounts for the others. Rs. 700 does not satisfy both fractional conditions when we back compute P and Q. Only Rs. 500 fits all three equations simultaneously. Common Pitfalls: Learners sometimes misinterpret the phrases and treat P as half of Q or R individually instead of half of the sum Q + R. Another mistake is to plug in approximate values without solving the equations exactly, which can quickly lead to inconsistent totals. The safest method is to write clear equations for all three statements and solve them systematically. Final Answer: The amount received by R is Rs. 500.

Question 15

Rohith starts a business by investing Rs. 3600. After some months, Puneeth joins him by investing Rs. 2400. At the end of one year, the profit is divided in the ratio 2 : 1 between Rohith and Puneeth. After how many months from the start did Puneeth…

Accepted Answer

Introduction / Context: This is a classic partnership question where one partner starts the business and another joins later. The profit ratio at the end of the year is known, and we must use this to determine the time for which the second partner was involved in the business. Given Data / Assumptions: Rohith invests Rs. 3600 at the start. Puneeth invests Rs. 2400 after some months and keeps it invested until the end of the year. Total business period is 12 months. The final profit ratio Rohith : Puneeth is 2 : 1. We must find after how many months Puneeth joined, that is, how long after Rohith started. Concept / Approach: Profit sharing in partnership is proportional to the products of capital and time. If one partner invests longer, he can receive a greater share even with similar capital. By letting the second partner's time be a variable, we can form a ratio equation equating capital time products to the known profit ratio and solve for the unknown number of months. Step-by-Step Solution: Step 1: Rohith invests Rs. 3600 for the full 12 months. Step 2: His capital time product = 3600 * 12 = 43,200. Step 3: Let Puneeth invest Rs. 2400 for x months. Step 4: Puneeth's capital time product = 2400 * x. Step 5: The profit ratio Rohith : Puneeth is 2 : 1, so capital time ratio must be 2 : 1. Step 6: Set up the proportion: 43,200 : 2400x = 2 : 1. Step 7: Cross multiply: 43,200 * 1 = 2 * 2400x, so 43,200 = 4800x. Step 8: Solve for x: x = 43,200 / 4800 = 9 months. Step 9: Puneeth invests for 9 months, so he must have joined after 12 - 9 = 3 months from the start. Verification / Alternative check: If Puneeth joins after 3 months, he invests for 9 months. Then his capital time product is 2400 * 9 = 21,600. Rohith's product is 43,200 as computed earlier. The ratio 43,200 : 21,600 simplifies by dividing both terms by 21,600 to 2 : 1, which matches the given profit ratio. This confirms that the joining time of 3 months is consistent with the profit division. Why Other Options Are Wrong: If Puneeth joined after 2, 2.5 or 3.5 months, his investment duration would be 10, 9.5 or 8.5 months respectively. These would give capital time products that do not produce the exact ratio 2 : 1 when compared with Rohith's 43,200. Therefore those joining times do not satisfy the required profit ratio. Common Pitfalls: One common error is to use the joining time directly as the variable in the proportion instead of using the actual duration of investment. Another mistake is to swap the ratio order, using 1 : 2 instead of 2 : 1 and ending up with the reciprocal of the true duration. Always define the variable clearly as the number of months of investment for the later partner, then convert to joining time at the end. Final Answer: Puneeth joined the business after 3 months from the start.

Question 16

A and B invest Rs. 10,000 each in a business. A keeps his capital for 8 months, while B keeps his capital invested for the full 12 months. If the total profit at the end of the year is Rs. 25,000, what are the respective shares of A and B?

Accepted Answer

Introduction / Context: This is a straightforward partnership problem where two partners invest the same amount but for different periods of time. The task is to find their individual shares in the final profit by using the idea of capital time products and proportional distribution. Given Data / Assumptions: A invests Rs. 10,000 for 8 months. B invests Rs. 10,000 for 12 months. Total profit at the end of the year is Rs. 25,000. We must find the profit shares of A and B. Concept / Approach: In partnership questions, profit is divided in proportion to the product of capital and time for each partner. Since both partners have equal capital but for different durations, the ratio of their shares will simply match the ratio of their investment periods. Once we know the ratio, we split the total profit according to that ratio. Step-by-Step Solution: Step 1: Calculate A's capital time product: 10,000 * 8 = 80,000. Step 2: Calculate B's capital time product: 10,000 * 12 = 1,20,000. Step 3: Ratio of A : B = 80,000 : 1,20,000. Step 4: Simplify this ratio by dividing both sides by 40,000 to get 2 : 3. Step 5: Total number of ratio parts = 2 + 3 = 5. Step 6: Total profit is Rs. 25,000, so each part is 25,000 / 5 = Rs. 5000. Step 7: A's share = 2 parts = 2 * 5000 = Rs. 10,000. Step 8: B's share = 3 parts = 3 * 5000 = Rs. 15,000. Verification / Alternative check: If A receives Rs. 10,000 and B receives Rs. 15,000, their total is 10,000 + 15,000 = Rs. 25,000, which equals the given total profit. The profit ratio 10,000 : 15,000 simplifies to 2 : 3, which matches the capital time ratio 80,000 : 1,20,000. This confirms that the distribution is correct and consistent with the partnership principle. Why Other Options Are Wrong: The pairs 15,000 and 10,000, 5000 and 20,000, and 20,000 and 5000 either reverse the ratio or produce totals that do not match Rs. 25,000 when combined, or do not satisfy the required 2 : 3 ratio from the capital time products. Therefore they cannot represent the correct distribution of profit. Common Pitfalls: One common mistake is to think that equal capital means equal profit, ignoring the fact that one partner invested for a longer period. Another error is to divide the profit only on the basis of time without checking the actual ratio correctly. Always multiply capital by time for each partner first, simplify the resulting ratio, and only then split the profit. Final Answer: The respective shares of A and B are Rs. 10,000 and Rs. 15,000.

Question 17

Three friends have dinner at a restaurant. When the bill arrives, Akshitha pays two thirds as much as Veena pays, and Veena pays half as much as Lasya pays. What fraction of the total bill is paid by Veena?

Accepted Answer

Introduction / Context: This is a ratio and fraction based aptitude question framed around sharing a restaurant bill. The goal is to use the given relationships between individual payments to express each person's contribution as a fraction of the total amount and then determine Veena's share as a single fraction. Given Data / Assumptions: Three friends are Akshitha, Veena and Lasya. Akshitha pays two thirds as much as Veena pays. Veena pays half as much as Lasya pays. The total bill is the sum of the amounts paid by all three friends. We must find the fraction of the total bill paid by Veena. Concept / Approach: We translate each comparative statement into an algebraic relation using variables. Then we express all payments in terms of a single base quantity, usually the largest amount. After that, we form the total bill and compute the fraction that corresponds to Veena's share. This type of structured algebra makes it easy to simplify the fraction correctly. Step-by-Step Solution: Step 1: Let Lasya's payment be L rupees. Step 2: Veena pays half as much as Lasya, so Veena's payment V = L / 2. Step 3: Akshitha pays two thirds as much as Veena, so Akshitha's payment A = (2 / 3) * V. Step 4: Substitute V = L / 2 into A: A = (2 / 3) * (L / 2) = L / 3. Step 5: Now express the total bill T as T = A + V + L. Step 6: Substitute A and V in terms of L: T = L / 3 + L / 2 + L. Step 7: Use a common denominator of 6: L / 3 = 2L / 6, L / 2 = 3L / 6, and L = 6L / 6. Step 8: So T = 2L / 6 + 3L / 6 + 6L / 6 = 11L / 6. Step 9: Veena's share V is L / 2, but we want this as a fraction of T = 11L / 6. Step 10: Fraction paid by Veena = V / T = (L / 2) / (11L / 6). Step 11: Simplify this: (L / 2) * (6 / 11L) = (6L) / (22L) = 6 / 22 = 3 / 11. Verification / Alternative check: We can choose a convenient value for L to verify. Let Lasya pay Rs. 22. Then Veena pays half of that, which is Rs. 11. Akshitha pays two thirds of Veena's amount, which is 2 / 3 * 11 ≈ 7.33. The total bill is 22 + 11 + 7.33 = 40.33. The fraction Veena pays is 11 / 40.33, which is very close to 0.2727. The fraction 3 / 11 is approximately 0.2727 as well, so the theoretical fraction matches this numerical check when we ignore rounding of Akshitha's share in rupees. In exact fractional form, the algebra gives a precise answer of 3 / 11. Why Other Options Are Wrong: 11 / 3 and 13 / 4 are greater than 1, so they cannot represent a fraction of a total bill. 2 / 13 is much smaller than 3 / 11 and does not match the algebraic calculation. Therefore it is also incorrect. Common Pitfalls: Learners may misread phrases like "two thirds as much as" and mistakenly treat them as "two thirds of the remaining bill". Another mistake is to forget to compute the fraction with respect to the total bill, instead of just comparing with one person's contribution. Always define variables clearly and form the ratio of the required amount to the total sum. Final Answer: Veena pays 3/11 of the total bill.

Question 18

Three milkmen rent a pasture. A grazes 24 cows for 4 months, B grazes 10 cows for 6 months, and C grazes 56 cows for 5 months. If A's share of the rent is Rs. 960, what is the total rent of the field?

Accepted Answer

Introduction / Context: This is a typical cows and pasture problem where rent is shared in proportion to the total grazing done by each person. Grazing is measured in terms of the product of the number of animals and the time they graze. From one person's known share of rent, we are asked to find the total rent of the pasture. Given Data / Assumptions: A grazes 24 cows for 4 months. B grazes 10 cows for 6 months. C grazes 56 cows for 5 months. A's share of rent is Rs. 960. Rent is shared in proportion to cows * months for each milkman. We must find the total rent of the field. Concept / Approach: The cost of using a common pasture is naturally proportional to the total grazing pressure applied, which is number of cows multiplied by the number of months. First we compute this product for each milkman, then form a ratio. Since we know A's rupee share and his part in the ratio, we can find the value of one ratio unit and from there compute the full rent. Step-by-Step Solution: Step 1: Compute A's grazing units: 24 cows * 4 months = 96 cow months. Step 2: Compute B's grazing units: 10 cows * 6 months = 60 cow months. Step 3: Compute C's grazing units: 56 cows * 5 months = 280 cow months. Step 4: Total grazing units = 96 + 60 + 280 = 436 cow months. Step 5: The ratio A : B : C is 96 : 60 : 280. Step 6: Sum of ratio parts is 436, matching the total cow months. Step 7: A's share of rent is Rs. 960, which corresponds to 96 parts. Step 8: Value of one part = 960 / 96 = Rs. 10. Step 9: Total rent = 436 parts * Rs. 10 per part = Rs. 4360. Verification / Alternative check: If the total rent is Rs. 4360 and each cow month is Rs. 10, then A pays 96 * 10 = Rs. 960, which matches the given information. B pays 60 * 10 = Rs. 600 and C pays 280 * 10 = Rs. 2800. Adding these gives 960 + 600 + 2800 = Rs. 4360, which is consistent with the total rent and the proportional distribution based on grazing units. Why Other Options Are Wrong: Rs. 4530, Rs. 4440 and Rs. 4280 do not give a per unit rent of exactly Rs. 10 when divided by 436, and they do not reproduce A's share as Rs. 960 when multiplied by A's 96 cow months. Therefore they are inconsistent with the given information and the proportional method and must be rejected. Common Pitfalls: A common mistake is to compare only the number of cows and ignore the different grazing durations. Another error is to attempt to use percentages instead of direct proportionality based on cow months. Always remember that in such problems, both the number of animals and the length of time they graze contribute equally to the cost share. Final Answer: The total rent of the field is Rs. 4360.

Question 19

P, Q and R enter into a partnership in the ratio 2 : 3 : 5. After 2 months, P increases his investment by 20% and Q increases his investment by 10%. If the total profit at the end of one year is Rs. 1,90,500, what amount does R receive as his share …

Accepted Answer

Introduction / Context: This partnership question involves three partners with an initial investment ratio, followed by changes in the investments of two partners after a certain period. We must account for both the capital amounts and the time periods to determine the effective share of each partner and then compute R's share from the total profit. Given Data / Assumptions: Initial investment ratio of P : Q : R is 2 : 3 : 5. After 2 months, P increases his investment by 20 percent. After 2 months, Q increases his investment by 10 percent. R keeps his investment unchanged for the whole year. Total duration is 12 months. Total profit at the end of the year is Rs. 1,90,500. We must find R's share of the profit. Concept / Approach: When investments change over time, we calculate effective capital time products for each segment. Investment shares in the final profit are proportional to the sum of these products. We first convert the percentage increases into new ratios for P and Q after 2 months, then combine the contributions over the full year to obtain an overall ratio including R. Finally we apply this ratio to the total profit to find R's share. Step-by-Step Solution: Step 1: Let the initial investments be 2x, 3x and 5x for P, Q and R. Step 2: For the first 2 months, investments are 2x, 3x and 5x. Step 3: After 2 months, P increases his investment by 20 percent, so new P investment is 2x * 1.2 = 2.4x. Step 4: After 2 months, Q increases his investment by 10 percent, so new Q investment is 3x * 1.1 = 3.3x. Step 5: R continues with 5x for the entire 12 months. Step 6: Capital time products for first 2 months: P = 2x * 2 = 4x, Q = 3x * 2 = 6x, R = 5x * 2 = 10x. Step 7: Capital time products for remaining 10 months: P = 2.4x * 10 = 24x, Q = 3.3x * 10 = 33x, R = 5x * 10 = 50x. Step 8: Total effective products: P = 4x + 24x = 28x, Q = 6x + 33x = 39x, R = 10x + 50x = 60x. Step 9: Overall ratio P : Q : R = 28 : 39 : 60. Step 10: Sum of ratio parts = 28 + 39 + 60 = 127. Step 11: R's fraction of the profit = 60 / 127. Step 12: Total profit is Rs. 1,90,500, so R's share = (60 / 127) * 1,90,500 = Rs. 90,000. Verification / Alternative check: Check that 1,90,500 multiplied by 60 and divided by 127 gives exactly 90,000. Now find the value of one ratio part: 1,90,500 / 127 = 1500. Then P receives 28 * 1500 = 42,000, Q receives 39 * 1500 = 58,500 and R receives 60 * 1500 = 90,000. Adding these shares gives 42,000 + 58,500 + 90,000 = 1,90,500, which matches the total profit and confirms that the ratio and R's share are correct. Why Other Options Are Wrong: Rs. 90,500, Rs. 87,500 and Rs. 88,900 do not correspond to 60 parts out of 127 when scaled to 1,90,500. They either break the total sum or produce non integer values for the other partners. Therefore, they are inconsistent with the calculated ratio and must be rejected. Common Pitfalls: A frequent mistake is to apply the percentage increases to the total profit instead of the capital. Another error is to assume that the original 2 : 3 : 5 ratio remains unchanged after the investment changes, which is not true. Always break the year into segments, compute capital time products for each segment and then add them for every partner before forming the final ratio. Final Answer: R receives Rs. 90,000 as his share of the profit.

Question 20

Karan starts a business by investing Rs. 29,000. After 5 months, Satish joins the business with a capital of Rs. 18,000. If at the end of the year the total profit is Rs. 16,970, what is Satish's share in the profit?

Accepted Answer

Introduction / Context: This question involves two partners where one invests from the beginning and the other joins later. The aim is to find the second partner's share of the profit using the concept of capital time products and proportional distribution of the total profit. Given Data / Assumptions: Karan invests Rs. 29,000 for the full 12 months. Satish invests Rs. 18,000, joining the business after 5 months. Satish's capital remains invested for the rest of the year. Total profit at the end of the year is Rs. 16,970. We must find Satish's share of this profit. Concept / Approach: In partnership problems, profit sharing is proportional to capital multiplied by time. When a partner joins late, his effective investment is smaller even if his capital is significant. We calculate capital time products for both partners, form a ratio, and then apply that ratio to the known total profit to find individual shares. Sometimes this leads to fractional values that are rounded to the nearest rupee in multiple choice options. Step-by-Step Solution: Step 1: Karan invests Rs. 29,000 for 12 months. Step 2: Karan's capital time product = 29,000 * 12 = 3,48,000. Step 3: Satish joins after 5 months, so he invests for 7 months. Step 4: Satish's capital time product = 18,000 * 7 = 1,26,000. Step 5: The ratio Karan : Satish based on capital time is 3,48,000 : 1,26,000. Step 6: Simplify this ratio by dividing both terms by 6000: 3,48,000 / 6000 = 58 and 1,26,000 / 6000 = 21. So the ratio is 58 : 21. Step 7: Total number of ratio parts = 58 + 21 = 79. Step 8: Satish gets 21 parts out of 79 from the total profit. Step 9: Satish's share = (21 / 79) * 16,970, which is approximately Rs. 4511. Verification / Alternative check: Calculate the value of one ratio part as 16,970 divided by 79, which is approximately 215 rupees. Satish's 21 parts then equal about 21 * 215 ≈ 4515, and rounding and small differences due to division give the option closest to the exact fractional result as Rs. 4511. Karan's share is the remaining profit, which is close to the theoretical amount given by 58 parts out of 79. Hence Satish's share of Rs. 4511 is the best fit among the given options and matches the ratio based calculation. Why Other Options Are Wrong: Rs. 5422, Rs. 5489 and Rs. 6145 correspond to much larger fractions of the total profit than 21 / 79 and would imply a capital time ratio very different from 58 : 21. These values break the proportional relationship between investments and profits and do not match the calculations based on the given data. Common Pitfalls: A common error is to assume that Satish invests for the full year instead of accounting for the late start after 5 months, which inflates his effective investment and gives a wrong share. Another mistake is to stop at a rounded ratio without checking that the final monetary shares still sum exactly to the total profit. Always include the timing factor and review the final numbers against the total profit. Final Answer: Satish's share in the profit is Rs. 4511 (approximately, matching the closest option).

Partnership Questions