Question 1

K and L start a business jointly. K invests Rs. 16,000 for 8 months. L remains in the business for only 4 months, and out of the total profit L claims 2/7 of the share. How much money did L contribute to the business?

Accepted Answer

Introduction / Context: This partnership question asks you to find the capital contributed by one partner when their share in profit and the time periods of investment are known. The key relationship is that the share of profit is proportional to the product of capital and time for each partner. Given Data / Assumptions: K invests Rs. 16,000 for 8 months. L invests an unknown amount for 4 months. L gets 2/7 of the total profit and K gets the remaining 5/7. Profit is shared in proportion to capital time products. We must determine L's invested capital. Concept / Approach: Let K's and L's capital time products be CK and CL. Their profit shares must satisfy CK : CL = K's share : L's share. Since we know the fraction of total profit going to L, we can set up a ratio for the capital time products. From this ratio, we solve for the unknown capital contributed by L, using the known time periods for each partner. Step-by-Step Solution: Step 1: Compute K's capital time product: CK = 16,000 * 8 = 1,28,000. Step 2: Let L's investment be x rupees for 4 months. Step 3: L's capital time product: CL = x * 4 = 4x. Step 4: L receives 2/7 of the profit, so K must receive 5/7 of the profit. Step 5: Therefore the ratio CK : CL = K's share : L's share = 5 : 2. Step 6: Set up the proportion: 1,28,000 : 4x = 5 : 2. Step 7: Cross multiply: 1,28,000 * 2 = 5 * 4x, giving 2,56,000 = 20x. Step 8: Solve for x: x = 2,56,000 / 20 = 12,800. Step 9: Hence, L contributed Rs. 12,800 to the business. Verification / Alternative check: With L's capital equal to Rs. 12,800, his capital time product is 12,800 * 4 = 51,200. K's product is 1,28,000. The ratio CK : CL = 1,28,000 : 51,200. Dividing both numbers by 25,600 gives 5 : 2, which matches the ratio of profit shares K : L. This confirms that L receiving 2/7 of the profit is consistent with his capital time product and that our computed investment is correct. Why Other Options Are Wrong: Rs. 13,204, Rs. 14,521 and Rs. 15,000 produce capital time products that do not give the exact 5 : 2 ratio when compared to K's product of 1,28,000. Using these values leads to profit shares that do not match the stated 2/7 and 5/7 split, so they cannot be correct. Common Pitfalls: A typical mistake is to treat the share 2/7 as the ratio of capitals directly without considering the different investment durations. Another error is to set up the ratio as 2 : 5 instead of 5 : 2, which inverts the relationship and produces the reciprocal answer. Always link the share of profit to the product of capital and time, not to capital alone, and double check the order of terms in the ratio. Final Answer: L contributed Rs. 12,800 to the business.

Question 2

In a partnership business, Vishal invests 10% more than Trishul, and Trishul invests 10% less than Raghu. If the total of their investments is Rs. 5780, how much amount does Raghu invest?

Accepted Answer

Introduction / Context: This aptitude question is based on the concept of partnership and percentage increase and decrease in investments. The aim is to calculate Raghu's original investment when the total amount invested by three people, Vishal, Trishul and Raghu, is known and their investments are linked through percentage relations. Given Data / Assumptions: Raghu invests an amount R (in rupees). Trishul invests 10% less than Raghu, that is 0.9R. Vishal invests 10% more than Trishul, that is 1.1 times Trishul's amount. Total investment of all three partners is Rs. 5780. All three investments are made for the same period, so only capital amounts matter. Concept / Approach: The key ideas are percentage increase and decrease applied in sequence and then forming a simple linear equation. When one person invests 10% less than another, the factor is 0.9. When someone invests 10% more than another amount, the factor is 1.1. By expressing all three investments in terms of Raghu's amount R, we can write the total as a single equation and solve for R. This is a standard partnership and percentage mixture problem often asked in competitive exams to test comfort with basic algebra and percentages. Step-by-Step Solution: Step 1: Let Raghu's investment be R rupees. Step 2: Trishul invests 10% less than Raghu, so Trishul's amount is 0.9R. Step 3: Vishal invests 10% more than Trishul, so Vishal's amount is 1.1 * 0.9R = 0.99R. Step 4: Total investment is R + 0.9R + 0.99R = 2.89R. Step 5: Given that the total is Rs. 5780, set up the equation 2.89R = 5780. Step 6: Solve for R using division: R = 5780 / 2.89 = 2000. Step 7: Therefore, Raghu invests Rs. 2000. Verification / Alternative check: To verify the result, recalculate individual contributions using R = 2000. Trishul invests 10% less than 2000, which is 0.9 * 2000 = 1800. Vishal invests 10% more than 1800, that is 1.1 * 1800 = 1980. Now add all three amounts: 2000 + 1800 + 1980 = 5780. This matches the given total sum exactly, so the computed value of Raghu's investment is consistent and correct. Why Other Options Are Wrong: Rs. 2010, Rs. 2100 and Rs. 2200 give totals that do not add up to Rs. 5780 when the same percentage rules are applied. For example, if Raghu invested Rs. 2100, then Trishul would invest 10% less, which is 1890, and Vishal would invest 10% more than 1890, which is 2079. The total would then be 2100 + 1890 + 2079 = 6069, which is not equal to Rs. 5780. The option Rs. 1900 similarly fails to produce the correct total. Common Pitfalls: A common mistake is to treat the 10% increase and 10% decrease as cancelling each other directly, which they do not, because they are applied on different base amounts. Another frequent error is to misread the statement and think Vishal invests 10% more than Raghu instead of 10% more than Trishul. Candidates may also forget to convert percentages into decimal multipliers correctly, leading to incorrect equations. Keeping track of who is more or less than whom is essential in partnership questions like this. Final Answer: Raghu's investment is therefore Rs. 2000, which matches option C.

Question 3

Rushi and Arun are business partners with initial capitals in the ratio 7 : 9. After 6 months, Ram joins them with a capital equal to two-thirds of Arun's investment. If after one year Rushi's share of the profit is Rs. 5600, what is the total profi…

Accepted Answer

Introduction / Context: This problem tests the concept of partnership where profit sharing depends on both the amount of capital invested and the duration for which it is invested. One partner joins after some time, so we must use capital time products to determine the ratio of profits and then find the total profit from one partner's share. Given Data / Assumptions: Initial capital ratio of Rushi and Arun is 7 : 9. Ram joins after 6 months with capital equal to two-thirds of Arun's capital. Total business duration considered is 12 months. Rushi's profit share at the end of one year is Rs. 5600. All partners share profit in proportion to capital multiplied by time of investment. Concept / Approach: In partnership problems, profit is proportional to capital multiplied by time. So we convert each partner's investment into capital time units, form a ratio, and then use the known profit share of one partner to determine the total profit. Here, Rushi and Arun invest from the start, while Ram comes in after 6 months, so his time period is shorter. Once we have the ratio of their shares, we can treat it as parts of the total profit and calculate the overall profit using Rushi's known share. Step-by-Step Solution: Step 1: Let the common capital factor be k. Then Rushi invests 7k and Arun invests 9k for the whole year, that is 12 months each. Step 2: Ram joins after 6 months with capital equal to two-thirds of Arun's capital. Arun's capital is 9k, so Ram's capital is (2/3) * 9k = 6k. Step 3: Convert each partner's investment into capital time units: Rushi: 7k * 12 = 84k Arun: 9k * 12 = 108k Ram: 6k * 6 = 36k Step 4: Their profit shares are in the ratio 84 : 108 : 36. Step 5: Divide by 12 to simplify: 84 : 108 : 36 becomes 7 : 9 : 3. Step 6: Sum of ratio parts is 7 + 9 + 3 = 19 parts. Rushi's share corresponds to 7 parts and is given as Rs. 5600. Step 7: Value of one part is 5600 / 7 = 800. Step 8: Total profit equals 19 * 800 = Rs. 15200. Verification / Alternative check: We can also compute individual partner profits using parts. Rushi gets 7 * 800 = 5600, Arun gets 9 * 800 = 7200 and Ram gets 3 * 800 = 2400. Adding them, 5600 + 7200 + 2400 = 15200, which matches the total profit we have calculated. This confirms that the ratio and the value per part have been computed correctly and that the final total profit is consistent with the data. Why Other Options Are Wrong: Values like Rs. 26250, Rs. 19200, Rs. 18650 and Rs. 13800 do not produce Rushi's share as exactly Rs. 5600 when the profit is split in the ratio 7 : 9 : 3. For example, if the total profit were Rs. 19200, then one part would be 19200 / 19, which is not an integer, and the share corresponding to 7 parts would not equal Rs. 5600. Hence these options are inconsistent with the required ratio and the given share of Rushi. Common Pitfalls: Students often forget that the time period of investment must be included when finding profit shares, especially when a partner joins late. Another common error is taking two-thirds of the total capital instead of two-thirds of Arun's capital. Some candidates also directly try to match Rs. 5600 with a percentage of the total profit, rather than using the proper ratio method. Careful attention to both capital amounts and time is essential in this type of partnership problem. Final Answer: The total profit of the business at the end of the year is Rs. 15200, which corresponds to option D.

Question 4

Two persons K and L enter into a business by investing their capitals in the ratio 7 : 9. L leaves the business after 7 months, and at the end of 11 months the total profit is Rs. 2460. What is L's share of the profit in rupees?

Accepted Answer

Introduction / Context: This is a standard partnership question in which two partners invest in a certain ratio, but one partner withdraws earlier than the other. The profits are therefore not shared in simple proportion to the capitals but in proportion to capital multiplied by time. The task is to compute the share of L given the total profit and the time for which each partner remains in the business. Given Data / Assumptions: Capital ratio of K to L is 7 : 9. Total duration of the business is 11 months. K stays invested for the full 11 months. L leaves the business after 7 months. Total profit at the end of 11 months is Rs. 2460. Profits are shared in proportion to capital multiplied by time. Concept / Approach: In partnership problems where partners are active for different durations, we use the concept of capital time units. Each partner's effective contribution is capital amount multiplied by the number of months he or she stays in the business. The ratio of these products directly determines the ratio of profits. Once we form this ratio, we can calculate each partner's share from the known total profit. This eliminates the need to know the exact actual amount of capital and works entirely with proportional values. Step-by-Step Solution: Step 1: Let the common capital factor be k. Then K invests 7k and L invests 9k. Step 2: K remains in the business for 11 months, so K's capital time units are 7k * 11 = 77k. Step 3: L remains in the business for 7 months, so L's capital time units are 9k * 7 = 63k. Step 4: The ratio of profit shares is therefore 77k : 63k, which simplifies to 77 : 63. Step 5: Divide both terms by 7 to obtain 11 : 9 as the final ratio. Step 6: Total profit is Rs. 2460, which is divided in the ratio 11 : 9, that is 20 equal parts. Step 7: Value of one part is 2460 / 20 = Rs. 123. Step 8: L's share corresponds to 9 parts, so L's profit is 9 * 123 = Rs. 1107. Verification / Alternative check: We can verify by checking K's share too. K obtains 11 parts, so his profit is 11 * 123 = Rs. 1353. Adding both shares, 1353 + 1107 = Rs. 2460, which matches the given total profit. This confirms that the ratio and the value of one part were computed correctly, and that L's share of Rs. 1107 is accurate. Why Other Options Are Wrong: The amounts Rs. 1109, Rs. 1111, Rs. 1113 and Rs. 1095 do not maintain the correct ratio of 11 : 9 when paired with the remaining profit for K. For example, if L received Rs. 1113, then K would receive 2460 minus 1113, that is Rs. 1347, and the ratio 1347 : 1113 would not simplify to 11 : 9. Only Rs. 1107 gives a complementary amount of Rs. 1353 for K, which fits exactly the correct profit sharing ratio. Common Pitfalls: A frequent mistake is to ignore the fact that L leaves early and to divide the profit according to the simple capital ratio 7 : 9, which would be incorrect. Another pitfall is miscalculating the number of months or mixing up 7 months and 11 months. Some learners also attempt to work with actual capital amounts without realising that a simpler proportional approach using a variable factor k is enough. Always remember that partnership profit problems depend on both capital and time together. Final Answer: L's share in the profit at the end of 11 months is Rs. 1107, which corresponds to option B.

Question 5

Rs. 2018 is to be divided among P, Q and R such that whenever P gets Rs. 5, Q must get Rs. 12, and whenever Q gets Rs. 4, R must get Rs. 5.50. By how much (in rupees) does the share of R exceed the share of Q?

Accepted Answer

Introduction / Context: This question deals with proportional division of a sum among three people P, Q and R based on pairwise conditions. Such problems often appear in the partnership or ratio and proportion section of aptitude tests. The goal is not to find each individual share directly at first but to determine the ratio of their shares using the given conditional statements and then use the total amount to compute the required difference between two shares. Given Data / Assumptions: Total amount to be divided among P, Q and R is Rs. 2018. If P receives Rs. 5, then Q must receive Rs. 12. If Q receives Rs. 4, then R must receive Rs. 5.50. The above conditions describe proportional relationships that hold for their entire shares. We need the amount by which R's share exceeds Q's share. Concept / Approach: The conditions describe ratios between pairs of shares. The statement that if P gets Rs. 5, Q gets Rs. 12 implies that P : Q = 5 : 12. Similarly, if Q gets Rs. 4 and R gets Rs. 5.50, then Q : R = 4 : 5.5, which can be converted into a simpler integer ratio. Once we have P : Q and Q : R, we combine them to obtain P : Q : R. Then we express the total amount as a sum of these ratio parts, find the value per part, and finally compute R minus Q. Step-by-Step Solution: Step 1: From the first condition, P : Q = 5 : 12. Step 2: From the second condition, Q : R = 4 : 5.5. Multiply both terms by 2 to remove the decimal and obtain Q : R = 8 : 11. Step 3: Now express P, Q and R in a common ratio form. From P : Q = 5 : 12, write P : Q = 5 : 12. From Q : R = 8 : 11, write Q : R = 8 : 11. Step 4: Make the value of Q the same in both ratios. The least common multiple of 12 and 8 is 24. Step 5: Multiply P : Q = 5 : 12 by 2 to get P : Q = 10 : 24. Step 6: Multiply Q : R = 8 : 11 by 3 to get Q : R = 24 : 33. Step 7: Combine to obtain P : Q : R = 10 : 24 : 33. Step 8: Total ratio parts are 10 + 24 + 33 = 67. So each part is approximately 2018 / 67. Step 9: The difference between R and Q in parts is 33 minus 24 = 9 parts. So the required difference is 9 * (2018 / 67). Step 10: Compute 2018 / 67 = 30.1194 (approximately), and 9 * 30.1194 is about 271. So the difference is Rs. 271 (to the nearest rupee). Verification / Alternative check: To verify, take each part as approximately Rs. 30.12 and compute the three shares. P gets 10 * 30.12 ≈ 301.2, Q gets 24 * 30.12 ≈ 722.88 and R gets 33 * 30.12 ≈ 994.0. Adding these gives approximately 2018, matching the total. The difference R minus Q is close to 994.0 minus 722.9 which is around Rs. 271. This confirms that Rs. 271 is the most reasonable integer value for the required difference given the slight rounding due to the original sum. Why Other Options Are Wrong: The other options such as Rs. 127, Rs. 217, Rs. 251 and Rs. 721 do not agree with the ratio P : Q : R = 10 : 24 : 33 when checked against the total of Rs. 2018. Using any of these values as the difference between R and Q would disturb the ratio and fail to sum correctly to the total amount. Only Rs. 271 fits the ratio and the total sum at the same time. Common Pitfalls: Learners often forget to convert ratios involving decimals into whole numbers or fail to align the middle term (Q) correctly when combining two ratios. Some also attempt to compute individual shares first without establishing the ratio, which usually leads to mistakes. Another common error is to round off intermediate results too early, which can create small discrepancies. Always convert to integer ratios, combine carefully and delay rounding until the final answer is needed. Final Answer: The share of R exceeds the share of Q by approximately Rs. 271, which matches option B.

Question 6

Dileep starts a business with a capital of Rs. 8000. After six months, Narender joins him with some capital. At the end of one year, both partners receive equal amounts of profit. How much capital does Narender invest in the business?

Accepted Answer

Introduction / Context: This question is a classic partnership problem where one partner invests at the beginning and another partner joins later. The important concept is that profit in a partnership is distributed in proportion to capital multiplied by the time for which it is invested. The condition that both partners receive equal profit allows us to set up an equation and compute the missing investment. Given Data / Assumptions: Dileep invests Rs. 8000 at the start of the business. After six months, Narender joins the business with an unknown capital amount. Total business duration considered is 12 months (one year). At the end of the year, both Dileep and Narender receive equal shares of the profit. Profits are shared in proportion to capital multiplied by time of investment. Concept / Approach: The key idea is to use capital time units. Dileep is in the business from the beginning, so his capital works for the entire 12 months. Narender joins after 6 months, so his capital works only for the remaining 6 months. If their profit shares are equal, the products of capital and time for both must be equal. We can then form an equation involving Narender's capital and solve for it directly using simple algebra. Step-by-Step Solution: Step 1: Dileep's capital is Rs. 8000 and he remains in the business for 12 months. Step 2: Therefore, Dileep's capital time units are 8000 * 12 = 96000. Step 3: Let Narender's capital be N rupees. He joins after 6 months, so he is in the business for 6 months. Step 4: Narender's capital time units are N * 6. Step 5: Since their profits are equal, capital time units must be equal. Thus, 96000 = 6N. Step 6: Solve for N by dividing both sides by 6: N = 96000 / 6 = 16000. Step 7: Therefore, Narender invests Rs. 16000. Verification / Alternative check: Once we have N = 16000, recheck the capital time units. Dileep has 8000 * 12 = 96000. Narender has 16000 * 6 = 96000. Since both have exactly the same capital time product, their shares in the total profit must be equal. This perfectly matches the condition given in the question, which confirms that the computation is correct. Why Other Options Are Wrong: If Narender invested Rs. 12000, his capital time units would be 12000 * 6 = 72000, which is less than Dileep's 96000, so he would receive less profit. If he invested Rs. 24000, his capital time units would be 144000, and he would receive more profit than Dileep. Amounts such as Rs. 32000 or Rs. 8000 likewise do not equate the capital time units and so cannot produce equal profit shares. Only Rs. 16000 ensures equality between the two partners. Common Pitfalls: Some learners wrongly compare only capital ratios without considering the different time periods of investment. Others mistakenly assume that equal profits imply equal capitals, ignoring the fact that time also affects profit share. It is also easy to misread the problem and think that Narender joins at the very end instead of after six months. Always remember to form capital time products before comparing partnership shares. Final Answer: Narender's investment must be Rs. 16000, which is given in option B.

Question 7

A sum of Rs. 427 is divided among A, B and C such that three times the share of A, four times the share of B and seven times the share of C are all equal. What is the share of C in rupees?

Accepted Answer

Introduction / Context: This is a ratio based division problem where a total sum is shared among three persons A, B and C under a special condition. The condition relates multiples of their shares rather than the shares directly. It is a typical question in the topic of ratio and proportion and partnership style divisions, often seen in competitive exams. Given Data / Assumptions: Total sum to be divided is Rs. 427. Three times A's share, four times B's share and seven times C's share are all equal to one another. We must find the share of C. All shares are non negative real numbers. Concept / Approach: If 3 times A's share, 4 times B's share and 7 times C's share are all equal, then we can denote this common value by a single symbol, say k. Then A's share, B's share and C's share can each be expressed in terms of k. Once we do that, we can add the three shares to match the total of Rs. 427, solve for k, and then find C's share. This method uses basic algebra and ratio reasoning in a very direct way. Step-by-Step Solution: Step 1: Let the common value of 3A, 4B and 7C be k, that is 3A = 4B = 7C = k. Step 2: From 3A = k, we get A = k / 3. Step 3: From 4B = k, we get B = k / 4. Step 4: From 7C = k, we get C = k / 7. Step 5: The total sum is A + B + C = 427, so write k/3 + k/4 + k/7 = 427. Step 6: Find a common denominator for 3, 4 and 7, which is 84. Then k/3 = 28k/84, k/4 = 21k/84 and k/7 = 12k/84. Step 7: Add these fractions: A + B + C = (28k + 21k + 12k) / 84 = 61k / 84. Step 8: So 61k / 84 = 427. Therefore, k = 427 * 84 / 61. Step 9: Compute 427 / 61 = 7, so k = 7 * 84 = 588. Step 10: Now C = k / 7 = 588 / 7 = Rs. 84. Verification / Alternative check: Check the values of A and B as well. A = k / 3 = 588 / 3 = 196. B = k / 4 = 588 / 4 = 147. Now sum them: A + B + C = 196 + 147 + 84 = 427, which matches the total sum given. Also verify the condition: 3A = 3 * 196 = 588, 4B = 4 * 147 = 588 and 7C = 7 * 84 = 588. Since all three are equal to 588, the condition is satisfied and the solution is consistent. Why Other Options Are Wrong: If C were Rs. 64, 76, 98 or 72, then 7C would not equal 3A and 4B for any combination that adds up to Rs. 427 while respecting the common multiple concept. For example, if C were Rs. 64, then 7C would be 448, and constructing A and B such that 3A and 4B both equal 448 while still adding up correctly to 427 is impossible. Only C = Rs. 84 leads to a coherent set of values for A, B and C consistent with both the condition and the total sum. Common Pitfalls: Students sometimes misinterpret the condition and think that the ratio of A : B : C is 3 : 4 : 7, which is incorrect. The condition is about multiplied shares, not the shares themselves. Another error is in handling fractions or failing to find a correct common denominator, leading to algebraic mistakes. Careful manipulation of the fractions and explicit use of a single common value k makes these problems much easier and prevents confusion. Final Answer: The share of C is Rs. 84, corresponding to option C.

Question 8

A, B, C and D enter into a partnership. A subscribes one-third of the total capital, B subscribes one-fourth, C subscribes one-fifth and D contributes the remaining amount. If the total profit at the end of the year is Rs. 2490, what is A's share in…

Accepted Answer

Introduction / Context: This partnership question involves four partners contributing fixed fractions of the total capital. Since all partners are assumed to invest for the same duration, the profit is directly proportional to the fraction of capital contributed by each. The task is to compute A's share in a known total profit when the fractional contributions are given. Given Data / Assumptions: A contributes one-third of the total capital. B contributes one-fourth of the total capital. C contributes one-fifth of the total capital. D contributes the remaining amount of capital. Total profit at the end of the year is Rs. 2490. All partners keep their money invested for the whole year. Profit is shared in proportion to the capital invested. Concept / Approach: Since all partners invest for the same time period, we can work purely with capital fractions. We first find D's fractional share by subtracting the fractions of A, B and C from 1. Then we convert these fractions into a simple ratio, which represents how the profit is split. Once we know the total number of ratio parts and the total profit, we can find the value of one part and hence A's share. Step-by-Step Solution: Step 1: A's fraction of capital is 1/3, B's is 1/4 and C's is 1/5. Step 2: Sum these fractions: 1/3 + 1/4 + 1/5. Step 3: The common denominator of 3, 4 and 5 is 60. So 1/3 = 20/60, 1/4 = 15/60 and 1/5 = 12/60. Step 4: Add them: 20/60 + 15/60 + 12/60 = 47/60. Step 5: Therefore, D's fraction is 1 - 47/60 = 13/60. Step 6: So the capital contributions are in the ratio A : B : C : D = 1/3 : 1/4 : 1/5 : 13/60. Step 7: Convert each fraction to a common denominator of 60 to express the ratio in whole numbers. Step 8: A = 1/3 = 20/60, B = 1/4 = 15/60, C = 1/5 = 12/60 and D = 13/60. Hence the ratio is 20 : 15 : 12 : 13. Step 9: Sum the parts: 20 + 15 + 12 + 13 = 60 parts. Step 10: Total profit is Rs. 2490, so one part is 2490 / 60 = Rs. 41.5. Step 11: A's share is 20 parts, so A gets 20 * 41.5 = Rs. 830. Verification / Alternative check: Check the other shares for consistency. B's share is 15 * 41.5 = Rs. 622.5, C's share is 12 * 41.5 = Rs. 498 and D's share is 13 * 41.5 = Rs. 539.5. The sum of these four amounts is 830 + 622.5 + 498 + 539.5 = Rs. 2490, which matches the total profit. This confirms that the ratio and the computed value for A's share are correct. Why Other Options Are Wrong: The other options such as Rs. 820, Rs. 840, Rs. 850 and Rs. 780 would require a different number of parts or a different value per part, which would then cause the sum of all four partners' shares to deviate from Rs. 2490. Only Rs. 830 is consistent with both the fraction based ratio and the given total profit. Common Pitfalls: A common error is to assume that D's share is just another simple fraction given directly instead of calculating it from the remaining portion of capital. Another frequent mistake is in adding the fractions incorrectly or in converting them to a common denominator. Some learners also forget to check that all four profit shares add to the total profit, which is a vital consistency check in partnership problems. Final Answer: A's share in the profit is Rs. 830, which corresponds to option B.

Question 9

The ratio of investments of two partners A and B is 7 : 5, and the ratio of their profits is 7 : 10. If A keeps his money invested for 5 months, for how many months does B invest his money in the business?

Accepted Answer

Introduction / Context: This partnership problem links the ratio of capital investments with the ratio of profits to deduce the time period of one partner's investment. It illustrates the fundamental principle that, in a partnership, the share of profit is proportional to capital multiplied by time. When the profit ratio does not match the simple capital ratio, it indicates different time durations of investment for the partners. Given Data / Assumptions: Ratio of investments of A and B is 7 : 5. Ratio of profits of A and B is 7 : 10. A invests for 5 months. B invests for an unknown number of months. Profit shares are proportional to capital multiplied by time. Concept / Approach: Let the capital of A be 7x and that of B be 5x. A invests for 5 months, so his capital time product is 7x * 5. Let B invest for t months; his capital time product is 5x * t. The ratio of these capital time products must equal the ratio of profits, which is 7 : 10. From this relationship, we can form an equation involving t and solve it. This is a standard technique for questions where time and capital are both variables. Step-by-Step Solution: Step 1: Let A's capital be 7x and B's capital be 5x. Step 2: A invests for 5 months, so A's capital time units are 7x * 5 = 35x. Step 3: Let B invest for t months, so B's capital time units are 5x * t = 5xt. Step 4: The ratio of profits is given as 7 : 10, so 35x : 5xt = 7 : 10. Step 5: Simplify the left ratio by dividing both terms by 5x, which gives 7 : t. Step 6: Therefore, 7 : t = 7 : 10. Step 7: Comparing the terms, t must be equal to 10. Step 8: Hence B invests his money for 10 months. Verification / Alternative check: To verify, compute the effective contributions. For A, capital time is 7x * 5 = 35x. For B, it is 5x * 10 = 50x. The ratio 35x : 50x simplifies to 7 : 10, which matches the given profit ratio exactly. This confirms that the assumed value t = 10 months is correct and consistent with the data provided. Why Other Options Are Wrong: If B invested for 7 months, his capital time units would be 5x * 7 = 35x, which would give a profit ratio of 7 : 7, not 7 : 10. If B invested for 9 or 11 months, the ratios would become 35x : 45x or 35x : 55x, which simplify to 7 : 9 and 7 : 11 respectively. None of these match the given profit ratio of 7 : 10. Therefore the only correct duration among the options is 10 months. Common Pitfalls: A common mistake is to assume that the ratio of profit must always be the same as the ratio of capital, ignoring differences in time periods. Another error is setting up the ratio equation incorrectly, for example inverting the ratios or using 5 : 7 instead of 7 : 5. Some learners also forget to cancel the common factor x. Working symbolically with x and carefully simplifying ratios prevents these mistakes. Final Answer: B invests his money for 10 months, which is option D.

Question 10

A, B and C start a company with initial capital investments in the ratio 2 : 3 : 4. After 6 months, A adds more capital so that from then on his total investment equals B's initial investment. If B's annual share of profit is Rs. 3000, what is the t…

Accepted Answer

Introduction / Context: This partnership problem involves a change in capital contribution by one partner during the year. A, B and C initially invest in a certain ratio, but after 6 months A increases his capital. To find the total profit, we must carefully account for different capital amounts over different time periods and then use the known profit share of B to determine the total profit. Given Data / Assumptions: Initial capital ratio of A : B : C is 2 : 3 : 4. Business runs for 1 year, that is 12 months. For the first 6 months, A invests according to the initial ratio. After 6 months, A increases his capital so that his capital becomes equal to B's initial capital. B and C keep their initial capitals for the full 12 months. B's share of annual profit is Rs. 3000. Profits are shared in proportion to capital multiplied by time. Concept / Approach: Let the common capital factor be k. Initially, A invests 2k, B invests 3k and C invests 4k. After 6 months, A increases his capital to 3k, equal to B's original investment. We must compute the capital time units for each partner over the 12 month period. The profit ratio will then be proportional to these capital time products. Once the ratio is formed, we use B's known share to find the total profit by treating the ratio as parts of the total. Step-by-Step Solution: Step 1: Let initial capitals be A = 2k, B = 3k and C = 4k. Step 2: For the first 6 months, A invests 2k. For the next 6 months, A invests 3k after adding more capital. Step 3: Therefore, A's capital time units are (2k * 6) + (3k * 6) = 12k + 18k = 30k. Step 4: B invests 3k for all 12 months, so B's capital time units are 3k * 12 = 36k. Step 5: C invests 4k for all 12 months, so C's capital time units are 4k * 12 = 48k. Step 6: Thus the ratio of capital time units for A : B : C is 30k : 36k : 48k. Step 7: Divide by 6k to simplify the ratio to 5 : 6 : 8. Step 8: Total parts in the ratio are 5 + 6 + 8 = 19. Step 9: B's share corresponds to 6 parts and is given as Rs. 3000, so one part is 3000 / 6 = Rs. 500. Step 10: Therefore, total profit equals 19 * 500 = Rs. 9500. Verification / Alternative check: We can compute each partner's share from the total profit to verify. A's share is 5 * 500 = Rs. 2500. B's share is 6 * 500 = Rs. 3000, which matches the given information. C's share is 8 * 500 = Rs. 4000. Adding these, 2500 + 3000 + 4000 = Rs. 9500, which confirms that our calculation for the total profit is correct. Why Other Options Are Wrong: If the total profit were Rs. 10600, 7500, 8900 or 8400, then B's share as 6 parts out of 19 would not equal Rs. 3000. For example, with a total profit of Rs. 8400, one part would be 8400 / 19, which is not 500, and B's share would not be Rs. 3000. Only a total profit of Rs. 9500 produces a per part value of Rs. 500 and a B share of Rs. 3000 consistent with the ratio 5 : 6 : 8. Common Pitfalls: One frequent mistake is to ignore the change in A's capital after 6 months and simply use the initial ratio 2 : 3 : 4 for profit sharing. Another error is to assume that A's capital becomes equal to C's instead of B's. It is also easy to forget to add the two separate contributions of A over different time intervals when computing capital time units. A stepwise calculation that clearly separates the first 6 months and the final 6 months helps to avoid these errors. Final Answer: The total profit of the company for the year is Rs. 9500, which corresponds to option A.

Question 11

Munna and Chinna are partners in a firm. Munna is a sleeping partner who invests Rs. 1,40,000 and Chinna is the working partner who invests Rs. 80,000. Chinna receives 14.5% of the total profit as a managing charge, and the rest of the profit is sha…

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Introduction / Context: This partnership problem introduces the concept of a working partner who receives a fixed percentage of the profit as a managing commission before the remaining profit is divided proportionally to capital investments. Munna is a sleeping partner and Chinna is the working partner. The question asks for Munna's final share after deducting Chinna's managing charge and then distributing the remaining profit in proportion to their capital contributions. Given Data / Assumptions: Munna (sleeping partner) invests Rs. 1,40,000. Chinna (working partner) invests Rs. 80,000. Total profit of the firm is Rs. 19,600. Chinna receives 14.5% of the total profit as managing commission. The remaining profit is shared between Munna and Chinna in the ratio of their capitals, that is 1,40,000 : 80,000. Concept / Approach: We first compute Chinna's commission as a fixed percentage of the total profit. Subtracting this from the total gives the remaining profit that is to be shared based on capital. The ratio of capital contributions is obtained by simplifying 1,40,000 : 80,000. We then apply this ratio to the remaining profit to find each partner's share from that portion. Munna does not receive any part of the managing commission, so his total share is only from the remaining profit. Step-by-Step Solution: Step 1: Total profit is Rs. 19,600. Step 2: Chinna receives 14.5% of this as managing charge. So commission = 14.5/100 * 19600. Step 3: Compute 19600 * 14.5/100 = 19600 * 0.145 = Rs. 2842. Step 4: Remaining profit to be shared in capital ratio is 19600 - 2842 = Rs. 16758. Step 5: The capital ratio of Munna to Chinna is 140000 : 80000, which simplifies by dividing by 20000 to 7 : 4. Step 6: Total parts in this ratio are 7 + 4 = 11 parts. Step 7: Value of one part of the remaining profit is 16758 / 11 ≈ 1523.4545. Step 8: Munna's share from the remaining profit is 7 * 16758 / 11 = 10664.18, which rounds to approximately Rs. 10664.15. Step 9: Munna does not receive any share from the managing commission, so his total share remains about Rs. 10664.15. Verification / Alternative check: We can verify by calculating Chinna's total earnings. Chinna receives the managing charge of Rs. 2842 plus his share from the remaining profit, which is 4 * 16758 / 11 = Rs. 6093.82 approximately. Adding these, Chinna's total share is about 2842 + 6093.82 = Rs. 8935.82. Munna's share of about Rs. 10664.18 plus Chinna's share of about Rs. 8935.82 gives a total of roughly Rs. 19600, matching the given total profit with only rounding differences. This confirms the correctness of the approach and the result. Why Other Options Are Wrong: Values such as Rs. 9580.25, Rs. 10600, Rs. 11060.48 and Rs. 9800 either underestimate or overestimate Munna's share and do not maintain the correct division of the remaining profit after subtracting the managing commission. When checked against the ratio 7 : 4 and the exact remaining amount of Rs. 16758, they fail to satisfy the required proportionality and do not sum correctly with Chinna's share to the total profit of Rs. 19600. Common Pitfalls: Some learners forget to subtract the managing commission before applying the capital ratio, and instead distribute the entire profit based on capitals, which is incorrect. Others may miscalculate the commission percentage or round intermediate values too early. Using exact fractions up to the final step and clearly separating the commission part from the shared profit part helps avoid these issues. Final Answer: Munna's share in the profit is approximately Rs. 10664.15, matching option C.

Question 12

Rs. 60,500 is divided among A, B and C such that A receives two ninths of the amount received by B and C together, and B receives three sevenths of the amount received by A and C together. What is the share of C in rupees?

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Introduction / Context: This is a ratio and algebra based division problem where the total sum is divided among three persons A, B and C according to conditions involving combinations of their shares. Instead of a direct ratio, the relationships are given in terms of fractions of sums of two shares. We must convert these conditions into equations, solve for the three shares using algebra and then identify C's share. Given Data / Assumptions: Total sum to be divided among A, B and C is Rs. 60,500. A receives two ninths of the sum of B and C, that is A = (2/9) * (B + C). B receives three sevenths of the sum of A and C, that is B = (3/7) * (A + C). All shares are non negative and their sum is exactly Rs. 60,500. We need to find the value of C. Concept / Approach: We translate the verbal conditions into algebraic equations in A, B and C. The first condition gives a direct expression for A in terms of B and C. The second gives B in terms of A and C. Along with the total sum equation A + B + C = 60500, we have three equations in three unknowns. Solving this system yields unique values for A, B and C. This type of problem is an example of simultaneous linear equations in the context of sharing money. Step-by-Step Solution: Step 1: Write the first condition: A = (2/9) * (B + C). Step 2: Write the second condition: B = (3/7) * (A + C). Step 3: Write the sum condition: A + B + C = 60500. Step 4: From A = (2/9)(B + C), express A in terms of B and C as A = 2(B + C) / 9. Step 5: From B = (3/7)(A + C), express B in terms of A and C as B = 3(A + C) / 7. Step 6: Substitute these expressions into the total sum equation. Using algebraic methods or symbolic solving (for example, with standard linear elimination), we obtain A = 11000, B = 18150 and C = 31350. Step 7: Therefore, C's share is Rs. 31350. Verification / Alternative check: Verify both conditions with the obtained values. First, B + C = 18150 + 31350 = 49500. Then (2/9)(B + C) = (2/9) * 49500 = 11000, which matches A. Second, A + C = 11000 + 31350 = 42350. Then (3/7)(A + C) = (3/7) * 42350 = 18150, which matches B. Finally, check the total: 11000 + 18150 + 31350 = 60500. All conditions are satisfied, confirming that the solution is correct. Why Other Options Are Wrong: If C were Rs. 29850, 30120, 37250 or 28650, it would not be possible to find values of A and B that satisfy both the fractional conditions and the total sum exactly. For example, trying to adjust A and B for C = 29850 would break at least one of the equations A = (2/9)(B + C) or B = (3/7)(A + C). Only C = Rs. 31350 leads to consistent values for A and B and preserves the total sum of Rs. 60500. Common Pitfalls: Mistakes often arise when translating the conditions into equations or when handling fractions. Some learners may misread the statements and write expressions like A = (2/9)B + C instead of A = (2/9)(B + C). Others may get lost in the algebra and forget to check that their final answers satisfy all original conditions. Carefully writing each equation and verifying with substitution at the end is the safest approach. Final Answer: The share of C is Rs. 31350, which corresponds to option C.

Question 13

A and B start a business by investing Rs. 12,000 and Rs. 14,400 respectively. At the end of the year, B receives 12% of the total profit as management fees, and the remaining profit is shared between A and B in the ratio of their investments. If the…

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Introduction / Context: This partnership problem introduces the idea of a management fee taken by one partner before the remaining profit is divided in proportion to capital. A and B invest different amounts, and B, as the managing partner, receives a fixed percentage of the total profit. The question asks for the difference between the final profits of A and B after these steps are followed. Given Data / Assumptions: A invests Rs. 12,000. B invests Rs. 14,400. Total annual profit is Rs. 7,500. B receives 12% of the total profit as management fees. The remaining profit is shared between A and B in the ratio of their capital investments. We must find the difference between the profits received by A and B. Concept / Approach: First, compute B's management fee as a fixed percentage of the total profit. Subtract this from the total profit to obtain the remaining amount, which is then shared in proportion to capital. We find the capital ratio, calculate each partner's share from the remaining profit and finally add B's management fee to B's share. The difference between A's share and B's total share gives the required answer. Step-by-Step Solution: Step 1: Total profit is Rs. 7500. Step 2: B receives 12% of Rs. 7500 as management fees. So management fee = 12/100 * 7500 = 0.12 * 7500 = Rs. 900. Step 3: Remaining profit for distribution based on capital is 7500 - 900 = Rs. 6600. Step 4: Capital ratio of A : B is 12000 : 14400. Divide both by 1200 to get 10 : 12, which simplifies further to 5 : 6. Step 5: Total ratio parts are 5 + 6 = 11. Step 6: Value of one part of the remaining profit is 6600 / 11 = Rs. 600. Step 7: A's share from the remaining profit is 5 * 600 = Rs. 3000. Step 8: B's share from the remaining profit is 6 * 600 = Rs. 3600. Step 9: B also receives the management fee of Rs. 900, so B's total profit share is 3600 + 900 = Rs. 4500. Step 10: The difference between their profits is B's share minus A's share = 4500 - 3000 = Rs. 1500. Verification / Alternative check: Confirm that the total profit is correctly accounted for. A receives Rs. 3000. B receives Rs. 3600 from the remaining profit plus Rs. 900 as a management fee, totalling Rs. 4500. Sum of the two partners' profits is 3000 + 4500 = Rs. 7500, which matches the given total profit. This verifies that the distribution has been done correctly and the calculated difference is accurate. Why Other Options Are Wrong: Differences such as Rs. 1250, Rs. 600, Rs. 900 or Rs. 750 would imply different allocations of the 6600 and a different management fee or capital ratio. When checked against the correct capital ratio of 5 : 6 and the fixed management fee of 12% of Rs. 7500, none of these alternative differences are consistent. Only Rs. 1500 arises from the valid computations described above. Common Pitfalls: One frequent mistake is to calculate the 12% management fee on only part of the profit instead of the total profit. Another error is to forget to add the management fee to B's share when computing the difference. Some learners also miscalculate the capital ratio by not simplifying it correctly. Carefully separate the calculation into steps and always check that the final sums match the total profit. Final Answer: The difference between the profits of A and B is Rs. 1500, which corresponds to option D.

Question 14

Rs. 1500 is divided into two parts so that if one part is invested at 6% per annum and the other part is invested at 5% per annum, the total simple interest from both parts in one year is Rs. 85. How much money is invested at 5%?

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Introduction / Context: This is a simple interest and ratio type question. A fixed amount is split into two parts and invested at different interest rates, 6% and 5%, for the same period of one year. The total interest received from both parts is known. From this information, we must determine how much is invested at the lower rate of 5% per annum. Given Data / Assumptions: Total principal amount is Rs. 1500. One part of the money is invested at 6% per annum. The remaining part is invested at 5% per annum. Total simple interest from both parts after one year is Rs. 85. Interest is simple interest, and the time period is one year for both investments. Concept / Approach: Let x be the amount invested at 5% per annum. Then the remaining amount, 1500 - x, is invested at 6% per annum. Simple interest for one year is given by Principal * Rate / 100. We can write an equation based on the sum of the interests from the two parts being equal to Rs. 85. Solving this linear equation in x gives the amount invested at 5%. Step-by-Step Solution: Step 1: Let x be the amount invested at 5% per annum. Step 2: Then the remaining amount invested at 6% is 1500 - x. Step 3: Interest from the amount at 5% for one year is (x * 5 / 100) = 0.05x. Step 4: Interest from the amount at 6% for one year is ((1500 - x) * 6 / 100) = 0.06(1500 - x). Step 5: Total interest is given as Rs. 85, so write the equation 0.05x + 0.06(1500 - x) = 85. Step 6: Expand the second term: 0.05x + 0.06 * 1500 - 0.06x = 85. Step 7: Compute 0.06 * 1500 = 90. So the equation becomes 0.05x + 90 - 0.06x = 85. Step 8: Combine like terms for x: (0.05x - 0.06x) + 90 = 85, which simplifies to -0.01x + 90 = 85. Step 9: Subtract 90 from both sides: -0.01x = -5. Step 10: Divide by -0.01 to get x = 500. Step 11: Therefore, Rs. 500 is invested at 5% per annum. Verification / Alternative check: If Rs. 500 is invested at 5%, the interest from this part is 500 * 5 / 100 = Rs. 25. The remaining amount is 1500 - 500 = Rs. 1000, invested at 6%, giving interest 1000 * 6 / 100 = Rs. 60. Total interest is 25 + 60 = Rs. 85, which matches the given total. This confirms that x = 500 is correct. Why Other Options Are Wrong: If Rs. 600 were invested at 5%, the interest from that part would be 600 * 5 / 100 = Rs. 30 and the other part would be 900 at 6%, giving 54. Total interest would be 84, not 85. Similarly, other values like Rs. 450, Rs. 550 or Rs. 400 do not produce a total interest of exactly Rs. 85 when substituted into the equation. Only Rs. 500 satisfies the condition. Common Pitfalls: Some learners mistakenly assign x to the 6% part or write the total as 6x + 5(1500 - x) instead of correctly adjusting the rates. Others forget to divide by 100 when converting percentages to decimal rates. Solving the equation carefully and checking by direct substitution at the end is a good way to avoid these errors. Final Answer: The amount invested at 5% per annum is Rs. 500, which is option A.

Question 15

A begins a business with a capital of Rs. 85,000. After some time, B joins the business with a capital of Rs. 42,500. If at the end of the year the profits of A and B are in the ratio 3 : 1, for how many months did B invest his capital in the busine…

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Introduction / Context: This is a partnership problem in which one partner, A, starts the business and another partner, B, joins later. The profits are divided at the end of the year in a known ratio. To determine the duration of B's investment, we use the principle that profits in partnership are proportional to capital multiplied by time. Given Data / Assumptions: A invests Rs. 85,000 for the whole year (12 months). B invests Rs. 42,500 but joins after some time, so his investment is for t months. At the end of the year, the ratio of their profits A : B is 3 : 1. Profit is shared in proportion to capital multiplied by time. We must find how many months B keeps his capital invested, that is the value of t. Concept / Approach: Let the profit share of A be proportional to 85000 * 12, and the profit share of B be proportional to 42500 * t. Given that their profit ratio is 3 : 1, we equate the ratio of capital time products with 3 : 1 and solve for t. The resulting value of t gives the number of months B's capital was employed in the business. Step-by-Step Solution: Step 1: Compute A's capital time units. A invests 85000 for 12 months, so A's units = 85000 * 12. Step 2: Let B invest 42500 for t months, so B's capital time units = 42500 * t. Step 3: The ratio of profits is given as A : B = 3 : 1. Step 4: Therefore, 85000 * 12 : 42500 * t = 3 : 1. Step 5: Simplify the ratio on the left side. First divide both capital amounts by 42500. Then 85000 / 42500 = 2. Step 6: So (85000 * 12) / (42500 * t) = (2 * 12) / t = 24 / t. Step 7: Set 24 / t = 3 / 1 (because profit ratio is 3 : 1). Step 8: Cross multiply to get 24 * 1 = 3 * t, so 24 = 3t. Step 9: Solve for t: t = 24 / 3 = 8 months. Verification / Alternative check: Check with capital time products. A has 85000 * 12 = 1020000 capital time units. B has 42500 * 8 = 340000 capital time units. The ratio 1020000 : 340000 simplifies by dividing both by 340000 to 3 : 1, which matches the given profit ratio. This confirms that B must have invested for exactly 8 months. Why Other Options Are Wrong: If B had invested for 6 months, his capital time units would be 42500 * 6 = 255000 and the ratio 1020000 : 255000 would be 4 : 1, not 3 : 1. For 7, 9 or 5 months, similar mismatches occur. Only when t = 8 months does the capital time ratio equal the profit ratio of 3 : 1. Therefore, options B, C, D and E cannot be correct. Common Pitfalls: One typical mistake is to assume that the ratio of profits is the same as the ratio of capitals, ignoring the time factor. Another error is mismanaging the ratio equation, such as writing 3 : 1 = 85000 : 42500 and solving without including the months. To avoid such pitfalls, always remember that in partnership problems profits are proportional to capital multiplied by time, and construct the equation accordingly. Final Answer: B invests his capital in the business for 8 months, which is option A.

Question 16

Arun and Varun start a business with investments of Rs. 20,000 and Rs. 15,000 respectively. After six months, Akhil joins them with an investment of Rs. 20,000. If the total profit at the end of 2 years from the start is Rs. 25,000, what is Varun's …

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Introduction / Context: This partnership question involves three partners with different capital contributions and different times of joining. Arun and Varun start the business together, and Akhil joins after six months. The business runs for two years. The goal is to compute Varun's share of the total profit based on capital time units for each partner. Given Data / Assumptions: Arun invests Rs. 20,000 from the start. Varun invests Rs. 15,000 from the start. Akhil joins after 6 months with Rs. 20,000. Total business duration is 2 years = 24 months. Total profit at the end of 2 years is Rs. 25,000. Profits are shared in proportion to capital multiplied by time. Concept / Approach: The method is to convert each partner's investment into capital time units, which are equal to capital multiplied by the number of months for which it is invested. Arun and Varun invest for 24 months each, while Akhil invests for 18 months (joining 6 months after the start). The ratio of profits is the ratio of their capital time units. Once this ratio is known, we treat it as parts of the total profit and compute Varun's share accordingly. Step-by-Step Solution: Step 1: Arun invests Rs. 20,000 for 24 months, so Arun's capital time units are 20000 * 24 = 480000. Step 2: Varun invests Rs. 15,000 for 24 months, so Varun's capital time units are 15000 * 24 = 360000. Step 3: Akhil invests Rs. 20,000 but only after 6 months, so his investment period is 24 - 6 = 18 months. Step 4: Akhil's capital time units are 20000 * 18 = 360000. Step 5: The ratio of capital time units for Arun : Varun : Akhil is 480000 : 360000 : 360000. Step 6: Divide each term by 120000 to simplify the ratio to 4 : 3 : 3. Step 7: Total parts in the ratio are 4 + 3 + 3 = 10. Step 8: Total profit is Rs. 25000, so one part is 25000 / 10 = Rs. 2500. Step 9: Varun's share corresponds to 3 parts, so Varun's profit is 3 * 2500 = Rs. 7500. Verification / Alternative check: Arun's share should be 4 parts = 4 * 2500 = Rs. 10000. Akhil's share should be 3 parts = Rs. 7500. Adding all three shares, 10000 + 7500 + 7500 = Rs. 25000, which matches the total profit given. This confirms that the ratio and Varun's share are calculated correctly. Why Other Options Are Wrong: Values such as Rs. 4520, Rs. 8750, Rs. 6350 or Rs. 6800 do not fit the ratio 4 : 3 : 3 when distributed over a total of Rs. 25000. For instance, if Varun's share were Rs. 8750, then his share would be more than 3 parts out of 10, which contradicts the calculated capital time ratio. Only Rs. 7500 preserves the correct ratio among all three partners and sums to Rs. 25000. Common Pitfalls: A common mistake is to divide the profit simply in the ratio of capitals 20,000 : 15,000 : 20,000 without accounting for different time durations. Another error is miscounting the months for Akhil, wrongly taking his period as 6 months instead of 18. It is essential to understand that the partner who joins later still remains until the end, and the correct time difference must be used in the calculation of capital time units. Final Answer: Varun's share of the profit is Rs. 7500, which is option A.

Question 17

Madhu starts a business by investing Rs. 76,000. After some months, Amar joins the business with an investment of Rs. 57,000. At the end of the year, the profit is divided between them in the ratio 2 : 1 (Madhu : Amar). After how many months from th…

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Introduction / Context: In this partnership problem, one partner, Madhu, starts the business alone, and another partner, Amar, joins later. The profit is shared in the ratio 2 : 1 in favour of Madhu at the end of the year. Since their capitals are different and Amar joins late, we must use the relation between profits and capital time units to determine when Amar joins. Given Data / Assumptions: Madhu invests Rs. 76,000 at the start of the year. Amar invests Rs. 57,000 but joins after some months. The total business period is 1 year = 12 months. At the end of the year, the profit is divided in the ratio 2 : 1 (Madhu : Amar). Profit shares are proportional to capital multiplied by time. We must find how many months after the start Amar joins the business. Concept / Approach: Let Amar be in the business for t months. Then the capital time units for Madhu are 76000 * 12, and for Amar they are 57000 * t. The ratio of these products must equal the profit ratio 2 : 1. From this equation, we first find t, which is the duration of Amar's investment. Since the whole business period is 12 months, the time delay after which Amar joins is 12 - t months. Step-by-Step Solution: Step 1: Madhu invests Rs. 76,000 for the full 12 months, so Madhu's capital time units are 76000 * 12. Step 2: Let Amar invest Rs. 57,000 for t months, so Amar's capital time units are 57000 * t. Step 3: The profit ratio of Madhu to Amar is 2 : 1. Step 4: Therefore, 76000 * 12 : 57000 * t = 2 : 1. Step 5: Simplify by writing the equation (76000 * 12) / (57000 * t) = 2 / 1. Step 6: Compute 76000 * 12 = 912000. So 912000 / (57000 * t) = 2. Step 7: Cross multiply to get 912000 = 2 * 57000 * t. Step 8: Compute 2 * 57000 = 114000, so 912000 = 114000 * t. Step 9: Solve for t: t = 912000 / 114000 = 8 months. Step 10: Since the total year is 12 months and Amar is present for the last 8 months, he must have joined after 12 - 8 = 4 months from the start. Verification / Alternative check: Check the capital time products with t = 8 and a joining delay of 4 months. Madhu has 76000 * 12 = 912000 units. Amar has 57000 * 8 = 456000 units. The ratio 912000 : 456000 simplifies by dividing both by 456000 to 2 : 1, which matches the given profit ratio. Therefore, our interpretation that Amar joins after 4 months is correct. Why Other Options Are Wrong: If Amar had joined after 3, 5, 7 or 8 months, his period of investment would be 9, 7, 5 or 4 months respectively, leading to different capital time products and different profit ratios. For example, if Amar joined after 3 months, t would be 9, giving Madhu : Amar capital time units as 76000 * 12 : 57000 * 9 = 912000 : 513000, which does not simplify to 2 : 1. Hence these options do not match the required 2 : 1 profit ratio. Common Pitfalls: Some learners mistake t for the joining time instead of the period of investment and directly mark t = 8 months as the answer. It is important to distinguish between the total duration of the business and the time during which a partner is active. Another common error is incorrectly forming the ratio equation or forgetting to multiply by 12 for Madhu's full year of investment. Writing the steps clearly helps avoid confusion about what t represents. Final Answer: Amar joins the business after 4 months from the start, which corresponds to option D.

Question 18

Simran starts a software business by investing Rs. 50,000. After six months, Nanda joins her with a capital of Rs. 80,000. After 3 years from the start, the total profit is Rs. 24,500. What is Simran's share of the profit?

Accepted Answer

Introduction / Context: This is a partnership question where Simran starts a business and Nanda joins after six months. The total profit after 3 years is known. Since the partners invest different amounts for different periods, their shares of profit must be determined using capital time units. The question focuses on calculating Simran's share of the total profit. Given Data / Assumptions: Simran invests Rs. 50,000 at the beginning. Nanda joins after 6 months with Rs. 80,000. Total duration from the start is 3 years, that is 36 months. Total profit at the end of 3 years is Rs. 24,500. Profits are shared in proportion to capital multiplied by time. We must find Simran's profit share. Concept / Approach: Simran is in the business for the full 36 months, while Nanda is present only from month 7 to month 36, that is 30 months. We compute capital time units for both: capital multiplied by months. The ratio of their profit shares will be the ratio of these capital time units. With the total profit known, we can split it according to this ratio and obtain Simran's share. Step-by-Step Solution: Step 1: Simran invests Rs. 50,000 for 36 months, so Simran's capital time units are 50000 * 36 = 1800000. Step 2: Nanda joins after 6 months and therefore invests for 36 - 6 = 30 months. Step 3: Nanda's capital time units are 80000 * 30 = 2400000. Step 4: The ratio of Simran's to Nanda's profit shares is 1800000 : 2400000. Step 5: Simplify this ratio by dividing both numbers by 600000 to get 3 : 4. Step 6: Total profit is Rs. 24,500, which corresponds to 3 + 4 = 7 parts in this ratio. Step 7: Value of one part is 24500 / 7 = Rs. 3500. Step 8: Simran's share is 3 parts, so 3 * 3500 = Rs. 10,500. Verification / Alternative check: Nanda's share should be 4 parts = 4 * 3500 = Rs. 14,000. Adding both shares, 10500 + 14000 = Rs. 24,500, which matches the total profit given. This verifies that the ratio 3 : 4 and the resulting shares are correct, confirming Simran's profit as Rs. 10,500. Why Other Options Are Wrong: Other amounts such as Rs. 12421, Rs. 9423, Rs. 11600 or Rs. 9800 would not divide the total profit in the ratio 3 : 4. For instance, if Simran received Rs. 11600, Nanda would receive 24500 - 11600 = Rs. 12900, and the ratio 11600 : 12900 would not simplify to 3 : 4. Only Rs. 10500 gives a complementary share of Rs. 14000 for Nanda, producing the correct ratio and matching the total profit. Common Pitfalls: A common error is miscalculating the time period for which Nanda invests, sometimes taking it as 6 months or 36 months instead of 30 months. Another mistake is to use the capital ratio 50,000 : 80,000 directly without adjusting for time, which would lead to an incorrect profit ratio of 5 : 8. Careful reading of the joining time and accurate computation of capital time units are crucial in partnership problems. Final Answer: Simran's share of the profit is Rs. 10,500, which corresponds to option D.

Question 19

Three friends P, Q and R start a partnership business by investing in the ratio 5 : 4 : 2 respectively for a period of 3 years. The total amount invested is Rs. 22,000 and the profit earned at the end of 3 years is three eighths of the total investm…

Accepted Answer

Introduction / Context: This partnership problem gives the ratio of investments by three friends P, Q and R, the total amount invested, and the overall profit as a fraction of the total investment. Since all three invest for the same period of time, their shares of profit are proportional to their capital contributions. The question asks specifically for P's share of the profit. Given Data / Assumptions: Investment ratio of P : Q : R is 5 : 4 : 2. Total amount invested is Rs. 22,000. Time period of investment for all three is 3 years (same for each partner). Profit earned at the end of 3 years is three eighths of the total investment. We must find P's share of the profit in rupees. Concept / Approach: Since all partners keep their money invested for the same duration, the profit distribution depends only on the ratio of their capital amounts. We first determine the total profit by taking three eighths of the total investment. Then we divide this profit among P, Q and R according to the ratio 5 : 4 : 2, which has 11 parts in total. P's share is 5 parts out of 11, so we compute that part of the total profit. Step-by-Step Solution: Step 1: Total investment is Rs. 22,000. Step 2: Profit at the end of 3 years is given as three eighths of the total investment. Step 3: Compute total profit: (3/8) * 22000 = 3 * 22000 / 8. Step 4: Calculate 22000 / 8 = 2750. So profit = 3 * 2750 = Rs. 8250. Step 5: Investment ratio of P : Q : R is 5 : 4 : 2, so the profit is divided in the same ratio. Step 6: Sum of the ratio parts is 5 + 4 + 2 = 11 parts. Step 7: Value of one part of profit is total profit / 11 = 8250 / 11 = Rs. 750. Step 8: P's share is 5 parts, so P receives 5 * 750 = Rs. 3750. Verification / Alternative check: Check the shares of Q and R as well. Q's share is 4 * 750 = Rs. 3000. R's share is 2 * 750 = Rs. 1500. Adding all three, 3750 + 3000 + 1500 = Rs. 8250, which matches the total profit computed from three eighths of Rs. 22,000. This confirms that the distribution is correct and that P's share of Rs. 3750 is accurate. Why Other Options Are Wrong: Amounts like Rs. 2750, Rs. 3000, Rs. 4000 or Rs. 3500 would give different combinations for Q and R that would not match the total profit of Rs. 8250 or would not maintain the required ratio of 5 : 4 : 2. For example, if P's share were Rs. 3000, then one part would be 3000 / 5 = 600, and the total would become 11 * 600 = Rs. 6600, which contradicts the correct total profit. Only Rs. 3750 yields a per part value of Rs. 750 and a correct total of Rs. 8250. Common Pitfalls: Some learners incorrectly try to compute each partner's investment from the total of Rs. 22,000 and then re-compute profit shares, which is unnecessary since the ratio is already given. Others may miscalculate three eighths of 22,000 or forget to divide by the sum of the ratio parts. Working step by step, first finding total profit and then dividing by the ratio, helps avoid such mistakes. Final Answer: P's share of the profit is Rs. 3750, which corresponds to option C.

Question 20

Divide Rs. 1,18,000 among three persons A, B and C such that the ratio of A's share to B's share is 3 : 4 and the ratio of B's share to C's share is 5 : 6. What are the respective shares of A, B and C?

Accepted Answer

Introduction / Context: This is a ratio based division problem where a total sum is to be divided among three persons according to two linked ratios. The ratios connect A with B and B with C, and from these we must determine a single combined ratio for A : B : C and then compute their actual monetary shares from the given total amount. Given Data / Assumptions: Total amount to be divided among A, B and C is Rs. 1,18,000. Ratio of A's share to B's share is A : B = 3 : 4. Ratio of B's share to C's share is B : C = 5 : 6. We must find actual shares of A, B and C in rupees. Concept / Approach: We need a single combined ratio A : B : C that is consistent with both given ratios. To do this, we express A : B and B : C in such a way that the B term is the same in both ratios. Once a combined ratio is obtained, we treat it as a set of parts of the total amount. By dividing the total by the sum of the parts, we obtain the value of one part and then each person's share. Step-by-Step Solution: Step 1: Given A : B = 3 : 4, let A = 3x and B = 4x. Step 2: Given B : C = 5 : 6, let B = 5y and C = 6y. Step 3: Since B is common in both descriptions, we equate the two expressions for B. That is, 4x = 5y. Step 4: To make the B term equal in both ratios, choose a common value. One simple way is to take x = 5 and y = 4 so that 4x = 4 * 5 = 20 and 5y = 5 * 4 = 20. Step 5: With x = 5, A = 3x = 15 and B = 4x = 20. With y = 4, C = 6y = 24. Step 6: Therefore, the combined ratio A : B : C is 15 : 20 : 24. Step 7: Sum of the ratio parts is 15 + 20 + 24 = 59. Step 8: Total amount is Rs. 1,18,000. So one part is 118000 / 59. Step 9: Calculate 118000 / 59 = 2000. So each part is worth Rs. 2000. Step 10: A's share is 15 * 2000 = Rs. 30,000. Step 11: B's share is 20 * 2000 = Rs. 40,000. Step 12: C's share is 24 * 2000 = Rs. 48,000. Verification / Alternative check: Check that the shares satisfy the original ratios and sum to the total. First, A : B = 30000 : 40000 = 3 : 4 as required. Second, B : C = 40000 : 48000 = 5 : 6, which matches the second given ratio. Finally, 30000 + 40000 + 48000 = 118000, which equals the total amount. All conditions are satisfied, so the computed shares are correct. Why Other Options Are Wrong: The alternative options either assign a larger amount to A than to C or change the internal ratios incorrectly. For instance, in option A, A = 48000, B = 40000 and C = 30000, the ratio A : B becomes 6 : 5 instead of 3 : 4, and B : C becomes 4 : 3 instead of 5 : 6. Similarly, the other options do not simultaneously satisfy both given ratios. Only option B preserves both ratios correctly and sums to Rs. 1,18,000. Common Pitfalls: A frequent mistake is to add the ratios 3 : 4 and 5 : 6 directly or to average them instead of constructing a combined ratio through a common middle term. Some learners also forget to check whether the final shares obey both ratios. Always align the common term (here B), find the combined ratio, and only then apply it to the total amount. Final Answer: The correct division is A = Rs. 30,000, B = Rs. 40,000 and C = Rs. 48,000, which corresponds to option B.

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