Question 1

Kishlay travels three equal distances at speeds of 10 km/h, 30 km/h and 2 km/h and takes a total time of 38 minutes. Find the total distance in kilometres.

Accepted Answer

Introduction / Context: This question tests the standard time, speed and distance concept where a person covers equal distances at different speeds. Many aptitude tests include such problems because they check understanding of the relationship between speed, time and distance, and how to handle situations where segments of a journey are equal in length but covered at different speeds. Here we need to use the fact that total time is the sum of times for each segment, and all three segments have the same distance. Given Data / Assumptions: The journey has three equal parts in terms of distance. Speeds on these three parts are 10 km/h, 30 km/h and 2 km/h. Total time taken for the whole journey is 38 minutes. We assume uniform speed on each segment and no waiting time between segments. Concept / Approach: The basic relationship is distance = speed * time, or equivalently time = distance / speed. When equal distances are travelled at different speeds, it is convenient to let the common distance of each segment be d kilometres. Then we express the time for each segment in hours, sum them, and equate this total to the given overall time converted into hours. Finally, we solve for d and multiply by the number of segments to get the total distance travelled by Kishlay. Step-by-Step Solution: Step 1: Let the distance of each segment be d kilometres. Step 2: Time for first segment at 10 km/h is d / 10 hours. Step 3: Time for second segment at 30 km/h is d / 30 hours. Step 4: Time for third segment at 2 km/h is d / 2 hours. Step 5: Total time in hours is 38 minutes = 38 / 60 hours. Step 6: Form the equation d / 10 + d / 30 + d / 2 = 38 / 60. Step 7: Take common denominator 30 on the left side: (3d + d + 15d) / 30 = 19d / 30. Step 8: So 19d / 30 = 38 / 60. Simplify the right side: 38 / 60 = 19 / 30. Step 9: Therefore 19d / 30 = 19 / 30, which implies d = 1 kilometre. Step 10: There are three equal segments, so total distance = 3 * d = 3 km. Verification / Alternative check: If each segment is 1 km, then time at 10 km/h is 1 / 10 hour, at 30 km/h is 1 / 30 hour and at 2 km/h is 1 / 2 hour. Converting to minutes, these are 6 minutes, 2 minutes and 30 minutes respectively. The total time is 6 + 2 + 30 = 38 minutes, which matches the given total time. This confirms that the total distance of 3 kilometres is consistent with the information in the question and therefore correct. Why Other Options Are Wrong: 2 km would give each segment as 2 / 3 km, and the total time would no longer add up to 38 minutes. 1 km would mean each segment is only 1 / 3 km, which clearly reduces total time too much. 4 km would make each segment longer than 1 km and would result in a total time greater than 38 minutes. Thus only 3 km satisfies the time condition exactly for the three speeds. Common Pitfalls: A common mistake is to take the simple average of the three speeds instead of correctly using the time formula for equal distances. Another frequent error is forgetting to convert minutes into hours before forming the equation, which leads to wrong proportions. Some learners also mistakenly assume equal times instead of equal distances. Carefully setting up the variable d for equal distances and correctly converting units avoids these errors in similar time, speed and distance questions. Final Answer: The total distance covered by Kishlay when he travels three equal distances at the given speeds in a total of 38 minutes is 3 km.

Question 2

A, B and C start a business and invest in the ratio 7 : 6 : 5. In the next year they increase their investments by 25%, 20% and 15% respectively. In what ratio should the profit earned only during the second year be distributed among them?

Accepted Answer

Introduction / Context: This question is a classic partnership and ratio problem where partners change their capital in the second year. The key idea is that profit for a particular period is shared in proportion to the effective capital in that period. Since we are asked about profit only for the second year, we need to focus solely on the revised investments after the percentage increases are applied to the original ratio 7 : 6 : 5. Given Data / Assumptions: Initial investment ratio of A, B and C is 7 : 6 : 5. In the second year A increases capital by 25%. B increases capital by 20%. C increases capital by 15%. We assume time period for the second year is the same for all three partners. Concept / Approach: Profit in a partnership is proportional to capital multiplied by time. Here time for the entire second year is the same for all partners, so profit sharing is directly proportional to the capital invested in that year alone. We first calculate the increased capital of each partner relative to the original ratio using percentage increase formulas. Then we convert these new values into a simple ratio by removing decimals and simplifying to the lowest terms, which gives the required sharing ratio for the second year profit. Step-by-Step Solution: Step 1: Take initial capitals as 7, 6 and 5 units for A, B and C. Step 2: A increases capital by 25%, so new capital of A = 7 * (1 + 25 / 100) = 7 * 1.25 = 8.75 units. Step 3: B increases capital by 20%, so new capital of B = 6 * (1 + 20 / 100) = 6 * 1.20 = 7.2 units. Step 4: C increases capital by 15%, so new capital of C = 5 * (1 + 15 / 100) = 5 * 1.15 = 5.75 units. Step 5: To remove decimals, multiply all three by 100: 875 : 720 : 575. Step 6: Simplify by dividing each term by 5 to get 175 : 144 : 115. Step 7: Therefore the second year profit must be divided in the ratio 175 : 144 : 115 for A, B and C. Verification / Alternative check: One can also assume specific initial amounts, for example 7000, 6000 and 5000. Then after the given increases, the second year amounts become 8750, 7200 and 5750. Their greatest common divisor is 50, so the ratio is 175 : 144 : 115 which matches the earlier calculation. Any consistent scaling of initial capitals leads to the same simplified ratio, confirming that the result is correct and independent of the absolute amount chosen initially. Why Other Options Are Wrong: The ratio 155 : 144 : 175 does not match the percentage increase pattern and comes from miscalculating A and C. The ratio 155 : 124 : 95 underestimates all partners and particularly distorts B and C values. The option 135 : 147 : 152 is inconsistent with the fact that A had the largest percentage increase and should have the highest proportion in the second year. Only 175 : 144 : 115 correctly reflects the adjusted capitals based on 25%, 20% and 15% increments. Common Pitfalls: Learners sometimes add the same absolute increment to each partner instead of calculating percentage increases separately. Another common mistake is to include the first year capital in the calculation even though the question asks only about profit in the second year. Forgetting to simplify ratios by clearing decimals also leads to different but equivalent looking numbers that do not match the answer choices. Careful handling of percentages and focusing only on the required year avoids these errors. Final Answer: The profit earned only during the second year should be distributed among A, B and C in the ratio 175 : 144 : 115.

Question 3

A, B and C enter into a partnership by investing a total of Rs 6000. A invests Rs 1000 and the remaining amount is invested by B and C in the ratio 2 : 3. If the annual profit is Rs 2400, what is the share of C in the profit?

Accepted Answer

Introduction / Context: This is a straightforward partnership question in which three partners together invest a fixed total capital. One partner has a specified amount, and the remaining capital is divided between the other two in a given ratio. Since all partners remain in the business for the same period, profit sharing is directly proportional to their capital investment. The task is to determine the share of profit received by partner C from the total annual profit. Given Data / Assumptions: Total investment by A, B and C together is Rs 6000. A invests Rs 1000. The remaining investment of Rs 5000 is shared by B and C in the ratio 2 : 3. Total annual profit is Rs 2400. All partners are assumed to invest for the same duration of one year. Concept / Approach: When all partners remain in the business for the same time, their shares in the profit are proportional to their invested capital. First we compute the individual capital of B and C using the given ratio. Then we form the capital ratio of A, B and C. Finally we apply this ratio to the total profit to find the share that belongs to C. The key is to correctly handle the division of the remaining capital in the ratio 2 : 3. Step-by-Step Solution: Step 1: Total capital is Rs 6000, with A investing Rs 1000. Step 2: Remaining capital for B and C together is 6000 - 1000 = Rs 5000. Step 3: B and C share this 5000 in ratio 2 : 3. Step 4: Sum of ratio parts = 2 + 3 = 5 parts. Step 5: Value of 1 part = 5000 / 5 = Rs 1000. Step 6: Capital of B = 2 * 1000 = Rs 2000. Step 7: Capital of C = 3 * 1000 = Rs 3000. Step 8: Capital ratio A : B : C = 1000 : 2000 : 3000 = 1 : 2 : 3. Step 9: Total parts in ratio = 1 + 2 + 3 = 6 parts. Step 10: Each part of profit = 2400 / 6 = Rs 400. Step 11: Share of C = 3 parts = 3 * 400 = Rs 1200. Verification / Alternative check: We can verify by computing all shares. A gets 1 part, which is Rs 400. B gets 2 parts, which is Rs 800. C gets 3 parts, which is Rs 1200. The sum of these shares is 400 + 800 + 1200 = Rs 2400, exactly matching the total profit. The internal ratios 400 : 800 : 1200 simplify to 1 : 2 : 3, confirming that the computed share of C is consistent with the capitals and the total profit. Why Other Options Are Wrong: Rs 600 corresponds to only 1.5 parts in the ratio, which does not match the 3 parts assigned to C. Rs 1800 would give C an excessively large share relative to his capital. Rs 1950 does not fit any simple multiple of the base part value and would make the total profit different from Rs 2400. Only Rs 1200 correctly fits the capital ratio and adds up to the given profit. Common Pitfalls: Students sometimes mistakenly divide the total profit only in the ratio 2 : 3 between B and C, ignoring A. Another mistake is to divide Rs 6000 directly in the ratio 1 : 2 : 3 without first checking that A invests exactly Rs 1000. Carefully separating the known capital of A from the remaining capital and then applying the ratio 2 : 3 helps avoid these errors. Final Answer: The share of C in the annual profit is Rs 1200.

Question 4

A starts a business with a capital of Rs 1800. After 4 months B joins with Rs 3600, and 6 months after the business started C joins with Rs 1200. If the total profit at the end of the year is Rs 1760, what is the share of A in the profit?

Accepted Answer

Introduction / Context: This problem involves a partnership where different partners join the business at different times and with different capitals. In such questions, profit is shared according to capital multiplied by time in months, often called money months. The aim is to compute the effective contribution of each partner and then find the share of profit belonging to partner A, given the total profit at the end of one year. Given Data / Assumptions: A invests Rs 1800 from the start of the business. B joins after 4 months with Rs 3600. C joins after 6 months from the start with Rs 1200. Total duration of the business is 12 months. Total profit at the end of the year is Rs 1760. Concept / Approach: In partnerships with varying joining times, each partner share is proportional to capital * time. We calculate money months for each partner, form a ratio, and then divide the total profit according to this ratio. The steps are to compute A, B and C contributions over their respective time spans, simplify the ratio and finally compute A share of the total profit of Rs 1760. Step-by-Step Solution: Step 1: A invests Rs 1800 for the entire 12 months. Step 2: Money months of A = 1800 * 12 = 21600. Step 3: B joins after 4 months, so B is in business for 12 - 4 = 8 months. Step 4: Money months of B = 3600 * 8 = 28800. Step 5: C joins after 6 months, so C is in business for 12 - 6 = 6 months. Step 6: Money months of C = 1200 * 6 = 7200. Step 7: Ratio of contributions A : B : C = 21600 : 28800 : 7200. Step 8: Divide all by 7200 to simplify, giving 3 : 4 : 1. Step 9: Total parts = 3 + 4 + 1 = 8 parts. Step 10: Each part of profit = 1760 / 8 = Rs 220. Step 11: Share of A = 3 parts = 3 * 220 = Rs 660. Verification / Alternative check: We can also find shares of B and C. Share of B = 4 * 220 = Rs 880, share of C = 1 * 220 = Rs 220. The total of all shares is 660 + 880 + 220 = Rs 1760, which matches the given total profit. This confirms that the ratio of 3 : 4 : 1 based on money months has been applied correctly and that A share of Rs 660 is accurate. Why Other Options Are Wrong: Rs 540 corresponds to 540 / 220 = 2.45 parts, which does not align with an integer ratio. Rs 840 would represent 3.82 parts and cannot be justified by any correct ratio of contributions. Rs 720 would be 3.27 parts, which again is inconsistent. Only Rs 660 lines up perfectly with 3 parts of 8 total parts in the profit sharing ratio derived from the actual capital and time contributions. Common Pitfalls: Learners sometimes ignore the months and divide profit only according to capital, which is wrong when partners join at different times. Another mistake is to miscount months, for example taking B contribution as 4 months instead of 8. Forgetting to simplify the ratio or using inconsistent units, such as mixing months and years, can also lead to incorrect answers. Using money months consistently and carefully counting the duration solves this type of partnership question reliably. Final Answer: The share of A in the profit at the end of the year is Rs 660.

Question 5

A and B enter into a business by investing Rs 4000 and Rs 5500 respectively. After six months A and B withdraw Rs 1000 and Rs 1500 respectively, and at that time C joins them with a capital of Rs 4x. If after one year three months C receives Rs 2250…

Accepted Answer

Introduction / Context: This question is a more advanced partnership problem where partners change their capitals during the course of the business and a new partner joins later. The profit share depends on both the amount of capital and the time for which each amount remains invested. We are given the total profit and the share of C, and we need to work backwards to determine the capital of C in terms of rupees, using the algebraic relation involving x. Given Data / Assumptions: A invests Rs 4000 initially. B invests Rs 5500 initially. After 6 months, A withdraws Rs 1000 and B withdraws Rs 1500. At the same time, C joins with capital of Rs 4x. Total duration of the business is 15 months (1 year and 3 months). Total profit is Rs 12250 and C share is Rs 2250. Concept / Approach: The key concept is money months or capital time product. For each partner we calculate capital * time over each phase, sum these contributions and then share the profit in that resultant ratio. Since C share and total profit are known, C share fraction gives the ratio of C effective contribution to total effective contributions. This leads to an equation in x, from which we calculate 4x and match with the given options to find C capital. Step-by-Step Solution: Step 1: Business runs for 15 months. First 6 months have initial capitals, next 9 months have changed capitals and include C. Step 2: For first 6 months, capital of A is 4000 and of B is 5500, C is not yet a partner. Step 3: Money months of A in first phase = 4000 * 6 = 24000. Step 4: Money months of B in first phase = 5500 * 6 = 33000. Step 5: For next 9 months, A capital is 4000 - 1000 = 3000, B capital is 5500 - 1500 = 4000 and C capital is 4x. Step 6: Money months of A in second phase = 3000 * 9 = 27000. Step 7: Money months of B in second phase = 4000 * 9 = 36000. Step 8: Money months of C in second phase = 4x * 9 = 36x. Step 9: Total money months of A = 24000 + 27000 = 51000. Step 10: Total money months of B = 33000 + 36000 = 69000. Step 11: Total money months of C = 36x. Step 12: Let total effective contribution S = 51000 + 69000 + 36x = 120000 + 36x. Step 13: C share fraction = 2250 / 12250. Simplify: 2250 / 12250 = 9 / 49. Step 14: So 36x / (120000 + 36x) = 9 / 49. Step 15: Cross multiply: 36x * 49 = 9 * (120000 + 36x). Step 16: Left side = 1764x, right side = 1080000 + 324x. Step 17: Rearranging gives 1764x - 324x = 1080000, so 1440x = 1080000. Step 18: Therefore x = 1080000 / 1440 = 750. Step 19: C capital = 4x = 4 * 750 = Rs 3000. Verification / Alternative check: With x = 750, C effective contribution is 36x = 27000. Total effective contribution S becomes 51000 + 69000 + 27000 = 147000. The fraction of profit for C is 27000 / 147000 = 27 / 147 = 9 / 49. Applying this to total profit 12250 gives 12250 * 9 / 49 = 2250, which matches the stated share. This confirms that C capital must be Rs 3000 and that the algebraic steps are consistent. Why Other Options Are Wrong: Rs 3600, Rs 3200 and Rs 4400 correspond to different values of x and would change the fraction 36x / (120000 + 36x). None of those values produce the exact fraction 9 / 49 required to give C share of Rs 2250 from Rs 12250. Therefore those options do not satisfy the effective contribution equation and must be rejected. Only Rs 3000 generates the correct ratio and profit share. Common Pitfalls: A frequent error is to ignore the mid year withdrawal and joining, and assume a simple ratio of capital only. Another mistake is to use 12 months instead of 15 months for the duration. Some learners also forget that C contributes only in the second phase, not for the entire period. Setting up money months for each partner separately in each phase and then forming a single equation based on the known profit share is the most reliable method for partnership questions of this style. Final Answer: The value of the investment of C is Rs 3000.

Question 6

Rs 3200 is divided among A, B and C in the ratio 3 : 5 : 8 respectively. What is the difference in rupees between the shares of B and C?

Accepted Answer

Introduction / Context: This is a basic ratio and proportion question often seen in aptitude exams. A fixed sum of money is divided among three people in a given ratio, and we are asked to find the difference between the shares of two of them. Such problems strengthen understanding of how to convert ratio parts into actual amounts and how to handle simple comparisons between those amounts. Given Data / Assumptions: Total amount to be divided is Rs 3200. The ratio of the shares of A, B and C is 3 : 5 : 8. We need the difference between the shares of B and C. All three receive only this amount and there are no other adjustments or conditions. Concept / Approach: When a sum is divided in a ratio, the ratio numbers represent proportional parts of the total. The standard method is to sum the ratio parts, divide the total amount by this sum to find the value of one part, and then multiply by the respective ratio numbers to find individual shares. Finally we find the difference between the specific shares required, which in this question are B and C amounts. Step-by-Step Solution: Step 1: Ratio of A : B : C is 3 : 5 : 8. Step 2: Sum of ratio parts = 3 + 5 + 8 = 16. Step 3: Value of one part = total amount / total parts = 3200 / 16. Step 4: Compute 3200 / 16 = Rs 200 per part. Step 5: Share of A = 3 * 200 = Rs 600. Step 6: Share of B = 5 * 200 = Rs 1000. Step 7: Share of C = 8 * 200 = Rs 1600. Step 8: Difference between shares of B and C = 1600 - 1000 = Rs 600. Verification / Alternative check: We can verify that the shares add up to the total amount. A share is 600, B share is 1000 and C share is 1600. Their sum is 600 + 1000 + 1600 = 3200, which equals the original total. Additionally, the ratio 600 : 1000 : 1600 simplifies by dividing each term by 200 to 3 : 5 : 8, which matches the given ratio. This confirms that the calculations are consistent and that the difference of Rs 600 is correct. Why Other Options Are Wrong: Rs 400 would correspond to a smaller difference and would contradict the derived amounts 1000 and 1600. Rs 800 would suggest that C has 800 more than B, which does not align with the ratio 3 : 5 : 8 when converted to actual values for a total of Rs 3200. Rs 900 is not consistent with any valid combination of shares under the given ratio. Only Rs 600 fits the correct calculation for the difference between Rs 1000 and Rs 1600. Common Pitfalls: Some learners divide the total by just one partner ratio number or confuse which shares to subtract. Others may incorrectly treat 3, 5 and 8 as percentages. Always remember to first sum the ratio numbers and then distribute the total proportionally. Clearly writing out each share before subtracting helps avoid sign errors and misinterpretation of the ratio. Final Answer: The difference between the shares of B and C is Rs 600.

Question 7

A, B and C invest amounts in the ratio 3 : 4 : 5 respectively. The three schemes offer compound interest at 20% per annum, 15% per annum and 10% per annum respectively. What will be the ratio of their amounts after 1 year?

Accepted Answer

Introduction / Context: This question combines ratio concepts with simple compound interest for a single year. Although the schemes are described as compound interest schemes, for exactly one year the amount is simply principal multiplied by 1 plus the interest rate. Our task is to find the ratio of the final amounts after one year when three different interest rates are applied to capitals in an initial ratio of 3 : 4 : 5. Given Data / Assumptions: Initial investment ratio A : B : C = 3 : 4 : 5. A earns compound interest at 20% per annum. B earns compound interest at 15% per annum. C earns compound interest at 10% per annum. We consider only 1 year duration for all three investments. Concept / Approach: For compound interest, amount after 1 year is given by A = P * (1 + R / 100). Since time is 1 year for all, we do not need higher powers. We treat the ratio capitals as actual principal values, multiply each by the appropriate factor and then form a ratio of the resulting amounts. Finally, we remove any decimals by scaling and reduce to the simplest ratio which matches one of the answer options. Step-by-Step Solution: Step 1: Assume investments are 3, 4 and 5 units for A, B and C respectively. Step 2: Amount for A after 1 year at 20% = 3 * (1 + 20 / 100) = 3 * 1.20 = 3.6 units. Step 3: Amount for B after 1 year at 15% = 4 * (1 + 15 / 100) = 4 * 1.15 = 4.6 units. Step 4: Amount for C after 1 year at 10% = 5 * (1 + 10 / 100) = 5 * 1.10 = 5.5 units. Step 5: Now we have ratio 3.6 : 4.6 : 5.5. Step 6: Multiply each term by 10 to remove decimals: 36 : 46 : 55. Step 7: There is no common factor among 36, 46 and 55, so 36 : 46 : 55 is already in simplest form. Verification / Alternative check: We can verify by choosing actual principal amounts, for example Rs 3000, Rs 4000 and Rs 5000. After one year their amounts would be 3000 * 1.20 = 3600, 4000 * 1.15 = 4600 and 5000 * 1.10 = 5500. The ratio 3600 : 4600 : 5500 simplifies by dividing all terms by 100 to 36 : 46 : 55, which matches the derived ratio. This confirms that the calculation is correct and independent of the chosen base principal units. Why Other Options Are Wrong: The ratio 3 : 15 : 25 does not reflect the effect of different interest rates and seems to treat interest as simple additions to the original ratio. The option 6 : 6 : 5 suggests B and C amounts are almost equal, which contradicts a higher initial capital for C. The ratio 12 : 23 : 11 bears no relation to the percentage increments 20%, 15% and 10%. Only 36 : 46 : 55 is consistent with multiplying each original capital by its correct amount factor. Common Pitfalls: Students may mistakenly add or subtract the percentage directly to the ratio values instead of using the multiplicative factor 1 + R / 100. Another common error is to convert interest to simple interest for multiple years even though only one year is involved. Forgetting to clear decimals properly can make it hard to match answer options. Using the simple formula for one year and scaling to remove decimals ensures an accurate and clean ratio. Final Answer: The ratio of their amounts after one year is 36 : 46 : 55.

Question 8

A, B and C together receive an amount of Rs 8400 and distribute it among themselves in the ratio 6 : 8 : 7 respectively. They save in the ratio 3 : 2 : 4 respectively and B saves Rs 400. What is the ratio of the expenditures of A, B and C respective…

Accepted Answer

Introduction / Context: This problem studies income, savings and expenditure using ratios. Three people share a total amount, then they save in a given ratio, and we know the actual saving of one of them. From this information we need to compute their expenditures and finally express those expenditures as a ratio. It combines basic ratio operations with the idea that expenditure equals income minus saving. Given Data / Assumptions: Total amount received by A, B and C together is Rs 8400. Income ratio (shares received) A : B : C = 6 : 8 : 7. Savings ratio A : B : C = 3 : 2 : 4. B saves Rs 400. Expenditure for each person = income share minus saving of that person. Concept / Approach: We first convert the income ratio into actual incomes by using the total Rs 8400. Next we use the savings ratio along with B actual saving to compute the savings of A and C. Once incomes and savings are known for each person, expenditure is simply income minus saving. The last step is expressing the three expenditure amounts as a simplified ratio. This follows the standard approach of using ratios to convert between totals, shares, savings and expenditures. Step-by-Step Solution: Step 1: Income ratio A : B : C is 6 : 8 : 7. Step 2: Sum of ratio parts = 6 + 8 + 7 = 21. Step 3: Value of one income part = 8400 / 21 = Rs 400. Step 4: Income of A = 6 * 400 = Rs 2400. Step 5: Income of B = 8 * 400 = Rs 3200. Step 6: Income of C = 7 * 400 = Rs 2800. Step 7: Savings are in ratio 3 : 2 : 4 for A : B : C. Step 8: Let savings of A, B and C be 3s, 2s and 4s respectively. We are told that B saves Rs 400, so 2s = 400. Step 9: From 2s = 400 we get s = 200. Step 10: Saving of A = 3 * 200 = Rs 600. Step 11: Saving of B = 2 * 200 = Rs 400. Step 12: Saving of C = 4 * 200 = Rs 800. Step 13: Expenditure of A = 2400 - 600 = Rs 1800. Step 14: Expenditure of B = 3200 - 400 = Rs 2800. Step 15: Expenditure of C = 2800 - 800 = Rs 2000. Step 16: Expenditure ratio A : B : C = 1800 : 2800 : 2000. Step 17: Divide all by 200 to simplify, giving 9 : 14 : 10. Verification / Alternative check: We can verify that incomes minus savings equal expenditures and everything balances. Total savings = 600 + 400 + 800 = Rs 1800. Total expenditure = 1800 + 2800 + 2000 = Rs 6600. Sum of total savings and total expenditure is 1800 + 6600 = Rs 8400, which matches the original total received. Also, the savings are indeed in the ratio 3 : 2 : 4 and B saving of 400 matches the given condition. Hence the expenditure ratio 9 : 14 : 10 is consistent. Why Other Options Are Wrong: The ratio 6 : 8 : 7 is the income ratio, not the expenditure ratio, so option 6 : 8 : 7 is incorrect. The ratio 8 : 6 : 7 does not match the expenditure values and simply reorders the income ratio. The ratio 12 : 7 : 9 does not correspond to 1800, 2800 and 2000 in any simple scaling. Only 9 : 14 : 10 matches the correctly derived expenditure amounts. Common Pitfalls: Learners sometimes confuse income and savings ratios and may try to divide the total Rs 8400 directly according to the savings ratio. Another common oversight is to assume that B saving is equal to one of the income parts rather than a separate quantity. Clearly distinguishing income, saving and expenditure, and working stepwise from income to saving to expenditure, prevents such mistakes. Final Answer: The ratio of the expenditures of A, B and C is 9 : 14 : 10.

Question 9

Rs 490 is divided among A, B and C such that the share of A is half of the share of B and three times the share of C. What is the share of C in rupees?

Accepted Answer

Introduction / Context: This question is a basic application of linear relationships between three shares. One person share is linked to the second and third shares through stated multiples. Using these relationships and the total sum, we can set up equations and solve for each individual share. The final objective is to find how much C receives from the total Rs 490. Given Data / Assumptions: Total amount to be distributed is Rs 490. Share of A is half of share of B. Share of A is three times share of C. We need to find share of C. Concept / Approach: We can let one share be a variable and express the others in terms of that variable using the relationships given. Then we use the fact that the sum of all three shares equals Rs 490. This leads to a simple equation which we solve to find the base share. Once A share is known, we compute C share using the relation that A is three times C. This approach avoids guesswork and provides a clear algebraic route to the answer. Step-by-Step Solution: Step 1: Let share of A be A rupees, share of B be B rupees and share of C be C rupees. Step 2: A is half of B, so A = B / 2, which implies B = 2A. Step 3: A is three times C, so A = 3C, which implies C = A / 3. Step 4: Total amount is A + B + C = 490. Step 5: Substitute B = 2A and C = A / 3 into the total: A + 2A + A / 3 = 490. Step 6: Combine like terms: A + 2A = 3A, so 3A + A / 3 = 490. Step 7: Express in terms of a single denominator: 3A = 9A / 3, so 9A / 3 + A / 3 = 10A / 3. Step 8: Therefore 10A / 3 = 490. Step 9: Multiply both sides by 3 to get 10A = 1470. Step 10: So A = 1470 / 10 = Rs 147. Step 11: C = A / 3 = 147 / 3 = Rs 49. Verification / Alternative check: Using A = 147 and C = 49, B = 2A = 294. The total sum is A + B + C = 147 + 294 + 49 = 490, which matches the given total amount. Also, A is half of B because 147 is half of 294 and A is three times C because 147 = 3 * 49. All conditions are satisfied, so the calculation for C share is correct. Why Other Options Are Wrong: Rs 147 is the share of A, not of C, and would make A and C equal which violates A = 3C. Rs 294 is the share of B and is twice A, not one third of A. Rs 245 would make the sum of shares exceed 490 or break the given relationships. Only Rs 49 fits every constraint and correctly represents C share. Common Pitfalls: Some learners accidentally treat statements like half and three times in the reverse direction, for example assuming B is half of A instead of A being half of B. Another mistake is to forget to include all three shares when forming the total. Carefully translating the language of the problem into equations and checking each relationship after solving helps avoid these issues. Final Answer: The share of C from the Rs 490 is Rs 49.

Question 10

A man has three sons, two daughters and a wife. They divide a sum of Rs 19000 among themselves such that each daughter gets 1.5 times the amount received by each son and the wife receives Rs 600 less than each son. What is the total amount received …

Accepted Answer

Introduction / Context: This question deals with distribution of money among family members with given conditions relating the amounts each member receives. The distribution involves multiple dependents and a spouse, and the relationships are expressed as multiples or differences relative to a son share. We are asked to compute the total received by all three sons combined from the total Rs 19000. Given Data / Assumptions: The family consists of three sons, two daughters and one wife. Total amount to be divided among them is Rs 19000. Each daughter gets 1.5 times the amount received by each son. The wife receives Rs 600 less than each son. We need to find total amount received by the three sons together. Concept / Approach: We start by assuming the amount received by each son is x rupees. Using the conditions, we express each daughter and the wife share in terms of x. Then we sum all these shares for all family members, set this equal to the total Rs 19000 and solve for x. Finally we multiply x by three to get the total amount received by the three sons together. This approach leverages simple algebra with linear equations. Step-by-Step Solution: Step 1: Let amount received by each son be x rupees. Step 2: Each daughter receives 1.5 times amount of each son, so each daughter gets 1.5x. Step 3: The wife receives x - 600 rupees. Step 4: There are 3 sons, so total for sons = 3x. Step 5: There are 2 daughters, so total for daughters = 2 * 1.5x = 3x. Step 6: Wife share is x - 600. Step 7: Sum of all shares = amount for sons + daughters + wife = 3x + 3x + (x - 600). Step 8: Combine like terms: 3x + 3x + x - 600 = 7x - 600. Step 9: This total must equal 19000, so 7x - 600 = 19000. Step 10: Add 600 to both sides to get 7x = 19600. Step 11: Therefore x = 19600 / 7 = Rs 2800. Step 12: Total amount received by 3 sons together = 3x = 3 * 2800 = Rs 8400. Verification / Alternative check: Using x = 2800, each son gets 2800, each daughter gets 1.5 * 2800 = 4200, and the wife gets 2800 - 600 = 2200. Sons receive 3 * 2800 = 8400, daughters receive 2 * 4200 = 8400, and wife receives 2200. The total is 8400 + 8400 + 2200 = 19000, exactly matching the given total. This confirms that x has been calculated correctly and that total amount for the three sons is indeed Rs 8400. Why Other Options Are Wrong: Rs 2800 is the amount received by one son, not by all three sons. Rs 3600 and Rs 5600 do not satisfy the equation formed by the conditions and would change the daughter and wife shares, leading to a total different from Rs 19000. Only Rs 8400 is consistent with all conditions and gives a correct distribution for all family members. Common Pitfalls: Some learners incorrectly apply the 1.5 factor to the total for all sons rather than to each son. Others may forget that the wife amount is x minus 600 and treat it as a fixed amount independent of x. Mistakes also occur when counting family members, for example using the wrong number of sons or daughters. Writing each share separately in terms of x and checking the final sum against the total greatly reduces such errors. Final Answer: The total amount received by the three sons together is Rs 8400.

Question 11

Two partners invest Rs 12500 and Rs 8500 respectively in a business. They agree that 60% of the profit will be divided equally between them and the remaining 40% will be treated as interest on capital in proportion to their investments. If one partn…

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Introduction / Context: This is a partnership profit sharing question that mixes two methods of distribution. A fixed percentage of profit is divided equally between partners, while the remaining percentage is distributed in the ratio of capital as a form of interest on capital. We are told the difference between final amounts received by the partners and must work backwards to determine the total profit for the business. Given Data / Assumptions: Two partners invest capitals Rs 12500 and Rs 8500. 60% of the profit is divided equally between them. 40% of the profit is shared in proportion to their capitals. One partner receives Rs 240 more than the other. We need to find the total profit. Concept / Approach: Let the total profit be P. Then 60% of P, or 0.6P, is shared equally, so each partner gets 0.3P from this part. The remaining 40% of P, or 0.4P, is shared in the ratio of capitals 12500 : 8500. Once we compute each partner share from this second part, we can express their total individual amounts and calculate the difference between them. Setting this difference equal to Rs 240 gives a simple equation that allows us to solve for P. Step-by-Step Solution: Step 1: Let total profit be P rupees. Step 2: 60% of P is shared equally, so each partner receives 0.3P from this part. Step 3: Remaining profit = 40% of P = 0.4P. Step 4: Capital ratio = 12500 : 8500. Divide both by 500 to simplify to 25 : 17. Step 5: From the 0.4P portion, partner 1 gets (25 / (25 + 17)) * 0.4P = (25 / 42) * 0.4P. Step 6: Partner 2 gets (17 / 42) * 0.4P. Step 7: Total share of partner 1 = 0.3P + (25 / 42) * 0.4P. Step 8: Total share of partner 2 = 0.3P + (17 / 42) * 0.4P. Step 9: Difference between their shares comes only from the second part and equals 0.4P * (25 / 42 - 17 / 42) = 0.4P * (8 / 42). Step 10: Simplify (8 / 42) to 4 / 21, so difference = 0.4P * 4 / 21 = 1.6P / 21. Step 11: Given the difference is Rs 240, so 1.6P / 21 = 240. Step 12: Multiply both sides by 21 to get 1.6P = 240 * 21 = 5040. Step 13: So P = 5040 / 1.6 = 3150. Verification / Alternative check: With total profit P = 3150, 60% part is 0.6 * 3150 = 1890, so each partner gets 945 from this part. The remaining 40% is 0.4 * 3150 = 1260. Dividing 1260 in the ratio 25 : 17 gives partner 1 share 1260 * 25 / 42 = 750 and partner 2 share 1260 * 17 / 42 = 510. Total for partner 1 = 945 + 750 = 1695, total for partner 2 = 945 + 510 = 1455. The difference is 1695 - 1455 = Rs 240, which matches the given difference, confirming that the total profit is Rs 3150. Why Other Options Are Wrong: If total profit were Rs 3250, Rs 4050 or Rs 3550, the difference computed using the same method would not equal 240. Each alternative total profit value produces a different difference between partner shares. Only P = Rs 3150 yields exactly Rs 240 difference, so the other options do not satisfy the conditions of the problem. Common Pitfalls: A common mistake is to distribute the entire profit only in proportion to capitals or only equally, ignoring the mixed rule. Another error is miscalculating the ratio part 25 : 17 or forgetting to divide the 0.4P portion correctly. Some learners also treat 60% and 40% as amounts rather than fractions of P. Keeping the two parts separate and carefully following the ratio calculations helps to avoid these errors. Final Answer: The total profit made in the business is Rs 3150.

Question 12

Rithika has Rs 8000. She invests some of this amount in savings scheme P for 3 years and the remaining amount in savings scheme Q for 2 years. Scheme P offers simple interest at 20% per annum, while scheme Q offers compound interest at 30% per annum…

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Introduction / Context: This question combines simple interest and compound interest concepts with an additional condition about withdrawing part of the investment later. Rithika splits her money between two schemes with different interest types and durations. We know the difference between the interests earned from the two schemes and must determine the original investments in each scheme, then adjust the investment in one scheme by debiting an amount and finally find the difference between the adjusted amounts. Given Data / Assumptions: Total money available to Rithika is Rs 8000. She invests x rupees in scheme P for 3 years at simple interest 20% per annum. She invests 8000 - x rupees in scheme Q for 2 years at compound interest 30% per annum, compounded annually. Interest from scheme P is Rs 930 more than interest from scheme Q. Later, Rs 1500 is debited from the amount invested in scheme P. We must find the difference between the amounts invested in P and Q after this debit. Concept / Approach: We first express the interest from each scheme in terms of x. For simple interest, interest = principal * rate * time. For compound interest, we compute the amount after 2 years and subtract the principal to get interest. Using the condition that interest from P exceeds interest from Q by Rs 930, we form an equation in x and solve it. Once we know x, we identify the investments in P and Q. Then we reduce the investment in P by Rs 1500 and compare the adjusted amounts to get the required difference. Step-by-Step Solution: Step 1: Let amount invested in P be x rupees, so amount in Q is 8000 - x rupees. Step 2: Simple interest from P for 3 years at 20% per annum is I_P = x * 20 / 100 * 3 = 0.6x. Step 3: For Q, principal is 8000 - x, rate is 30% per annum, time is 2 years and interest is compound. Step 4: Amount in Q after 2 years is A_Q = (8000 - x) * (1 + 30 / 100)^2 = (8000 - x) * 1.3^2 = (8000 - x) * 1.69. Step 5: Compound interest from Q is I_Q = A_Q - (8000 - x) = (8000 - x) * (1.69 - 1) = (8000 - x) * 0.69. Step 6: We are told that interest from P is Rs 930 more than interest from Q, so 0.6x = 0.69 * (8000 - x) + 930. Step 7: Expand the right side: 0.69 * (8000 - x) = 0.69 * 8000 - 0.69x = 5520 - 0.69x. Step 8: So the equation becomes 0.6x = 5520 - 0.69x + 930 = 6450 - 0.69x. Step 9: Bring x terms together: 0.6x + 0.69x = 6450, so 1.29x = 6450. Step 10: Therefore x = 6450 / 1.29 = 5000. Step 11: So original investment in P is Rs 5000 and in Q is 8000 - 5000 = Rs 3000. Step 12: Now Rs 1500 is debited from the amount in P, so adjusted investment in P is 5000 - 1500 = Rs 3500. Step 13: Adjusted investment in Q remains Rs 3000. Step 14: Difference between amounts in P and Q after debit = 3500 - 3000 = Rs 500. Verification / Alternative check: With x = 5000, interest from P is I_P = 0.6 * 5000 = Rs 3000. For Q, principal is 3000. Amount after 2 years at 30% compound interest is 3000 * 1.3^2 = 3000 * 1.69 = 5070, so interest I_Q = 5070 - 3000 = Rs 2070. The difference I_P - I_Q = 3000 - 2070 = Rs 930, which matches the condition in the question. This confirms that x = 5000 is correct and that the later debit of Rs 1500 from P leading to a difference of Rs 500 between P and Q is consistent. Why Other Options Are Wrong: If the final difference between investments in P and Q were Rs 200, Rs 300 or Rs 400, it would imply different initial investments and would break the interest difference relation of Rs 930. Only the initial split 5000 and 3000 satisfies the interest condition, and after debiting Rs 1500 from P the resulting difference is exactly Rs 500. Therefore the other numerical options cannot match all the constraints of the problem. Common Pitfalls: A common mistake is to treat scheme Q as simple interest rather than compound interest, leading to a different interest equation. Another pitfall is to forget that Q runs for only 2 years while P runs for 3 years. Some learners may also misread the phrase about debiting Rs 1500 and incorrectly apply this change before computing the interest relation. Carefully distinguishing original investments, interest calculations and later adjustments helps to avoid these errors. Final Answer: After debiting Rs 1500 from the investment in scheme P, the difference between the amounts invested in P and Q is Rs 500.

Question 13

U and V start a business by investing Rs 184000 and Rs 224000 respectively. If the share of U in the profit at the end of the year is Rs 20700, what is the total profit earned by them together?

Accepted Answer

Introduction / Context: This is a direct partnership proportion question where two partners invest for the same duration. When the time period is equal for all partners, profit is shared in the ratio of their capitals. We are given the capital investments of U and V and the profit share of U, and we need to find the total profit for the year. Given Data / Assumptions: U invests Rs 184000. V invests Rs 224000. Both investments are for the same time period of one year. U share in the profit is Rs 20700. We must find total profit earned by U and V together. Concept / Approach: When partners invest for equal times, profit share is directly proportional to capital. So the ratio of U and V profits is the same as the ratio of their capitals, 184000 : 224000. We simplify this ratio, treat U share as corresponding to the first term, and then scale the ratio to find the full profit corresponding to the sum of both terms. This standard ratio scaling approach makes the calculation simple and transparent. Step-by-Step Solution: Step 1: Capital ratio U : V = 184000 : 224000. Step 2: Divide both terms by 8000 to simplify: 184000 / 8000 = 23 and 224000 / 8000 = 28. Step 3: So profit ratio U : V = 23 : 28. Step 4: Let common profit factor be k, so profit of U = 23k and profit of V = 28k. Step 5: Given profit of U is 20700, so 23k = 20700. Step 6: Therefore k = 20700 / 23 = 900. Step 7: Total profit = 23k + 28k = 51k. Step 8: Substitute k = 900, total profit = 51 * 900 = 45900. Verification / Alternative check: We can compute V share as 28k = 28 * 900 = 25200. Adding U and V shares gives 20700 + 25200 = 45900, which matches the total profit obtained from 51k. The ratio 20700 : 25200 simplifies to 23 : 28 when divided by 900, confirming that the profit shares are correctly proportional to the capitals 184000 and 224000. This double check validates that the total profit is Rs 45900. Why Other Options Are Wrong: Total profit of Rs 43200, Rs 52300 or Rs 56400 would lead to different shares for U and V that are not in the ratio 23 : 28 and would not give U exactly Rs 20700. Since the profit ratio must match the capital ratio and U share is fixed, only one total profit value can satisfy both conditions. That value is Rs 45900. Common Pitfalls: A common mistake is to subtract capitals or to compute some percentage difference instead of working directly with the ratio. Another error is solving 23k = 20700 incorrectly or approximating without proper division. Using a clear ratio approach, simplifying 184000 : 224000 correctly, and then solving for k ensures that such questions are handled accurately and quickly in exams. Final Answer: The total profit earned by U and V together at the end of the year is Rs 45900.

Question 14

A profit of Rs 103500 has to be divided between three partners A, B and C in the ratio 11 : 7 : 5. How much amount in rupees does B receive?

Accepted Answer

Introduction / Context: This question is a direct application of ratio distribution for profits among three partners. The total profit is given along with the ratio of shares. We are asked to find the amount received by one specific partner, B. Such problems build fluency in converting ratio parts into actual monetary values by using simple proportion methods. Given Data / Assumptions: Total profit to be divided is Rs 103500. Ratio of profit shares of A, B and C is 11 : 7 : 5. We must find the share of B. All three partners are assumed to be entitled to profit exactly in this ratio with no other adjustments. Concept / Approach: The ratio 11 : 7 : 5 represents how the total profit is split among A, B and C. The sum of the ratio parts tells us into how many equal parts the total profit is conceptually divided. Each partner share is then the product of the number of parts associated with that partner and the value of one part. For B we simply take 7 parts out of the total number of parts. Step-by-Step Solution: Step 1: Ratio A : B : C is 11 : 7 : 5. Step 2: Sum of ratio parts = 11 + 7 + 5 = 23. Step 3: Value of one part = total profit / total parts = 103500 / 23. Step 4: Compute 103500 / 23 = 4500. Step 5: B share equals 7 parts. Step 6: Share of B = 7 * 4500 = Rs 31500. Verification / Alternative check: We can also compute shares of A and C to validate the distribution. Share of A = 11 * 4500 = Rs 49500. Share of C = 5 * 4500 = Rs 22500. The total of all three shares is 49500 + 31500 + 22500 = 103500, which exactly matches the given total profit. Also, the ratio 49500 : 31500 : 22500 simplifies to 11 : 7 : 5 when divided by 4500, confirming that the calculations preserve the intended ratio. Why Other Options Are Wrong: Rs 22500 is the share of C, not B, based on 5 parts of the ratio. Rs 49500 is the share of A, corresponding to 11 parts. Rs 50500 does not match any partner share and would make the total exceed the given 103500 or change the ratio. Only Rs 31500 corresponds to 7 parts out of 23 and fits the given ratio and total. Common Pitfalls: Learners sometimes forget to sum the ratio correctly or miscalculate the value of one part. Another frequent error is misidentifying which ratio number belongs to which partner. Writing the ratio clearly, labelling each term with A, B and C, and checking that the final shares add up to the total profit helps avoid such mistakes. Final Answer: B receives a profit share of Rs 31500.

Question 15

A, B and C together start a business. Three times the investment of A is equal to four times the investment of B, and the capital of B is twice the capital of C. What is the ratio of their shares in the profit?

Accepted Answer

Introduction / Context: This partnership question focuses on expressing several capital relationships in a simple ratio form. When time of investment is the same for all partners, profit is directly proportional to the invested capital. By translating the given equalities between the capitals of A, B and C into a ratio, we can find the ratio of their shares in the profit without any explicit profit value being given. Given Data / Assumptions: Three times the investment of A equals four times the investment of B. Capital of B is twice the capital of C. All three invest for the same duration. Profit shares are directly proportional to their capitals. Concept / Approach: We assign variables to capitals of A, B and C and then use the given equations to relate them. Solving these equations gives each capital in terms of a single variable, leading to a clean ratio. Since profit shares are proportional to capitals, the final capital ratio is the same as the profit ratio we are asked to find. This is a standard technique for handling chained ratio conditions. Step-by-Step Solution: Step 1: Let investments be A, B and C for the three partners. Step 2: We are told that 3A = 4B. Step 3: We are also told that B = 2C. Step 4: From B = 2C, write C = B / 2. Step 5: From 3A = 4B, write A = 4B / 3. Step 6: Now express A, B and C all in terms of C for easier comparison. Step 7: Since B = 2C, substitute in A = 4B / 3 to get A = 4 * (2C) / 3 = 8C / 3. Step 8: So A : B : C = 8C / 3 : 2C : C. Step 9: Multiply all terms by 3 to clear the denominator: 8C : 6C : 3C. Step 10: Cancel C to get capital ratio A : B : C = 8 : 6 : 3. Step 11: Because time is equal, profit ratio is also 8 : 6 : 3. Verification / Alternative check: We can pick specific numerical values consistent with this ratio. For example, let C = 3 units, then B = 2C = 6 units and A must satisfy 3A = 4B = 24, so A = 8 units. This gives A : B : C = 8 : 6 : 3 which matches the derived ratio. Any scaling of these numbers preserves all given conditions, confirming that 8 : 6 : 3 is the correct and unique ratio for their capitals and profits. Why Other Options Are Wrong: Ratios 8 : 3 : 6, 3 : 8 : 6 and 3 : 6 : 8 do not satisfy the relationships 3A = 4B and B = 2C simultaneously. Checking them quickly shows that they fail either the multiple condition between A and B or the relation between B and C. Only the ratio 8 : 6 : 3 preserves both constraints and therefore is the only possible profit ratio. Common Pitfalls: A common mistake is to misinterpret the order of partners in the ratio or to incorrectly manipulate the equations, for example writing 3A / 4B = 1 and then guessing values. Another issue is forgetting to clear denominators properly, which leads to fractional ratios that are not simplified. Stepwise substitution and scaling as done above avoids confusion and ensures a consistent ratio. Final Answer: The ratio of their shares in the profit is 8 : 6 : 3 for A, B and C respectively.

Question 16

A and B start an enterprise together, with A as the active partner. A invests Rs 4000 and invests an additional Rs 2000 after 8 months. B invests Rs 5000 and withdraws Rs 2000 after 9 months. Being the active partner, A also takes an allowance of Rs…

Accepted Answer

Introduction / Context: This partnership question involves changing capitals over time and an active partner allowance taken from the profit. A invests more capital partway through the year, while B withdraws some capital after nine months. Additionally, A receives a fixed monthly allowance from the profit for managing the business. We need to find B final share of the profit after accounting for A allowance and capital time effect for both partners. Given Data / Assumptions: Total duration of the business is 12 months. A initially invests Rs 4000. A invests an additional Rs 2000 after 8 months, so his capital becomes Rs 6000 for the last 4 months. B invests Rs 5000 from the start and withdraws Rs 2000 after 9 months, so his capital becomes Rs 3000 for the last 3 months. A, being active partner, takes allowance of Rs 100 per month, that is 100 * 12 = Rs 1200 from the profit. Total profit before distribution is Rs 6700. Concept / Approach: We first deduct A allowance of Rs 1200 from the total profit to get the remaining profit to be shared in proportion to effective capital time contributions, or money months, of A and B. We compute money months separately for each phase where capitals remain constant. These values give us the ratio of A and B for profit sharing. Applying this ratio to the remaining profit yields the individual shares, from which we identify B part. Step-by-Step Solution: Step 1: Total profit is Rs 6700. A takes allowance of Rs 100 per month for 12 months, that is 100 * 12 = Rs 1200. Step 2: Remaining profit to be shared according to capital is 6700 - 1200 = Rs 5500. Step 3: For A, first 8 months capital is 4000, so money months = 4000 * 8 = 32000. Step 4: For last 4 months, A capital is 4000 + 2000 = 6000, so money months = 6000 * 4 = 24000. Step 5: Total money months of A = 32000 + 24000 = 56000. Step 6: For B, first 9 months capital is 5000, so money months = 5000 * 9 = 45000. Step 7: For last 3 months, B capital is 5000 - 2000 = 3000, so money months = 3000 * 3 = 9000. Step 8: Total money months of B = 45000 + 9000 = 54000. Step 9: Ratio of contributions A : B = 56000 : 54000. Step 10: Divide both terms by 2000, giving 28 : 27 as the simplified ratio. Step 11: Total ratio parts = 28 + 27 = 55 parts. Step 12: Share per part of the remaining profit = 5500 / 55 = Rs 100. Step 13: A share from remaining profit = 28 * 100 = Rs 2800, while B share = 27 * 100 = Rs 2700. Verification / Alternative check: We can confirm that the total distribution is consistent. A receives allowance Rs 1200 plus profit share Rs 2800, totalling Rs 4000. B receives Rs 2700. Combined, these sums are 4000 + 2700 = Rs 6700, which matches the original total profit. The ratio of 56000 : 54000 simplifies correctly to 28 : 27, verifying the basis for the profit split. This confirms B share is correctly calculated as Rs 2700. Why Other Options Are Wrong: If B share were Rs 3350, Rs 3250 or Rs 2800, the total distribution would no longer match Rs 6700 after also accounting for A allowance and capital contribution. These values would break the 28 : 27 ratio of money months or lead to an incorrect residual profit share for A. Only Rs 2700 fits both the ratio and the total profit condition. Common Pitfalls: One common error is to ignore A allowance and share the entire Rs 6700 by the 28 : 27 ratio. Another is to mistakenly treat capital changes as happening at the start of the year rather than after 8 or 9 months. Learners may also forget to break the year into segments with different capitals. Carefully separating allowance, subtracting it, and then computing money months for each phase prevents these mistakes. Final Answer: The share of B in the profit is Rs 2700.

Question 17

A sum of Rs 15525 is divided among Sunil, Anil and Jamil so that if Rs 22, Rs 35 and Rs 48 are subtracted from their respective shares, the remaining amounts are in the ratio 7 : 10 : 13. What would be the ratio of their shares if instead Rs 16, Rs …

Accepted Answer

Introduction / Context: This problem requires handling a sum divided among three persons with conditions involving both subtraction and addition of different amounts. Initially, subtracting certain sums yields a given ratio. We must first use this information to determine the original individual shares. Then we apply a different set of additions to those original shares and find the new ratio. This tests linear equation setup and careful manipulation of constants in distribution problems. Given Data / Assumptions: Total sum divided among Sunil, Anil and Jamil is Rs 15525. When Rs 22, Rs 35 and Rs 48 are subtracted from their shares respectively, the remaining amounts are in the ratio 7 : 10 : 13. We are asked for the ratio of their shares if Rs 16, Rs 77 and Rs 37 are added to their original shares. There are no other conditions on time or profit; this is a pure division problem. Concept / Approach: We denote the original shares by variables and use the given ratio after subtraction to express them in terms of a common factor. This relation, combined with the total sum condition, allows us to solve for that factor and thus obtain concrete values of each share. Then we add the new constants Rs 16, Rs 77 and Rs 37 to these shares and form a new ratio. Finally, we simplify this ratio to its lowest integer form to match the answer options. Step-by-Step Solution: Step 1: Let original shares of Sunil, Anil and Jamil be S, A and J respectively. Step 2: After subtracting Rs 22, Rs 35 and Rs 48, we are told (S - 22) : (A - 35) : (J - 48) = 7 : 10 : 13. Step 3: Let common factor be k, so S - 22 = 7k, A - 35 = 10k and J - 48 = 13k. Step 4: Therefore S = 7k + 22, A = 10k + 35 and J = 13k + 48. Step 5: Total sum S + A + J = 15525, so (7k + 22) + (10k + 35) + (13k + 48) = 15525. Step 6: Combine like terms: (7k + 10k + 13k) + (22 + 35 + 48) = 30k + 105. Step 7: So 30k + 105 = 15525. Step 8: Subtract 105 from both sides: 30k = 15525 - 105 = 15420. Step 9: k = 15420 / 30 = 514. Step 10: Now compute original shares. S = 7 * 514 + 22 = 3598 + 22 = 3620. Step 11: A = 10 * 514 + 35 = 5140 + 35 = 5175. Step 12: J = 13 * 514 + 48 = 6682 + 48 = 6730. Step 13: Check total: 3620 + 5175 + 6730 = 15525, which is correct. Step 14: Now add new amounts to original shares: Sunil gets S + 16 = 3620 + 16 = 3636. Step 15: Anil gets A + 77 = 5175 + 77 = 5252. Step 16: Jamil gets J + 37 = 6730 + 37 = 6767. Step 17: New ratio is 3636 : 5252 : 6767. Step 18: To simplify, find common factor. All three numbers are divisible by 101. Step 19: 3636 / 101 = 36, 5252 / 101 = 52 and 6767 / 101 = 67. Step 20: So the new ratio is 36 : 52 : 67. Verification / Alternative check: We can verify the initial condition. After subtracting the earlier amounts, we have 3620 - 22 = 3598, 5175 - 35 = 5140 and 6730 - 48 = 6682. Dividing each by 7, 10 and 13 respectively, we find 3598 / 7 = 514, 5140 / 10 = 514 and 6682 / 13 = 514, confirming that a common factor k = 514 works and the ratio is correct. This consistency shows that the original shares are accurate and that the added amounts lead to the rightly simplified ratio 36 : 52 : 67. Why Other Options Are Wrong: Ratios 9 : 13 : 17 and 18 : 26 : 35 do not match the calculated values 3636, 5252 and 6767 when simplified, and cannot come from a simple common factor division. Option None of these is invalid because we found a valid ratio that matches option 36 : 52 : 67 exactly. Therefore 36 : 52 : 67 is the only ratio consistent with both the subtraction and addition conditions. Common Pitfalls: Some learners may mistakenly add or subtract the constants directly to the ratio numbers rather than to the actual shares. Others may skip checking that the original shares satisfy the total sum condition before moving to the second part of the question. Handling the algebra carefully, writing all equations explicitly and verifying both initial and modified conditions helps avoid these mistakes. Final Answer: After adding the new amounts to their original shares, the ratio of the sums for Sunil, Anil and Jamil is 36 : 52 : 67.

Question 18

Income of A is Rs 140 more than income of B, and income of C is Rs 80 more than income of D. The ratio of incomes of A and C is 2 : 3 and the ratio of incomes of B and D is 1 : 2. What are the incomes of A, B, C and D respectively?

Accepted Answer

Introduction / Context: This question involves four related incomes with both difference conditions and ratio conditions. Income of A is related to B by a fixed difference, and income of C is related to D by another fixed difference. At the same time, the ratios A : C and B : D are given. We need to solve this system of relationships to find the exact incomes of A, B, C and D. This is essentially a simultaneous equations problem with ratios. Given Data / Assumptions: Income of A is Rs 140 more than income of B. Income of C is Rs 80 more than income of D. Ratio of incomes A : C is 2 : 3. Ratio of incomes B : D is 1 : 2. We must determine incomes of A, B, C and D. Concept / Approach: We translate each word statement into an algebraic equation. We use one variable to represent income of B and express income of D in terms of B using the ratio B : D. Then we express A in terms of B using the difference condition, and C in terms of D using the other difference condition. Finally, we apply the ratio A : C = 2 : 3 to connect these expressions and solve for B. Once B is known, A, C and D follow directly from their relationships with B. Step-by-Step Solution: Step 1: Let income of B be B and income of D be D. Step 2: Given B : D = 1 : 2, so D = 2B. Step 3: A is Rs 140 more than B, so A = B + 140. Step 4: C is Rs 80 more than D, so C = D + 80. Step 5: Substitute D = 2B into C expression to get C = 2B + 80. Step 6: Ratio A : C is given as 2 : 3, so (A / C) = 2 / 3. Step 7: Substitute A = B + 140 and C = 2B + 80 into the ratio equation: (B + 140) / (2B + 80) = 2 / 3. Step 8: Cross multiply: 3(B + 140) = 2(2B + 80). Step 9: Expand both sides: 3B + 420 = 4B + 160. Step 10: Rearrange to solve for B: 420 - 160 = 4B - 3B gives 260 = B. Step 11: So income of B is Rs 260. Step 12: Income of D is D = 2B = 2 * 260 = Rs 520. Step 13: Income of A is A = B + 140 = 260 + 140 = Rs 400. Step 14: Income of C is C = D + 80 = 520 + 80 = Rs 600. Verification / Alternative check: Check each condition with A = 400, B = 260, C = 600 and D = 520. Difference A - B = 400 - 260 = 140, which matches the first condition. Difference C - D = 600 - 520 = 80, matching the second condition. Ratio A : C = 400 : 600 simplifies by dividing by 200 to 2 : 3. Ratio B : D = 260 : 520 simplifies by dividing by 260 to 1 : 2. All given relationships hold, so these values are consistent and correct. Why Other Options Are Wrong: Other options either fail the difference condition, the ratio condition, or both. For example, in option A, 260 and 120 do not have a difference of 140, and the ratios also fail. Similarly, in option B and option D, at least one of the given ratios or differences between incomes is incorrect. Only option C fulfils A - B = 140, C - D = 80, A : C = 2 : 3 and B : D = 1 : 2 simultaneously. Common Pitfalls: Some learners may try to work with multiple variables without reducing them using ratios, which makes the algebra heavier. Others may accidentally reverse the ratio B : D and set B = 2D, which changes the relationships. Careful reading of the ratios and differences and using a small number of variables with systematic substitution helps avoid errors and reach the correct incomes quickly. Final Answer: The incomes of A, B, C and D are Rs 400, Rs 260, Rs 600 and Rs 520 respectively.

Question 19

A starts a cement trading business by investing Rs 5 lakhs. After 2 months B joins the business by investing Rs 10 lakhs, and 4 months after B joins, C also joins by investing Rs 20 lakhs. One year after A started the business the total profit is Rs…

Accepted Answer

Introduction / Context: This question describes a partnership where partners join at different times with different capitals. The business runs for one year from the time A starts. B and C join later with larger capitals. Profit is to be shared according to capital multiplied by the time for which each capital remained invested. We must calculate B effective contribution in money months and use it to find B share from the total profit of Rs 350000. Given Data / Assumptions: A invests Rs 5 lakhs at the start of the business. B joins after 2 months investing Rs 10 lakhs. C joins 4 months after B, which is 6 months after A, investing Rs 20 lakhs. Total time from A start to profit calculation is 12 months. Total profit after 1 year is Rs 350000. Concept / Approach: The key idea is money months. We treat capitals in lakhs or rupees consistently over the periods they are invested. A is in the business for the full 12 months, B for 10 months and C for 6 months. We compute capital * time for each partner, which gives the relative weights for sharing the profit. Profit is then divided in the ratio of these money months. Finally, we apply this ratio to the total profit to find B share. Step-by-Step Solution: Step 1: Use capitals in lakhs for simplicity. A capital = 5, B capital = 10, C capital = 20. Step 2: Time for A is full year = 12 months. Step 3: Time for B is from month 2 to month 12, that is 10 months. Step 4: Time for C is from month 6 to month 12, that is 6 months. Step 5: Money months for A = 5 * 12 = 60 units. Step 6: Money months for B = 10 * 10 = 100 units. Step 7: Money months for C = 20 * 6 = 120 units. Step 8: Ratio of contributions A : B : C = 60 : 100 : 120. Step 9: Simplify ratio by dividing all terms by 20, giving 3 : 5 : 6. Step 10: Total ratio parts = 3 + 5 + 6 = 14 parts. Step 11: Value of one part of profit = 350000 / 14. Step 12: Compute 350000 / 14 = 25000. Step 13: B share corresponds to 5 parts, so B share = 5 * 25000 = Rs 125000. Verification / Alternative check: We can find shares of A and C to confirm correctness. A share = 3 * 25000 = Rs 75000. B share = 5 * 25000 = Rs 125000. C share = 6 * 25000 = Rs 150000. Sum of all shares is 75000 + 125000 + 150000 = 350000, which matches the total profit. The ratio 75000 : 125000 : 150000 simplifies by dividing by 25000 to 3 : 5 : 6, confirming that money month ratio has been applied correctly. Why Other Options Are Wrong: If B share were Rs 75000, Rs 100000 or Rs 150000, the corresponding ratio among A, B and C would not match 3 : 5 : 6 given the fixed total profit. For example, Rs 150000 for B would make B equal to or larger than C in a way inconsistent with the capital time contributions. Only Rs 125000 ensures all shares are proportional to 3 : 5 : 6 and sum to Rs 350000. Common Pitfalls: Learners may mistakenly use 12 months for all three partners or miscount the months between joining times. Another frequent error is forgetting to simplify the money months correctly before extracting the ratio. Some also misread the phrase 4 months after B joins and assume C joins 4 months from the start. Carefully counting months from A start and forming capital * time products solves such partnership timing questions reliably. Final Answer: B share of the profit from the cement trading business is Rs 125000.

Question 20

A, B and C invest in a business in the ratio 3 : 6 : 5. A and C are working partners, while B is a sleeping partner whose share of profit is only three-fourths of what it would have been if he were a working partner. The firm earns a profit of Rs. 5…

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Introduction / Context: This partnership question tests understanding of how profit is shared when one partner is a sleeping partner whose share is reduced relative to working partners. The capital contribution ratio among A, B and C is given, and B does not work in the business. Only a part of the total profit is actually distributed, and we are asked to find C's share of this distributed profit. Such questions are important for exams on business mathematics and quantitative aptitude because they combine ratios, proportional reasoning and interpretation of conditions about working and sleeping partners. Given Data / Assumptions: - Capital ratio of A : B : C is 3 : 6 : 5.- A and C are working partners.- B is a sleeping partner and will receive only three-fourths of his normal profit share.- Total profit earned by the firm is Rs. 50,000.- Half of the profit is reinvested and half, that is Rs. 25,000, is distributed among the partners.- We need to find C's share in the distributed profit. Concept / Approach: The basic idea is that profit sharing is proportional to the product of capital and any adjustment factors, such as working or sleeping partner status. First, we interpret the ratio 3 : 6 : 5 as the base ratio of profits if all three were active working partners. Since B is a sleeping partner and receives only three-fourths of his normal share, we reduce B's share by multiplying his part of the ratio by 3/4. This produces an effective profit-sharing ratio. Finally, we apply this adjusted ratio to the portion of profit that is actually distributed, which is Rs. 25,000, and compute C's share. Step-by-Step Solution: Step 1: Base capital ratio is A : B : C = 3 : 6 : 5.Step 2: If B were a working partner, his profit share would be proportional to 6 parts. As a sleeping partner, he receives only three-fourths of that share.Step 3: Adjust B's ratio part: effective B part = 6 * (3/4) = 4.5.Step 4: The effective profit-sharing ratio becomes A : B : C = 3 : 4.5 : 5.Step 5: Sum of ratio parts = 3 + 4.5 + 5 = 12.5.Step 6: Out of Rs. 50,000 total profit, only half is distributed: distributed profit = 50,000 / 2 = Rs. 25,000.Step 7: C's share = (C's ratio part / total parts) * distributed profit.Step 8: C's share = (5 / 12.5) * 25,000.Step 9: 5 / 12.5 = 0.4, so C's share = 0.4 * 25,000 = Rs. 10,000. Verification / Alternative check: Check each partner's share using the effective ratio.Total units = 12.5; each unit = 25,000 / 12.5 = Rs. 2,000.A's share = 3 * 2,000 = Rs. 6,000.B's share = 4.5 * 2,000 = Rs. 9,000 (already reduced for sleeping status).C's share = 5 * 2,000 = Rs. 10,000.Sum = 6,000 + 9,000 + 10,000 = Rs. 25,000, which matches the distributed amount, so the calculation is consistent. Why Other Options Are Wrong: - 20000: This would imply C receives almost the entire distributed profit, which contradicts the given ratios and reduction for B.- 6000: This equals A's share, not C's, and comes from using 3 ratio parts instead of 5.- 9000: This is B's reduced sleeping-partner share, not C's share. Common Pitfalls: - Forgetting that only half of the total profit is distributed and not applying the ratio directly to Rs. 50,000.- Reducing B's ratio incorrectly by subtracting one-fourth instead of multiplying his share by three-fourths.- Adjusting the total profit instead of adjusting B's ratio part, which can make the ratio inconsistent.- Ignoring that A and C remain working partners whose shares are not reduced. Final Answer: The amount of profit that C receives from the distributed portion of Rs. 25,000 is Rs. 10,000.

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