Partnership shares profit according to capital committed over time, when the agreement says returns follow investment. Mixture questions track the amount of each component and its concentration or price. Both topics are weighted-average problems beneath their worded surface.
1. Capital-time weights
If partners invest different capital amounts for different durations, their profit shares are proportional to capital × time. A's ₹10,000 for 12 months has weight 120,000 rupee-months; B's ₹15,000 for 8 months also has weight 120,000. They share equally, even though their capital amounts differ. State the common time unit before multiplying.
When a partner changes investment midway, split the timeline into intervals. Add each interval's capital-time product. A salary or commission paid to a managing partner, if specified, is usually removed from the distributable profit first; then divide the remainder according to the capital-time ratio.
Worked example 1. A invests ₹20,000 for 12 months; B invests ₹30,000 after four months, for eight months. Their weights are 240,000 and 240,000, so they divide a ₹9,000 distributable profit equally at ₹4,500 each.
2. Component amounts in a mixture
A concentration of 30% means 30 units of the named component per 100 units of mixture. To combine mixtures, add the component amounts and total amounts separately. Mixing 10 L of 20% acid with 15 L of 40% acid gives 2+6=8 L acid in 25 L total, or 32% acid. The unweighted average of 20% and 40% is 30%, but the volumes differ.
Worked example 2. Mix 8 L of 25% solution with 12 L of 50% solution. Pure substance amounts are 2 L and 6 L. The 20 L mixture contains 8 L pure substance, so its strength is 40%.
3. Alligation as a difference rule
If two component strengths are low L and high H, and desired mean is M with L<M<H, the quantities are in the ratio (H−M):(M−L), low component to high component. This follows by equating the excess from the high component with the deficit from the low. The rule is shorthand for a weighted-average equation, not a separate law.
Worked example 3. In what ratio should 20% and 50% solutions be mixed for 32%? Low:high = (50−32):(32−20)=18:12=3:2. Check: (3×20+2×50)/5=32%.
Alligation also works for prices per equal unit: cheap and costly goods combine to a target mean price when sold without other charges. It does not directly account for a later profit margin, evaporation or unequal unit sizes. Model those changes separately first.
4. Replacement and repeated removal
When x litres are removed from a well-mixed vessel of volume V and replaced with pure water, the fraction of the original substance remaining after one operation is 1−x/V. After n identical operations, original substance remaining is initial amount × (1−x/V)^n. The mixture must be thoroughly mixed before each removal; otherwise the removed amount may not have the current concentration.
Worked example 4. A 20 L vessel contains pure milk. Remove 5 L and replace it with water twice, mixing each time. Milk remaining =20×(15/20)^2=11.25 L. Water is 8.75 L, so the final milk fraction is 56.25%.
5. Profit after dilution or mixing
When cost and sale prices enter a mixture problem, keep quantities and values distinct. Find the total cost of all components, then the total quantity, then the sale revenue. A ratio of milk to water alone cannot determine profit without the relevant unit costs and sale price. Treat free water as zero cost only if the question allows it; real handling costs are outside a simplified exam model unless stated.
Recall before practice
1. Explain why a partner's weight is capital multiplied by time. 2. Reconstruct alligation from the weighted-average equation. 3. State the assumption needed for repeated-removal formulae. 4. Check why the final mixture strength must lie between its component strengths.
Chapter 10 practice — 50 questions
Choose one option for each item. Keep a separate answer list; explanations follow this set.
1. Arjun started a firm with ₹10,000. Mohan joined at the same time with a certain capital. If the annual profit was divided between Arjun and Mohan in the ratio 1 : 5, how much did Mohan invest?
A. ₹10,000 B. ₹55,000 C. ₹45,000 D. ₹50,000
2. A 110-litre milk-water mixture contains 40% water. How much pure water must be added so that water becomes 50% of the new mixture?
A. 132 litres B. 22 litres C. 11 litres D. 33 litres
3. A 70-litre alcohol-water mixture contains 10% water. How much pure water must be added so that water becomes 30% of the new mixture?
A. 27 litres B. 14 litres C. 20 litres D. 90 litres
4. Deepak and Rohit invest ₹80,000 and ₹1,00,000 respectively in a business for one year. If the total profit at the end of the year is ₹27,000, what is the share of Rohit?
A. ₹13,500 B. ₹33,750 C. ₹12,000 D. ₹15,000
5. Sunita started a venture with ₹55,000. Priya joined at the same time with a certain capital. If the annual profit was divided between Sunita and Priya in the ratio 11 : 6, how much did Priya invest?
A. ₹25,000 B. ₹30,000 C. ₹55,000 D. ₹35,000
6. A 20-litre milk-water mixture contains 40% water. How much pure water must be added so that water becomes 60% of the new mixture?
A. 12 litres B. 30 litres C. 4 litres D. 10 litres
7. Geeta, Amit and Arjun invest capitals in the ratio 2 : 5 : 7 for 12, 10 and 6 months respectively. In what ratio should the profit be divided among them?
A. 6 : 5 : 3 B. 12 : 25 : 21 C. 2 : 5 : 7 D. 21 : 25 : 12
8. A 70-litre alcohol-water mixture contains 25% water. How much pure water must be added so that water becomes 30% of the new mixture?
A. 10 litres B. 75 litres C. 5 litres D. 12 litres
9. From 200 litres of pure petrol, 40 litres are drawn out and replaced with water. This process is repeated so that it is carried out twice in all. How much petrol is there in the final mixture?
A. 160 litres B. 72 litres C. 120 litres D. 128 litres
10. A vessel contains a mixture of milk and water in the ratio 2 : 3. If the total quantity of the mixture is 25 litres, find the difference between the quantities of milk and water in it.
A. 15 litres B. 20 litres C. 10 litres D. 5 litres
11. Kavita and Rahul invest ₹60,000 and ₹80,000 respectively in a venture for one year. If the total profit at the end of the year is ₹8,400, what is the share of Kavita?
A. ₹6,300 B. ₹4,200 C. ₹3,600 D. ₹4,800
12. Ravi started a venture with ₹40,000. Suresh joined at the same time with a certain capital. If the annual profit was divided between Ravi and Suresh in the ratio 2 : 1, how much did Suresh invest?
A. ₹25,000 B. ₹15,000 C. ₹40,000 D. ₹20,000
13. Asha started a venture with ₹25,000. Suresh joined at the same time with a certain capital. If the annual profit was divided between Asha and Suresh in the ratio 5 : 13, how much did Suresh invest?
A. ₹60,000 B. ₹25,000 C. ₹70,000 D. ₹65,000
14. A 110-litre milk-water mixture contains 40% water. How much pure water must be added so that water becomes 60% of the new mixture?
A. 165 litres B. 66 litres C. 22 litres D. 55 litres
15. Ravi started a venture with ₹60,000. Some months later Rekha joined with ₹70,000. At the end of the year the profits of Ravi and Rekha were in the ratio 24 : 7. After how many months did Rekha join?
A. 3 months B. 9 months C. 8 months D. 10 months
16. Priya started a firm with ₹1,10,000. After 4 months, Meena joined with ₹80,000. If the profit at the end of the year is ₹29,400, what is Priya's share?
A. ₹19,800 B. ₹14,700 C. ₹9,600 D. ₹20,400
17. Priya, Rahul and Meena invest ₹1,50,000, ₹1,80,000 and ₹2,40,000 respectively in a business for one year. If the total profit at the end of the year is ₹14,250, what is the share of Meena?
A. ₹6,000 B. ₹4,750 C. ₹3,750 D. ₹4,500
18. A container holds 27 litres of pure milk. 9 litres are drawn out and replaced with water, and this is done twice in all. What is the ratio of milk to water in the container now?
A. 5 : 4 B. 4 : 5 C. 4 : 9 D. 1 : 2
19. Mohan, Kavita and Vikas invest ₹30,000, ₹40,000 and ₹50,000 respectively in a shop for the same period. If Kavita's share of the annual profit is ₹2,400, what is the total profit?
A. ₹4,800 B. ₹7,200 C. ₹5,760 D. ₹9,600
20. Sanjay started a venture with ₹90,000. Some months later Suresh joined with ₹1,00,000. At the end of the year the profits of Sanjay and Suresh were in the ratio 18 : 5. After how many months did Suresh join?
A. 3 months B. 10 months C. 9 months D. 8 months
21. Two vessels contain milk and water in the ratios 4 : 5 and 5 : 1 respectively. If quantities in the ratio 2 : 1 from the two vessels are mixed, what is the ratio of milk to water in the resulting mixture?
A. 31 : 23 B. 3 : 2 C. 13 : 11 D. 23 : 31
22. Five containers A to E hold milk-water mixtures. The total volume and the ratio of milk to water in each is: A: 63 L (7 : 2); B: 190 L (3 : 7); C: 100 L (7 : 3); D: 48 L (5 : 1); E: 90 L (2 : 3). What is the quantity of milk in container C?
A. 30 litres B. 65 litres C. 70 litres D. 100 litres
23. Rohit, Priya and Rahul invest ₹50,000, ₹75,000 and ₹1,25,000 respectively in a shop for the same period. If Rahul's share of the annual profit is ₹7,500, what is the total profit?
A. ₹15,000 B. ₹7,500 C. ₹22,500 D. ₹37,500
24. Sanjay buys 80 kg of salt in all, some at ₹40 per kg and the rest at ₹65 per kg, paying ₹4,575 in total. How much salt was bought at ₹65 per kg?
A. 40 kg B. 55 kg C. 25 kg D. 60 kg
25. Priya and Sunita invest ₹20,000 and ₹50,000 respectively to start a firm together. At the end of one year the total profit is ₹7,000. What is the difference between the shares of Priya and Sunita?
A. ₹5,000 B. ₹7,000 C. ₹3,000 D. ₹2,000
26. From 216 litres of pure spirit, 36 litres are drawn out and replaced with water. This process is repeated so that it is carried out three times in all. How much water is there in the final mixture?
A. 91 litres B. 108 litres C. 66 litres D. 125 litres
27. In what ratio must rice costing ₹35 per kg be mixed with rice costing ₹95 per kg so that the mixture costs ₹83 per kg?
A. 4 : 1 B. 1 : 5 C. 1 : 4 D. 1 : 2
28. A vessel contains 42 litres of a mixture of acid and water in the ratio 5 : 2. How much acid must be added so that the ratio of acid to water becomes 5 : 1?
A. 35 litres B. 5 litres C. 28 litres D. 30 litres
29. Priya and Ravi invest ₹20,000 and ₹50,000 in a business. Priya is the working partner and receives 12% of the profit for managing the business; the remaining profit is divided in the ratio of their capitals. If the total profit is ₹70,000, what is Ravi's share?
A. ₹26,000 B. ₹50,000 C. ₹44,000 D. ₹35,000
30. Mohan, Anita and Pooja invested for 4, 6 and 12 months respectively and their profits were in the ratio 2 : 4 : 5. What was the ratio of their capitals?
A. 2 : 4 : 5 B. 5 : 8 : 6 C. 2 : 6 : 15 D. 6 : 8 : 5
31. Five containers A to E hold milk-water mixtures. The total volume and the ratio of milk to water in each is: A: 91 L (2 : 5); B: 162 L (4 : 5); C: 45 L (2 : 3); D: 117 L (2 : 7); E: 180 L (4 : 5). What is the quantity of water in container B?
A. 72 litres B. 162 litres C. 117 litres D. 90 litres
32. Arjun started a venture with ₹50,000. Some months later Asha joined with ₹30,000. At the end of the year the profits of Arjun and Asha were in the ratio 20 : 3. After how many months did Asha join?
A. 9 months B. 3 months C. 10 months D. 8 months
33. A vessel contains 27 litres of a mixture of milk and water in the ratio 5 : 4. How much milk must be added so that the ratio of milk to water becomes 5 : 2?
A. 10 litres B. 15 litres C. 20 litres D. 30 litres
34. Anita buys 85 kg of sugar in all, some at ₹80 per kg and the rest at ₹100 per kg, paying ₹7,900 in total. How much sugar was bought at ₹100 per kg?
A. 11 kg B. 60 kg C. 30 kg D. 55 kg
35. Sunita, Rohit and Vikas invest ₹5,000, ₹10,000 and ₹15,000 respectively in a venture for the same period. If Rohit's share of the annual profit is ₹2,400, what is the total profit?
A. ₹9,600 B. ₹7,920 C. ₹7,200 D. ₹4,800
36. Five containers A to E hold milk-water mixtures. The total volume and the ratio of milk to water in each is: A: 119 L (5 : 2); B: 60 L (4 : 1); C: 54 L (5 : 4); D: 66 L (5 : 1); E: 162 L (7 : 2). What is the quantity of milk in container C?
A. 39 litres B. 30 litres C. 54 litres D. 24 litres
37. A vessel contains 27 litres of a mixture of acid and water in the ratio 2 : 1. How much acid must be added so that the ratio of acid to water becomes 4 : 1?
A. 18 litres B. 16 litres C. 20 litres D. 36 litres
38. From 25 litres of pure wine, 10 litres are drawn out and replaced with water. This process is repeated so that it is carried out twice in all. How much wine is there in the final mixture?
A. 5 litres B. 15 litres C. 16 litres D. 9 litres
39. Meena started a business with ₹60,000. After 5 months, Arjun joined with ₹80,000. If the profit at the end of the year is ₹16,000, what is Meena's share?
A. ₹10,000 B. ₹8,000 C. ₹9,000 D. ₹7,000
40. Arjun started a firm with ₹50,000. Some months later Rohit joined with ₹1,40,000. At the end of the year the profits of Arjun and Rohit were in the ratio 10 : 21. After how many months did Rohit join?
A. 2 months B. 3 months C. 9 months D. 4 months
41. A acid-water mixture contains 20% water. When 57 litres of pure water is added, water becomes 50% of the new mixture. What was the initial quantity of the mixture?
A. 38 litres B. 152 litres C. 57 litres D. 95 litres
42. A container holds 36 litres of pure milk. 6 litres are drawn out and replaced with water, and this is done twice in all. What is the ratio of milk to water in the container now?
A. 2 : 1 B. 25 : 11 C. 11 : 25 D. 25 : 36
43. Geeta buys 110 kg of pulses in all, some at ₹45 per kg and the rest at ₹85 per kg, paying ₹6,950 in total. What is the ratio of the quantity bought at ₹45 per kg to that bought at ₹85 per kg?
A. 5 : 6 B. 17 : 9 C. 6 : 5 D. 9 : 17
44. Vikas started a venture with ₹40,000. After 4 months, Neha joined with a certain capital. If out of the annual profit Vikas got ₹12,000 and Neha got ₹13,000, how much did Neha invest?
A. ₹70,000 B. ₹60,000 C. ₹65,000 D. ₹1,30,000
45. Sunita and Rekha invested ₹1,00,000 and ₹30,000 respectively in a business. After 6 months, Sunita added 1/4 of the initial capital, while Rekha's capital remained unchanged. If the profit at the end of the year is ₹95,000, what is Rekha's share?
A. ₹47,500 B. ₹25,000 C. ₹75,000 D. ₹20,000
46. A acid-water mixture contains 15% water. When 15 litres of pure acid is added, water becomes 10% of the new mixture. What was the initial quantity of the mixture?
A. 40 litres B. 30 litres C. 15 litres D. 45 litres
47. Deepak started a venture with ₹20,000. Rekha joined after 2 months with ₹30,000, and Vikas joined after 7 months from the start with ₹30,000. If the total profit at the end of the year is ₹46,000, find Rekha's share.
A. ₹16,000 B. ₹10,000 C. ₹17,250 D. ₹20,000
48. Pooja and Anita invested ₹60,000 and ₹1,00,000 respectively in a business. After 8 months, Pooja withdrew 1/3 of the capital, while Anita's capital remained unchanged. If the profit at the end of the year is ₹1,15,000, what is Anita's share?
A. ₹57,500 B. ₹71,875 C. ₹75,000 D. ₹40,000
49. Amit and Mohan invest ₹60,000 and ₹1,50,000 in a business. Amit is the working partner and receives 12% of the profit for managing the business; the remaining profit is divided in the ratio of their capitals. If the total profit is ₹1,22,500, what is Amit's total share?
A. ₹14,700 B. ₹45,500 C. ₹35,000 D. ₹30,800
50. A alcohol-water mixture contains 25% water. When 440 litres of pure alcohol is added, water becomes 5% of the new mixture. What was the initial quantity of the mixture?
A. 110 litres B. 440 litres C. 220 litres D. 550 litres
Chapter 10 — Answers and explanations
After checking the key, re-solve any miss without looking at the formula.
1. D. Let Mohan's capital be x. Profit ratio = ₹10,000 × 12 : x × 12 = 1 : 5. So x = (120000 × 5) / (1 × 12) = ₹50,000. For the same period, capitals are in the same ratio as profits.
APAR26-10-04 | Find a capital from the profit ratio | Easy
2. B. Fix the component that does not change: milk = 60% of 110 = 66 L stays fixed and becomes 50% of the new total ⇒ new total = 66 × 100/50 = 132 L. Quantity added = 132 − 110 = 22 L. (Taking 10% of 110 = 11 L ignores that the total also grows.)
APAR26-10-26 | Adding a pure component to change the percentage | Easy
3. C. Fix the component that does not change: alcohol = 90% of 70 = 63 L stays fixed and becomes 70% of the new total ⇒ new total = 63 × 100/70 = 90 L. Quantity added = 90 − 70 = 20 L. (Taking 20% of 70 = 14 L ignores that the total also grows.)
APAR26-10-27 | Adding a pure component to change the percentage | Easy
4. D. For the same period, profit is shared in the ratio of capitals: ₹80,000 : ₹1,00,000 = 4 : 5. Sum of parts = 9; one part = ₹27,000/9 = ₹3,000. Rohit's share = 5 × ₹3,000 = ₹15,000. (Dividing the profit equally, ₹13,500 each, ignores the capitals.)
APAR26-10-05 | Simple partnership: share of profit | Easy
5. B. Let Priya's capital be x. Profit ratio = ₹55,000 × 12 : x × 12 = 11 : 6. So x = (660000 × 6) / (11 × 12) = ₹30,000. For the same period, capitals are in the same ratio as profits.
APAR26-10-07 | Find a capital from the profit ratio | Easy
6. D. Fix the component that does not change: milk = 60% of 20 = 12 L stays fixed and becomes 40% of the new total ⇒ new total = 12 × 100/40 = 30 L. Quantity added = 30 − 20 = 10 L. (Taking 20% of 20 = 4 L ignores that the total also grows.)
APAR26-10-31 | Adding a pure component to change the percentage | Easy
7. B. Profit ∝ capital × time. Required ratio = 2 × 12 : 5 × 10 : 7 × 6 = 24 : 50 : 42 = 12 : 25 : 21. (Using the capital ratio 2 : 5 : 7 alone ignores the different periods.)
APAR26-10-06 | Profit ratio from capitals and periods | Easy
8. C. Fix the component that does not change: alcohol = 75% of 70 = 52.5 L stays fixed and becomes 70% of the new total ⇒ new total = 52.5 × 100/70 = 75 L. Quantity added = 75 − 70 = 5 L. (Taking 5% of 70 = 3.5 L ignores that the total also grows.)
APAR26-10-30 | Adding a pure component to change the percentage | Easy
9. D. petrol left after n replacements = V × (1 − x/V)^n = 200 × (1 − 40/200)^2 = (4/5)^2 = 128 L. (Simply removing 40 L 2 times would leave 120 L of petrol — wrong, because each later draw removes a mixture.)
APAR26-10-29 | Repeated replacement: quantity left | Easy
10. D. Total parts = 2 + 3 = 5; one part = 25/5 = 5 L. Milk = 2 × 5 = 10 L, water = 3 × 5 = 15 L. Difference = 5 L. (Dividing by 3 instead of the sum of parts is the common slip.)
APAR26-10-32 | Component of a mixture from its ratio | Easy
11. C. For the same period, profit is shared in the ratio of capitals: ₹60,000 : ₹80,000 = 3 : 4. Sum of parts = 7; one part = ₹8,400/7 = ₹1,200. Kavita's share = 3 × ₹1,200 = ₹3,600. (Dividing the profit equally, ₹4,200 each, ignores the capitals.)
APAR26-10-01 | Simple partnership: share of profit | Easy
12. D. Let Suresh's capital be x. Profit ratio = ₹40,000 × 12 : x × 12 = 2 : 1. So x = (480000 × 1) / (2 × 12) = ₹20,000. For the same period, capitals are in the same ratio as profits.
APAR26-10-02 | Find a capital from the profit ratio | Easy
13. D. Let Suresh's capital be x. Profit ratio = ₹25,000 × 12 : x × 12 = 5 : 13. So x = (300000 × 13) / (5 × 12) = ₹65,000. For the same period, capitals are in the same ratio as profits.
APAR26-10-03 | Find a capital from the profit ratio | Easy
14. D. Fix the component that does not change: milk = 60% of 110 = 66 L stays fixed and becomes 40% of the new total ⇒ new total = 66 × 100/40 = 165 L. Quantity added = 165 − 110 = 55 L. (Taking 20% of 110 = 22 L ignores that the total also grows.)
APAR26-10-28 | Adding a pure component to change the percentage | Easy
15. B. Let Rekha invest for t months. Profit ratio = ₹60,000 × 12 : ₹70,000 × t = 24 : 7. So 720000 × 7 = 70000 × t × 24 ⇒ t = 3 months. Rekha joined after 12 − 3 = 9 months. (Answering 3 confuses the time invested with the time of joining.)
APAR26-10-10 | Find when the second partner joined | Medium
16. A. Profit is shared in the ratio (capital × months). Priya : Meena = ₹1,10,000 × 12 : ₹80,000 × 8 = 1320000 : 640000 = 33 : 16. One part = ₹29,400/49 = ₹600; Priya's share = 33 × ₹600 = ₹19,800. (Meena's money worked for 8 months, not 4; ignoring time altogether gives ₹17,021.05.)
APAR26-10-15 | Compound partnership: late joiner | Medium
17. A. For the same period, profit is shared in the ratio of capitals: ₹1,50,000 : ₹1,80,000 : ₹2,40,000 = 5 : 6 : 8. Sum of parts = 19; one part = ₹14,250/19 = ₹750. Meena's share = 8 × ₹750 = ₹6,000. (Dividing the profit equally, ₹4,750 each, ignores the capitals.)
APAR26-10-12 | Simple partnership: share of profit | Medium
18. B. milk left = V × (1 − x/V)^n = 27 × (1 − 9/27)^2 = (2/3)^2 = 12 L; water = 27 − 12 = 15 L. Ratio = 12 : 15 = 4 : 5. (Subtracting 9 L 2 times, leaving 9 L, ignores that later draws remove a mixture.)
APAR26-10-43 | Repeated replacement: final ratio | Medium
19. B. Capital ratio = 3 : 4 : 5 (sum 12). Kavita gets 4/12 of the profit, so 4 parts = ₹2,400 ⇒ one part = ₹600 and total profit = 12 × ₹600 = ₹7,200. (Multiplying ₹2,400 by 3 assumes equal shares.)
APAR26-10-19 | Simple partnership: total profit from one share | Medium
20. C. Let Suresh invest for t months. Profit ratio = ₹90,000 × 12 : ₹1,00,000 × t = 18 : 5. So 1080000 × 5 = 100000 × t × 18 ⇒ t = 3 months. Suresh joined after 12 − 3 = 9 months. (Answering 3 confuses the time invested with the time of joining.)
APAR26-10-18 | Find when the second partner joined | Medium
21. A. Take 36 L from the first (LCM-friendly) and 18 L from the second. Milk = 36 × 4/9 + 18 × 5/6 = 16 + 15 = 31; water = 20 + 3 = 23. Ratio = 31 : 23 = 31 : 23. (Adding ratio terms directly, 3 : 2, is wrong.)
APAR26-10-40 | Combining two mixtures | Medium
22. C. Split the total of the container in its ratio: milk in C = 100 × 7/(7 + 3) = 70 L. (Using the other component's part, 30 L, is the usual mix-up.)
APAR26-10-33 | Mixture table: quantity of a component | Medium
23. A. Capital ratio = 2 : 3 : 5 (sum 10). Rahul gets 5/10 of the profit, so 5 parts = ₹7,500 ⇒ one part = ₹1,500 and total profit = 10 × ₹1,500 = ₹15,000. (Multiplying ₹7,500 by 3 assumes equal shares.)
APAR26-10-08 | Simple partnership: total profit from one share | Medium
24. B. Let x kg be at ₹40; then (80 − x) kg is at ₹65. 40x + 65(80 − x) = 4575 ⇒ 5200 − 25x = 4575 ⇒ x = 25 kg, so the rest = 55 kg. Quantity at ₹65 = 55 kg. (Splitting 80 kg equally, 40 kg each, ignores the total cost.)
APAR26-10-45 | Quantities from total weight and total cost | Medium
25. C. Profit ratio = capital ratio = 2 : 5, sum 7; one part = ₹7,000/7 = ₹1,000. Difference = (5 − 2) parts = 3 × ₹1,000 = ₹3,000.
APAR26-10-13 | Simple partnership: difference of shares | Medium
26. A. spirit left after n replacements = V × (1 − x/V)^n = 216 × (1 − 36/216)^3 = (5/6)^3 = 125 L. Water = 216 − 125 = 91 L. (Simply removing 36 L 3 times would leave 108 L of spirit — wrong, because each later draw removes a mixture.)
APAR26-10-34 | Repeated replacement: water in the mixture | Medium
27. C. By the rule of alligation, (quantity of cheaper) : (quantity of dearer) = (dearer price − mean) : (mean − cheaper price) = (95 − 83) : (83 − 35) = 12 : 48 = 1 : 4. (Writing the differences the other way round gives 4 : 1.)
APAR26-10-42 | Alligation: ratio for a target mean price | Medium
28. D. acid = 42 × 5/7 = 30 L, water = 12 L. Water stays 12 L; for the ratio 5 : 1, acid must be 12 × 5/1 = 60 L. Quantity added = 30 L. (Applying the new ratio to the old total 42 L is the standard error.)
APAR26-10-41 | Adding a pure component to change the ratio | Medium
29. C. Management share = 12% of ₹70,000 = ₹8,400. Remaining = ₹61,600, divided 2 : 5 ⇒ one part = ₹8,800. Priya gets ₹8,400 + 2 × ₹8,800 = ₹26,000; Ravi gets 5 × ₹8,800 = ₹44,000. (Splitting the whole ₹70,000 by capital gives ₹50,000 — the 12% must be removed first.)
APAR26-10-16 | Working partner with a share of profit as salary | Medium
30. D. Capital = profit ÷ time. Ratio = 2/4 : 4/6 : 5/12; multiplying by the LCM 12 gives 6 : 8 : 5. (Multiplying profit by time, 2 : 6 : 15, is the reverse operation and is wrong.)
APAR26-10-20 | Capital ratio from profit ratio and periods | Medium
31. D. Split the total of the container in its ratio: water in B = 162 × 5/(4 + 5) = 90 L. (Using the other component's part, 72 L, is the usual mix-up.)
APAR26-10-37 | Mixture table: quantity of a component | Medium
32. A. Let Asha invest for t months. Profit ratio = ₹50,000 × 12 : ₹30,000 × t = 20 : 3. So 600000 × 3 = 30000 × t × 20 ⇒ t = 3 months. Asha joined after 12 − 3 = 9 months. (Answering 3 confuses the time invested with the time of joining.)
APAR26-10-11 | Find when the second partner joined | Medium
33. B. milk = 27 × 5/9 = 15 L, water = 12 L. Water stays 12 L; for the ratio 5 : 2, milk must be 12 × 5/2 = 30 L. Quantity added = 15 L. (Applying the new ratio to the old total 27 L is the standard error.)
APAR26-10-38 | Adding a pure component to change the ratio | Medium
34. D. Let x kg be at ₹80; then (85 − x) kg is at ₹100. 80x + 100(85 − x) = 7900 ⇒ 8500 − 20x = 7900 ⇒ x = 30 kg, so the rest = 55 kg. Quantity at ₹100 = 55 kg. (Splitting 85 kg equally, 42.5 kg each, ignores the total cost.)
APAR26-10-39 | Quantities from total weight and total cost | Medium
35. C. Capital ratio = 1 : 2 : 3 (sum 6). Rohit gets 2/6 of the profit, so 2 parts = ₹2,400 ⇒ one part = ₹1,200 and total profit = 6 × ₹1,200 = ₹7,200. (Multiplying ₹2,400 by 3 assumes equal shares.)
APAR26-10-14 | Simple partnership: total profit from one share | Medium
36. B. Split the total of the container in its ratio: milk in C = 54 × 5/(5 + 4) = 30 L. (Using the other component's part, 24 L, is the usual mix-up.)
APAR26-10-44 | Mixture table: quantity of a component | Medium
37. A. acid = 27 × 2/3 = 18 L, water = 9 L. Water stays 9 L; for the ratio 4 : 1, acid must be 9 × 4/1 = 36 L. Quantity added = 18 L. (Applying the new ratio to the old total 27 L is the standard error.)
APAR26-10-36 | Adding a pure component to change the ratio | Medium
38. D. wine left after n replacements = V × (1 − x/V)^n = 25 × (1 − 10/25)^2 = (3/5)^2 = 9 L. (Simply removing 10 L 2 times would leave 5 L of wine — wrong, because each later draw removes a mixture.)
APAR26-10-35 | Repeated replacement: quantity left | Medium
39. C. Profit is shared in the ratio (capital × months). Meena : Arjun = ₹60,000 × 12 : ₹80,000 × 7 = 720000 : 560000 = 9 : 7. One part = ₹16,000/16 = ₹1,000; Meena's share = 9 × ₹1,000 = ₹9,000. (Arjun's money worked for 7 months, not 5; ignoring time altogether gives ₹6,857.14.)
APAR26-10-09 | Compound partnership: late joiner | Medium
40. B. Let Rohit invest for t months. Profit ratio = ₹50,000 × 12 : ₹1,40,000 × t = 10 : 21. So 600000 × 21 = 140000 × t × 10 ⇒ t = 9 months. Rohit joined after 12 − 9 = 3 months. (Answering 9 confuses the time invested with the time of joining.)
APAR26-10-17 | Find when the second partner joined | Medium
41. D. Let the initial quantity be V. The acid content does not change: 80% of V = 50% of (V + 57) ⇒ V = 95 L. Check: acid = 80% of 95 = 76 L stays fixed and becomes 50% of the new total ⇒ new total = 76 × 100/50 = 152 L, so 57 L was added.
APAR26-10-50 | Adding a pure component (find original quantity) | Difficult
42. B. milk left = V × (1 − x/V)^n = 36 × (1 − 6/36)^2 = (5/6)^2 = 25 L; water = 36 − 25 = 11 L. Ratio = 25 : 11 = 25 : 11. (Subtracting 6 L 2 times, leaving 24 L, ignores that later draws remove a mixture.)
APAR26-10-46 | Repeated replacement: final ratio | Difficult
43. C. Mean price = ₹6,950/110 = ₹63.18 per kg. Alligation: cheaper : dearer = (85 − 63.18) : (63.18 − 45) = 21.82 : 18.18 = 6 : 5. Check: 60 kg × 45 + 50 kg × 85 = ₹6,950.
APAR26-10-47 | Quantities from total weight and total cost (ratio) | Difficult
44. C. Let Neha's capital be x. Profit ratio = ₹40,000 × 12 : x × 8 = 12 : 13. So x = (480000 × 13) / (12 × 8) = ₹65,000. (Ignoring the 4-month gap gives ₹43,333.33, which is wrong.)
APAR26-10-25 | Find a capital from the profit ratio (time-weighted) | Difficult
45. D. Sunita's capital-months = ₹1,00,000 × 6 + ₹1,25,000 × 6 = 1350000; Rekha's = ₹30,000 × 12 = 360000. Ratio = 15 : 4, sum 19; one part = ₹95,000/19 = ₹5,000. Rekha's share = 4 × ₹5,000 = ₹20,000. (Ignoring the change gives ₹21,923.08; applying it to the whole year gives ₹18,387.1.)
APAR26-10-22 | Compound partnership: capital changed midway | Difficult
46. B. Let the initial quantity be V. The water content does not change: 15% of V = 10% of (V + 15) ⇒ V = 30 L. Check: water = 15% of 30 = 4.5 L stays fixed and becomes 10% of the new total ⇒ new total = 4.5 × 100/10 = 45 L, so 15 L was added.
APAR26-10-49 | Adding a pure component (find original quantity) | Difficult
47. D. Ratio of (capital × months): ₹20,000 × 12 : ₹30,000 × 10 : ₹30,000 × 5 = 240000 : 300000 : 150000 = 8 : 10 : 5. Sum = 23; one part = ₹46,000/23 = ₹2,000. Rekha's share = 10 × ₹2,000 = ₹20,000. (Using capitals alone gives ₹17,250.)
APAR26-10-23 | Compound partnership: three partners joining at different times | Difficult
48. C. Pooja's capital-months = ₹60,000 × 8 + ₹40,000 × 4 = 640000; Anita's = ₹1,00,000 × 12 = 1200000. Ratio = 8 : 15, sum 23; one part = ₹1,15,000/23 = ₹5,000. Anita's share = 15 × ₹5,000 = ₹75,000. (Ignoring the change gives ₹71,875; applying it to the whole year gives ₹82,142.86.)
APAR26-10-24 | Compound partnership: capital changed midway | Difficult
49. B. Management share = 12% of ₹1,22,500 = ₹14,700. Remaining = ₹1,07,800, divided 2 : 5 ⇒ one part = ₹15,400. Amit gets ₹14,700 + 2 × ₹15,400 = ₹45,500; Mohan gets 5 × ₹15,400 = ₹77,000. (Splitting the whole ₹1,22,500 by capital gives ₹35,000 — the 12% must be removed first.)
APAR26-10-21 | Working partner with a share of profit as salary | Difficult
50. A. Let the initial quantity be V. The water content does not change: 25% of V = 5% of (V + 440) ⇒ V = 110 L. Check: water = 25% of 110 = 27.5 L stays fixed and becomes 5% of the new total ⇒ new total = 27.5 × 100/5 = 550 L, so 440 L was added.
APAR26-10-48 | Adding a pure component (find original quantity) | Difficult