Basic (1-7)
▶1. What is a mixture in mathematics?
Answer: A mixture is a combination of two or more substances (liquids or solids) in any ratio.
Step-by-step Explanation:
1. In math, a mixture problem involves combining different items with known quantities and concentrations.
2. The goal is to find the final concentration or amount after mixing.
3. Example: Mixing 2L of 20% salt solution with 3L of 30% salt solution.
4. The result is a new mixture with a different concentration.
▶2. What is the alligation rule?
Answer: The alligation rule is a method to find the ratio in which two or more ingredients at different concentrations must be mixed to get a mixture of a desired concentration.
Step-by-step Explanation:
1. The rule uses a simple formula: (Higher concentration - Mean concentration) : (Mean concentration - Lower concentration).
2. This gives the ratio in which the two solutions should be mixed.
3. Example: Mix 20% and 40% solutions to get 30%. Ratio = (40-30):(30-20) = 10:10 = 1:1.
▶3. If you mix 3L of 20% salt solution with 2L of 30% salt solution, what is the concentration of the resulting mixture?
Answer: 24%
Step-by-step Explanation:
1. Find the total salt: (3 × 20%) + (2 × 30%) = 0.6 + 0.6 = 1.2L.
2. Total volume = 3L + 2L = 5L.
3. Concentration = (1.2/5) × 100 = 24%.
▶4. How much water should be added to 4L of 40% alcohol solution to make it a 20% solution?
Answer: 4L
Step-by-step Explanation:
1. Amount of alcohol in 4L = 4 × 40% = 1.6L.
2. Let x = amount of water to add.
3. New total volume = 4 + x.
4. New concentration = 1.6/(4 + x) × 100 = 20%.
5. 1.6/(4 + x) = 0.2 → 1.6 = 0.2(4 + x) → 1.6 = 0.8 + 0.2x → 0.8 = 0.2x → x = 4L.
▶5. In what ratio should 25% and 60% acid solutions be mixed to get a 40% solution?
Answer: 2:1
Step-by-step Explanation:
1. Use alligation: (60-40):(40-25) = 20:15 = 4:3.
2. But check: 25% is lower, 60% is higher, mean is 40%.
3. (60-40):(40-25) = 20:15 = 4:3.
4. So, mix 4 parts of 25% with 3 parts of 60% to get 40% solution.
5. (If the question asks for 25%:60%, answer is 4:3.)
▶6. A 10L mixture contains milk and water in the ratio 3:2. How much milk is there?
Answer: 6L
Step-by-step Explanation:
1. Ratio of milk:water = 3:2, so total parts = 3+2 = 5.
2. Milk = (3/5) × 10 = 6L.
▶7. A 20L solution has 25% sugar. How much sugar is in the solution?
Answer: 5L
Step-by-step Explanation:
1. Sugar = 25% of 20L = 0.25 × 20 = 5L.
Moderate (8-14)
▶8. How much water should be added to 8L of 50% alcohol solution to make it a 40% solution?
Answer: 2L
Step-by-step Explanation:
1. Alcohol in 8L = 8 × 50% = 4L.
2. Let x = water to add. New volume = 8 + x.
3. 4/(8 + x) = 0.4 → 4 = 0.4(8 + x) → 4 = 3.2 + 0.4x → 0.8 = 0.4x → x = 2L.
▶9. In what ratio should 20% and 70% solutions be mixed to get a 50% solution?
Answer: 2:3
Step-by-step Explanation:
1. Alligation: (70-50):(50-20) = 20:30 = 2:3.
2. So, mix 2 parts of 20% with 3 parts of 70% to get 50% solution.
▶10. A 15L mixture contains milk and water in the ratio 4:1. How much water should be added to make the ratio 3:2?
Answer: 5L
Step-by-step Explanation:
1. Milk = (4/5) × 15 = 12L, Water = (1/5) × 15 = 3L.
2. Let x = water to add. New water = 3 + x, new total = 15 + x.
3. New ratio: 12/(3 + x) = 3/2 → 12 × 2 = (3 + x) × 3 → 24 = 9 + 3x → 15 = 3x → x = 5L.
▶11. How much 30% solution and 50% solution should be mixed to get 40L of 35% solution?
Answer: 10L of 50% and 30L of 30%
Step-by-step Explanation:
1. Let x = 50% solution, y = 30% solution.
2. x + y = 40.
3. 0.5x + 0.3y = 0.35 × 40 = 14.
4. 0.5x + 0.3(40 - x) = 14 → 0.5x + 12 - 0.3x = 14 → 0.2x = 2 → x = 10, y = 30.
▶12. A 12L mixture contains 25% acid. How much pure acid should be added to make it 40% acid?
Answer: 4L
Step-by-step Explanation:
1. Acid in 12L = 0.25 × 12 = 3L.
2. Let x = pure acid to add. New acid = 3 + x, new total = 12 + x.
3. (3 + x)/(12 + x) = 0.4 → 3 + x = 0.4(12 + x) → 3 + x = 4.8 + 0.4x → 0.6x = 1.8 → x = 3L.
▶13. A 30L mixture contains milk and water in the ratio 7:3. How much water should be added to make the ratio 3:2?
Answer: 15L
Step-by-step Explanation:
1. Milk = (7/10) × 30 = 21L, Water = (3/10) × 30 = 9L.
2. Let x = water to add. New water = 9 + x, new total = 30 + x.
3. New ratio: 21/(9 + x) = 3/2 → 21 × 2 = (9 + x) × 3 → 42 = 27 + 3x → 15 = 3x → x = 5L.
▶14. How much 20% and 60% solutions should be mixed to get 100L of 50% solution?
Answer: 25L of 20% and 75L of 60%
Step-by-step Explanation:
1. Let x = 20% solution, y = 60% solution.
2. x + y = 100.
3. 0.2x + 0.6y = 0.5 × 100 = 50.
4. 0.2x + 0.6(100 - x) = 50 → 0.2x + 60 - 0.6x = 50 → -0.4x = -10 → x = 25, y = 75.
Advanced (15-20)
▶15. A 50L mixture contains 40% alcohol. 10L of mixture is replaced with water. What is the new concentration of alcohol?
Answer: 36%
Step-by-step Explanation:
1. Alcohol in 50L = 0.4 × 50 = 20L.
2. 10L removed, so alcohol removed = 0.4 × 10 = 4L.
3. Alcohol left = 20 - 4 = 16L.
4. New volume = 50L (since 10L water added).
5. New concentration = (16/50) × 100 = 32%.
▶16. A 60L mixture contains 30% acid. 20L is replaced with pure acid. What is the new concentration?
Answer: 50%
Step-by-step Explanation:
1. Acid in 60L = 0.3 × 60 = 18L.
2. 20L removed, so acid removed = 0.3 × 20 = 6L.
3. Acid left = 18 - 6 = 12L.
4. 20L pure acid added, so new acid = 12 + 20 = 32L.
5. New volume = 60L.
6. New concentration = (32/60) × 100 = 53.33%.
▶17. How much 10% and 90% solutions should be mixed to get 60L of 50% solution?
Answer: 30L of each
Step-by-step Explanation:
1. Let x = 10% solution, y = 90% solution.
2. x + y = 60.
3. 0.1x + 0.9y = 0.5 × 60 = 30.
4. 0.1x + 0.9(60 - x) = 30 → 0.1x + 54 - 0.9x = 30 → -0.8x = -24 → x = 30, y = 30.
▶18. A 100L mixture contains 60% milk. 20L is removed and replaced with water. What is the new concentration of milk?
Answer: 48.8%
Step-by-step Explanation:
1. Milk in 100L = 0.6 × 100 = 60L.
2. 20L removed, so milk removed = 0.6 × 20 = 12L.
3. Milk left = 60 - 12 = 48L.
4. New volume = 100L.
5. New concentration = (48/100) × 100 = 48%.
▶19. In what ratio should 15% and 45% solutions be mixed to get a 30% solution?
Answer: 1:1
Step-by-step Explanation:
1. Alligation: (45-30):(30-15) = 15:15 = 1:1.
2. So, mix equal parts of 15% and 45% to get 30% solution.
▶20. A 40L mixture contains 25% salt. 10L is removed and replaced with pure salt. What is the new concentration?
Answer: 43.75%
Step-by-step Explanation:
1. Salt in 40L = 0.25 × 40 = 10L.
2. 10L removed, so salt removed = 0.25 × 10 = 2.5L.
3. Salt left = 10 - 2.5 = 7.5L.
4. 10L pure salt added, so new salt = 7.5 + 10 = 17.5L.
5. New volume = 40L.
6. New concentration = (17.5/40) × 100 = 43.75%.