1️⃣ In what ratio should water and milk be mixed so that the resulting mixture is 50% milk?
Step-by-Step Explanation
50% milk means 50% water.
The ratio of water to milk is 50:50.
This simplifies to 1:1.
Answer: 1:1 ✅
2️⃣ A mixture contains milk and water in the ratio 2:1. What is the percentage of milk in the mixture?
Step-by-Step Explanation
Total parts = 2 (milk) + 1 (water) = 3.
Percentage of milk = (2/3) * 100%.
Percentage of milk = 66.67%.
Answer: 66.67% ✅
3️⃣ If 1 liter of water is added to 4 liters of milk, what is the ratio of water to milk?
Step-by-Step Explanation
Water = 1 liter.
Milk = 4 liters.
Ratio of water to milk = 1:4.
Answer: 1:4 ✅
4️⃣ A mixture contains two liquids A and B in the ratio 3:2. If the total quantity of the mixture is 10 liters, how much is liquid A?
Step-by-Step Explanation
Total parts = 3 (A) + 2 (B) = 5.
Quantity of A = (3/5) * 10 liters.
Quantity of A = 6 liters.
Answer: 6 liters ✅
5️⃣ A mixture contains 20% alcohol. How much water should be added to 1 liter of this mixture to make the concentration 10%?
Step-by-Step Explanation
Alcohol in 1 liter = 20% of 1 liter = 0.2 liters.
Let 'x' be the total volume of the new mixture.
10% of x = 0.2 liters.
x = 0.2 / 0.1 = 2 liters.
Water added = 2 liters - 1 liter = 1 liter.
Answer: 1 liter ✅
6️⃣ In what ratio must a grocer mix two varieties of pulses costing ₹15 and ₹20 per kg respectively, so as to get a mixture costing ₹16.50 per kg?
Step-by-Step Explanation
Using the alligation method:
Costlier price (20) - Mean price (16.50) = 3.50.
Mean price (16.50) - Cheaper price (15) = 1.50.
Ratio = 1.50 : 3.50.
Ratio = 15 : 35.
Ratio = 3 : 7.
Answer: 3:7 ✅
7️⃣ A container contains a mixture of two liquids A and B in the ratio 7:5. If 9 liters of mixture are drawn off and the container is filled with 9 liters of B, the ratio changes to 7:9. How many liters of liquid A were there initially?
Step-by-Step Explanation
Let the initial quantities be 7x and 5x.
Liquid A removed = (7/12) * 9 = 21/4 liters.
Liquid B removed = (5/12) * 9 = 15/4 liters.
(7x - 21/4) / (5x - 15/4 + 9) = 7/9.
(28x - 21) / (20x - 15 + 36) = 7/9.
9(28x - 21) = 7(20x + 21).
252x - 189 = 140x + 147.
112x = 336.
x = 3.
Initial quantity of A = 7x = 7 * 3 = 21 liters.
Answer: 21 liters ✅
8️⃣ A vessel contains milk and water in the ratio 4:3. If 6 liters of water are added, the ratio becomes 8:7. What is the quantity of milk in the vessel?
Step-by-Step Explanation
Let initial quantities be 4x and 3x.
4x / (3x + 6) = 8/7.
28x = 24x + 48.
4x = 48.
x = 12.
Quantity of milk = 4x = 4 * 12 = 48 liters.
Answer: 48 liters ✅
9️⃣ Two vessels A and B contain milk and water mixed in the ratio 5:3 and 2:3 respectively. In what ratio should these mixtures be mixed to form a new mixture containing half milk and half water?
Step-by-Step Explanation
Milk in A = 5/8, Milk in B = 2/5, Milk in mixture = 1/2.
Using alligation: (5/8 - 1/2) / (1/2 - 2/5).
(5/8 - 4/8) / (5/10 - 4/10).
(1/8) / (1/10).
10/8 = 5/4.
Water in A = 3/8, Water in B = 3/5, Water in mixture = 1/2.
Using alligation: (3/8 - 1/2) / (1/2 - 3/5).
(3/8 - 4/8) / (5/10 - 6/10).
(-1/8) / (-1/10).
10/8 = 5/4.
The inverse ratio is 4:5, however the milk to water ratio is 9:4.
Answer: 9:4 ✅
🔟 A jar contains a mixture of milk and water in the ratio 3:1. A part of the mixture is replaced by water and now the ratio becomes 1:1. What fraction of the mixture was replaced?