Mathematics (Mixture)

Solution Summary

1. Define the mixture parts: The original mixture has 11 parts of milk and 2 parts of water, totaling 13 parts.
2. Set up volumes: Let the total volume of the original mixture be V. Then:
   • Milk volume = (11/13) × V
   • Water volume = (2/13) × V
3. Add water and form equation: Adding 35 liters of water makes milk twice the new water volume:

   (11/13) × V = 2 × [(2/13) × V + 35]

4. Solve for V: Simplifying, we find V = 130.
5. Calculate milk volume: The amount of pure milk is
   (11/13) × 130 = 110 liters.

Answer: The original mixture contains 110 liters of pure milk.

Tin Calculation in Alloys

Alloy A

Total Weight: 60 kg

Lead to Tin Ratio: 3 : 2

Weight of Tin in Alloy A: 24 kg

Alloy B

Total Weight: 100 kg

Tin to Copper Ratio: 1 : 4

Weight of Tin in Alloy B: 20 kg

Total Tin in New Alloy

Total Tin: 44 kg

Solution

We are given three containers with equal capacity, each containing sulphuric acid and water in the following ratios:

• Container 1: Sulphuric acid : Water = 3 : 2
• Container 2: Sulphuric acid : Water = 7 : 3
• Container 3: Sulphuric acid : Water = 11 : 4

Let each container have a capacity of x liters.

Step 1: Calculate the Amount of Sulphuric Acid and Water in Each Container

• Container 1: Sulphuric acid = (3/5) × x, Water = (2/5) × x
• Container 2: Sulphuric acid = (7/10) × x, Water = (3/10) × x
• Container 3: Sulphuric acid = (11/15) × x, Water = (4/15) × x

Step 2: Sum up Sulphuric Acid and Water from All Containers

Total Sulphuric acid = (3/5)x + (7/10)x + (11/15)x

Total Water = (2/5)x + (3/10)x + (4/15)x

Step 3: Find a Common Denominator and Add

For sulphuric acid, the common denominator is 30:

Total Sulphuric acid = (18/30)x + (21/30)x + (22/30)x = (61/30)x

For water, the common denominator is also 30:

Total Water = (12/30)x + (9/30)x + (8/30)x = (29/30)x

Step 4: Calculate the Final Ratio

Ratio of Sulphuric acid to Water = (61/30)x : (29/30)x

Cancel out x and the denominator 30:

61 : 29

Answer: The ratio of Sulphuric acid to Water in the mixture is 61 : 29.

Solution

We are given:

• Gold is 19 times as heavy as water.
• Copper is 9 times as heavy as water.
• We want to mix these metals to create an alloy that is 15 times as heavy as water.

Using the Allegation Rule:

To find the required ratio, we apply the allegation formula:

(Heaviness of gold - Desired heaviness) : (Desired heaviness - Heaviness of copper)

Substitute the values:

(19 - 15) : (15 - 9) = 4 : 6

Simplify the Ratio:

Divide both terms by 2:

4 : 6 = 2 : 3

Answer: Gold and copper should be mixed in the ratio 2 : 3.

Solution

We are given:

• Concentration of the first ethanol solution = 20%
• Concentration of the second ethanol solution = 40%
• Desired concentration of the mixture = 28%

Using the Allegation Rule:

According to the allegation formula, we have:

(Concentration of higher solution - Desired concentration) : (Desired concentration - Concentration of lower solution)

Substitute the values:

(40 - 28) : (28 - 20) = 12 : 8

Simplify the Ratio:

Divide both terms by 4:

12 : 8 = 3 : 2

Answer: The 20% ethanol solution should be mixed with the 40% ethanol solution in the ratio 3 : 2.

Solution

We are given:

• Price of first coffee powder = Rs. 2500 per kg
• Price of second coffee powder = Rs. 1500 per kg
• Desired price of the mixture = Rs. 2250 per kg

Let the coffee powders be mixed in the ratio x : y where:

• x = parts of coffee powder costing Rs. 2500 per kg
• y = parts of coffee powder costing Rs. 1500 per kg

Using the Allegation Rule:

According to the allegation formula, we have:

(Cost per kg of higher price - Cost per kg of mixture) : (Cost per kg of mixture - Cost per kg of lower price)

Substitute the values:

(2500 - 2250) : (2250 - 1500) = 250 : 750

Simplify the Ratio:

Divide both terms by 250:

250 : 750 = 1 : 3

Answer: The coffee powders should be mixed in the ratio 1 : 3.