Understanding Mixture and Ratio Problems

This problem involves combining two different mixtures, each containing milk and juice in a specific ratio. We are told that equal volumes of these two mixtures are mixed together, and we need to find the resulting ratio of milk to juice in the final combined mixture.

Let's break down the problem:

• Mixture 1 has a milk to juice ratio of 2 : 1.
• Mixture 2 has a milk to juice ratio of 4 : 5.
• Equal volumes of Mixture 1 and Mixture 2 are mixed.
• We need to find the final milk to juice ratio in the resulting mixture.

Calculating Components in Equal Volumes

To solve this mixture problem, it's helpful to assume a specific volume for the equal quantities being mixed. A good choice for the volume is a value that is a multiple of the sum of the ratio parts for each mixture. This makes calculating the amounts of milk and juice easier.

For Mixture 1, the ratio is 2 : 1. The total number of parts is $2 + 1 = 3$.

For Mixture 2, the ratio is 4 : 5. The total number of parts is $4 + 5 = 9$.

The least common multiple (LCM) of 3 and 9 is 9. So, let's assume we take 9 units of volume from Mixture 1 and 9 units of volume from Mixture 2.

Amount of Milk and Juice in Mixture 1 (9 units)

In Mixture 1, the milk to juice ratio is 2:1, which means milk constitutes $\frac{2}{2+1} = \frac{2}{3}$ of the mixture, and juice constitutes $\frac{1}{2+1} = \frac{1}{3}$ of the mixture.

• Amount of Milk in 9 units of Mixture 1: $\frac{2}{3} \times 9 = 6$ units
• Amount of Juice in 9 units of Mixture 1: $\frac{1}{3} \times 9 = 3$ units

Amount of Milk and Juice in Mixture 2 (9 units)

In Mixture 2, the milk to juice ratio is 4:5, which means milk constitutes $\frac{4}{4+5} = \frac{4}{9}$ of the mixture, and juice constitutes $\frac{5}{4+5} = \frac{5}{9}$ of the mixture.

• Amount of Milk in 9 units of Mixture 2: $\frac{4}{9} \times 9 = 4$ units
• Amount of Juice in 9 units of Mixture 2: $\frac{5}{9} \times 9 = 5$ units

Finding the Resulting Ratio

When the two equal volumes (9 units each) are mixed, the total volume of the resulting mixture is $9 + 9 = 18$ units. The total amount of milk in the resulting mixture is the sum of the milk from both mixtures, and the total amount of juice is the sum of the juice from both mixtures.

• Total amount of Milk in the resulting mixture: Amount of Milk from Mixture 1 + Amount of Milk from Mixture 2 $= 6 + 4 = 10$ units
• Total amount of Juice in the resulting mixture: Amount of Juice from Mixture 1 + Amount of Juice from Mixture 2 $= 3 + 5 = 8$ units

The ratio of milk to juice in the resulting mixture is the total milk divided by the total juice.

Resulting Ratio = Total Milk : Total Juice $= 10 : 8$

This ratio can be simplified by dividing both numbers by their greatest common divisor, which is 2.

Simplified Resulting Ratio = $\frac{10}{2} : \frac{8}{2} = 5 : 4$

Thus, the ratio of milk to juice in the resulting mixture is 5 : 4.

Summary of Results

Revision Table: Key Concepts in Mixture Problems

Additional Information on Mixture Problems

Mixture problems are common in quantitative aptitude. They often involve combining substances with different ratios or concentrations. The key is usually to calculate the total amount of each component in the final mixture and then form the ratio.

• Weighted Average Method: For some mixture problems (especially those involving concentrations), a weighted average approach can be used. However, for simple ratio mixing like this with equal volumes, directly adding the component amounts is straightforward.
• Unequal Volumes: If unequal volumes were mixed, you would calculate the amounts of each component based on the specific volume taken from each mixture before adding them together. For example, if 18 units of Mixture 1 and 9 units of Mixture 2 were mixed.
• Ratio and Proportion: Mixture problems heavily rely on the concepts of ratio and proportion. Understanding how to work with fractions derived from ratios is crucial.
• Units: While we used "units" here, the concept works regardless of the actual volume units (liters, gallons, etc.) because we are dealing with ratios of equal volumes.

Practice with different types of mixture problems will help build confidence in handling variations like adding a pure substance to a mixture or removing a part of the mixture.