Solving Alloy Mixture Ratio Problems

This problem involves mixing two different alloys, each with a specific ratio of lead and tin, and determining the ratio of lead and tin in the resulting mixture.

Analyzing Alloy X Composition

We are given 25 kg of alloy X. The ratio of lead to tin in alloy X is 1 ∶ 2.

• The total number of parts in the ratio is \(1 + 2 = 3\).
• The proportion of lead in alloy X is \(\frac{1}{3}\).
• The proportion of tin in alloy X is \(\frac{2}{3}\).

Let's calculate the actual amount of lead and tin in 25 kg of alloy X:

• Amount of Lead in X \(= \frac{1}{3} \times 25 \text{ kg} = \frac{25}{3} \text{ kg}\)
• Amount of Tin in X \(= \frac{2}{3} \times 25 \text{ kg} = \frac{50}{3} \text{ kg}\)

Analyzing Alloy Y Composition

We are given 125 kg of alloy Y. The ratio of lead to tin in alloy Y is 2 ∶ 3.

• The total number of parts in the ratio is \(2 + 3 = 5\).
• The proportion of lead in alloy Y is \(\frac{2}{5}\).
• The proportion of tin in alloy Y is \(\frac{3}{5}\).

Let's calculate the actual amount of lead and tin in 125 kg of alloy Y:

• Amount of Lead in Y \(= \frac{2}{5} \times 125 \text{ kg} = 2 \times 25 \text{ kg} = 50 \text{ kg}\)
• Amount of Tin in Y \(= \frac{3}{5} \times 125 \text{ kg} = 3 \times 25 \text{ kg} = 75 \text{ kg}\)

Calculating Lead and Tin in the Mixture

When alloy X and alloy Y are mixed, the total amount of lead in the mixture is the sum of the lead from X and the lead from Y. Similarly, the total amount of tin is the sum of the tin from X and the tin from Y.

• Total amount of Lead in the mixture \(= \text{Lead in X} + \text{Lead in Y} = \frac{25}{3} \text{ kg} + 50 \text{ kg}\)
• To add these, find a common denominator: \(\frac{25}{3} + \frac{50 \times 3}{3} = \frac{25 + 150}{3} = \frac{175}{3} \text{ kg}\)

• Total amount of Tin in the mixture \(= \text{Tin in X} + \text{Tin in Y} = \frac{50}{3} \text{ kg} + 75 \text{ kg}\)
• To add these, find a common denominator: \(\frac{50}{3} + \frac{75 \times 3}{3} = \frac{50 + 225}{3} = \frac{275}{3} \text{ kg}\)

Determining the Ratio in the Mixture

The ratio of lead to tin in the mixture is the total amount of lead divided by the total amount of tin.

Ratio (Lead : Tin) in mixture \(= (\text{Total Lead}) : (\text{Total Tin})\)

Ratio \(= \frac{175}{3} : \frac{275}{3}\)

We can multiply both sides of the ratio by 3 to remove the denominator:

Ratio \(= 175 : 275\)

To simplify this ratio, we find the greatest common divisor (GCD) of 175 and 275. Both numbers are divisible by 25.

• \(175 \div 25 = 7\)
• \(275 \div 25 = 11\)

So, the simplified ratio of lead to tin in the mixture is 7 ∶ 11.

Summary of Results

Ratio of Lead : Tin in mixture \(= \frac{175}{3} : \frac{275}{3} = 175 : 275 = 7 : 11\).

Revision Table: Key Concepts for Alloy Mixture Problems

Additional Information: Ratio and Proportion in Mixtures

Problems involving mixing different substances with varying compositions are common in quantitative aptitude. Understanding how to work with ratios and proportions is crucial. When mixing alloys, you sum the individual amounts of each metal from the constituent alloys to find the total amount of that metal in the final mixture. The ratio in the mixture is then determined by the total quantities of the components.

Remember that ratios represent relative amounts, not absolute amounts. To calculate the actual quantity of a component, you need the total quantity of the mixture or the specific part.

For example, a 1 ∶ 2 lead-tin alloy means that for every 1 kg of lead, there are 2 kg of tin. In a 3 kg sample of this alloy, there would be 1 kg of lead and 2 kg of tin.