Alloy X contains 70% copper and 30% zinc. Alloy Y contains 40% copper, 25% zinc and 35% aluminium. Alloy X and Y are mixed in the ratio of 1 ∶ 3. What is the ratio of copper and zinc in the newly formed alloy?
38 ∶ 21
Understanding the Alloy Mixing Problem
This problem requires us to determine the final ratio of copper and zinc in a new alloy formed by mixing two different alloys, Alloy X and Alloy Y, in a specific proportion. The initial composition of each alloy is given as percentages of its constituent metals.
Composition of the Starting Alloys
Let's first understand the metallic content of the alloys we are mixing.
Alloy X Composition
Alloy X is composed of:
- Copper: \(70\%\)
- Zinc: \(30\%\)
Alloy Y Composition
Alloy Y contains three metals:
- Copper: \(40\%\)
- Zinc: \(25\%\)
- Aluminium: \(35\%\)
Mixing Alloys X and Y
Alloy X and Alloy Y are mixed in the ratio of \(1:3\). To make calculations simple, let's assume we take 1 unit weight of Alloy X and 3 unit weights of Alloy Y. The total weight of the resulting new alloy will be \(1 + 3 = 4\) units.
Calculating Component Amounts in the New Alloy
Now, we calculate the actual amount (in units of weight) of copper and zinc contributed by each alloy based on the mixing ratio.
From 1 unit of Alloy X:
- Amount of Copper from X = \(70\%\) of 1 = \(0.70 \times 1 = 0.70\) units
- Amount of Zinc from X = \(30\%\) of 1 = \(0.30 \times 1 = 0.30\) units
From 3 units of Alloy Y:
- Amount of Copper from Y = \(40\%\) of 3 = \(0.40 \times 3 = 1.20\) units
- Amount of Zinc from Y = \(25\%\) of 3 = \(0.25 \times 3 = 0.75\) units
- Amount of Aluminium from Y = \(35\%\) of 3 = \(0.35 \times 3 = 1.05\) units
The total amount of each metal in the new alloy is the sum of the amounts contributed by Alloy X and Alloy Y.
Copper Content in the Mixture
Total Copper in the new alloy = Copper from X + Copper from Y
Total Copper = \(0.70\) units + \(1.20\) units = \(1.90\) units
Zinc Content in the Mixture
Total Zinc in the new alloy = Zinc from X + Zinc from Y
Total Zinc = \(0.30\) units + \(0.75\) units = \(1.05\) units
Aluminium Content in the Mixture
Total Aluminium in the new alloy = Aluminium from X + Aluminium from Y
Total Aluminium = \(0\) units + \(1.05\) units = \(1.05\) units (Though not required for the final ratio, it's part of the composition).
| Component | Amount from Alloy X (1 unit) | Amount from Alloy Y (3 units) | Total Amount in New Alloy (4 units) |
|---|---|---|---|
| Copper | \(0.70\) units | \(1.20\) units | \(1.90\) units |
| Zinc | \(0.30\) units | \(0.75\) units | \(1.05\) units |
| Aluminium | \(0\) units | \(1.05\) units | \(1.05\) units |
Determining the Copper to Zinc Ratio
The question asks for the ratio of copper to zinc in the newly formed alloy. This is calculated by taking the total amount of copper and the total amount of zinc in the mixture.
Ratio of Copper : Zinc = Total Copper : Total Zinc
Ratio = \(1.90 : 1.05\)
Simplifying the Final Ratio
To express the ratio \(1.90 : 1.05\) in the simplest form, we can first eliminate the decimals by multiplying both numbers by 100.
Ratio = \(190 : 105\)
Now, we find the greatest common divisor (GCD) of 190 and 105 to simplify the ratio. Both numbers are divisible by 5.
- \(190 \div 5 = 38\)
- \(105 \div 5 = 21\)
The simplified ratio of copper to zinc in the newly formed alloy is \(38 : 21\).
Revision Table: Key Concepts in Alloy Mixing Calculations
| Concept | Description | How it Applies Here |
|---|---|---|
| Percentage Composition | Shows the proportion of each element in an alloy by weight. | Used to find the amount of copper, zinc, and aluminium in given quantities of Alloy X and Alloy Y. |
| Mixing Ratio | Indicates the relative amounts (by weight or volume) of components being mixed. | The \(1:3\) ratio of Alloy X to Alloy Y determined the base quantities used for calculation (1 unit of X, 3 units of Y). |
| Total Amount of a Component | The sum of the amounts of that component contributed by all parts of the mixture. | Total copper is the sum of copper from Alloy X and Alloy Y. Total zinc is the sum of zinc from Alloy X and Alloy Y. |
| Ratio Simplification | Expressing a ratio in its simplest form by dividing both parts by their greatest common divisor. | The ratio \(1.90 : 1.05\) was simplified to \(38 : 21\). |
Additional Information: Understanding Metal Alloys
An alloy is a mixture of metals or a mixture of a metal and another element. Alloying is done to enhance the properties of pure metals, such as strength, hardness, durability, or corrosion resistance. Common examples include:
- Brass: A common alloy of copper and zinc, known for its workability and appearance. The varying ratios of copper and zinc result in different types of brass.
- Bronze: Traditionally an alloy of copper and tin, stronger and more durable than brass.
- Steel: Primarily an alloy of iron and carbon, with varying amounts of other elements like manganese, chromium, vanadium, and tungsten.
- Pewter: Historically composed of tin, with the addition of lead, antimony, and copper. Modern pewter is lead-free.
Calculations involving mixing alloys, like the one demonstrated, are fundamental in metallurgy and manufacturing to control the final properties of the material.
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