Question

A metal alloy is made from copper, zinc and steel in the ratio 3:4:1. (a) Calculate the amount of copper in a \(30 \mathrm{~kg}\) block of the alloy. (b) \(10 \mathrm{~kg}\) of copper is added to an existing \(40 \mathrm{~kg}\) block of the alloy to form a new alloy. Calculate the ratio of copper, zinc and steel in the new alloy.

Short answer

Answer: The amount of copper in the 30 kg block is $\frac{90}{8}$ kg. Question 2: If 10 kg of copper is added to a 40 kg block of the alloy, what is the new ratio of copper, zinc, and steel in the new alloy? Answer: The new ratio of copper, zinc, and steel in the alloy is 5:4:1.

Step-by-step solution

(Step 1: Find the total ratio amount)

The given ratio of copper, zinc, and steel is \(3:4:1\). Let's find the total ratio amount by adding the parts of the ratio: Total ratio amount: \(3 + 4 + 1 = 8\)

(Step 2: Find the copper proportion)

Since the total weight of the alloy is \(30 \mathrm{~kg}\) and we know the total ratio amount, we can find the copper proportion by multiplying the copper ratio (3) with the proportion of the total weight over the total ratio amount: Amount of copper: \(3 (\frac{30}{8}) = \frac{90}{8} \mathrm{~kg}\) So, the block has \(\frac{90}{8} \mathrm{~kg}\) of copper. (a) The amount of copper in a \(30 \mathrm{~kg}\) block of the alloy is \(\frac{90}{8}\) kg.

(Step 3: Find the new amounts of each metal)

First, we will find the amounts of copper, zinc, and steel in the \(40 \mathrm{~kg}\) block of the alloy, using the same method as in step 2: Amount of copper: \(3(\frac{40}{8}) = 15 \mathrm{~kg}\) Amount of zinc: \(4(\frac{40}{8}) = 20 \mathrm{~kg}\) Amount of steel: \(1(\frac{40}{8}) = 5 \mathrm{~kg}\) Next, we will add the \(10 \mathrm{~kg}\) of additional copper: New amount of copper: \(15 \mathrm{~kg} + 10 \mathrm{~kg} = 25 \mathrm{~kg}\)

(Step 4: Calculate the new ratio)

To find the new ratio, we will divide each new amount by the greatest common divisor (GCD) of the three amounts (25, 20, and 5): GCD(25, 20, 5) = 5 New ratio of copper, zinc, and steel: Copper: \(\frac{25}{5} = 5\) Zinc: \(\frac{20}{5} = 4\) Steel: \(\frac{5}{5} = 1\) So, the new ratio is \(5:4:1\). (b) The ratio of copper, zinc, and steel in the new alloy is \(5:4:1\).

Key concepts

Ratio Calculation

When it comes to understanding the composition of an alloy—or any mixture—the concept of ratio calculation is fundamental. A ratio is a way to show the relationship between two or more quantities, denoting how many times one quantity is contained within another. In the context of metals in an alloy, ratios enable us to determine the proportion of each metal in the mixture.

Let's consider the exercise where an alloy is composed of copper, zinc, and steel in a ratio of 3:4:1. This means that for every 8 parts (since 3 + 4 + 1 = 8) of the alloy, there are 3 parts copper, 4 parts zinc, and 1 part steel. To calculate how much copper is in a 30 kg block of this alloy, we need to determine what portion of the block is made up of copper. We do this by dividing the total weight of the block by the total number of parts in the ratio and then multiplying by the number of copper parts:
Proportion of copper = Total mass of alloy x (Copper ratio / Total ratio) = 30 kg x (3/8) = \( \frac{90}{8} \) kg

• The resulting value \( \frac{90}{8} \) kg represents the weight of copper in the alloy block.

By mastering this ratio calculation, students can understand and solve problems related to the compositions of mixtures and alloys.

Proportion Problems

In any scenario where we must adjust the quantities of ingredients in a mixture, proportion problems come into play. These include cases where the composition of an alloy changes, such as when additional copper is added to the mix as in the exercise provided.

To solve proportion problems, it’s usually helpful to first determine the original amounts of each component, as was done for the 40 kg block of alloy. After altering the alloy's composition by adding more of one metal, we then compare the new quantities to each other to establish a new ratio. In this case, 10 kg of additional copper changes the balance of the mixture.

Once we have the new weights of copper, zinc, and steel, the strategy involves finding the greatest common divisor (GCD) of the quantities and using it to simplify the new ratio:
New ratio = \( \frac{New weight of copper}{GCD} \) : \( \frac{New weight of zinc}{GCD} \) : \( \frac{New weight of steel}{GCD} \)

• This process helps ensure that the quantities are expressed in their simplest form, which in the example problem, resulted in a new ratio of 5:4:1.

Tackling proportion problems requires attention to these steps to correctly alter and interpret the composition of mixtures and alloys.

Metal Alloy Composition

Understanding metal alloy composition involves grasping how different metals are combined in fixed proportions to create a material with specific properties. Alloys are made by melting and mixing two or more metals, and the ratio of each metal can significantly affect the alloy's characteristics such as strength, malleability, and resistance to corrosion.

In the given exercise, we see an alloy made from copper, zinc, and steel. Each metal contributes unique qualities to the final product. Copper is known for its conductivity, zinc for its corrosion resistance, and steel for its durability. Keeping track of each metal's proportion is crucial when the alloy's composition is strategically designed for certain applications.

Calculating the metal alloy composition in a real-world context helps ensure the desired physical properties of the alloy are achieved. Slight variations in the ratio can lead to significant differences in performance, making the concepts of ratio and proportion critical for those in materials science and metallurgy, as well as anyone curious about how everyday metal objects get their diverse attributes.