Question

Brass is an alloy composed of \(55 \%\) copper and \(45 \%\) zinc by weight. You have 25 ounces of copper. How many ounces of zinc do you need to make brass? (See Example 2.)

Short answer

Zinc needed is \(\frac{45 \times 25}{55}\) ounces.

Step-by-step solution

Establish the Proportion

Given the ratio of copper to zinc is 55 to 45, it can be rewritten as \(\frac{55}{45}=\frac{25}{x}\), where x represents the weight of zinc needed.

Cross-Multiply to Solve for x

Cross multiplying gives \(55x = 45 \times 25\). Now, isolate x (the weight of zinc) by dividing both side of the equation by 55.

Perform Final Calculation

x = \(\frac{45 \times 25}{55}\). Calculate the result to get the weight of zinc needed.

Key concepts

Mathematical Proportions

When we talk about proportions in mathematics, we're referring to a relationship that states that two ratios are equivalent. They arise from the concept of 'ratio', which measures how many times one number contains another by division. For example, if we have two lengths, A and B, wherein A is twice as long as B, we can express this as a ratio of A to B being 2:1.

In the context of alloys, like with the brass problem, we use proportions to maintain a specific ratio of components to get the desired material properties. Brass, which consists of copper and zinc, should adhere to the established ratio of those metals to qualify as brass. In our exercise, the ratio is 55 parts of copper for every 45 parts zinc. When we have a certain amount of one component, such as 25 ounces of copper, and we need to find out how much of the other component (zinc) we need to make brass, we set up a proportion to solve for the unknown amount.

It's essential to understand that when dealing with proportions in a practical context, such as creating an alloy, the ratios must be used based on the same units (in this case, weight in ounces) to ensure accuracy.

Cross Multiplication

Cross multiplication is a tool that's especially handy when it comes to solving proportions. Once we have our proportion, like \frac{55}{45}=\frac{25}{x}\ where 'x' is our unknown, cross multiplication can be our next step to find 'x'. To cross multiply means to multiply the numerator of one ratio by the denominator of the other ratio across the equation.

Here's how we can apply it to our problem:
\[ 55 \times x = 45 \times 25 \]
We multiply 55 by 'x' and 45 by 25. You'll notice that it forms a cross when you draw lines to multiply these pairs, hence the term 'cross multiplication'. What you get is an equation that will help you isolate and determine 'x', which stands for the number of ounces of zinc needed in our problem.

Understanding the cross multiplication process is vital because it's a method that's used beyond just alloy proportions – it applies to any proportional reasoning question, like scaling recipes, determining distances on maps, or adjusting in any situation where keeping a certain ratio is necessary.

Algebraic Equations

Algebraic equations are the backbone of problem-solving in algebra. They consist of symbols and numbers that show the relationship between quantities. 'x', for example, is a common symbol for an unknown value that we are trying to find.

In our alloy problem, the equation we get from cross multiplication is \(55x = 45 \times 25\). The goal now is to solve this algebraic equation for 'x'. This involves isolating 'x' on one side of the equation to find its value. We do this by performing algebraic operations that maintain the balance of the equation. We divide both sides by 55, simplifying the equation to \(x = \frac{45 \times 25}{55}\).

The final calculation will give us the exact amount of zinc needed to maintain the correct proportion in the brass alloy. Learning to solve algebraic equations is key as they are used to model real-life situations and help in understanding and solving various practical problems.