Question

Brass is a mixture of copper and zinc in the ratio 8:3. Calculate the weight of zinc in \(20 \mathrm{~kg}\) of brass.

Short answer

Answer: Approximately 5.45 kg of zinc.

Step-by-step solution

Determine the Total Parts

In the ratio 8:3, there are a total of 8 parts copper and 3 parts zinc. Therefore, there are (8 + 3) = 11 total parts in the mixture.

Find the Fraction of Zinc in the Brass

Since there are 3 parts zinc and 11 total parts, the fraction of zinc in the brass is \(\frac{3}{11}\).

Calculate the Weight of Zinc in 20 kg of Brass

To find the weight of zinc in 20 kg of brass, multiply the fraction of zinc by the total weight of brass: $$\frac{3}{11} \times 20 \mathrm{~kg} = \frac{60}{11} \mathrm{~kg} \approx 5.45 \mathrm{~kg}$$ So, there are approximately 5.45 kg of zinc in 20 kg of brass.

Key concepts

Mixtures

A mixture is essentially a combination of two or more substances where each retains its unique properties. In our example, brass is the mixture containing copper and zinc. These metals are blended physically in a specific ratio, namely 8 parts of copper to 3 parts of zinc. Understanding the ratio is crucial when working with mixtures as it dictates the proportion of each component within the whole.

• Physical Combination: Each part remains distinct and does not change its inherent properties. For brass, this means both copper and zinc maintain their separateness yet contribute collectively to form the mixture.
• Proportionality: The given ratio provides the foundation for calculating quantities of each component based on their respective proportions.

Understanding mixtures highlights how individual ingredients contribute to a final product. Knowing about such combinations allows for precise calculations in various applications, from chemistry to culinary arts.

Fractions

Fractions are mathematical expressions representing parts of a whole. They are used broadly to solve ratio and proportion problems, especially those involving mixtures. In this exercise, the fraction sheds light on the specific amount of zinc in the brass mixture.

• Numerator and Denominator: The numerator (the top number) represents the parts of zinc, and the denominator (the bottom number) represents the total parts in the mixture. Thus, \( \frac{3}{11} \) indicates three parts zinc out of eleven parts total.
• Application: This fraction is used to compute the percentage of zinc present in any given brass quantity. It aids in transforming the ratio into a workable calculation format, letting us easily determine the precise weight of zinc.

Grasping how fractions relate to real-world contexts allows you to handle everyday problems involving parts and wholes efficiently.

Weight Calculations

Weight calculations involve determining the mass of a specific component in a mixture or compound. This concept is crucial when you're required to pinpoint exact quantities within a whole, such as in our brass example.

• Using the Fraction: Once the fraction representing the component is identified, it gets multiplied by the total weight to find the component’s mass. For the 20 kg brass, the expression \( \frac{3}{11} \times 20 \) leads to the weight of zinc.
• Conducting Precise Calculations: Through proper multiplication, you arrive at an exact or approximate weight value, herein calculated as approximately 5.45 kg of zinc.

By mastering weight calculations, you'll seamlessly convert ratios and fractions into tangible, practical results, helping in fields like mixtures analysis or materials science.