Question

A metal alloy contains \(15.8 \%\) nickel \((\mathrm{m} / \mathrm{m}) \%\). What mass of the metal alloy would contain \(36.5 \mathrm{~g}\) of nickel?

Short answer

The metal alloy would have a mass of approximately 230.76 g.

Step-by-step solution

Understand the Problem

The problem asks us to determine the total mass of a metal alloy that contains 36.5 grams of nickel. We know that the alloy is composed of 15.8% nickel by mass.

Set Up the Proportion

We know the percentage composition by mass of nickel is 15.8%, which means for every 100 grams of the alloy, 15.8 grams are nickel. Using this as a basis, we can set up the following proportion equation: \( \frac{15.8}{100} = \frac{36.5}{x} \), where \(x\) is the total mass of the alloy that contains 36.5 grams of nickel.

Solve the Proportion Equation

To solve for \(x\), the total mass, cross-multiply:\[ 15.8x = 3650 \] Now, solve for \(x\) by dividing both sides by 15.8:\[ x = \frac{3650}{15.8} \approx 230.76 \]

Verify the Solution

Recalculate using the obtained total mass of the alloy (230.76 grams) to ensure it contains 36.5 grams of nickel. Calculate 15.8% of 230.76 and verify if it equals 36.5 grams: \[ 0.158 \times 230.76 \approx 36.5 \] Since the calculation holds true, our solution is verified.

Key concepts

Alloy Composition

Alloys are mixtures of metals that, when combined, offer certain desirable properties such as strength, lightness, or corrosion resistance. Imagine alloys as the cake where each ingredient changes the flavor and texture.
One crucial element when discussing alloys is their composition. This refers to the specific proportion each metal represents in the total mixture.
For example, in our problem, the alloy contains nickel as one of its main components. The percentage composition, like the 15.8% nickel in this case, describes how much of the total mass of the alloy comes from nickel.

Understanding the composition helps determine the characteristics an alloy will offer. Knowing these proportions allows for solving practical problems such as the amount of material required for specific purposes, as we've seen in the exercise.

Nickel Content Calculation

When calculating the nickel content, we need to focus on how much of it we find in a given metal alloy, which is clearly defined by its percentage composition. The stated percentage lets us determine the exact amount of nickel present.
For our problem-solving approach, we assume if 15.8% of the alloy is nickel, then per 100 grams of the alloy, 15.8 grams are nickel. This is the basis of our calculation.

• We start by establishing a proportion based on this percentage.
• In our exercise, we use the formula: \( \frac{15.8}{100} = \frac{36.5}{x} \)
• This equation sets up a relationship between the known percentage and the desired mass of the alloy.

Solving this proportion correctly allows us to estimate the total mass of the alloy needed to reach the specific nickel content set in the problem.

Mass Percentage Problem Solving

Mass percentage problem solving often involves using algebra to understand relationships between the components and the whole mixture. This can help in fields ranging from chemistry to industry.
The first step is setting up a proportion that establishes a relationship between the parts of a mixture and its total mass.
Using a simple cross-multiplication technique, as shown in the equation \( 15.8x = 3650 \), we solve for the unknown, which in this case is the total mass of the alloy.
After finding \( x \), the verification step ensures our calculations are correct. By recalculating using the obtained total mass, we ensure that the desired component matches its specified percentage in the mixture.
Once reassured, it confirms the accuracy of our solution and grasps the concept of mass percentage problem-solving effectively.