Problem 15 - Basic Engineering Mathematics

Question

An alloy is made up of metals A and B in the ratio \(2.5: 1\) by mass. How much of \(\mathrm{A}\) has to be added to \(6 \mathrm{~kg}\). of \(\mathrm{B}\) to make the alloy?

Step-by-Step Solution

Answer

Add 15 kg of A to 6 kg of B.

Step 1: Understand the Ratio

The alloy is made up of metals A and B in the ratio 2.5:1 by mass. This means that for every part of B, there are 2.5 parts of A in the alloy.

Step 2: Set up the Equation

If we let the mass of A be denoted by \(x\), then according to the ratio, \( \frac{x}{6} = 2.5 \), because there are 6 kg of B.

Step 3: Solve for A

To find \(x\), we solve the equation \( \frac{x}{6} = 2.5 \). By multiplying both sides by 6, we find \(x = 2.5 \times 6 = 15\).

Key Concepts

Algebra

Algebra is a branch of mathematics that uses variables to find answers to problems. It involves forming equations to express relationships. In the given problem, algebra helps us use the given mass ratio of an alloy to determine the needed amount of one component. The main goal in this algebraic problem is to solve for the unknown variable, which represents the amount of metal A in the alloy.

In the context of this exercise, we use an algebraic equation to find the mass that should be added. Here are some key benefits of using algebra:

• It allows us to represent complex words and ideas numerically.
• Algebraic equations can solve real-life problems efficiently.
• These equations form the basis for broad applications across science and engineering.

By setting the mass of metal A as \(x\) and using the given ratio to create an equation \( \frac{x}{6} = 2.5 \), algebra helps us calculate the precise quantity needed.

Mass Ratio

A mass ratio expresses the quantitative relationship between two or more substances. In any alloy, it tells us how much of one base material is mixed with another. The mass ratio helps maintain specific properties in a material by dictating the proportion of its components.

For this alloy, the given mass ratio is 2.5:1 for metals A and B. This means for every 1 unit of mass of metal B, we need 2.5 units of metal A. Using this information is crucial in determining how much of each metal is required to form the alloy. Knowing mass ratios is essential for:

• Ensuring the correct composition of alloys.
• Monitoring the quality and characteristics of materials.
• Avoiding wastage of resources by adding precise quantities.

Understanding and implementing mass ratios is a fundamental practice in manufacturing and material engineering.

Problem Solving

Problem-solving involves identifying solutions to complex scenarios through logical reasoning. In mathematics, problem-solving includes understanding the problem, devising a plan, carrying out the plan, and reflecting on the solution. For our problem of mixing an alloy, this process is unfolded as follows:

First, we identify that we need to calculate how much of metal A to add. By understanding the given ratio of metals A and B and having the mass of B, a plan is made by setting up the equation \( \frac{x}{6} = 2.5 \). Solving this, we multiply both sides by 6 to isolate \(x\), resulting in \(x = 15\).

• Breaking down problems into smaller, more manageable steps is vital in problem-solving.
• Checking the solution helps ensure that it meets the problem's requirements.
• Reflecting on the process can help improve future problem-solving skills.

Effective problem-solving skills are universally applicable and valuable in tackling various challenges, whether mathematical or real-world.