Problem 41 - Algebra 1: Concepts and Skills

Question

Use \(a=\frac{p}{100} b\). Complete the sentence: Write a percent equation for the statement “a is 300 percent of b.” Then choose one set of values for a, b, and p that make the equation true.

Step-by-Step Solution

Answer

The percent equation is \(a = 3b\). A possible set of values satisfying this equation is \(a = 3\), \(b = 1\), and \(p = 300\).

Step 1: Create a Percent Equation

The given statement can be written as a percent equation: 'a is 300 percent of b' translates to \(a = \frac{300}{100}b\), or more simplified, \(a = 3b\).

Step 2: Find Appropriate Unit Values

Next, suitable values for a, b, and p have to be selected that will satisfy the equation. Any values which allow \(a = 3b\) to be true can be picked, for instance, \(a = 3\), \(b = 1\), and \(p = 300\).

Key Concepts

Percentage Calculation

Understanding percentage calculations is essential in solving problems involving discounts, interest rates, and proportions. The statement given "a is 300 percent of b" can be broken down to understand how percentages work. Here, the percentage (p) is 300. When calculating the percentage of a number, you follow the formula:

• Convert the percentage into a fraction by dividing it by 100; hence, 300% becomes \( \frac{300}{100} \) or simply 3.
• Multiply this fraction by the base number (in this case, 'b') to find the result.

If a number is said to be 100% of another, it is equal to that number. If it is 300%, it means it is equivalent to 3 times that number. This conversion is a handy way to visualize percentages in a more tangible form.

Equation Solving

Solving equations is a fundamental skill in algebra that allows you to find unknown values. In this exercise, the equation derived from the statement is \( a = 3b \). Here are the steps to solve such an equation:

• Identify the known values and the unknown values.
• If given specific values for one or two variables, substitute these into the equation.
• Rearrange the equation to isolate the unknown variable. For example, if you know 'b', you can find 'a' by multiplying ‘b’ by 3 without changing the equation's balance.

The idea is simple: maintain balance in the equation while isolating the unknown quantity to find its value. This skill becomes more effective with practice, as it forms the base for tackling more complex problems.

Algebraic Expressions

Algebraic expressions like \( a = 3b \) are a way to represent relationships between variables using numbers, variables, and arithmetic operations. They are pivotal in making sense of mathematical problems. Here's how you can work with them efficiently:

• Recognize the variables: 'a', 'b', and 'p' in our context.
• Understand the constants involved, such as numbers like 3 or 100, which remain the same throughout your calculations.
• Apply arithmetic operations and rules describing the relationship, such as multiplication or division. For instance, identifying that 'a' is 3 times 'b' using the concept of multiplication.
• Use expressions to model real-world situations like calculating total costs, making predictions, or understanding growth patterns.

By interpreting algebraic expressions correctly, you can manipulate them to solve equations, helping you discover unknowns and verify solutions in various scenarios.