Problem 18 - Prealgebra

Question

Solve each of the following problems. 16 is \(20 \%\) of what number?

Step-by-Step Solution

Answer

16 is 20% of 80.

Step 1: Identify the Variables

We need to find the number from which 16 is 20%. Let's represent this unknown number as \(x\). We know that 16 is 20% of \(x\), so we can set up an equation to express this relationship.

Step 2: Set Up the Equation

We know that 20% or \(0.2\) of \(x\) equals 16. We can write this equation as: \[0.2x = 16\]

Step 3: Solve for x

To solve for \(x\), divide both sides of the equation by 0.2. \[x = \frac{16}{0.2}\]

Step 4: Calculation

Perform the division to find the value of \(x\). \[x = 80\] So, 16 is 20% of 80.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in math that involves finding the value of an unknown variable. In our problem, we need to find the number, represented by \(x\), from which 16 is 20%. To solve any equation, follow these general steps:

• Identify the variable you need to find. Here, it's \(x\).
• Set up a mathematical expression or equation that represents the problem. For this problem, it's \(0.2x = 16\).
• Solve the equation using algebraic techniques, such as addition, subtraction, multiplication, or division, whatever is necessary to isolate the variable.

Solving for \(x\) requires dividing both sides by 0.2, which gives us \(x = \frac{16}{0.2}\). This division tells us how many times 16 can be obtained when we take 20% of \(x\). Performing the calculation yields \(x = 80\).
By following these steps, you can solve a wide variety of equations and find unknown quantities in similar percentage problems.

Basic Algebra

Basic algebra involves using symbols, usually letters, to represent numbers in mathematical expressions and equations. In this exercise, you encountered the variable \(x\) as an unknown.Algebra also involves understanding the operations and rules that govern how to manipulate these symbols to maintain equality:

• Variables like \(x\) help us write problems in a generic form that can be solved through a series of logical steps.
• Operations on equations are centered around preserving the balance of the equation. So, whatever you do to one side, you must do to the other. This is why dividing both sides by 0.2 works to isolate \(x\).
• Algebraic manipulations like solving for \(x\) from \(0.2x = 16\) provide a foundation for understanding more complex mathematical concepts.

Practice with algebraic equations enhances problem-solving skills and helps develop mathematical thinking. Whether for percentages, proportions, or other relationships, algebra is a powerful tool. It's like being able to see "behind the scenes" of math problems and to understand the principles of how numbers relate.

Mathematical Reasoning

Mathematical reasoning is the critical thinking process that involves analyzing problems and identifying the logical steps needed to solve them. In percentage problems like this one, it allows you to understand and characterize the problem's essential components.One aspect of reasoning involves translating a word problem into a mathematical equation. Given that 16 is 20% of some number, you reason to find what operation and relationships exist. This involves recognizing percentages as fractions or decimals and setting up the correct equation, \(0.2x = 16\).
Here's how mathematical reasoning unfolds in a similar context:

• Identifying what the problem is asking: What number is 16 a part of?
• Understanding that percentages translate to decimals in algebraic expressions.
• Deciding on the correct algebraic manipulation to isolate the variable, ensuring logical steps yield the answer.

The beauty of mathematical reasoning is in its applicability to diverse problems. As you enhance your reasoning skills, you develop the ability to approach problems methodically and with confidence, translating real-life scenarios into solvable mathematical questions. It not only helps in math class but in daily decision-making situations too.