Problem 58 - Elementary Algebra

Question

Answer the question by setting up and solving an appropriate equation. 26 is what percent of 20 ?

Step-by-Step Solution

Answer

26 is 130% of 20.

Step 1: Understand the Problem

We need to find out what percent 26 is of 20. This means we should interpret the phrase '26 is what percent of 20?' as an equation.

Step 2: Set Up the Equation

To find what percent 26 is of 20, we use the formula where the part is equal to 26 and the whole is equal to 20. Let \( x \) represent the percentage:\[ x \times 20 = 26 \]

Step 3: Solve the Equation

We need to isolate \( x \) in the equation \( x \times 20 = 26 \). To do this, divide both sides of the equation by 20:\[ x = \frac{26}{20} \]

Step 4: Simplify the Fraction

Simplify \( \frac{26}{20} \) to find the proportional decimal:\[ x = 1.3 \]

Step 5: Convert to Percentage

Convert the decimal to a percentage by multiplying by 100:\[ x = 1.3 \times 100 = 130\% \]

Key Concepts

Understanding Algebraic Equations

Algebraic equations are mathematical expressions that represent relationships between variables and constants. When we say "26 is what percent of 20?", we're dealing with two known values (26 and 20), and an unknown percentage we need to find.
To set up an algebraic equation for this problem, we use a variable, often represented by \( x \), to stand for the unknown percentage. Our goal is to express the situation as \( x \cdot 20 = 26 \).

• Here, \( 26 \) is the part we are comparing, and \( 20 \) is the whole or original amount.
• We need an equation to show that multiplying this unknown percentage (\( x \)) and the whole (\( 20 \)) equals the part (\( 26 \)).

This format makes it easier to isolate the variable and solve for the unknown, ultimately determining the percentage through further steps.

Approaching Problem Solving

Problem-solving in mathematics involves interpreting the task and translating words into mathematical expressions. Every math problem, such as finding what percentage 26 is of 20, requires some logical steps:

• Understanding the Question: Recognize what you need to find. Here, it's the percent representation of one number relative to another.
• Formulating a Plan: Identify the formula or method needed to address the question, often setting up an algebraic equation.
• Executing the Plan: Apply mathematical operations and solve the problem step by step. This involves manipulation of the equation to solve for the unknown variable.
• Reviewing the Solution: Check if the result makes sense in the context of the problem. For example, does the percentage seem reasonable given the numbers involved?

By following these steps, you can address similar percent problems efficiently and effectively.

Converting Decimals to Percentages

Converting a decimal to a percentage is a fundamental skill in mathematics. Once you solve the algebraic equation and simplify the fraction, you might find that the value isn't in a conventional percentage form.
To convert a decimal to a percentage, you follow a simple step:
Multiply the decimal by 100.

• This is because "percent" means "per hundred" and multiplying by 100 shifts the decimal point two places to the right, converting it into a percentage.

For instance, in our problem, after solving \( x = \frac{26}{20} \), we reduced it to the decimal \( x = 1.3 \).

• Converting 1.3 to a percentage involves multiplying 1.3 by 100, giving us \( 130\% \).

Remember, decimals and percentages are just two different ways of representing the same value, so knowing how to switch between them broadens your problem-solving toolkit.