Problem 18 - Prealgebra

Question

Solve each problem using the percent equation. 43.6 is what percent of \(20 ?\)

Step-by-Step Solution

Answer

43.6 is 218% of 20.

Step 1: Understand the Problem

The problem asks us to find out what percent 43.6 is of 20. This can be translated into the equation format: "43.6 = x% of 20." This means we need to solve for 'x.'

Step 2: Express the Problem as an Equation

The percent equation is given as \( a = \frac{x}{100} \cdot b \). Here, \( a = 43.6 \) and \( b = 20 \). Plug these values into the equation: \( 43.6 = \frac{x}{100} \cdot 20 \).

Step 3: Solve for x

First, isolate \( x \) by multiplying both sides by 100: \[ 4360 = x \times 20 \]. Then, divide both sides by 20 to solve for \( x \):\[ x = \frac{4360}{20} \].

Step 4: Calculate the Final Value

Perform the division: \( x = \frac{4360}{20} = 218 \).

Step 5: Interpret the Result

The calculated result, 218, tells us that 43.6 is 218% of 20.

Key Concepts

Prealgebra

Prealgebra forms the foundation of all mathematical concepts taught in school. It involves basic arithmetic operations and introduces variables, which represent unknown quantities. When solving problems involving percentages, prealgebra helps us translate words into mathematical expressions. In this exercise, think of the number 43.6 and the word 'percent' together. Also, remember that percent means 'per hundred'.

• First, identify the known numbers: 43.6 (the part) and 20 (the whole).
• Then, represent 'what percent' with the variable \( x \).

This leads to the equation where 43.6 equals \( x \) percent of 20. Using a variable helps us organize our steps and calculate the percent. Understanding the relationship between part, whole, and percent is crucial in solving such problems efficiently.

Problem Solving

When solving problems, especially those involving the percent equation, it's important to follow a structured approach. This involves understanding the problem, setting up the equation, and solving for the unknown. Here are some steps that can help:

• Begin by thoroughly understanding what the problem is asking. In this case, we're asked to find what percent 43.6 is of 20.
• Translate the word problem into a mathematical equation. This step requires identifying the known information and arranging it correctly.
• Apply basic arithmetic and algebraic techniques to isolate the variable and solve the equation. For the given problem, isolating \( x \) involves undoing the operations on it by performing inverse operations such as multiplication and division.

By organizing these steps, we can effectively tackle similar problems. A clear logical approach prevents mistakes and ensures a full understanding of the problem-solving process.

Mathematical Equations

Equations are fundamental to solving mathematical problems as they represent relationships between quantities. In the percent equation \( a = \frac{x}{100} \cdot b \), \( a \) is the part or portion of the whole \( b \), and \( x \) represents the percentage.

• Understanding the equation structure is crucial: it tells us that one side of the equation represents values based on percentage operations.
• Translating the given numbers into this equation involves identifying what each part represents. Here, 43.6 is the part, 20 is the whole, and \( x \) is the percent we want to find.
• Isolating the variable requires algebraic manipulation, such as rearranging the equation by multiplying or dividing as necessary to solve for \( x \).

These processes lay down the basis for understanding more complex equations. They are essential skills for any student learning prealgebra and problem-solving methodologies.