Problem 73 - Elementary Algebra

Question

Find the solution. Nine percent of a number is \(77.4 .\) What is the number?

Step-by-Step Solution

Answer

Answer: The unknown number is 860.

Step 1: Set up the equation

Let the unknown number be x. We are given that 9% of x is 77.4. We can write this as an equation: \(0.09x = 77.4\)

Step 2: Solve for the unknown number

To solve for x, we can divide both sides of the equation by 0.09: \(x = \frac{77.4}{0.09}\)

Step 3: Calculate the result

Dividing 77.4 by 0.09, we get: \(x = 860\) So, the unknown number is 860.

Key Concepts

Percentages in Algebra

Understanding percentages in algebra is a key skill in problem-solving. A percentage represents a fraction with a denominator of 100. When we say "9% of a number," we mean to find 9 parts out of 100 of that number.
This can be expressed algebraically where you convert the percentage into its decimal form.

• To convert a percentage to a decimal, divide the percentage by 100. For example, 9% becomes 0.09.
• This conversion allows us to set up equations using the percentage expressed as a decimal.

When given that a percentage of an unknown number is equal to a certain value, like 9% of a number being 77.4, this concept allows us to formulate an algebraic equation to find the unknown number.

Equation Setup

The first and crucial step in solving algebra problems involving percentages is to correctly set up an equation.
Based on the given problem, "9% of a number is 77.4," you need to represent this statement with an equation.

• First, let the unknown number be represented by a variable, commonly "x." This variable represents the mystery number we are trying to uncover.
• The phrase "9% of a number" translates to "0.09 times the number," or mathematically, "0.09x."
• Setting this equal to the given value results in the equation: \(0.09x = 77.4\).

This equation effectively captures all the information given in the word problem and prepares us for the next step: solving for the unknown value.

Solving Linear Equations

After setting up your equation, the next step is to solve it, which often involves isolating the variable on one side of the equation. Here’s how you do it:
Given the equation \(0.09x = 77.4\), your goal is to solve for \(x\), which means you need to get \(x\) by itself on one side of the equation.

• To isolate \(x\), divide both sides of the equation by 0.09. This step simplifies the equation to \(x = \frac{77.4}{0.09}\).
• Once you perform the division, you get \(x = 860\).

Congratulations! You have determined that the unknown number is 860. Solving linear equations like this is vital as it helps to find unknowns in many types of problems, often just requiring clear logical steps and basic arithmetic operations.