Problem 82 - Elementary Algebra

Question

In the following exercises, translate and solve. \(12.3 \%\) of what number is \(\$ 92.25 ?\)

Step-by-Step Solution

Answer

\(750\)

Step 1: Understand the problem

We need to find a number such that 12.3% of it equals $92.25.

Step 2: Translate the problem into an equation

Let's denote the number we need to find as \(x\). The problem statement '12.3% of \(x\) is \(\$92.25\)' can be written as an equation: \[ 0.123 \times x = 92.25 \]

Step 3: Solve the equation

To solve for \(x\), divide both sides of the equation by 0.123: \[ x = \frac{92.25}{0.123} \]

Step 4: Calculate the value

Use a calculator to find the value of \[ x = \frac{92.25}{0.123} \]. This gives \[ x \approx 750 \]

Key Concepts

solving percentage equations

Solving percentage equations involves changing percentages into decimals. This is the key first step. A percentage like 12.3% is the same as 0.123.
Next, when a problem says '12.3% of what number is \(92.25?', we need to create an equation. We denote the unknown number as 𝑥.
Therefore, 12.3% of 𝑥 can be written as 0.123𝑥. The problem now reads: 0.123𝑥 = 92.25. To find 𝑥, you divide both sides of the equation by 0.123:
\
\
\[ x = \frac{92.25}{0.123} \]
\
\
Finally, use a calculator to perform the division and find 𝑥. In this case, 𝑥 ≈ 750. So, 12.3% of 750 is \)92.25.
This process simplifies percentage problems effectively.

basic algebra

Basic algebra helps in solving equations like the percentage problem. Start by understanding the variables and constants in the equation.
The unknown number we need is typically represented as 𝑥, making it easier to form the equation.
In the given problem: \ \ \ \[ 0.123 x = 92.25 \] \ \ \ Here, 0.123 is a constant (the percentage converted into a decimal), and 92.25 is another constant. We solve for 𝑥 (the unknown variable).
\
Follow these basic steps in algebra:

• Identify the variable (𝑥)
• Set up the equation
• Isolate the variable by performing operations on both sides of the equation

In this problem, you isolate 𝑥 by dividing both sides by 0.123. Basic algebra is powerful in solving various equations systematically.

translating word problems

Translating word problems into mathematical equations is a crucial skill.
First, read the problem carefully. Identify the key information and what is being asked.
In our exercise, the phrase '12.3% of what number is $92.25?' gives clues to form the equation. The percentage (12.3%) tells us to use multiplication, and 'of what number' indicates our unknown variable (𝑥).

• Translate '12.3%' to 0.123 for mathematical convenience.
• Restate the problem: 0.123𝑥 = 92.25

Now, solve this equation to find the unknown number. By practicing translating word problems frequently, you will develop a keen ability to understand and solve them more efficiently.