Problem 66 - Fundamentals of Mathematics

Question

(Section 7.7) 11.62 is \(83 \%\) of what number?

Step-by-Step Solution

Answer

11.62 is 83% of 14.

Step 1: Understand the problem

The problem asks for the original number that 11.62 represents when it is 83% of that number. This is a percentage problem where a part of the total is given, and we need to find the total.

Step 2: Set up the equation

To find the unknown total (let's call it 'x'), translate the problem into an equation. The statement "11.62 is 83% of what number" can be written as:\[ 11.62 = 0.83x \]

Step 3: Solve for the unknown

We need to solve the equation for \(x\). To do this, divide both sides of the equation by 0.83:\[ x = \frac{11.62}{0.83} \]

Step 4: Calculate the answer

Perform the division to find the value of \(x\):\[ x = 14 \]

Step 5: Verify the calculation

Check that 83% of the calculated number, 14, is indeed 11.62. Calculate 0.83 times 14, and ensure it equals 11.62. This confirms that the calculation is correct.

Key Concepts

Percentage Calculation

Percentage calculations are widely used in everyday math. In this specific problem, the task was to determine what total number corresponds to a given percentage. Let's break it down.
Percentage means out of 100. So, 83% is the same as saying 83 out of 100. When this percentage is linked to a certain value, 11.62 in our exercise, it indicates that the 11.62 is 83% of some total number.
You can translate percentage scenarios into an equation for easier handling. The process involves converting the percentage into its decimal form. This is done by dividing the percentage number by 100. For example:

• 83% becomes 0.83 in decimal form by calculating 83/100.

This conversion is crucial because it allows us to set up an equation to solve for the unknown number with simple algebra.

Equation Solving

Equation solving is often needed in mathematics to find an unknown value. In this problem, we needed to find the total number when 11.62 is 83% of it.
We set up the equation as follows: \[ 11.62 = 0.83x \] Here, \(x\) represents the unknown value we are solving for.
To isolate \(x\), we perform operations to both sides of the equation. To solve equations like these, following steps typically help:

• Identify the operation used with the variable \(x\), here it is multiplication with 0.83.
• To isolate \(x\), do the opposite operation. Divide both sides by 0.83.

The goal is to get \(x\) by itself on one side of the equation, which gives you the solution for \(x\). This algebraic manipulation is essential in solving many mathematical problems.

Division Operation

The division operation is one of the four basic arithmetic operations, crucial for this type of problem-solving.
In our example, after setting up the equation \[ 11.62 = 0.83x \], we divided both sides by 0.83 to solve for \(x\). This step can be broken down as:

• Perform the division by taking the given value, 11.62 in this case.
• Divide it by 0.83, which is the decimal representation of the percentage.
• This calculation provides the solution: \(x = \frac{11.62}{0.83} = 14\).

Division is the operation of determining how many times one number is contained in another. It's crucial in finding unknowns, especially when percentages or rates are involved. By mastering division, you can tackle a wide range of mathematical problems with confidence.