Problem 24 - Prealgebra

Question

Solve each of the following problems. 6 is \(3 \%\) of what number?

Step-by-Step Solution

Answer

6 is \(3\%\) of 200.

Step 1: Understand the Problem

We know that 6 is equivalent to \(3\%\) of some number. Our goal is to find this unknown number.

Step 2: Translate Percentage to Decimal

Convert \(3\%\) to a decimal by dividing by 100. This gives us \(0.03\).

Step 3: Set Up the Equation

Set up the equation based on the information provided: \[0.03 \times x = 6\] where \(x\) is the unknown number.

Step 4: Solve for the Unknown Number

Solve the equation for \(x\) by dividing both sides by \(0.03\): \[x = \frac{6}{0.03}\]

Step 5: Calculate the Division

Perform the division: \[x = 200\]

Step 6: Verify the Solution

Verify the solution by calculating \(3\%\) of 200 and ensuring it equals 6: \(0.03 \times 200 = 6\). This confirms our solution is correct.

Key Concepts

Decimal Conversion

When we're dealing with percentage problems, decimal conversion is an essential concept to grasp. A percentage, like \(3\%\), represents a part per hundred. To convert a percentage into a decimal, you divide by 100. This is because 100% equals 1 in decimal form. So, for any percentage \(p\), the decimal form is \(\frac{p}{100}\).
For example, converting \(3\%\) to a decimal involves a simple division:

• \(3\% = \frac{3}{100} = 0.03\).

Understanding this conversion makes it possible to work with percentages in equations. Decimal conversion simplifies complicated percentage calculations and makes it easier to use in algebraic expressions.

Equation Solving

After converting the percentage into a decimal, the next step is setting up an equation. Let's say we know that \(6\) is \(3\%\) of an unknown number, \(x\). Our goal is to find \(x\).
We convert \(3\%\) to \(0.03\) as a decimal. The equation from the problem is:

• \(0.03 \times x = 6\).

In equation solving, you want to isolate the variable, which in this case is \(x\). To solve for \(x\), divide both sides by \(0.03\):

• \(x = \frac{6}{0.03}\).

Solve the division to find \(x\). For this exercise, the math tells us that \(x = 200\).
This solution process involves understanding the equation and performing simple arithmetic to find the answer.

Verification of Solution

After finding a solution, it's crucial to verify its correctness. Verification ensures that we haven't made errors in our calculation process. For this percentage problem, we want to check if \(3\%\) of \(200\) is indeed \(6\).
The verification process involves substituting \(x = 200\) back into the equation we formed:

• Calculate \(0.03 \times 200\).

This results in \(6\), confirming our solution. Verification involves re-calculating or re-assessing the problem using a different approach, helping ensure accuracy and reliability in mathematics.
This step reassures you that your calculations and conversion were correct, making it a vital part of solving math problems.