Problem 20 - Prealgebra

Question

Solve each of the following problems. 11 is \(25 \%\) of what number?

Step-by-Step Solution

Answer

11 is 25% of 44.

Step 1: Understand the Problem

We need to find a number, say \( x \), where \( 11 \) is \( 25\% \) of \( x \). This means we are looking for a number that, when \( 25\% \) of it is calculated, gives us \( 11 \).

Step 2: Convert Percent to Decimal

To work with a percentage, it's easier to convert \( 25\% \) to a decimal. We convert a percentage to a decimal by dividing by \( 100 \).\[25\% = \frac{25}{100} = 0.25\]

Step 3: Set Up the Equation

Since \( 0.25 \) of \( x \) equals \( 11 \), we write the equation:\[0.25 \times x = 11\]

Step 4: Solve for x

To find \( x \), divide both sides of the equation by \( 0.25 \):\[x = \frac{11}{0.25}\]\( \frac{11}{0.25} \) simplifies to \( 11 \div 0.25 = 44 \). Therefore, \( x = 44 \).

Step 5: Verify the Solution

Check that \( 25\% \) of \( 44 \) is indeed \( 11 \). Calculate:\[0.25 \times 44 = 11\]The calculation is correct, confirming our solution is accurate.

Key Concepts

Prealgebra

Prealgebra is the foundation of all advanced math topics, introducing learners to basic arithmetic operations like addition, subtraction, multiplication, and division. It sets the stage for algebraic concepts by teaching how numbers and operations interact in equations. The beauty of prealgebra is its simplicity, allowing you to easily spot patterns and relationships between numbers.
Prealgebra often involves word problems that require translating everyday situations into mathematical equations. This may involve recognizing keywords that indicate operations, such as "of" typically meaning multiplication in percentage problems. By mastering prealgebra, you'll develop critical problem-solving skills that are essential for tackling more complex equations later on.
Whether it's percentages, fractions, or basic factorizations, prealgebra teaches how to manipulate numbers in a logical way to find a solution.

Solving Equations

Solving equations is a vital skill in algebra where the goal is to find the value of the unknown variable. In our example exercise, we set up the equation to find the value of \( x \).
The main steps in solving any equation include:

• Isolating the variable on one side of the equation.
• Performing inverse operations to maintain equality.

In the step-by-step solution, once we express "11 is 25% of what number?", we form the equation \(0.25 \times x = 11\).
By dividing both sides by \(0.25\), we isolate \(x\) and solve for it. This technique of solving allows you to see the relationship between numbers and how they work together through operations like multiplication, division, addition, or subtraction.
Such practice strengthens your ability to handle more complex algebraic expressions in the future.

Converting Percentages to Decimals

Converting percentages into decimals is an essential skill, especially when working with equations involving percentage values, as it makes calculations manageable.
To convert a percentage to a decimal:

• Divide the percentage by 100.
• For example, converting 25% gives us \( \frac{25}{100} = 0.25 \).

This conversion is a straightforward move but a crucial one. It transforms what can be a complicated percentage operation into a simple multiplication within the equation.
Understanding this conversion ensures clarity when solving percentage problems and forms a bridge between different forms of numerical representation seen often in math and everyday life. Converting percentages precisely and efficiently can aid in interpreting data, budgeting, and planning, making it a highly practical tool.