90 Of What Number Is 36

Finding the number of which 90% is 36 involves a blend of basic algebra and percentage calculations, crucial skills applicable in everyday scenarios like calculating discounts, understanding financial statements, and determining statistical data. This article will guide you through the process, providing a clear understanding and practical steps to solve such problems.

Understanding the Basics

Before diving into the solution, let’s clarify a few key concepts:

• Percentage: A percentage is a way of expressing a number as a fraction of 100. Per cent means "per hundred." For example, 50% means 50 out of 100.
• Algebraic Equation: An algebraic equation is a mathematical statement that two expressions are equal. It typically contains variables (unknowns) and constants.

With these basics in mind, we can translate the problem "90% of what number is 36" into an algebraic equation.

Translating the Problem into an Equation

The problem can be expressed as:

90% of X = 36

Where X represents the unknown number we are trying to find. To solve this, we need to convert the percentage into a decimal.

Converting Percentage to Decimal

To convert a percentage to a decimal, divide it by 100. Thus:

90% = 90 / 100 = 0.9

Now, we can rewrite our equation as:

0.9 * X = 36

This equation is now ready to be solved for X.

Step-by-Step Solution

Here’s how to solve the equation 0.9 * X = 36 step by step:

1. Isolate the Variable: To find the value of X, we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 0.9.
2. Divide Both Sides:

   0.9 * X / 0.9 = 36 / 0.9

3. Simplify: The 0.9 on the left side cancels out, leaving:

   X = 36 / 0.9

4. Perform the Division: Divide 36 by 0.9:

   X = 40

Therefore, 90% of 40 is 36.

Verification

To ensure our solution is correct, we can substitute X with 40 in the original equation:

0.9 * 40 = 36

36 = 36

Since the equation holds true, our solution is correct.

Alternative Method: Using Fractions

Another approach to solving this problem is by using fractions instead of decimals.

Converting Percentage to Fraction

To convert 90% to a fraction, express it as a fraction with a denominator of 100:

90% = 90 / 100

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 10:

90 / 100 = 9 / 10

Rewriting the Equation with Fractions

Now, rewrite the original equation using the fraction:

(9 / 10) * X = 36

Solving for X

To solve for X, multiply both sides of the equation by the reciprocal of the fraction (10/9):

(10 / 9) * (9 / 10) * X = 36 * (10 / 9)

The fractions on the left side cancel out, leaving:

X = 36 * (10 / 9)

Multiply 36 by 10:

X = 360 / 9

Divide 360 by 9:

X = 40

Again, we find that X = 40.

Common Mistakes to Avoid

When solving percentage problems, it's easy to make mistakes. Here are some common pitfalls to avoid:

• Incorrect Conversion: Failing to convert the percentage to a decimal or fraction properly. Always remember to divide the percentage by 100.
• Misinterpreting the Problem: Not correctly identifying what the problem is asking. Ensure you understand which value is the percentage, which is the part, and which is the whole.
• Algebraic Errors: Making mistakes while manipulating the equation, such as incorrect division or multiplication.
• Not Verifying the Solution: Forgetting to check your answer by substituting it back into the original equation.

Real-World Applications

Understanding how to solve percentage problems is valuable in many real-world scenarios:

• Calculating Discounts: Retailers often offer discounts expressed as percentages. Knowing how to calculate these discounts helps you determine the actual price you'll pay.
• Financial Planning: Understanding percentages is essential for calculating interest rates, investment returns, and loan payments.
• Data Analysis: Percentages are widely used in statistics to represent proportions and changes in data sets.
• Sales and Marketing: Businesses use percentages to track sales growth, market share, and conversion rates.
• Everyday Math: From tipping at restaurants to splitting bills, percentages are a part of daily life.

Practice Problems

To reinforce your understanding, here are a few practice problems:

1. 60% of what number is 42?
2. 25% of what number is 15?
3. 120% of what number is 72?
4. 15% of what number is 9?
5. 80% of what number is 100?

Solutions

1. 70
2. 60
3. 60
4. 60
5. 125

Advanced Applications

Beyond basic calculations, understanding percentages can be applied to more complex problems, such as:

Compound Interest

Compound interest involves earning interest not only on the initial principal but also on the accumulated interest from previous periods. The formula for compound interest is:

A = P (1 + r/n)^(nt)

Where:

• A = the future value of the investment/loan, including interest
• P = the principal investment amount (the initial deposit or loan amount)
• r = the annual interest rate (as a decimal)
• n = the number of times that interest is compounded per year
• t = the number of years the money is invested or borrowed for

Understanding percentages is crucial for calculating the annual interest rate (r) and interpreting the final amount (A).

Calculating Percentage Change

Percentage change is used to describe the degree to which a quantity changes over time. It is often used in economics, finance, and statistics. The formula for percentage change is:

Percentage Change = ((New Value - Old Value) / Old Value) * 100

For example, if a stock's price increases from $50 to $75, the percentage change is:

((75 - 50) / 50) * 100 = (25 / 50) * 100 = 50%

Weighted Averages

Weighted averages are used when some values contribute more to the average than others. Each value is assigned a weight, which is a percentage of the total. The weighted average is calculated by multiplying each value by its weight and then summing the results.

For example, if a student scores 80% on a test worth 60% of their grade and 90% on a test worth 40% of their grade, the weighted average is:

(80 * 0.6) + (90 * 0.4) = 48 + 36 = 84%

The Role of Technology

Today, calculators and software tools can quickly solve percentage problems. However, understanding the underlying principles is still essential. Relying solely on technology without grasping the fundamentals can lead to errors if the input is incorrect or if the tool is used improperly.

Calculators

Calculators, both physical and digital, can perform percentage calculations with ease. Most calculators have a percent key (%) that simplifies the process.

Spreadsheet Software

Spreadsheet software like Microsoft Excel and Google Sheets are powerful tools for handling more complex percentage-related calculations. These programs allow you to create formulas, analyze data sets, and generate charts and graphs based on percentage changes and distributions.

Online Tools

Numerous websites and apps offer percentage calculators and tools. These can be useful for quick calculations or for verifying your work.

Conclusion

Solving the problem "90% of what number is 36" is a straightforward application of basic algebra and percentage concepts. By converting the percentage to a decimal or fraction, setting up an algebraic equation, and solving for the unknown variable, we can find the answer. Whether you're calculating discounts, analyzing financial data, or solving everyday math problems, a solid understanding of percentages is an invaluable skill. Remember to practice regularly, avoid common mistakes, and leverage technology wisely to enhance your problem-solving abilities.