75 Of What Number Is 36

Finding the number when you know a percentage can seem tricky, but with a clear approach, it becomes a straightforward process. This article will provide a step-by-step guide to solving the question "75% of what number is 36?" We will explore the underlying concepts, different methods to arrive at the solution, and offer examples to solidify your understanding. Whether you're a student brushing up on your math skills or just curious, this comprehensive guide will equip you with the knowledge to tackle similar problems confidently.

Understanding the Basics of Percentages

Percentages are a fundamental concept in mathematics, used to express a number as a fraction of 100. The word "percent" comes from the Latin per centum, meaning "out of one hundred." Understanding how percentages relate to fractions and decimals is crucial for solving problems like the one we're addressing.

• Percentage to Fraction: To convert a percentage to a fraction, divide the percentage by 100. For example, 75% is equivalent to 75/100, which can be simplified to 3/4.

• Percentage to Decimal: To convert a percentage to a decimal, divide the percentage by 100. For example, 75% is equivalent to 0.75.

• The Basic Percentage Formula: The fundamental formula that underlies all percentage problems is:

  (Part / Whole) = (Percentage / 100)

  Where:

  • Part is the specific amount you're considering.
  • Whole is the total amount or the number you're trying to find.
  • Percentage is the percentage relationship between the part and the whole.

Setting Up the Equation: "75% of What Number is 36?"

The question "75% of what number is 36?" is a classic percentage problem. To solve it, we need to translate the words into a mathematical equation. Here's how we break down the sentence:

• "75%" translates to 75/100 or 0.75.
• "of" in this context means multiplication.
• "what number" represents the unknown value we are trying to find. Let's call it "x."
• "is" means equals (=).
• "36" is the result.

Therefore, we can rewrite the question as an equation:

75% of x = 36

Or, in mathematical terms:

0. 75 * x = 36

Now that we have the equation, we can proceed to solve for x.

Step-by-Step Solution: Solving for the Unknown Number

To find the value of x in the equation 0.75 * x = 36, we need to isolate x on one side of the equation. This can be done by dividing both sides of the equation by 0.75.

Step 1: Write down the equation:

0. 75 * x = 36

Step 2: Divide both sides by 0.75:

(0.75 * x) / 0.75 = 36 / 0.75

Step 3: Simplify:

x = 36 / 0.75

Step 4: Perform the division:

x = 48

Therefore, 75% of 48 is 36.

Alternative Methods to Solve the Problem

While dividing by the decimal form of the percentage is a common method, there are other ways to solve this problem. These methods can provide a deeper understanding and alternative approaches.

Method 1: Using Fractions

Instead of converting the percentage to a decimal, we can use its fractional equivalent. As mentioned earlier, 75% is equal to 3/4. So, the equation becomes:

(3/4) * x = 36

To solve for x, we need to multiply both sides of the equation by the reciprocal of 3/4, which is 4/3.

Step 1: Write down the equation:

(3/4) * x = 36

Step 2: Multiply both sides by 4/3:

(4/3) * (3/4) * x = 36 * (4/3)

Step 3: Simplify:

x = 36 * (4/3)

Step 4: Perform the multiplication:

x = (36 * 4) / 3

x = 144 / 3

x = 48

Method 2: Using Proportions

Another method involves setting up a proportion. A proportion is an equation that states that two ratios are equal. In this case, we can set up the following proportion:

75 / 100 = 36 / x

This proportion states that 75 is to 100 as 36 is to x. To solve for x, we can cross-multiply:

Step 1: Write down the proportion:

75 / 100 = 36 / x

Step 2: Cross-multiply:

75 * x = 36 * 100

Step 3: Simplify:

75x = 3600

Step 4: Divide both sides by 75:

x = 3600 / 75

Step 5: Perform the division:

x = 48

Real-World Examples and Applications

Understanding how to solve percentage problems is not just an academic exercise; it has practical applications in everyday life. Here are a few examples:

• Calculating Discounts: When shopping, understanding percentages helps you calculate discounts quickly. For example, if an item is 25% off and the original price is $60, you can calculate the discount amount as 0.25 * $60 = $15.
• Determining Sales Tax: When making a purchase, you often need to calculate the sales tax. If the sales tax rate is 8% and your purchase is $50, the sales tax amount is 0.08 * $50 = $4.
• Analyzing Financial Data: Percentages are frequently used in finance to analyze data. For example, you might want to calculate the percentage increase in your investment portfolio or the percentage of your income that goes towards rent.
• Understanding Statistics: Percentages are used extensively in statistics to represent data. For example, you might see statistics reported as "60% of respondents agreed with the statement."

Common Mistakes to Avoid

When working with percentages, it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

• Forgetting to Convert to Decimal or Fraction: One of the most common mistakes is forgetting to convert the percentage to a decimal or fraction before performing calculations. Remember that you can't directly use the percentage number in calculations without converting it first.
• Misinterpreting "of": The word "of" in percentage problems usually means multiplication, but it's essential to read the problem carefully to ensure you understand the relationship between the numbers.
• Incorrectly Setting Up the Equation: Setting up the equation incorrectly can lead to the wrong answer. Make sure you understand what the problem is asking before you start solving it.
• Rounding Errors: Rounding errors can occur when you round numbers prematurely during calculations. To minimize rounding errors, it's best to round your final answer only.

Advanced Percentage Concepts

Once you have a solid understanding of basic percentage problems, you can explore more advanced concepts, such as:

• Percentage Increase and Decrease: Calculating the percentage increase or decrease between two values. The formula for percentage increase is [(New Value - Old Value) / Old Value] * 100, and the formula for percentage decrease is [(Old Value - New Value) / Old Value] * 100.
• Compound Interest: Understanding how compound interest works, where interest is earned not only on the principal but also on the accumulated interest.
• Weighted Averages: Calculating weighted averages, where different values are assigned different weights or importance.
• Percentage Change: Understanding percentage change, which is the extent to which a variable gains or loses value.

Practice Problems

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

1. 20% of what number is 10?
2. 150% of what number is 60?
3. 60 is 25% of what number?
4. 12% of what number is 9?
5. 85% of what number is 170?

Answers:

1. 50
2. 40
3. 240
4. 75
5. 200

Conclusion

Mastering percentage problems is a valuable skill that extends far beyond the classroom. By understanding the basic principles, setting up equations correctly, and avoiding common mistakes, you can confidently tackle a wide range of percentage-related questions. Remember to practice regularly and apply these concepts to real-world scenarios to reinforce your understanding. Whether you're calculating discounts, analyzing financial data, or interpreting statistics, a solid grasp of percentages will serve you well in many aspects of life.