12 Is 15 Of What Number

Unraveling the Mystery: 12 is 15% of What Number?

Understanding percentages is a fundamental skill with applications in everyday life, from calculating discounts to understanding financial reports. One common type of percentage problem is figuring out the whole when you know a part and the percentage that part represents. The question "12 is 15% of what number?" falls into this category. This article will break down the process of solving this type of problem, explore the underlying mathematical principles, provide multiple solution methods, and delve into real-world examples to solidify your understanding.

Understanding the Core Concept: Percentages

At its heart, a percentage is simply a way of expressing a fraction or a ratio as a part of 100. The word "percent" comes from the Latin "per centum," meaning "out of one hundred." Therefore, 15% means 15 out of every 100, which can be written as the fraction 15/100 or the decimal 0.15.

When we say "12 is 15% of what number?", we're essentially saying that 12 represents 15 parts out of a total 100 parts of some unknown number. Our goal is to find that unknown number, the whole value.

Method 1: The Algebraic Equation

The most direct and reliable method for solving percentage problems is to translate the word problem into an algebraic equation. This approach allows for a clear and structured solution. Here's how:

1. Identify the unknowns and knowns:

   • We know the part: 12
   • We know the percentage: 15% (which is 0.15 as a decimal)
   • We need to find the whole: Let's represent this unknown number with the variable x.

2. Translate the words into an equation:

   • "12 is 15% of what number?" translates to: 12 = 0.15 * x

3. Solve for x:

   • To isolate x, we need to divide both sides of the equation by 0.15:
     • 12 / 0.15 = x
     • x = 80

Therefore, 12 is 15% of 80.

Explanation:

This method leverages the fundamental relationship between percentages, parts, and wholes. By converting the percentage into a decimal and setting up a simple algebraic equation, we can easily solve for the unknown whole. The key is to accurately represent the relationship described in the problem using mathematical symbols.

Method 2: The Proportion Method

Another effective way to solve percentage problems is by using proportions. A proportion is an equation that states that two ratios are equal. In this case, we can set up a proportion that relates the known part and percentage to the unknown whole.

1. Set up the proportion:

   • We know that 12 is to x (the unknown whole) as 15 is to 100. This can be written as:
     • 12 / x = 15 / 100

2. Cross-multiply:

   • To solve the proportion, we cross-multiply:
     • 12 * 100 = 15 * x
     • 1200 = 15x

3. Solve for x:

   • Divide both sides of the equation by 15:
     • 1200 / 15 = x
     • x = 80

Again, we find that 12 is 15% of 80.

Explanation:

The proportion method relies on the concept of equivalent ratios. By setting up a proportion that reflects the relationship between the part, the whole, and the percentage, we can use cross-multiplication to solve for the unknown whole. This method can be particularly helpful for visual learners.

Method 3: The Unitary Method

The unitary method involves finding the value of one unit (in this case, 1%) and then scaling that value up to find the value of the whole (100%).

1. Find the value of 1%:

   • If 12 represents 15%, then we can find the value of 1% by dividing 12 by 15:

     • 12 / 15 = 0.8

   • Therefore, 1% of the unknown number is 0.8.

2. Find the value of 100% (the whole):

   • Since 1% is 0.8, then 100% is 0.8 multiplied by 100:
     • 0.8 * 100 = 80

Thus, 12 is 15% of 80.

Explanation:

The unitary method breaks down the problem into smaller, more manageable steps. By first finding the value of a single percentage unit, we can then easily calculate the value of the entire whole. This method is particularly useful for building a conceptual understanding of percentages.

Method 4: Using a Calculator

For a quick and efficient solution, you can use a calculator. This method directly translates the algebraic equation approach into calculator steps.

1. Understand the equation: Remember, we're solving for x in the equation 12 = 0.15 * x.

2. Rearrange the equation: To isolate x, we need to divide 12 by 0.15.

3. Input into the calculator: Enter "12 / 0.15" into your calculator.

4. The answer: The calculator will display "80".

Therefore, 12 is 15% of 80.

Explanation:

While this method is the fastest, it's important to understand the underlying mathematical principles to avoid blindly relying on the calculator. Make sure you grasp the relationship between the numbers and the percentage before using this method.

Real-World Examples

Let's explore some real-world scenarios where you might encounter this type of percentage problem:

• Sales and Discounts: Imagine a store is having a sale, and an item is marked down by a certain percentage. If you know the amount of the discount and the percentage discount, you can calculate the original price of the item. For instance, if an item is discounted by $12, which represents a 15% discount, you can find the original price using the methods described above.

• Financial Investments: If your investment has grown by $12, which represents a 15% return on your initial investment, you can calculate the initial investment amount using the same principles.

• Exam Scores: If you answered 12 questions correctly on a test, and this represents 15% of the total questions, you can determine the total number of questions on the test.

• Ingredients in a Recipe: Suppose a recipe calls for 12 grams of a specific spice, and this spice represents 15% of the total weight of all the spices in the recipe. You can then calculate the total weight of all the spices.

Deep Dive: The Importance of "Of"

The word "of" in percentage problems is a crucial indicator of multiplication. When you see "15% of what number?", it signifies that you need to multiply 15% (expressed as a decimal or fraction) by the unknown number. This understanding is fundamental to correctly setting up the equation or proportion needed to solve the problem.

Common Mistakes to Avoid

• Confusing the Part and the Whole: One of the most common mistakes is incorrectly identifying which number represents the part and which represents the whole. Always carefully read the problem to determine which value represents a portion of the unknown number.

• Forgetting to Convert the Percentage: When using algebraic equations or proportions, it's essential to convert the percentage into a decimal or fraction. For example, 15% must be converted to 0.15 or 15/100 before being used in calculations.

• Incorrectly Setting up the Proportion: Ensure that the proportion is set up correctly, with corresponding values in the correct positions. For example, if you're comparing the part to the whole, make sure you maintain that relationship on both sides of the equation.

• Relying Solely on Calculators Without Understanding: While calculators are helpful, relying on them without understanding the underlying mathematical principles can lead to errors. Always try to understand the problem conceptually before using a calculator.

Beyond the Basics: Advanced Percentage Problems

While the question "12 is 15% of what number?" is a fundamental percentage problem, there are more complex scenarios that build upon these basic principles. These include:

• Percentage Increase and Decrease: Calculating percentage increases (e.g., price increases, population growth) and percentage decreases (e.g., discounts, depreciation).

• Compound Interest: Understanding how interest accrues over time, with interest earned on both the principal and previously earned interest.

• Percentage Change: Determining the percentage change between two values, often used to track trends and performance.

Mastering the basic concepts of percentages, as outlined in this article, is essential for tackling these more advanced problems.

Practice Problems

To solidify your understanding, try solving these practice problems:

1. 25 is 20% of what number?
2. 8 is 5% of what number?
3. 45 is 30% of what number?
4. 10 is 2% of what number?
5. 60 is 75% of what number?

Solutions to Practice Problems

1. 25 is 20% of what number?

   • Using the algebraic equation method: 25 = 0.20 * x => x = 25 / 0.20 = 125
   • Answer: 125

2. 8 is 5% of what number?

   • Using the proportion method: 8 / x = 5 / 100 => 800 = 5x => x = 800 / 5 = 160
   • Answer: 160

3. 45 is 30% of what number?

   • Using the unitary method: 45 / 30 = 1.5 (1% is 1.5) => 1.5 * 100 = 150
   • Answer: 150

4. 10 is 2% of what number?

   • Using the calculator method: 10 / 0.02 = 500
   • Answer: 500

5. 60 is 75% of what number?

   • Using the algebraic equation method: 60 = 0.75 * x => x = 60 / 0.75 = 80
   • Answer: 80

Conclusion

The question "12 is 15% of what number?" is a classic example of a percentage problem that can be solved using various methods. Whether you prefer algebraic equations, proportions, the unitary method, or a calculator, understanding the underlying principles is key to finding the correct answer. By mastering these techniques and practicing with real-world examples, you can confidently tackle any percentage problem that comes your way. Remember to carefully identify the part, the percentage, and the whole, and to choose the method that best suits your learning style. With consistent practice, you'll develop a strong intuition for percentages and their applications in everyday life.