12 Of What Number Is 15

Understanding fractions, percentages, and their relationships to whole numbers is a fundamental concept in mathematics. The question "12 of what number is 15" delves into these core ideas, requiring a clear understanding of how to translate word problems into mathematical equations and solve for the unknown. This article provides a comprehensive guide to solving this problem, exploring the underlying principles and offering various approaches to reach the solution.

Decoding the Problem

The phrase "12 of what number is 15" can seem confusing at first glance. Breaking down the sentence into its mathematical components is essential. Here's how we can interpret it:

• "of" generally indicates multiplication.
• "what number" represents the unknown value we are trying to find, which we can denote as 'x'.
• "is" signifies equality.

Therefore, the problem translates to the equation: (1/2) * x = 15. Our goal is to isolate 'x' and find its value.

Methods to Solve: (1/2) * x = 15

Several methods can be employed to solve the equation (1/2) * x = 15. We'll explore some of the most common and effective approaches:

1. Isolating the Variable

This is the most direct method. To isolate 'x', we need to undo the multiplication by (1/2). The inverse operation of multiplying by (1/2) is dividing by (1/2). Remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of (1/2) is 2/1 or simply 2.

• Step 1: Multiply both sides of the equation by 2.

  (1/2) * x * 2 = 15 * 2

• Step 2: Simplify.

  x = 30

Therefore, 1/2 of 30 is 15.

2. Using Proportions

Proportions provide another way to approach this problem. We can set up a proportion relating the fraction (1/2) to the given value (15) and the unknown whole number (x).

• Step 1: Set up the proportion.

  (1/2) = (15/x)

• Step 2: Cross-multiply.

  1 * x = 2 * 15

• Step 3: Simplify.

  x = 30

This method confirms that 1/2 of 30 is indeed 15.

3. Thinking in Terms of the Whole

Another way to conceptualize the problem is to recognize that if 15 represents one-half of a number, then the whole number must be twice the value of 15.

• Step 1: Recognize that 15 is half the number.

• Step 2: Multiply 15 by 2 to find the whole number.

  15 * 2 = 30

This approach provides a quick and intuitive way to solve the problem.

4. Converting to Decimal Form

We can convert the fraction (1/2) into its decimal equivalent, which is 0.5. The equation then becomes:

0. 5 * x = 15

To solve for 'x', we divide both sides of the equation by 0.5:

• Step 1: Divide both sides by 0.5.

  x = 15 / 0.5

• Step 2: Simplify.

  x = 30

This method utilizes decimal representation to arrive at the same solution.

Verifying the Solution

It's crucial to verify the solution to ensure accuracy. To do this, we substitute the value of 'x' (which is 30) back into the original equation:

(1/2) * 30 = 15

15 = 15

Since the equation holds true, our solution is correct.

Expanding the Concept: Different Fractions and Percentages

The principles used to solve "12 of what number is 15" can be applied to a wide range of similar problems involving different fractions and percentages. Let's explore some variations:

Example 1: Using a Different Fraction

Problem: 1/4 of what number is 20?

Solution:

• Equation: (1/4) * x = 20

• Isolate x: Multiply both sides by 4.

  x = 20 * 4

• Simplify:

  x = 80

Therefore, 1/4 of 80 is 20.

Example 2: Using Percentages

Problem: 25% of what number is 50?

Solution:

• Convert percentage to a decimal: 25% = 0.25

• Equation: 0.25 * x = 50

• Isolate x: Divide both sides by 0.25.

  x = 50 / 0.25

• Simplify:

  x = 200

Therefore, 25% of 200 is 50.

Example 3: A More Complex Fraction

Problem: 3/5 of what number is 27?

Solution:

• Equation: (3/5) * x = 27

• Isolate x: Multiply both sides by the reciprocal of (3/5), which is (5/3).

  x = 27 * (5/3)

• Simplify:

  x = (27 * 5) / 3 x = 135 / 3 x = 45

Therefore, 3/5 of 45 is 27.

Real-World Applications

Understanding how to solve these types of problems has numerous real-world applications, including:

• Finance: Calculating discounts, interest rates, and investment returns often involves finding a percentage of a certain amount.
• Cooking: Recipes frequently require scaling ingredients up or down, which involves understanding fractions and proportions.
• Retail: Determining sale prices and calculating markups relies on understanding percentages and their relationship to the original price.
• Statistics: Analyzing data and interpreting results often involves working with fractions and percentages.

Common Mistakes to Avoid

While solving these problems, it's important to be aware of common mistakes that students often make:

• Misinterpreting the wording: Carefully read the problem and identify the key components, such as the fraction or percentage, the known value, and the unknown value.
• Incorrectly setting up the equation: Ensure that the equation accurately represents the relationship described in the problem.
• Performing the wrong operation: Remember to use the inverse operation to isolate the variable. If the variable is being multiplied, divide both sides of the equation. If the variable is being divided, multiply both sides.
• Forgetting to convert percentages to decimals: When working with percentages, always convert them to decimals or fractions before performing calculations.
• Not verifying the solution: Always check your answer by substituting it back into the original equation to ensure that it holds true.

Advanced Applications and Extensions

The concepts explored in this article can be extended to more advanced mathematical topics, such as:

• Algebraic equations: Solving more complex algebraic equations often involves manipulating fractions, percentages, and variables.
• Ratio and proportion problems: Many ratio and proportion problems can be solved using the same principles as the problems discussed in this article.
• Calculus: Understanding fractions and percentages is essential for working with derivatives and integrals in calculus.

The Importance of Conceptual Understanding

While memorizing formulas and procedures can be helpful, it's crucial to develop a conceptual understanding of the underlying principles. This will allow you to solve a wider range of problems and apply your knowledge to new situations.

Encourage students to visualize the problem, draw diagrams, and use manipulatives to develop a deeper understanding of fractions, percentages, and their relationships to whole numbers.

Conclusion

The problem "12 of what number is 15" provides a valuable opportunity to reinforce fundamental mathematical concepts. By understanding the relationship between fractions, percentages, and whole numbers, and by mastering the techniques for solving these types of problems, students can develop a strong foundation for future success in mathematics and other fields. This article has provided several methods to approach this problem, along with examples, real-world applications, and common mistakes to avoid. Remember that practice and a strong conceptual understanding are key to mastering these skills.