12 Of What Number Is 42

12 of What Number is 42? Unlocking the Power of Algebra

This seemingly simple question, "12 of what number is 42?", is actually a gateway to understanding fundamental algebraic concepts. It's a perfect introduction to problem-solving strategies that extend far beyond basic arithmetic. This article will not only answer the question but delve into the various methods to solve it, explain the underlying mathematical principles, and showcase its relevance in real-world applications.

Understanding the Problem: Translating Words into Equations

The first step in tackling any word problem is to translate the words into a mathematical equation. Let's break down the sentence "12 of what number is 42":

• "12 of": This indicates multiplication. We're taking 12 of something.
• "what number": This is our unknown value, often represented by a variable, like 'x'.
• "is": This signifies equality.
• "42": This is the result of the multiplication.

Therefore, the sentence translates to the following algebraic equation:

12 * x = 42

Method 1: Solving for 'x' through Division

The most straightforward method to solve for 'x' is to isolate it on one side of the equation. Since 'x' is being multiplied by 12, we perform the inverse operation—division—on both sides of the equation:

12 * x / 12 = 42 / 12

This simplifies to:

x = 3.5

Therefore, 12 of 3.5 is 42.

Method 2: Utilizing Proportions

Proportions offer another powerful approach to solving this problem. A proportion is a statement that two ratios are equal. We can set up a proportion as follows:

12 / 100 = 42 / x

This represents 12 parts out of 100 (or 12%) being equivalent to 42 parts out of an unknown total (x).

To solve this proportion, we can cross-multiply:

12 * x = 42 * 100

12x = 4200

Now, divide both sides by 12:

x = 4200 / 12

x = 350

This result, 350, represents the total number where 12% (or 12 parts out of 100) equals 42.

This demonstrates how the context of the problem is important. If the problem were framed as "12% of what number is 42", the answer is 350, not 3.5. This highlights the significance of accurately interpreting word problems.

Method 3: Understanding Percentages

If we interpret "12 of" as meaning "12 percent of," the problem changes slightly. Then the equation would be:

0.12x = 42

Dividing both sides by 0.12:

x = 42 / 0.12

x = 350

This reinforces the importance of clearly understanding the language used in the problem statement.

Real-World Applications: Where This Matters

The seemingly simple equation 12x = 42 has wide-ranging applications in various fields:

1. Business and Finance:

• Calculating percentages: Determining profit margins, discounts, tax rates, interest rates, and more. For instance, if a company makes a 12% profit on a certain item, and that profit is $42, this equation helps calculate the item's total sales revenue.
• Scaling up or down: Adjusting recipes, production quotas, or investment portfolios based on proportions. If a recipe calls for 12 units of ingredient A, and you need to make a batch yielding 42 units of the final product, you can use this to determine the required amount of ingredient A.
• Cost analysis: Calculating the cost per unit when the total cost of 12 units is known. If 12 items cost $42, this equation helps find the cost of one item.

2. Science and Engineering:

• Measurements and conversions: Converting units of measurement, scaling down experimental data, analyzing ratios in chemical reactions.
• Data analysis: Extrapolating from sample data to estimate larger populations or trends.

3. Everyday Life:

• Sharing items: Dividing resources equally or proportionally among a group of people. If you have 42 cookies and want to give 12 cookies to one friend, this equation can help you determine the total amount of friends to distribute the cookies to.
• Recipe adjustments: Increasing or decreasing recipe quantities based on the number of servings desired.
• Budgeting: Allocating a certain percentage of your budget to different expenses.

Expanding the Concept: Beyond Simple Equations

The problem "12 of what number is 42?" serves as a foundational example. More complex word problems often involve multiple variables, multiple steps, and may require applying additional mathematical concepts. However, the core principles of translating words into equations and then solving for unknowns remain consistent.

Mastering Word Problems: Tips and Strategies

Solving word problems effectively requires a systematic approach. Here are some helpful strategies:

• Read carefully: Thoroughly read and understand the problem before attempting to solve it. Identify the known and unknown quantities.
• Translate into an equation: Convert the words into a mathematical expression or equation that represents the problem.
• Define variables: Clearly define the variables representing the unknown quantities.
• Solve the equation: Use algebraic techniques to isolate the variable and solve for its value.
• Check your answer: Always check your solution against the original word problem to ensure it makes sense in the context.
• Practice regularly: Consistent practice is key to mastering word problems.

Conclusion: The Power of Problem-Solving

The seemingly simple question, "12 of what number is 42?", unlocks a world of mathematical understanding. It demonstrates the practical application of algebra in various contexts, from business and finance to everyday life. By mastering the techniques outlined above, you'll be well-equipped to tackle increasingly complex problems and improve your problem-solving skills. Remember to practice regularly, break down complex problems into smaller, manageable steps, and always check your solutions to ensure accuracy. The ability to translate real-world scenarios into mathematical equations is a valuable skill that extends far beyond the classroom.