51 Is 85 Of What Number

51 is 85% of What Number? A Comprehensive Guide to Solving Percentage Problems

Percentage problems are a fundamental part of mathematics, appearing in various contexts from everyday finances to complex scientific calculations. Understanding how to solve these problems is crucial for anyone looking to improve their numerical skills and problem-solving abilities. This article will delve deep into the question, "51 is 85% of what number?", providing a step-by-step solution, exploring different methods, and expanding on the broader concept of percentage calculations. We will also examine practical applications and offer tips for mastering these types of problems.

Understanding the Problem: Deconstructing the Question

The question "51 is 85% of what number?" presents a classic percentage problem. We're given a part (51), the percentage (85%), and we need to find the whole. This is a common scenario in many real-world applications, such as calculating the original price of an item after a discount, determining the total number of students based on a percentage attending a specific event, or finding the overall population based on a surveyed sample.

Key Terms:

• Part: The portion of the whole that is given (in this case, 51).
• Percentage: The portion of the whole expressed as a fraction of 100 (in this case, 85%).
• Whole: The total amount we need to find. This is our unknown value.

Method 1: Using the Percentage Formula

The most straightforward method to solve this problem is by using the basic percentage formula:

Part = (Percentage/100) * Whole

Let's substitute the known values into the formula:

51 = (85/100) * Whole

Now, we need to solve for the "Whole":

1. Convert the percentage to a decimal: 85% = 85/100 = 0.85

2. Rewrite the equation: 51 = 0.85 * Whole

3. Isolate the "Whole": Divide both sides of the equation by 0.85:

   Whole = 51 / 0.85

4. Calculate the result: Whole = 60

Therefore, 51 is 85% of 60.

Method 2: Using Proportions

Another effective method involves setting up a proportion. A proportion is an equation that states that two ratios are equal. We can represent the problem as follows:

51/x = 85/100

Where 'x' represents the whole number we are trying to find.

To solve this proportion:

1. Cross-multiply: 51 * 100 = 85 * x

2. Simplify: 5100 = 85x

3. Isolate 'x': Divide both sides by 85:

   x = 5100 / 85

4. Calculate the result: x = 60

Again, we find that 51 is 85% of 60.

Method 3: Using the Unitary Method

The unitary method focuses on finding the value of one unit (1%) and then scaling it up to find the total.

1. Find 1%: If 85% is 51, then 1% would be 51/85.

2. Calculate 1%: 51/85 = 0.6

3. Find 100%: To find the whole (100%), multiply 1% by 100: 0.6 * 100 = 60

Therefore, 51 is 85% of 60.

Practical Applications: Real-World Examples

Percentage problems are prevalent in various aspects of life. Here are some real-world examples demonstrating the practical application of solving "51 is 85% of what number?":

• Sales and Discounts: A store offers an 85% discount on an item, and the discounted price is $51. The original price of the item is $60.

• Surveys and Statistics: In a survey of 85% of a school's population, 51 students expressed preference for a particular program. The total number of students in the school is 60.

• Test Scores: A student scored 51 points on a test that represents 85% of the total possible points. The total points possible on the test is 60.

• Financial Calculations: An investment grows by 85%, resulting in a gain of $51. The initial investment was $60.

Mastering Percentage Problems: Tips and Tricks

Mastering percentage problems requires practice and a thorough understanding of the underlying concepts. Here are some helpful tips:

• Practice Regularly: The more you practice, the more comfortable you'll become with different problem types.

• Visualize the Problem: Try to picture the problem visually. Imagine a pie chart representing the whole and its parts.

• Understand the Formula: Thoroughly understand the basic percentage formula and its variations.

• Check Your Work: Always check your calculations to ensure accuracy. You can do this by working backward from your answer.

• Use Different Methods: Try solving the same problem using different methods (formula, proportion, unitary method) to improve your understanding and problem-solving skills.

• Utilize Online Resources: Numerous online resources and calculators can help you check your work and deepen your understanding.

Conclusion: Expanding Your Mathematical Toolkit

Solving percentage problems, such as determining what number 51 represents as 85%, is a crucial skill applicable to various areas of life. By understanding the different methods – the formula, proportion, and unitary method – you can approach such problems with confidence and efficiency. Remember that consistent practice and a deep understanding of the underlying concepts are essential for mastering this fundamental mathematical skill. This comprehensive guide should empower you to confidently tackle similar percentage problems, improving your mathematical proficiency and enhancing your ability to solve real-world challenges involving percentages. With practice and consistent application of these methods, you'll become increasingly adept at navigating the world of percentages.