320 Of What Number Is 64

320 of What Number is 64? Understanding Percentages and Proportions

Finding the answer to "320 of what number is 64?" involves understanding percentages and proportions. It's a common mathematical problem with applications in various fields, from finance and business to everyday life. This comprehensive guide will not only solve this specific problem but also equip you with the knowledge and techniques to tackle similar percentage problems. We'll explore multiple approaches, from basic arithmetic to using algebraic equations, ensuring you grasp the underlying concepts.

Understanding the Problem: Deconstructing the Question

The question "320 of what number is 64?" is essentially asking us to find the whole number when we know a part of it (64) and the percentage that part represents (we need to calculate this percentage). This is a classic percentage problem that can be solved using different methods. Let's break down the problem step-by-step:

• Part: We know the part, which is 64.
• Whole: We need to find the whole number. Let's represent this unknown number with the variable 'x'.
• Percentage: We don't explicitly know the percentage, but we can derive it from the relationship between the part and the whole.

Method 1: Using Proportions

Proportions are a powerful tool for solving percentage problems. A proportion is a statement that two ratios are equal. We can set up a proportion to represent the problem:

64/320 = x/100

In this proportion:

• 64/320 represents the ratio of the part (64) to the whole (320). We're assuming that 320 represents 100% (or the whole number).
• x/100 represents the unknown percentage (x) out of 100%.

To solve this proportion, we can cross-multiply:

64 * 100 = 320 * x

6400 = 320x

Now, divide both sides by 320 to isolate 'x':

x = 6400 / 320

x = 20

Therefore, 64 is 20% of 320. This means that 320 is the whole number. But the original question asks "320 of what number is 64?", implying that 320 is a part of a larger number. We've identified that 320 represents 20% of the larger whole. Let's use this information.

If 20% is 320, then 1% is 320/20 = 16.

Therefore 100% (the whole number) is 16 * 100 = 1600

Therefore, 320 is 20% of 1600.

Method 2: Using Algebraic Equations

We can also approach this problem using algebraic equations. Let's represent the unknown number as 'x'. The problem can be translated into the following equation:

(320/x) * 100 = 64

This equation states that the percentage that 320 represents in relation to x is equal to 64. Solving for 'x':

32000/x = 64

Multiply both sides by x:

32000 = 64x

Now, divide both sides by 64:

x = 32000 / 64

x = 500

Therefore, 320 is 64% of 500. This solution assumes a different interpretation of the problem where 64 is the value representing a percentage of a larger whole (500).

Method 3: Understanding the Relationship Between the Numbers

The initial interpretation of "320 of what number is 64?" might lead to confusion. One must carefully consider the phrasing. The phrase "320 of what number" could be misleading. The problem presents a ratio of 64:320 that is then scaled up to find the total value, which we did using the proportion method. If the question was "What percentage of 320 is 64?", the answer is different. The issue boils down to understanding which number is representing the percentage of which.

This highlights the importance of precise wording and careful problem interpretation.

Applying these methods to similar problems

Let’s try another example: "150 is what percent of 750?"

Method 1: Proportions

150/750 = x/100

15000 = 750x

x = 15000 / 750

x = 20

Therefore, 150 is 20% of 750.

Method 2: Algebraic Equation

(150/750) * 100 = x

0.2 * 100 = x

x = 20

Again, 150 is 20% of 750.

Conclusion: Mastering Percentage Problems

Solving percentage problems like "320 of what number is 64?" requires a clear understanding of ratios, proportions, and algebraic equations. While the phrasing of the original question might seem ambiguous, both methodologies we used led to distinct, yet valid solutions, depending on the interpretation of the problem. Remember that careful reading and precise interpretation are key to solving these types of problems. By mastering these techniques, you’ll be well-equipped to handle similar percentage problems across various mathematical applications. The key is to identify the known values (part, whole, or percentage) and use the appropriate method—proportions or algebraic equations—to find the unknown. Practicing different types of percentage problems will further solidify your understanding and make you more proficient in solving them. Always double-check your work and consider the context of the problem to ensure the solution is logical and makes sense within the given scenario.