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Chapter 1: Problem 95

Use the definition of percent to convert to fractions. $$ 512 \% $$

Short Answer

Expert verified
512% as a fraction is \( \frac{128}{25} \).

Step by step solution

01

Understand the Concept of Percent

The term 'percent' means 'per hundred'. Therefore, when we see 512%, it signifies 512 out of 100.
02

Convert Percent to Fraction

To convert 512% to a fraction, write the percent as 512 per 100. The fraction form will be \( \frac{512}{100} \).
03

Simplify the Fraction

To simplify \( \frac{512}{100} \), find the greatest common divisor (GCD) of 512 and 100. The GCD is 4. Divide both the numerator and denominator by 4 to get \( \frac{512 \div 4}{100 \div 4} = \frac{128}{25} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Percent
The concept of percent is an essential building block in math. The word "percent" comes from the Latin phrase "per centum," which translates to "by the hundred." So, when we talk about something like 512%, we are considering a ratio or a fraction of 512 out of 100.

This makes it easier to work with comparisons, as it standardizes things to a base of 100. Think of percent as a tool that helps us communicate how large or small a number is in relation to a base of 100. For instance, saying "512%" indicates you have 512 parts for every 100 parts, essentially showing how many times more than 100 you have.
Converting Percents to Fractions
Whenever you need to convert a percent to a fraction, keep in mind that percent means "out of 100." This makes the conversion process straightforward. For any given percent, you can translate it directly into a fraction by using its percent value as the numerator and 100 as the denominator.

For example, with 512%, you write it as a fraction:
  • The numerator is 512.
  • The denominator is 100.
Thus, the fraction is expressed as \( \frac{512}{100} \). Converting to a fraction is a crucial skill as it allows for easier calculations and comparisons in math.
Finding the Greatest Common Divisor (GCD)
When working with fractions, especially when simplifying, the greatest common divisor (GCD) is incredibly helpful. The GCD is the largest number that can divide both the numerator and the denominator without leaving a remainder.

In our example, with the fraction \( \frac{512}{100} \), identifying the GCD allows us to simplify the fraction efficiently. You can find the GCD by listing the factors of both the numerator and the denominator, or use a method such as the Euclidean algorithm. Here, the GCD of 512 and 100 is 4.

Knowing the GCD is fundamental when simplifying fractions, ensuring that both top and bottom numbers are reduced to their smallest possible whole numbers.
Simplification of Fractions
Simplification is the process of reducing a fraction to its simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). Simplifying fractions makes them easier to work with mathematically.

For \( \frac{512}{100} \), you've found that the GCD is 4:
  • Divide 512 by 4 to get 128.
  • Divide 100 by 4 to get 25.
Thus, \( \frac{512}{100} \) simplifies to \( \frac{128}{25} \).

Simplified fractions are often more concise and neat, making mathematical operations clearer and results easier to interpret.

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