Problem 87 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand Percentage Definition

A percentage is a way of expressing a number as a fraction of 100. Therefore, 512% can be written as \( \frac{512}{100} \). This conversion uses the principle that 'percent' means 'per hundred'.

Step 2: Simplify the Fraction

Now, simplify \( \frac{512}{100} \). To do this, we need to find the greatest common divisor (GCD) of 512 and 100, which is 4. Now divide both the numerator and the denominator by 4 to simplify the fraction.

Step 3: Perform the Division

Divide the numerator (512) and the denominator (100) by their GCD, which is 4. \[ \frac{512 \div 4}{100 \div 4} = \frac{128}{25} \]Thus, 512% as a fraction in simplest form is \( \frac{128}{25} \).

Key Concepts

Percentage Definition

A percentage is a mathematical concept used to express a number as a fraction of 100. When you see a percentage, it literally means "per hundred." For example, if you have 512%, this is the same as saying 512 per 100.
It's a handy way of comparing sizes relative to a whole. To convert a percentage into a fraction, place the number over 100. For instance, to convert 512% to a fraction, you take the percentage number 512 and write it over 100: \( \frac{512}{100} \). This is the primary step in moving from percent to fraction.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible. To simplify a fraction, you divide both the top number (numerator) and the bottom number (denominator) by the greatest common divisor (GCD).
Simplifying helps to make fractions easier to read and use. For example, with \( \frac{512}{100} \), you simplify by dividing both 512 and 100 by their GCD. Simplified fractions not only look cleaner, but they're also easier to interpret and use in calculations.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. Finding the GCD is crucial when simplifying fractions.
To find the GCD of 512 and 100, you can use the method of listing divisors or the Euclidean algorithm. In this case, the GCD is 4.
Once you have the GCD, you divide both the numerator and the denominator of your fraction by it. For \( \frac{512}{100} \), divide both 512 and 100 by 4 to get \( \frac{128}{25} \). This process ensures that the fraction is in its simplest form, making it more intuitive to work with.