Problem 56 - Algebra 1: Concepts and Skills

Question

Write the percent as a fraction or as a mixed number in simplest form. (Skills Review p. 768 ) $$ 500 \% $$

Step-by-Step Solution

Answer

500% can be expressed as the mixed number 5.

Step 1: Convert the percentage to a fraction

First, to convert a percentage to a fraction, we use the definition of percent, which means 'per 100'. Therefore, we convert 500% to a fraction by placing it over 100, making it \( \frac{500}{100} \).

Step 2: Simplify the fraction

The fraction \( \frac{500}{100} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 100. This results in \( \frac{500 \div 100}{100 \div 100} = \frac{5}{1} \).

Step 3: Convert to mixed number

The fraction \( \frac{5}{1} \) can also be expressed as a mixed number. As the numerator is larger than the denominator, this fraction is also an improper fraction. We divide 5 by 1 to get the whole number 5. Therefore, 500% is equal to the mixed number 5.

Key Concepts

Converting Percent to Fraction

When you come across a percentage and need to convert it into a fraction, the process is straightforward. The word 'percent' comes from the Latin phrase 'per centum', which means 'by the hundred.' This is why we directly associate percentages with the number 100, which is the base value for percentages. To convert a percentage to a fraction:

• Write the percentage number without the percent sign.
• Place that number over 100. This gives you a fraction that represents the original percentage.

For example, converting 500% to a fraction involves writing it as \( \frac{500}{100} \) since the idea is that 500% is 500 per 100.

Simplifying Fractions

A simplified fraction is one where the numerator (top number) and the denominator (bottom number) are as small as possible and have no common divisors besides one. Simplifying a fraction makes it easier to understand and work with.

To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

In our example, the greatest common divisor of 500 and 100 is 100. Dividing both by this number, we get \( \frac{500 \div 100}{100 \div 100} = \frac{5}{1} \), which is the simplest form of the fraction.

Mixed Numbers

Mixed numbers are numbers consisting of an integer (whole number) and a proper fraction—one where the numerator is less than the denominator. They are used to express values greater than one that have a fractional component.

To convert an improper fraction, which has a numerator larger than or equal to the denominator, into a mixed number, you divide the numerator by the denominator:

• The quotient (result of division) becomes the whole number part.
• The remainder becomes the numerator of the proper fraction.
• The original denominator stays the same.

For instance, converting the improper fraction \( \frac{5}{1} \) to a mixed number, we simply divide 5 by 1, resulting in 5, which is both the quotient and the entire mixed number since there is no remainder.

Improper Fractions

Improper fractions have numerators that are larger than or equal to the denominators. They represent values equal to or greater than one whole.

Converting improper fractions to mixed numbers helps to grasp the size of the value they represent, but sometimes working with them in their improper form is necessary for calculations.

For example, if you have \( \frac{5}{1} \) from converting 500%, you’re working with an improper fraction. Because the numerator (5) is greater than the denominator (1), it could also be expressed as a whole number or a mixed number—in this case, simply 5, since 5 divided by 1 is 5.