Problem 52 - Prealgebra

Question

Change each percent to a fraction in lowest terms. $$66 \frac{2}{3} \%$$

Step-by-Step Solution

Answer

\(66 \frac{2}{3}\%\) is \(\frac{2}{3}\) as a fraction in lowest terms.

Step 1: Convert Mixed Number Percent to Improper Fraction

First, convert the mixed number \(66 \frac{2}{3}\)% into an improper fraction. To do that, multiply the whole number 66 by the denominator 3, and then add the numerator 2: \(66 \times 3 + 2 = 198 + 2 = 200\). So, \(66 \frac{2}{3} = \frac{200}{3}\).

Step 2: Convert Percent to Decimal

To convert the percent to a decimal, divide the fraction you found by 100, because percent means 'per hundred'. This is the same as multiplying by \(\frac{1}{100}\). Hence, \(\frac{200}{3} \times \frac{1}{100} = \frac{200}{300}\).

Step 3: Simplify the Fraction

Simplify \(\frac{200}{300}\) by finding the greatest common divisor (GCD). The GCD of 200 and 300 is 100. Divide both the numerator and the denominator by 100: \(\frac{200 \div 100}{300 \div 100} = \frac{2}{3}\). Thus, \(66 \frac{2}{3} \%\) as a fraction in lowest terms is \(\frac{2}{3}\).

Key Concepts

Understanding Improper Fractions

An improper fraction is a type of fraction where the numerator, or the top number, is larger than or equal to the denominator, which is the bottom number. This means the value of an improper fraction is greater than or equal to 1. It's what happens when, instead of having a piece of something, you have a whole or more than one whole.
For example, think about the fraction \(\frac{200}{3}\). Here, 200 is the numerator and 3 is the denominator. Since 200 is much larger than 3, this is an improper fraction.
Converting a mixed number to an improper fraction is an essential skill, especially when dealing with percentages that include mixed numbers. By multiplying the whole number by the denominator and adding the numerator (as done in the initial step of the solution provided), you can easily convert any mixed number into an improper fraction.

Exploring Mixed Numbers

A mixed number consists of a whole number and a proper fraction. It's a compact and intuitive way to express fractions greater than one without feeling overwhelmed by large numerators and denominators. For instance, the number \(66\frac{2}{3}\) combines 66 whole parts with an extra two-thirds of a part.
To convert a mixed number to an improper fraction, you need to follow these simple steps:

• Multiply the whole number by the denominator.
• Add the result to the numerator.
• Place the total over the original denominator.

These steps will swiftly change a mixed number like \(66\frac{2}{3}\) into its improper form, \(\frac{200}{3}\), as we observed in the solution. Mixed numbers are user-friendly as they easily convey larger quantities, but converting them into improper fractions is crucial for further mathematical operations.

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator are as small as possible yet still retain the value of the fraction. This is done by finding the greatest common divisor (GCD) of the numerator and the denominator, which is the largest number that divides both evenly.
In our example, once we had the improper fraction \(\frac{200}{300}\), simplifying it was our next step. The GCD of 200 and 300 is 100, so dividing both the numerator and the denominator by 100 simplifies the fraction to \(\frac{2}{3}\).
Simplification is not only about reducing numbers but also about making the fraction easier to understand and work with. It's especially important in cases like this where the original percentage needs to be expressed as the most straightforward fraction possible. Keep in mind, any time you have a fraction, you should check if it can be simplified for clarity and convenience.