Problem 20 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. $$16 \frac{2}{3} \%$$

Step-by-Step Solution

Answer

The simplest fraction form is \(\frac{1}{6}\) and the decimal form is approximately \(0.167\).

Step 1: Convert Mixed Percent to Improper Fraction

The given percent is \(16 \frac{2}{3}\%\). First, convert the mixed number into an improper fraction: \[16 \frac{2}{3} = \frac{50}{3}\].

Step 2: Express the Improper Fraction as a Fraction

Since we want to express the percent as a fraction, write the improper fraction over 100: \[\frac{50}{3} \times \frac{1}{100} = \frac{50}{300}\].

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{50}{300}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 50: \[ \frac{50}{300} = \frac{1}{6} \].

Step 4: Express the Percent as a Decimal

Convert the percent into a decimal by dividing the percent by 100: \[\frac{50}{3} \div 100 = \frac{50}{300} = 0.1666\ldots\]. This decimal is an infinitely repeating decimal, which can be rounded to \(0.167\) for simplification.

Key Concepts

Fractions

Fractions are a way to represent numbers that are not whole. They are written in the form \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator. The numerator tells how many parts we have, and the denominator tells how many parts make up a whole. Whenever you have a fraction, you are essentially dividing: the numerator divided by the denominator. This is crucial when converting percents to fractions, since percents represent parts out of 100.
In the problem above, the percent \(16 \frac{2}{3}\%\) was converted into an improper fraction, \(\frac{50}{3}\), to make calculations easier. Instead of working directly with the percent, it was helpful to consider this special fraction first. Afterward, the fraction was placed over 100 to find the equivalent fraction value for the percentage. Understanding these concepts allows you to manipulate numbers more efficiently in math.

Decimals

Decimals are another way of expressing parts of a whole. They are often more intuitive than fractions because they align with our base-ten number system. When a percentage or fraction is changed into a decimal, you get a straightforward representation of that number.
To create a decimal value from a fraction, you divide the numerator by the denominator. For instance, in the initial example, we derived the fraction \(\frac{1}{6}\) from simplifying \(\frac{50}{300}\). When you perform this division, \(\frac{1}{6}\) becomes approximately \(0.1666\ldots\) (a repeating decimal).

• Repeating decimals continue indefinitely. They are often shown with a bar above the repeating part.
• For practical purposes, such decimals are rounded — for example, \(0.1666\ldots\) may be approximated as \(0.167\).

The key thing is to remember how decimals connect with fractions; you're just expressing the same value in a different format, often with decimals being easier to grasp at a glance.

Mixed Numbers

Mixed numbers are numbers that include both a whole number and a fraction part. They are helpful when discussing values greater than one but not whole numbers, providing more precision.
When you convert a percentage that is a mixed number, like \(16 \frac{2}{3}\%\), into another form, the first step is to convert it into an "improper fraction". An improper fraction has a numerator larger than the denominator. This makes it easier to handle mathematically, especially when making conversions.

• For example, converting \(16 \frac{2}{3}\) to an improper fraction gives \(\frac{50}{3}\).
• Next, this improper fraction can be used to create an equivalent percent form by setting it over 100.

By understanding mixed numbers and their relationship to improper fractions, you'll find it simpler to navigate and convert through different mathematical contexts.

Simplification

Simplification is a crucial mathematical process where you reduce fractions to their simplest form. This means that the numerator and the denominator share no common factors, except for 1. It makes handling the number easier and reduces complexity in calculations.
To simplify \(\frac{50}{300}\), the greatest common divisor (GCD) of the numerator and the denominator was determined to be 50. Dividing both by this number simplified the fraction to \(\frac{1}{6}\). This step made future calculations easier, such as converting the number to a decimal.

• Identify the greatest common divisor between the numerator and the denominator.
• Divide both parts of the fraction by this number to simplify.

This simplification is not only mathematically essential but also optimizes the fraction for subsequent operations like decimal conversion.