Problem 27 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. $$223 \%$$

Step-by-Step Solution

Answer

223% is \( \frac{223}{100} \) or 2\( \frac{23}{100} \) and as a decimal, it is 2.23.

Step 1: Convert Percent to Fraction

To start, recognize that 223% is equivalent to the fraction \( \frac{223}{100} \) since percent means per hundred.

Step 2: Simplify Fraction (If Necessary)

Check if \( \frac{223}{100} \) can be simplified. Since 223 and 100 have no common factors other than 1, the fraction is already in its simplest form.

Step 3: Convert to Mixed Number

Convert \( \frac{223}{100} \) into a mixed number by dividing 223 by 100. This results in 2 whole parts and a remainder of 23, giving \( 2\frac{23}{100} \).

Step 4: Convert Percent to Decimal

Convert 223% to a decimal by dividing 223 by 100, resulting in 2.23.

Key Concepts

Understanding Fractions

Fractions represent a part of a whole. They're expressed as the ratio of two numbers, the numerator over the denominator. Consider a cake cut into equal pieces: if you have two pieces out of five, you have the fraction \( \frac{2}{5} \).
The numerator (top number) shows how many parts you have, and the denominator (bottom number) indicates how many parts the whole is divided into. The fraction \( \frac{223}{100} \) tells us that we have 223 parts out of 100, which usually means a quantity greater than one whole part. This is a crucial step in converting a percent to a fraction, where percent means "per hundred."

• Always express the fraction with 100 as the denominator when converting from percent.
• Remember to simplify if possible by finding common factors.

Understanding fractions helps us see how percentages relate to whole numbers.

Converting to Decimals

Decimals extend our ability to describe parts of wholes in a format that's compatible with measurement and everyday calculations. They're based on the number ten.
When converting a fraction to a decimal, like turning \( \frac{223}{100} \) into a decimal, you divide the numerator by the denominator.
The result, 2.23, shows us it's greater than two whole units but not quite three. Here are simple points to remember:

• Percent to decimal conversion involves dividing by 100.
• Fractions like \( \frac{1}{2} \) convert to decimals such as 0.5.

Decimals can make certain mathematical processes, like addition and subtraction, clearer and more intuitive.

What Are Mixed Numbers?

Mixed numbers combine whole numbers and fractions. They look like this: \(2\frac{23}{100}\). It's a natural way to express quantities that involve whole parts and fractional parts together.
To convert an improper fraction like \( \frac{223}{100} \) to a mixed number, identify how many whole groups fit into the fraction. In this case, 100 goes into 223 twice, leaving a remainder of 23. This becomes \(2\frac{23}{100}\).
Some key points to remember:

• Divide the numerator by the denominator to find the whole number part.
• The remainder becomes the new numerator, over the original denominator.

Mixed numbers offer a clear way to understand and express quantities that aren't whole but close.

The Simplification Process

Simplification is about making fractions as straightforward as possible by reducing them to their lowest terms. This involves finding the greatest common factor (GCF) between the numerator and the denominator.
In our case, we determined that \( \frac{223}{100} \) cannot be simplified further because 223 and 100 have no common factors besides 1. Simplified fractions are easier to understand and work with in calculations.

• Check for common factors to simplify.
• A prime number in the numerator might mean the fraction is already in simplest form.
• Simplification doesn't change the value, just makes it clearer.

Remember that simplification can provide the most readable and user-friendly form of a fraction for mathematical operations.