Problem 38 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a fraction or a mixed number in simplest form. $$ \frac{2}{9}+\frac{8}{9} $$

Step-by-Step Solution

Answer

The sum of \(\frac{2}{9}\) and \(\frac{8}{9}\) is \(1\frac{1}{9}\).

Step 1: Add the fractions

As both fractions \(\frac{2}{9}\) and \(\frac{8}{9}\) have the same denominator, add the numerators directly. This yields \(\frac{2+8}{9}\), which equals to \(\frac{10}{9}\).

Step 2: Simplify the Fraction into a Mixed Number

\(\frac{10}{9}\) is an improper fraction, because the numerator is greater than the denominator. It can be transformed into a mixed number. 10 divided by 9 gives 1 with a remainder of 1. This results in the mixed number \(1 \frac{1}{9}\).

Key Concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This means that the fraction represents a value greater than or equal to one whole. Improper fractions are common when adding fractions, especially when summing up values that together exceed the denominator.

To identify an improper fraction, just look at the relationship between the numerator and the denominator. If the numerator is greater, you've got an improper fraction on your hands. They might look daunting, but they're actually very useful, especially in mathematics involving division or when wanting to convert into mixed numbers.

• Numerator greater or equal to the denominator
• Represents value >= 1
• Can be converted to mixed numbers

Knowing how to handle improper fractions is important for simplifying them or changing their form, like into mixed numbers, to make calculations easier to understand.

Common Denominators

To add or subtract fractions, having a common denominator is key. The denominator is the bottom part of a fraction that shows how many equal parts make up a whole. When fractions share the same denominator, you can directly add or subtract the numerators (top numbers).

In the example given, both fractions, \( \frac{2}{9} \) and \( \frac{8}{9} \), already have the same denominator, 9. This makes it straightforward: you simply add the numerators together while keeping the denominator the same.

• Ensures fractions are like-sized parts
• Simplifies addition or subtraction of fractions

If the fractions didn't have a common denominator, you would need to find one, usually by finding a least common multiple (LCM) of the denominators. Once equalized, you can easily perform the arithmetic operation with the fractions.

Mixed Numbers

A mixed number is a way to represent a number that consists of a whole number and a fraction combined. It provides a clearer understanding of a quantity that is more than one whole but not an exact whole number.

To convert an improper fraction to a mixed number, you divide the numerator by the denominator. The quotient (the number of times the denominator fits into the numerator) is the whole number part, and any remainder becomes the numerator of the fraction part.

For example, with \( \frac{10}{9} \): divide 10 by 9 which gives 1 with a remainder of 1. This translates to \( 1 \frac{1}{9} \).

• Consists of a whole number and a fraction
• Useful for clearer understanding of values
• Formed by converting an improper fraction

Mixed numbers are particularly useful in real-life scenarios, like measurements or when needing to express a number practically. They make complex fractions easier to interpret and apply.