Problem 2 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. \(8 \frac{2}{9}+1 \frac{5}{9}\)

Step-by-Step Solution

Answer

The result is \(9 \frac{7}{9}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert each mixed number into an improper fraction. For \(8 \frac{2}{9}\), multiply the whole number \(8\) by the denominator \(9\) and add the numerator \(2\): \(8 \times 9 + 2 = 72 + 2 = 74\). Thus, \(8 \frac{2}{9} = \frac{74}{9}\). For \(1 \frac{5}{9}\), multiply \(1\) by \(9\) and add \(5\): \(1 \times 9 + 5 = 9 + 5 = 14\). Thus, \(1 \frac{5}{9} = \frac{14}{9}\).

Step 2: Add the Improper Fractions

Now, add the improper fractions \(\frac{74}{9}\) and \(\frac{14}{9}\). Since they have the same denominator, only the numerators are added: \(74 + 14 = 88\). So \(\frac{74}{9} + \frac{14}{9} = \frac{88}{9}\).

Step 3: Convert Back to Mixed Number

Convert the improper fraction \(\frac{88}{9}\) back to a mixed number. Divide \(88\) by \(9\): \(88 \div 9 = 9\) with a remainder of \(7\). Thus, \(\frac{88}{9}\) is equivalent to \(9 \frac{7}{9}\).

Key Concepts

Understanding Improper Fractions

Improper fractions are fractions where the numerator is larger than or equal to the denominator. They may look a little intimidating at first because they represent quantities greater than one, but they are straightforward once you understand them. Improper fractions can be compared to mixed numbers, which combine a whole number with a proper fraction.

Let's take an example: the improper fraction \( \frac{74}{9} \). Here, 74 is the numerator, and 9 is the denominator. Since 74 is much larger than 9, this tells us we have more than one whole. Improper fractions are especially useful in mathematical operations because they simplify the process of adding, subtracting, multiplying, and dividing fractions by maintaining a single fractional entity without switching between whole numbers and fractions.

They are often converted from mixed numbers to perform calculations, allowing for seamless integration into mathematical equations without changing denominators or adding whole numbers separately.

Converting Mixed Numbers to Improper Fractions

Converting mixed numbers to improper fractions is a fundamental skill in fraction operations. It involves a simple process that allows you to deal effortlessly with additions, subtractions, and more.

To convert a mixed number such as \(8 \frac{2}{9}\) into an improper fraction, follow these steps:

• First, multiply the whole number part by the denominator of the fractional part. Here, you multiply 8 by 9 to get 72.
• Next, add the numerator of the fractional part to this product. In this case, you add 2 to 72, resulting in 74.
• Now, place this sum as the numerator over the original denominator to create the improper fraction: \(\frac{74}{9}\).

By converting mixed numbers into improper fractions, you streamline calculations and make them more mathematically manageable; this is especially useful when adding or subtracting fractions across multiple steps without combining whole numbers separately.

Addition of Fractions

The addition of fractions can initially seem complicated; however, if approached step-by-step, it becomes quite manageable.

Adding fractions, especially improper ones, is easier when they share the same denominator because you only need to add the numerators together. For instance, with fractions like \(\frac{74}{9}\) and \(\frac{14}{9}\):

• Since both have 9 as the denominator, preserve it.
• Add the numerators: 74 plus 14 equals 88.
• Combine them into a single fraction: \(\frac{88}{9}\).

The result is still an improper fraction, which can then be converted back into a mixed number to present the final solution. This whole process flow highlights how understanding improper fractions and their operations simplify solving real-world problems involving fractions. Once you get the hang of it, adding fractions follows a smooth and repetitive process that becomes second nature.