Problem 79 - Prealgebra

Question

The following problems review addition and subtraction with fractions and mixed numbers. $$3 \frac{1}{2}+5 \frac{1}{2}$$

Step-by-Step Solution

Answer

The sum of \(3 \frac{1}{2}\) and \(5 \frac{1}{2}\) is 9.

Step 1: Convert Mixed Numbers to Improper Fractions

We start by converting the mixed numbers to improper fractions. A mixed number, such as \(3 \frac{1}{2}\), can be converted by multiplying the whole number by the denominator and then adding the numerator. Therefore, \(3 \times 2 + 1 = 7\), so \(3 \frac{1}{2} = \frac{7}{2}\). Similarly, \(5 \times 2 + 1 = 11\), so \(5 \frac{1}{2} = \frac{11}{2}\).

Step 2: Add the Improper Fractions

Both fractions have the same denominator, so we can add them directly. This gives us: \(\frac{7}{2} + \frac{11}{2} = \frac{7 + 11}{2} = \frac{18}{2}\).

Step 3: Simplify the Result

The fraction \(\frac{18}{2}\) simplifies to a whole number by dividing the numerator by the denominator: \(\frac{18}{2} = 9\). This means the sum of the original mixed numbers is \(9\).

Key Concepts

Mixed Numbers

A mixed number combines a whole number with a fraction. Think of a pizza cut into halves: if you have 3 whole pizzas and half of another one, you'd describe that as \(3 \frac{1}{2}\). To perform mathematical operations involving mixed numbers, it's often necessary to convert them to improper fractions, as this allows for easier calculations.

• To convert, multiply the whole number by the fraction's denominator.
• Add the result to the numerator of the fraction.
• For example, to convert \(3 \frac{1}{2}\) into an improper fraction: \(3 \times 2 + 1 = 7\), making it \(\frac{7}{2}\).

This step ensures you can easily add or subtract, as we'll discuss further.

Improper Fractions

Improper fractions are fractions where the numerator is equal to or larger than the denominator. They represent a quantity greater than or equal to one whole. For example, in \(\frac{7}{2}\), the numerator 7 tells you how many parts you have, and the denominator 2 tells you what type of part — in this case, halves.

• This format is beneficial for addition and subtraction because it allows the computation of precise results by focusing on one type of part.
• After converting mixed numbers to improper fractions, adding them becomes simpler since having the same denominator eliminates the need for finding a common one.
• For example, adding \(\frac{7}{2}\) and \(\frac{11}{2}\) directly: \(\frac{7 + 11}{2} = \frac{18}{2}\).

As seen, improper fractions streamline calculations, especially with direct addition.

Fraction Simplification

Once you perform operations on improper fractions, the result might still need some refining. This refinement comes from fraction simplification, or the process of reducing fractions to their simplest form. It involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by this number to simplify the fraction.
• For example, for \(\frac{18}{2}\), the GCD is 2. Thus, simplification gives: \(\frac{18}{2} = 9\).

In some cases, simplification might yield a whole number, as shown above, making it clear and concise as a solution. This process highlights why simplification is an integral part of working with fractions.