Problem 8 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. \(9 \frac{1}{4}+5 \frac{3}{4}\)

Step-by-Step Solution

Answer

The result is 15.

Step 1: Convert mixed numbers into improper fractions

To make addition or subtraction easier, first convert each mixed number into an improper fraction. For \(9 \frac{1}{4}\), the improper fraction is \(\frac{37}{4}\) because \(9 \times 4 + 1 = 37\). For \(5 \frac{3}{4}\), the improper fraction is \(\frac{23}{4}\) because \(5 \times 4 + 3 = 23\).

Step 2: Add the improper fractions

Since the denominators are the same, add the numerators directly. So, \(\frac{37}{4} + \frac{23}{4} = \frac{(37 + 23)}{4}\), which equals \(\frac{60}{4}\).

Step 3: Simplify the resulting fraction

Convert \(\frac{60}{4}\) into a whole or mixed number by dividing the numerator by the denominator. Here, \(60 \div 4 = 15\), so \(\frac{60}{4}\) simplifies to 15.

Key Concepts

Improper Fractions

Improper fractions are a way to express fractions where the numerator is larger than the denominator. This is often used when working with mixed numbers in mathematical operations like addition or subtraction.
Mixed numbers are a combination of a whole number and a proper fraction. For instance, in the expression \( 9 \frac{1}{4} \), 9 is the whole number and \( \frac{1}{4} \) is the fraction. To convert a mixed number to an improper fraction, you multiply the whole number by the denominator and add it to the numerator of the fractional part.

• Example: To convert \( 9 \frac{1}{4} \) into an improper fraction, compute \( 9 \times 4 + 1 = 37 \) resulting in \( \frac{37}{4} \).
• Similarly, for \( 5 \frac{3}{4} \), compute \( 5 \times 4 + 3 = 23 \) resulting in \( \frac{23}{4} \).

Using improper fractions allows us to handle operations more systematically since it turns mixed numbers into expressions that are easier to add or subtract directly.

Addition of Fractions

Adding fractions involves combining the numerators while keeping the common denominator.
This is straightforward when the fractions already share the same denominator. You simply add the numerators together and place the sum over the same denominator.
For example, if you have \( \frac{37}{4} + \frac{23}{4} \), the addition becomes a simple:

• Combine numerators: \( 37 + 23 = 60 \).
• Use the same denominator: \( 4 \).
• The result is \( \frac{60}{4} \).

This process relies on both fractions having similar denominators, which simplifies the calculation significantly compared to dealing with different denominators. If they were different, finding a common denominator would be necessary first.

Simplifying Fractions

The final step in many fraction problems is simplification, which involves reducing the fraction to its simplest form.
For fractions like \( \frac{60}{4} \), simplification may convert it to either a simpler fraction or, in some cases, a whole number if the numerator divides evenly by the denominator.

• To simplify \( \frac{60}{4} \), divide 60 by 4.
• This division gives 15 with no remainder, showing \( 15 \) is a whole number.
• Therefore, \( \frac{60}{4} = 15 \).

This step not only makes the final answer clearer but also checks the solution for potential errors.
Simplifying reduces complexity and allows for better understanding and presentation of numerical results.