Problem 4 - Prealgebra

Question

Change each mixed number to an improper fraction. $$7 \frac{1}{2}$$

Step-by-Step Solution

Answer

The mixed number \(7 \frac{1}{2}\) as an improper fraction is \(\frac{15}{2}\).

Step 1: Understand the Mixed Number

The mixed number given is \(7 \frac{1}{2}\), where 7 is the whole number and \(\frac{1}{2}\) is the fractional part. Our goal is to convert this mixed number into an improper fraction.

Step 2: Multiply Whole Number by Denominator

For the mixed number \(7 \frac{1}{2}\), multiply the whole number by the denominator of the fractional part: \(7 \times 2 = 14\).

Step 3: Add the Numerator

Take the result from Step 2 and add the numerator of the fractional part: \(14 + 1 = 15\).

Step 4: Form the Improper Fraction

Place the sum from Step 3 over the original denominator: the improper fraction is \(\frac{15}{2}\).

Key Concepts

Improper Fractions

Improper fractions might initially sound a bit confusing, but they are just a type of fraction. An improper fraction is simply a fraction where the numerator (the top part) is greater than or equal to the denominator (the bottom part). This is different from a proper fraction, where the numerator is less than the denominator. For example, in the improper fraction \( \frac{15}{2} \):

• 15 is the numerator
• 2 is the denominator
• The numerator is larger than the denominator

Improper fractions are commonly used in arithmetic and algebra because they are easier to work with than mixed numbers during calculations. They can be easily converted back to mixed numbers if needed, making them a flexible tool for mathematical operations.

Fractions

Fractions are a fundamental concept in mathematics, representing parts of a whole. A fraction consists of two parts:

• The numerator, which tells you how many parts we have
• The denominator, which tells you how many parts make up a whole

Understanding fractions is crucial as they are used in various mathematical operations and everyday life scenarios. Fractions can be categorized as:

• Proper Fractions: Where the numerator is less than the denominator, like \( \frac{3}{4} \).
• Improper Fractions: Where the numerator is greater than the denominator, like \( \frac{15}{2} \).
• Mixed Numbers: A combination of a whole number and a fractional part, like \( 7\frac{1}{2} \).

When we work with fractions, it’s essential to understand these categories and how they relate to each other.

Convert Mixed Numbers to Improper Fractions

Converting mixed numbers to improper fractions is a straightforward process and can be done in a few simple steps. This can help when performing operations like addition or subtraction with fractions or mixed numbers. Here’s how it works:1. **Identify the whole number and the fractional part** of the mixed number. For example, in \( 7\frac{1}{2} \), 7 is the whole number, and \( \frac{1}{2} \) is the fractional part.2. **Multiply the whole number by the denominator of the fraction.** This tells you how many parts the whole number represents if each whole is split into equal-sized parts defined by the denominator. With \( 7 \times 2 = 14 \), it shows that 7 wholes are equivalent to 14 halves.3. **Add the numerator of the fractional part** to the result from step 2. This step adds any additional parts above the whole numbers. So, \( 14 + 1 = 15 \).4. **Write this result as the new numerator**, keeping the original denominator from the fractional part. So, the mixed number \( 7\frac{1}{2} \) becomes the improper fraction \( \frac{15}{2} \).By practicing these steps, you can easily and accurately switch between mixed numbers and improper fractions, making calculations easier.