Problem 131 - Introductory Algebra for College Students

Question

Explain how to convert a mixed number to an improper fraction and give an example.

Step-by-Step Solution

Answer

The process for converting a mixed number to an improper fraction involves multiplying the whole number by the denominator, adding the numerator, and putting that value over the original denominator. The mixed number \(3 \frac{1}{2}\) thus converts to the improper fraction \( \frac{7}{2}\).

Step 1: Understanding Mixed Numbers and Improper Fractions

A mixed number is a whole number and a fraction combined into one 'mixed' number. An example of a mixed number is \(3 \frac{1}{2}\). An improper fraction, on the other hand, is a fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). An example would be \( \frac{7}{2}\).

Step 2: Conversion Process

To convert a mixed number into an improper fraction, follow these steps: First, multiply the whole number part by the denominator of the fractional part. Second, add the numerator of the fractional part to the result obtained. The resultant value will be the numerator of the improper fraction. The denominator of the improper fraction remains the same as the fractional part of the mixed number.

Step 3: Example

Given the mixed number \(3 \frac{1}{2}\), we convert it following the steps in the previous step. First, multiply the whole number (3) by the denominator of the fractional part (2) which equals to 6. Then, add the numerator of the fractional part (1) to the result, which gives us 7. Therefore, the numerator of the improper fraction is 7 and the denominator remains as 2. Hence, the improper fraction equivalent to the mixed number \(3 \frac{1}{2}\) is \( \frac{7}{2}\).