Problem 22 - Fundamentals of Mathematics

Question

Convert each improper fraction to a mixed number. $$\frac{15}{2}$$

Step-by-Step Solution

Answer

The mixed number is \( 7 \frac{1}{2} \).

Step 1: Understand Improper and Mixed Fractions

An improper fraction has a numerator larger than its denominator, like \( \frac{15}{2} \). A mixed number includes a whole number and a proper fraction (fraction where the numerator is smaller than the denominator).

Step 2: Divide the Numerator by the Denominator

To convert \( \frac{15}{2} \) into a mixed number, divide 15 by 2. This gives a quotient of 7 and a remainder of 1.\[ 15 \div 2 = 7 \text{ R }1 \]

Step 3: Form the Mixed Number

Combine the quotient and the remainder to form the mixed number. The quotient (7) becomes the whole number, and the remainder (1) becomes the numerator of the fractional part with the original denominator (2):\[ 7 \frac{1}{2} \]

Key Concepts

Mixed Number

A mixed number is a numerical expression that combines a whole number and a fraction. It's helpful when dealing with quantities larger than what a single unit fraction can express. For example, instead of writing the improper fraction \( \frac{15}{2} \), we can convert it to the mixed number \( 7 \frac{1}{2} \). This means we have 7 full units and an additional half of a unit. Converting improper fractions to mixed numbers makes it easier to visualize and understand the size and value of the fraction. This is particularly handy in real-life scenarios, like measuring ingredients in cooking or partitioning objects evenly.

Numerator and Denominator

In any fraction, the top part is called the **numerator**, and the bottom part is the **denominator**. The numerator represents how many parts of the whole we have, while the denominator tells us how many parts the whole is divided into. For example, in \( \frac{15}{2} \), the numerator is 15, indicating that we have 15 parts of a whole. The denominator is 2, which means the whole is divided into 2 equal parts. When converting an improper fraction to a mixed number, we see how the numerator's division by the denominator can also be visualized as full parts plus remaining parts.

Division in Fractions

Understanding division is crucial in fractions, especially when working with improper fractions. To convert an improper fraction like \( \frac{15}{2} \) to a mixed number, we perform a division: 15 divided by 2.

• The quotient, 7, represents the whole number of times 2 fits into 15.
• The remainder, 1, is what's left over after dividing.

This process explains why our improper fraction becomes \( 7 \frac{1}{2} \) as a mixed number. The initial division helps break down the fraction into more understandable components — the integer ratio (7) and the leftover fractional part (\( \frac{1}{2} \)). Mastering division in this context helps not only in converting fractions but also aids in grasping the deeper connections between numbers.