Summary of Understanding and Operating with Fractions

## Introduction **Fractions** help us represent parts of a whole. When you share a pizza with friends and take a slice, you're using fractions without even realizing it. In this material, you'll learn what fractions are, how to read them, compare them, and use them in daily life. ### What is a Fraction? > A fraction is a way to show how many parts of a whole we have; it's made up of a **numerator** (the top part) and a **denominator** (the bottom part). - Example: The fraction $\frac{3}{4}$ means 3 parts out of 4 equal parts. ## Parts of a Fraction ### Numerator and Denominator - **Numerator**: the top number, which tells us how many parts we're considering. Example: in $\frac{2}{5}$, the numerator is $2$. - **Denominator**: the bottom number, which tells us how many equal parts the whole has been divided into. In $\frac{2}{5}$, the denominator is $5$. > Definition: The numerator represents the number of parts we have; the denominator represents the total number of equal parts an object is divided into. ### Types of Fractions - **Proper Fraction**: where the numerator is smaller than the denominator, for example $\frac{3}{8}$. - **Improper Fraction**: where the numerator is greater than or equal to the denominator, for example $\frac{7}{4}$ or $\frac{4}{4}$. - **Mixed Number**: a combination of a whole number and a proper fraction, for example $1\,\frac{3}{4}$. | Type | Condition | Example | |---|---:|---:| | Proper Fraction | numerator < denominator | $\frac{3}{8}$ | | Improper Fraction | numerator \geq denominator | $\frac{7}{4}$ | | Mixed Number | whole number + fraction | $1\,\frac{3}{4}$ | ## How to Read and Write Fractions - Read $\frac{5}{6}$ as "five-sixths". - To convert an improper fraction to a mixed number: divide the numerator by the denominator. For example, for $\frac{11}{4}$, $11 \div 4 = 2$ remainder $3$, so $\frac{11}{4} = 2\,\frac{3}{4}$. ## Simplifying Fractions - Simplifying means writing the fraction using the smallest possible numbers that have the same value. - To simplify, divide the numerator and denominator by the same number (a common divisor). Example: $$\frac{6}{8}$$ Divide by $2$: $$\frac{6}{8} = \frac{3}{4}$$ > Definition: A fraction is in its simplest form when the numerator and denominator have no common divisors greater than 1. ## Equivalent Fractions - Two fractions are equivalent if they represent the same part of the whole. - Multiply or divide the numerator and denominator by the same number to get equivalent fractions. Example: $$\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$$ Because $1 \times 2 = 2$ and $2 \times 2 = 4$, etc. ## Comparing Fractions - If they have the same denominator, the fraction with the larger numerator is greater: $\frac{5}{8} > \frac{3}{8}$. - If they have the same numerator, the fraction with the smaller denominator is greater: $\frac{3}{4} > \frac{3}{6}$. - For different denominators, find a common denominator or convert them to equivalent fractions, and then compare. Example: Compare $\frac{2}{3}$ and $\frac{3}{5}$. Find a common denominator: $15$. $$\frac{2}{3} = \frac{10}{15}$$ $$\frac{3}{5} = \frac{9}{15}$$ Since $10 > 9$, then $\frac{2}{3} > \frac{3}{5}$. ## Adding and Subtracting Fractions 1. If they have the same denominator: add or subtract the numerators and keep the denominator. - Example: $$\frac{3}{10} + \frac{4}{10} = \frac{7}{10}$$ 2. If they have different denominators: find a common denominator, convert the fractions, and then add or subtract. - Example: $$\frac{1}{4} + \frac{1}{6}$$ Common denominator: $12$ $$\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}$$ Add: $$\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$$ > Definition: A common denominator is a number that serves as the denominator for two or more fractions, allowing them to be added or subtracted. ## Multiplying Fractions - Multiply the numerators and multiply the denominators. - Simplify the result if possible. Example: $$\frac{2}{3} \times \frac{