Pre-Algebra Unit 4 Review: Fractions

What Are Fractions?

• Fractions represent parts of a whole or parts of a group
• Consist of a numerator (top number) and a denominator (bottom number)
• Numerator indicates the number of parts being considered
• Denominator indicates the total number of equal parts in the whole or group
• Can be written in the form $\frac{a}{b}$, where $a$ is the numerator and $b$ is the denominator
• Proper fractions have a numerator smaller than the denominator ($\frac{3}{4}$)
• Improper fractions have a numerator greater than or equal to the denominator ($\frac{5}{3}$)

Types of Fractions

• Proper fractions have a numerator smaller than the denominator ($\frac{2}{5}$)
• Improper fractions have a numerator greater than or equal to the denominator ($\frac{7}{3}$)
  • Can be converted to a mixed number by dividing the numerator by the denominator
• Mixed numbers consist of a whole number and a proper fraction ($3\frac{1}{4}$)
  • Can be converted to an improper fraction by multiplying the whole number by the denominator and adding the numerator
• Unit fractions have a numerator of 1 ($\frac{1}{6}$)
• Reciprocals are fractions that, when multiplied together, equal 1 ($\frac{2}{3}$ and $\frac{3}{2}$)
• Like fractions have the same denominator ($\frac{3}{8}$ and $\frac{5}{8}$)
• Unlike fractions have different denominators ($\frac{1}{4}$ and $\frac{2}{3}$)

Equivalent Fractions

• Fractions that represent the same value, even though they may look different
• Can be obtained by multiplying or dividing both the numerator and denominator by the same non-zero number
  • $\frac{2}{4} = \frac{2 \times 2}{4 \times 2} = \frac{4}{8}$
• Simplifying fractions involves dividing both the numerator and denominator by their greatest common factor (GCF)
  • $\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$
• Finding equivalent fractions is useful when adding, subtracting, or comparing fractions with different denominators
• Cross multiplication can be used to check if two fractions are equivalent
  • If $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$

Comparing and Ordering Fractions

• Fractions can be compared using various methods
• If two fractions have the same denominator, compare their numerators
  • The fraction with the larger numerator is greater
• If two fractions have the same numerator, compare their denominators
  • The fraction with the smaller denominator is greater
• To compare fractions with different numerators and denominators, find a common denominator
  • Multiply both the numerator and denominator of each fraction by the denominator of the other fraction
  • Compare the resulting numerators
• Fractions can be ordered from least to greatest or greatest to least
• Mixed numbers should be converted to improper fractions before comparing or ordering

Adding and Subtracting Fractions

• To add or subtract fractions with the same denominator, add or subtract the numerators and keep the denominator the same
  • $\frac{3}{7} + \frac{2}{7} = \frac{3 + 2}{7} = \frac{5}{7}$
• To add or subtract fractions with different denominators, find the least common denominator (LCD) first
  • Multiply both the numerator and denominator of each fraction by the appropriate factor to obtain the LCD
  • Add or subtract the resulting numerators and keep the LCD as the denominator
• When adding or subtracting mixed numbers, add or subtract the whole numbers and fractions separately
  • If the resulting fraction is improper, convert it to a mixed number
• Simplify the final answer if possible

Multiplying Fractions

• To multiply fractions, multiply the numerators together and multiply the denominators together
  • $\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}$
• Simplify the fractions before multiplying, if possible
• When multiplying a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same
• To multiply mixed numbers, convert them to improper fractions first, then multiply as usual
• Multiplying fractions always results in a smaller value (unless multiplying by 1 or a reciprocal)

Dividing Fractions

• To divide fractions, multiply the first fraction by the reciprocal of the second fraction
  • $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$
• The reciprocal of a fraction is obtained by flipping the numerator and denominator
• When dividing a fraction by a whole number, multiply the denominator by the whole number
• To divide mixed numbers, convert them to improper fractions first, then proceed with division
• Dividing by a fraction is equivalent to multiplying by its reciprocal

Fractions in Real Life

• Fractions are used in various real-life situations
• Cooking and baking often involve measuring ingredients using fractions (cups, tablespoons)
• Carpentry and construction use fractions to measure lengths, widths, and heights (inches, feet)
• Time can be expressed using fractions (half an hour, quarter past)
• Probability is often expressed as a fraction (chances of winning a lottery)
• Discounts and sales are sometimes represented as fractions (25% off, half-price)
• Fractions are used in sports to describe parts of a game or playing field (half-time, quarter-court)
• Music uses fractions to indicate note durations (whole note, half note, quarter note)