Mixed Fractions Cheatsheet

Cheatsheet Content

What is a Mixed Fraction? A mixed fraction (also called a mixed number) combines a whole number and a proper fraction . A proper fraction has a numerator smaller than its denominator (e.g., $\frac{1}{2}$, $\frac{3}{4}$). Example: $3\frac{1}{2}$ means "three and one-half". Converting Mixed Fractions to Improper Fractions An improper fraction has a numerator greater than or equal to its denominator (e.g., $\frac{7}{2}$, $\frac{5}{3}$). Steps: Multiply the whole number by the denominator of the fraction. Add the numerator of the fraction to the result from Step 1. This is your new numerator. Keep the original denominator. Formula: $A\frac{B}{C} = \frac{(A \times C) + B}{C}$ Example: Convert $3\frac{1}{2}$ to an improper fraction. Multiply whole number by denominator: $3 \times 2 = 6$ Add numerator: $6 + 1 = 7$ Keep denominator: $\frac{7}{2}$ So, $3\frac{1}{2} = \frac{7}{2}$ Converting Improper Fractions to Mixed Fractions Steps: Divide the numerator by the denominator. The quotient is the whole number. The remainder becomes the new numerator. The denominator stays the same. Example: Convert $\frac{17}{3}$ to a mixed fraction. Divide $17 \div 3$. Quotient is $5$, remainder is $2$. Whole number is $5$. New numerator is $2$. Denominator is $3$. So, $\frac{17}{3} = 5\frac{2}{3}$ Adding and Subtracting Mixed Fractions Method 1: Convert to Improper Fractions Steps: Convert all mixed fractions to improper fractions. Find a common denominator for the improper fractions (if necessary). Add or subtract the numerators. Simplify the result and convert back to a mixed fraction if desired. Example (Addition): $2\frac{1}{3} + 1\frac{1}{2}$ Convert: $\frac{7}{3} + \frac{3}{2}$ Common denominator (6): $\frac{14}{6} + \frac{9}{6}$ Add: $\frac{23}{6}$ Convert back: $3\frac{5}{6}$ Example (Subtraction): $3\frac{1}{4} - 1\frac{2}{3}$ Convert: $\frac{13}{4} - \frac{5}{3}$ Common denominator (12): $\frac{39}{12} - \frac{20}{12}$ Subtract: $\frac{19}{12}$ Convert back: $1\frac{7}{12}$ Method 2: Separate Whole and Fractional Parts Steps: Add/subtract the whole numbers. Add/subtract the fractional parts (find a common denominator). Combine the results. If the fractional part is improper, convert it and add to the whole number. Borrow from the whole number if the second fraction is larger in subtraction. Example (Addition): $2\frac{1}{3} + 1\frac{1}{2}$ Whole numbers: $2 + 1 = 3$ Fractions: $\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}$ Combine: $3\frac{5}{6}$ Example (Subtraction with Borrowing): $3\frac{1}{4} - 1\frac{2}{3}$ Cannot subtract $\frac{2}{3}$ from $\frac{1}{4}$ directly. Borrow 1 from $3$: $3\frac{1}{4} = 2 + 1 + \frac{1}{4} = 2 + \frac{4}{4} + \frac{1}{4} = 2\frac{5}{4}$ Now: $2\frac{5}{4} - 1\frac{2}{3}$ Whole numbers: $2 - 1 = 1$ Fractions: $\frac{5}{4} - \frac{2}{3} = \frac{15}{12} - \frac{8}{12} = \frac{7}{12}$ Combine: $1\frac{7}{12}$ Multiplying Mixed Fractions Steps: Always convert mixed fractions to improper fractions first. Multiply the numerators together. Multiply the denominators together. Simplify the resulting improper fraction (reduce to lowest terms and convert to mixed fraction if desired). Formula: $A\frac{B}{C} \times D\frac{E}{F} = \frac{(A \times C) + B}{C} \times \frac{(D \times F) + E}{F}$ Example: $1\frac{1}{2} \times 2\frac{1}{3}$ Convert: $\frac{3}{2} \times \frac{7}{3}$ Multiply numerators: $3 \times 7 = 21$ Multiply denominators: $2 \times 3 = 6$ Result: $\frac{21}{6}$ Simplify: $\frac{7}{2} = 3\frac{1}{2}$ Dividing Mixed Fractions Steps: Always convert mixed fractions to improper fractions first. Keep the first fraction, change the division sign to multiplication, and flip the second fraction (reciprocal). Multiply the fractions (as shown above). Simplify the result. Formula: $A\frac{B}{C} \div D\frac{E}{F} = \frac{(A \times C) + B}{C} \div \frac{(D \times F) + E}{F} = \frac{(A \times C) + B}{C} \times \frac{F}{(D \times F) + E}$ Example: $3\frac{1}{2} \div 1\frac{1}{4}$ Convert: $\frac{7}{2} \div \frac{5}{4}$ Keep, Change, Flip: $\frac{7}{2} \times \frac{4}{5}$ Multiply: $\frac{28}{10}$ Simplify: $\frac{14}{5} = 2\frac{4}{5}$