-
Changing Improper Fractions to Mixed Numbers:
Divide the numerator by the denominator. The quotient is the whole number, and the remainder over the denominator is the fraction part.
Example: \( \frac{11}{3} = 3 \frac{2}{3} \)
-
Changing Mixed Numbers to Improper Fractions:
Multiply the whole number by the denominator, add the numerator, and put the result over the denominator.
Example: \( 2 \frac{5}{6} = \frac{2 \times 6 + 5}{6} = \frac{17}{6} \)
-
Adding Fractions with Like Denominators:
Add the numerators, keep the same denominator.
Example: \( \frac{3}{8} + \frac{2}{8} = \frac{5}{8} \)
-
Subtracting Fractions with Like Denominators:
Subtract the numerators, keep the same denominator.
Example: \( \frac{7}{10} - \frac{3}{10} = \frac{4}{10} = \frac{2}{5} \)
-
Adding or Subtracting Fractions with Unlike Denominators:
Find a common denominator, rewrite each fraction, then add or subtract.
Example: \( \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \)
-
Adding Mixed Numbers with Unlike Denominators:
Convert mixed numbers to improper fractions or add whole numbers and fractions separately (after finding a common denominator).
Example: \( 1\frac{1}{2} + 2\frac{2}{3} = \frac{3}{2} + \frac{8}{3} = \frac{9}{6} + \frac{16}{6} = \frac{25}{6} = 4 \frac{1}{6} \)
-
Subtracting Mixed Numbers with Unlike Denominators:
Find a common denominator, borrow if necessary, then subtract.
Example: \( 3\frac{1}{4} - 1\frac{1}{2} = \frac{13}{4} - \frac{3}{2} = \frac{13}{4} - \frac{6}{4} = \frac{7}{4} = 1\frac{3}{4} \)
-
Estimating Sums and Differences of Fractions and Mixed Numbers:
Round each fraction or mixed number to the nearest whole number or benchmark fraction, then add or subtract.
Example: \( 2\frac{5}{6} \approx 3 \), \( 1\frac{1}{4} \approx 1 \), so \( 2\frac{5}{6} + 1\frac{1}{4} \approx 4 \)
-
Multiplying Fractions and Whole Numbers:
Multiply the whole number by the numerator, keep the denominator.
Example: \( 4 \times \frac{3}{5} = \frac{12}{5} = 2\frac{2}{5} \)
-
Multiplying Fractions: Reciprocals:
The reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). Multiplying a number by its reciprocal gives 1.
Example: \( \frac{2}{3} \times \frac{3}{2} = 1 \)
-
Multiplying Fractions and Mixed Numbers: Reducing:
Convert mixed numbers to improper fractions, multiply, and reduce (simplify) the result.
Example: \( 1\frac{1}{2} \times \frac{4}{9} = \frac{3}{2} \times \frac{4}{9} = \frac{12}{18} = \frac{2}{3} \)
-
Dividing Fractions by Whole Numbers:
Multiply by the reciprocal of the whole number.
Example: \( \frac{3}{5} \div 2 = \frac{3}{5} \times \frac{1}{2} = \frac{3}{10} \)
-
Dividing Whole Numbers by Fractions:
Multiply the whole number by the reciprocal of the fraction.
Example: \( 4 \div \frac{2}{3} = 4 \times \frac{3}{2} = \frac{12}{2} = 6 \)
-
Dividing Fractions by Fractions:
Multiply by the reciprocal of the divisor.
Example: \( \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} \)
-
Dividing Mixed Numbers:
Convert to improper fractions, then multiply by the reciprocal.
Example: \( 2\frac{1}{2} \div 1\frac{1}{4} = \frac{5}{2} \div \frac{5}{4} = \frac{5}{2} \times \frac{4}{5} = 2 \)
-
Ratios:
A ratio compares two quantities, often written as \( a:b \), \( a \) to \( b \), or \( \frac{a}{b} \).
Example: The ratio of 8 to 12 is \( 8:12 \) or \( 2:3 \).
-
Proportions and Cross-Multiplying:
A proportion is an equation that states two ratios are equal. Cross-multiplying helps solve for unknowns.
Example: If \( \frac{a}{b} = \frac{c}{d} \), then \( a \times d = b \times c \)
-
Ratio Tables:
A table of equivalent ratios.
Example:
-
Rates:
A rate is a ratio comparing two different units.
Example: 60 miles in 2 hours is a rate of 30 miles/hour.
-
Problem-Solving with Proportions:
Use proportions to solve real-world problems, such as finding missing values in similar figures or recipes.