Decimals and Percents
Convert fluently among fractions, decimals, and percents. Perform operations with decimals including rounding, and interpret percent as a rate per hundred in context.
A lesson of Pre-Algebra, in Number Sense & Operations, about 10 minutes.
Fractions: Operations and Applications
By the end of this lesson, you will be able to add, subtract, multiply, and divide fractions and mixed numbers, convert between fractions and decimals, and apply fractions to real-world problems.
Fraction Fundamentals
Fraction: $\dfrac{a}{b}$ where $a$ is the numerator and $b$ is the denominator ($b \neq 0$).
Equivalent fractions: $\dfrac{1}{2} = \dfrac{2}{4} = \dfrac{4}{8}$ — multiply or divide numerator and denominator by the same nonzero value.
Simplest form (lowest terms): when numerator and denominator share no common factors (GCF = 1).
Simplify $\dfrac{18}{24}$:
$\text{GCF}(18, 24) = 6$; $\dfrac{18 \div 6}{24 \div 6} = \dfrac{3}{4}$
Mixed numbers and improper fractions:
- Mixed number: $2\dfrac{3}{5}$ (whole + fraction)
- Improper fraction: $\dfrac{13}{5}$ (numerator ≥ denominator)
Convert mixed → improper: $2\dfrac{3}{5} = \dfrac{2 \times 5 + 3}{5} = \dfrac{13}{5}$
Convert improper → mixed: $\dfrac{17}{4} = 4\dfrac{1}{4}$ (17 ÷ 4 = 4 remainder 1)
Adding and Subtracting Fractions
Same denominator: add/subtract numerators, keep denominator.
$$\frac{3}{7} + \frac{2}{7} = \frac{5}{7}$$
Different denominators: find the LCD, convert to equivalent fractions, then add/subtract.
Example: $\dfrac{3}{4} + \dfrac{5}{6}$
LCD of 4 and 6: LCM = 12
$\dfrac{3}{4} = \dfrac{9}{12}$ and $\dfrac{5}{6} = \dfrac{10}{12}$
$\dfrac{9}{12} + \dfrac{10}{12} = \dfrac{19}{12} = 1\dfrac{7}{12}$
Mixed numbers — example: $3\dfrac{1}{2} - 1\dfrac{3}{4}$
Convert: $\dfrac{7}{2} - \dfrac{7}{4}$
LCD = 4: $\dfrac{14}{4} - \dfrac{7}{4} = \dfrac{7}{4} = 1\dfrac{3}{4}$
Multiplying Fractions
Rule: multiply numerators and multiply denominators; simplify.
$$\frac{3}{5} \times \frac{2}{7} = \frac{6}{35}$$
Mixed numbers: convert to improper fractions first.
$$2\frac{1}{3} \times 1\frac{1}{2} = \frac{7}{3} \times \frac{3}{2} = \frac{21}{6} = \frac{7}{2} = 3\frac{1}{2}$$
Cross-canceling (simplify before multiplying):
$$\frac{4}{9} \times \frac{3}{8} = \frac{\cancel{4}^1}{9} \times \frac{3}{\cancel{8}^2} = \frac{1 \times 3}{9 \times 2} = \frac{3}{18} = \frac{1}{6}$$
Dividing Fractions
keep the first fraction, change division to multiplication, flip the second fraction (multiply by the reciprocal).