1.3 Fractions

Understanding Fractions

Fractions let you express parts of a whole and show up constantly in algebra. Getting comfortable with simplifying, operating on, and converting fractions now will make nearly every later topic smoother.

Simplification of Fractions

To simplify a fraction, you divide the numerator and denominator by their greatest common factor (GCF), the largest number that divides both evenly.

Example: Simplify $\frac{12}{18}$

1. Find the GCF of 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The GCF is 6.
2. Divide both numerator and denominator by 6:

$\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$

A fraction is in lowest terms when the numerator and denominator share no common factor other than 1.

Basic Operations with Fractions

Addition and Subtraction require a common denominator. You can't add or subtract fractions that represent different-sized pieces.

1. Find the least common denominator (LCD), the smallest number both denominators divide into evenly.
2. Convert each fraction to an equivalent fraction with that LCD.
3. Add or subtract the numerators. Keep the denominator the same.

Example: $\frac{1}{3} + \frac{1}{4}$

• The LCD of 3 and 4 is 12.
• $\frac{1}{3} = \frac{4}{12}$ and $\frac{1}{4} = \frac{3}{12}$
• $\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$

Multiplication is more straightforward: multiply numerators together and denominators together, then simplify.

$\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}$

Division means multiplying by the reciprocal of the second fraction. The reciprocal flips the numerator and denominator (the reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$).

$\frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9}$

Order of Operations with Fractions

Fractions follow the same PEMDAS rules as any other numbers: Parentheses, Exponents, Multiplication/Division (left to right), then Addition/Subtraction (left to right).

The most common mistake is adding before multiplying. Always scan the expression for multiplication and division first.

Example: $\frac{1}{2} + \frac{1}{3} \times \frac{3}{4}$

1. Multiplication first: $\frac{1}{3} \times \frac{3}{4} = \frac{3}{12} = \frac{1}{4}$
2. Then addition: $\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}$

Working with Fractions in Algebra

Fractions in Algebraic Expressions

When an algebraic expression contains fractions, substitute the given values for the variables and then simplify using the same fraction rules.

Example: If $x = 2$ and $y = 3$, evaluate $\frac{x}{y} + \frac{y}{x}$

1. Substitute: $\frac{2}{3} + \frac{3}{2}$
2. Find the LCD of 3 and 2, which is 6.
3. Convert: $\frac{4}{6} + \frac{9}{6}$
4. Add: $\frac{13}{6}$

Notice how step 2 in the original expression (finding a common denominator) is the same skill you'd use with plain numbers. The algebra just tells you which numbers to plug in.

Conversions Between Fraction Types

Improper fraction to mixed number:

1. Divide the numerator by the denominator.
2. The quotient is the whole number part; the remainder becomes the new numerator over the same denominator.

• $\frac{17}{4}$: 17 ÷ 4 = 4 remainder 1, so $\frac{17}{4} = 4\frac{1}{4}$

Mixed number to improper fraction:

1. Multiply the whole number by the denominator, then add the numerator.
2. Put that result over the original denominator.

• $3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5}$

Fraction to decimal: Divide the numerator by the denominator.

• $\frac{3}{4} = 3 \div 4 = 0.75$

Decimal to fraction:

1. Write the decimal as a fraction with a power of 10 in the denominator (10, 100, 1000, etc., depending on the number of decimal places).
2. Simplify.

• $0.6 = \frac{6}{10} = \frac{3}{5}$

Types of Fractions and Proportions

Rational numbers can be written as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. Examples: $\frac{3}{4}$, $\frac{-5}{2}$, and even whole numbers like $7 = \frac{7}{1}$.

Common fractions have whole-number numerators and denominators, like $\frac{1}{2}$ or $\frac{3}{4}$. These are the fractions you work with most often.

Complex fractions have a fraction in the numerator, the denominator, or both. To simplify one, treat the main fraction bar as division:

$\frac{\frac{1}{2}}{\frac{3}{4}} = \frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}$

Proportions state that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$. You can solve a proportion by cross-multiplying, which gives you $ad = bc$. From there, solve for the unknown variable using basic algebra.