Lesson 2: Operations on Fractions

New Concept

Just as with numbers, we can multiply, divide, add, and subtract algebraic fractions. This lesson introduces the rules for each operation, focusing on factoring to simplify products and quotients, and finding common denominators for sums and differences.

What’s next

You'll begin by mastering multiplication and division with interactive examples. Then, we'll move on to practice cards for adding and subtracting like fractions.

Property

To multiply two fractions, multiply their numerators together and multiply their denominators together. If $b \neq 0$ and $d \neq 0$, then

To multiply algebraic fractions:

1. Factor each numerator and denominator completely.
2. If any factor appears in both a numerator and a denominator, divide out that factor.
3. Multiply the remaining factors of the numerator and the remaining factors of the denominator.
4. Reduce the product if necessary.

Examples

• To multiply $\frac{4}{5} \cdot \frac{2}{3}$, we calculate $\frac{4 \cdot 2}{5 \cdot 3} = \frac{8}{15}$.
• To multiply $\frac{5x}{6} \cdot \frac{9}{10x^2}$, we factor and cancel: $\frac{5x}{2 \cdot 3} \cdot \frac{3 \cdot 3}{2 \cdot 5x \cdot x} = \frac{3}{4x}$.
• To multiply $\frac{x+3}{2x-8} \cdot \frac{x-4}{x^2-9}$, we factor first: $\frac{x+3}{2(x-4)} \cdot \frac{x-4}{(x-3)(x+3)} = \frac{1}{2(x-3)}$.

Explanation

Multiplying fractions means finding a part of a part. Multiply the tops (numerators) to get the new number of pieces, and multiply the bottoms (denominators) to find the new total size. Always cancel common factors first to simplify!

Property

To divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction. If $b, c, d \neq 0$, then

To divide one fraction by another:

1. Take the reciprocal of the second fraction and change the division to multiplication.
2. Follow the rules for multiplication of fractions.

Examples

• To divide $\frac{2}{7} \div \frac{3}{5}$, we change to multiplication: $\frac{2}{7} \cdot \frac{5}{3} = \frac{10}{21}$.
• To divide $\frac{14a^2}{3b} \div \frac{7a}{6}$, we find the reciprocal and multiply: $\frac{14a^2}{3b} \cdot \frac{6}{7a} = \frac{2 \cdot 7a \cdot a}{3b} \cdot \frac{2 \cdot 3}{7a} = \frac{4a}{b}$.
• To divide $\frac{x-5}{4x^2} \div \frac{x^2-25}{8x}$, we flip, factor, and cancel: $\frac{x-5}{4x^2} \cdot \frac{8x}{(x-5)(x+5)} = \frac{2}{x(x+5)}$.

Explanation

Dividing by a fraction is the same as multiplying by its reciprocal. Just flip the second fraction upside down and multiply. This helps find how many times the second fraction can fit into the first one. It’s the classic “keep, change, flip” method.

Property

Fractions with the same denominator are called like fractions. To add or subtract like fractions, combine the numerators and keep the denominator the same. If $c \neq 0$, then

To add or subtract like fractions:

1. Add or subtract the numerators.
2. Keep the same denominator.
3. Reduce the sum or difference if necessary.

Examples

• To add $\frac{3x-2}{x+1} + \frac{x+6}{x+1}$, we combine numerators: $\frac{(3x-2)+(x+6)}{x+1} = \frac{4x+4}{x+1} = \frac{4(x+1)}{x+1} = 4$.
• To subtract $\frac{5a-1}{a-3} - \frac{2a+8}{a-3}$, distribute the negative: $\frac{5a-1-(2a+8)}{a-3} = \frac{5a-1-2a-8}{a-3} = \frac{3a-9}{a-3} = \frac{3(a-3)}{a-3} = 3$.
• To subtract $\frac{y^2-3y}{y-5} - \frac{2y+20}{y-5}$, we get $\frac{y^2-3y-(2y+20)}{y-5} = \frac{y^2-5y-20}{y-5}$.

Explanation

You can only add or subtract things of the same kind. The denominator tells you the 'kind' of fraction (e.g., fifths). So, you combine the numerators (how many) and keep the denominator (the kind) unchanged. Be careful to distribute the negative sign to all terms in the second numerator when subtracting.

Property

If an algebraic fraction cannot be reduced, you can simplify it by dividing the denominator into the numerator. If the denominator is a monomial, divide the monomial into each term of the numerator. The result is the sum of a polynomial and a simpler algebraic fraction.

Examples

• To divide $\frac{12x^4 - 8x^2}{4x}$, we divide each term: $\frac{12x^4}{4x} - \frac{8x^2}{4x} = 3x^3 - 2x$.
• To divide $\frac{15y^5 + 5y^3 + 2}{5y^2}$, we get $\frac{15y^5}{5y^2} + \frac{5y^3}{5y^2} + \frac{2}{5y^2} = 3y^3 + y + \frac{2}{5y^2}$.
• To divide $\frac{18a^3b^2 - 9ab^3 + ab}{3ab}$, we calculate $\frac{18a^3b^2}{3ab} - \frac{9ab^3}{3ab} + \frac{ab}{3ab} = 6a^2b - 3b^2 + \frac{1}{3}$.

Explanation

This is like using the distributive property for division. You break one large, complex fraction into several smaller, simpler fractions by dividing each part of the numerator by the denominator. This makes simplifying much more manageable.