Adding and Subtracting Rational Expressions with Like & Unlike Denominators

Chapter 18: Adding and Subtracting Rational Expressions with Like Denominators

Concept Explanation:

When dealing with rational expressions that have like denominators, you can add or subtract the numerators while keeping the denominator the same.

For example:

$\frac{a}{d} + \frac{b}{d} = \frac{a + b}{d}$

$\frac{a}{d} - \frac{b}{d} = \frac{a - b}{d}$

Common Mistakes:

Neglecting to Simplify: Always check if you can simplify the result.
Incorrectly Adding/Subtracting Numerators: Ensure that you perform the correct operation on the numerators.

Hard Questions:

Q1: Add $\frac{2x}{5} + \frac{3x}{5}$ .

Step-by-Step Solution:

Since the denominators are the same, add the numerators:

$\frac{2x + 3x}{5} = \frac{5x}{5} = x$

Answer:

The result is $x$ .

Q2: Subtract $\frac{5a}{12} - \frac{2a}{12}$ .

Step-by-Step Solution:

Since the denominators are the same, subtract the numerators:

$\frac{5a - 2a}{12} = \frac{3a}{12} = \frac{a}{4}$

Answer:

The result is $\frac{a}{4}$ .

Q3: Add $\frac{3y}{8} + \frac{4y}{8}$ .

Step-by-Step Solution:

Since the denominators are the same, add the numerators:

$\frac{3y + 4y}{8} = \frac{7y}{8}$

Answer:

The result is $\frac{7y}{8}$ .

Chapter 19: Adding and Subtracting Rational Expressions with Unlike Denominators

Concept Explanation:

To add or subtract rational expressions with unlike denominators, you must first find the least common denominator (LCD), rewrite each expression with that denominator, and then perform the addition or subtraction.

Common Mistakes:

Forgetting to Find the LCD: Always find the LCD before proceeding with the operations.
Incorrectly Combining Denominators: Make sure to rewrite each term properly before combining.

Hard Questions:

Q1: Add $\frac{1}{3} + \frac{1}{4}$ .

Step-by-Step Solution:

The LCD of 3 and 4 is 12.
Rewrite the fractions:

$\frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}$

Add:

$\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$

Answer:

The sum is $\frac{7}{12}$ .

Q2: Subtract $\frac{5}{6} - \frac{1}{2}$ .

Step-by-Step Solution:

The LCD of 6 and 2 is 6.
Rewrite the second fraction:

$\frac{1}{2} = \frac{3}{6}$

Subtract:

$\frac{5}{6} - \frac{3}{6} = \frac{2}{6} = \frac{1}{3}$

Answer:

The difference is $\frac{1}{3}$ .

Q3: Add $\frac{2}{5} + \frac{3}{10}$ .

Step-by-Step Solution:

The LCD of 5 and 10 is 10.
Rewrite the first fraction:

$\frac{2}{5} = \frac{4}{10}$

Add:

$\frac{4}{10} + \frac{3}{10} = \frac{7}{10}$

Answer:

The sum is $\frac{7}{10}$ .

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