Adding and subtracting with like denominators Knowledge-Based Study

Study notes

1. What Are Like Denominators?

   In a fraction, the bottom number is called the denominator. It tells us how many equal parts the whole is divided into. The top number is the numerator, which indicates how many of those parts are being considered.

   Fractions have like denominators (or common denominators) when their bottom numbers are identical. For example, $\frac{2}{7}$ and $\frac{3}{7}$ share the like denominator of 7.

   Because both fractions refer to equal-sized parts, we can combine or separate them directly without changing the size of the parts.

   The Addition Rule

   When adding fractions that have the same denominator, add only the numerators and keep the denominator unchanged:

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

   Step-by-Step Procedure

   1. Verify that the denominators are the same.
   2. Add the numerators together.
   3. Write the sum over the common denominator.
   4. Simplify the resulting fraction to its simplest form, if possible.

   Worked Example

   Calculate $\frac{3}{8} + \frac{1}{8}$:

   $$\frac{3}{8} + \frac{1}{8} = \frac{3 + 1}{8} = \frac{4}{8}$$

   Both 4 and 8 can be divided by their greatest common divisor, 4:

   $$\frac{4 \div 4}{8 \div 4} = \frac{1}{2}$$

   The Subtraction Rule

   When subtracting fractions with the same denominator, subtract the second numerator from the first numerator and keep the denominator the same:

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

   Step-by-Step Procedure

   1. Check that both fractions have identical denominators.
   2. Subtract the numerators: first numerator minus second numerator.
   3. Keep the shared denominator.
   4. Reduce the resulting fraction to simplest form when applicable.

   Worked Example

   Calculate $\frac{7}{10} - \frac{3}{10}$:

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

   Divide numerator and denominator by 2 to simplify:

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

   Important Points to Remember

   • Never add or subtract denominators: A common error is writing $\frac{1}{5} + \frac{2}{5} = \frac{3}{10}$. This is incorrect because the size of the pieces (fifths) has not changed. The correct answer is $\frac{3}{5}$.
   • Proper Simplification: Always check whether the numerator and denominator share any common factors greater than 1.
   • Improper Fractions: If the result has a numerator greater than or equal to the denominator, such as $\frac{5}{4}$, it can remain as an improper fraction or be written as a mixed number ($1\frac{1}{4}$) depending on context.