Adding and subtracting with unlike denominators Knowledge-Based Study

Study notes

1. Why Can't We Add or Subtract Directly?

   The denominator of a fraction indicates the total number of equal parts into which a whole is divided. The numerator counts how many of those parts we have.

   When fractions have the same denominator, their parts are the same size, allowing direct addition or subtraction of the numerators:

   $$\frac{1}{5} + \frac{2}{5} = \frac{3}{5}$$

   However, when denominators are different, such as in $\frac{1}{2}$ and $\frac{1}{3}$, the parts represent different sizes. One-half represents a larger slice than one-third. Therefore, we cannot directly combine or subtract their numerators without first expressing both fractions in terms of equal-sized parts.

   Least Common Multiple (LCM)

   To add or subtract fractions with unlike denominators, we must rewrite them as equivalent fractions that share a common denominator.

   The most efficient common denominator is the Least Common Denominator (LCD), which is the least common multiple (LCM) of the given denominators.

   Steps to find the LCD:

   1. List the multiples of each denominator or use prime factorization.
   2. Identify the smallest positive multiple shared by all denominators.

   Example: Find the LCD for $\frac{3}{4}$ and $\frac{1}{6}$.

   • Multiples of 4: 4, 8, 12, 16, 20...
   • Multiples of 6: 6, 12, 18, 24...

   The least common denominator is 12.

   Step-by-Step Addition

   Once the common denominator is determined, follow these three steps:

   1. Multiply the numerator and denominator of each fraction by the factor needed to obtain the common denominator.
   2. Add the new numerators while keeping the common denominator unchanged.
   3. Simplify the resulting fraction to simplest form, or convert it to a mixed number if required.

   Worked Example:

   Compute: $$\frac{2}{5} + \frac{1}{4}$$

   Step 1: Find the LCD of 5 and 4. Since 5 and 4 share no common factors other than 1, their least common multiple is $5 \times 4 = 20$.

   Step 2: Rewrite each fraction with denominator 20:

   $$\frac{2 \times 4}{5 \times 4} = \frac{8}{20}$$

   $$\frac{1 \times 5}{4 \times 5} = \frac{5}{20}$$

   Step 3: Add the numerators:

   $$\frac{8}{20} + \frac{5}{20} = \frac{8 + 5}{20} = \frac{13}{20}$$

   The fraction $\frac{13}{20}$ is in simplest form.

   Step-by-Step Subtraction

   Subtraction follows the same fundamental rule: find a common denominator, convert each fraction, and subtract the numerators.

   Worked Example:

   Compute: $$\frac{5}{6} - \frac{3}{8}$$

   Step 1: Find the LCD of 6 and 8.

   • Multiples of 6: 6, 12, 18, 24, 30...
   • Multiples of 8: 8, 16, 24, 32...

   The LCD is 24.

   Step 2: Convert each fraction:

   $$\frac{5 \times 4}{6 \times 4} = \frac{20}{24}$$

   $$\frac{3 \times 3}{8 \times 3} = \frac{9}{24}$$

   Step 3: Subtract the numerators:

   $$\frac{20}{24} - \frac{9}{24} = \frac{20 - 9}{24} = \frac{11}{24}$$

   Since 11 is prime and not a divisor of 24, $\frac{11}{24}$ is fully simplified.

   Key Points to Remember

   • Never add or subtract denominators: A common error is writing $\frac{1}{3} + \frac{1}{2} = \frac{2}{5}$. Denominators identify the unit size; they do not get added or subtracted.
   • Always simplify: If the numerator and denominator share a common factor greater than 1, divide both by their greatest common divisor (GCD). For example, $\frac{6}{8}$ simplifies to $\frac{3}{4}$.
   • Improper fractions: If your sum has a numerator greater than or equal to the denominator (such as $\frac{7}{4}$), it represents a value greater than or equal to 1. This can be expressed either as an improper fraction $\frac{7}{4}$ or as a mixed number $1\frac{3}{4}$.