Subtracting fractions may seem intimidating, but it’s actually quite simple.
When you subtract fractions, you’re essentially finding the difference between two values that are represented as fractions.
To do this, you subtract the numerators once the denominators are the same.
The numerator is the top number in a fraction, and the denominator is the bottom number.
$$ \frac{\text{numerator}}{\text{denominator}} $$
To subtract fractions, you must have a common denominator. If the denominators are not the same, you’ll need to find a common denominator before you can subtract the fractions. Once you have a common denominator, you can subtract the numerators and write the answer over the common denominator.
Understanding Fractions
Fractions are a way of representing parts of a whole. They are used to describe quantities that are not whole numbers. A fraction consists of two parts: the numerator and the denominator. The numerator represents the number of parts that are being considered, while the denominator represents the total number of parts that make up the whole.
Types of Fractions
- Proper fraction: numerator < denominator, e.g., $\frac{1}{2}$
- Improper fraction: numerator ≥ denominator, e.g., $\frac{5}{3}$
- Mixed number: whole number + proper fraction, e.g., $3\frac{1}{2}$
Understanding the different types of fractions is important when working with fractions, as it helps you to identify and compare fractions more easily.
You can read more about fractions here.
Subtracting Fractions with the Same Denominator
If denominators are the same, subtract numerators only.
$$ \frac{3}{5} - \frac{2}{5} = \frac{3 - 2}{5} = \frac{1}{5} $$
Subtracting Fractions with Different Denominators
Step 1: Find the Least Common Denominator (LCM)
Example: $\frac{3}{8} - \frac{1}{12}$
Multiples:
- 8: {8, 16, 24, …}
- 12: {12, 24, 36, …}
- LCM = 24
Step 2: Convert to Equivalent Fractions
$$ \frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}, \quad \frac{1}{12} = \frac{1 \times 2}{12 \times 2} = \frac{2}{24} $$
Step 3: Subtract
$$ \frac{9}{24} - \frac{2}{24} = \frac{7}{24} $$
Subtracting Mixed Fractions
Convert to Improper Fractions
Example:
- $2\frac{1}{3} = \frac{7}{3}$
- $3\frac{2}{5} = \frac{17}{5}$
Find LCM of 3 and 5 = 15
Convert:
- $\frac{7}{3} = \frac{35}{15}$
- $\frac{17}{5} = \frac{51}{15}$
Subtract: $$ \frac{35}{15} - \frac{51}{15} = \frac{-16}{15} = -1\frac{1}{15} $$
Alternatively: Separate Whole and Fraction Parts
Rewrite: $2\frac{1}{3} - 3\frac{2}{5}$ as $2 + \frac{1}{3} - 3 - \frac{2}{5}$
Operate on fractions: $$ \frac{1}{3} - \frac{2}{5} = \frac{5 - 6}{15} = \frac{-1}{15} $$
Whole numbers: $$ 2 - 3 = -1 $$
Final result: $$ -1 - \frac{1}{15} = -1\frac{1}{15} $$
Subtracting Fractions Summary
- Make denominators the same.
- Subtract numerators.
- Simplify the result.
For mixed numbers, either:
- Convert to improper fractions first, or
- Subtract whole numbers and fractions separately.