Lesson 13: Adding and Subtracting Fractions and Mixed Numbers

New Concept

This course builds a complete understanding of arithmetic, from fundamental operations to complex problem-solving with fractions, decimals, and percents.

What’s next

This card introduces the big picture. Next, we’ll dive into a key skill: adding and subtracting fractions through worked examples and visual models.

Property

To add or subtract fractions that have common denominators, add or subtract the numerators and leave the denominators unchanged. For example, $\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$.

Examples

• When adding, combine the numerators: $\frac{3}{4} + \frac{3}{4} = \frac{6}{4}$, which simplifies to $1\frac{1}{2}$.
• When subtracting, find the difference of the numerators: $\frac{5}{8} - \frac{2}{8} = \frac{3}{8}$.
• For a quick addition, $\frac{2}{7} + \frac{4}{7} = \frac{6}{7}$.

Explanation

Think of it like counting pizza slices of the same size! If you have 3 eighths of a pizza and your friend gives you 2 more eighths, you just add the number of slices. The size of the slice (the denominator) stays the same. You now have 5 eighths. It’s that simple!

Property

When two fractions have the same denominator, we say they have common denominators. To create them, find an equivalent fraction by multiplying by a form of 1, such as $\frac{a}{b} \cdot \frac{c}{c} = \frac{ac}{bc}$.

Examples

• To add $\frac{2}{3} + \frac{3}{4}$, use 12 as the common denominator: $\frac{2}{3} \cdot \frac{4}{4} + \frac{3}{4} \cdot \frac{3}{3} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12}$.
• To subtract $\frac{3}{4} - \frac{1}{6}$, the least common denominator is 12: $\frac{3}{4} \cdot \frac{3}{3} - \frac{1}{6} \cdot \frac{2}{2} = \frac{9}{12} - \frac{2}{12} = \frac{7}{12}$.
• To add $\frac{1}{2} + \frac{1}{5}$, the common denominator is 10: $\frac{1}{2} \cdot \frac{5}{5} + \frac{1}{5} \cdot \frac{2}{2} = \frac{5}{10} + \frac{2}{10} = \frac{7}{10}$.

Explanation

You can't add fractions with different slice sizes! To make it work, you must find a 'common denominator' so the pieces are the same size. A quick trick is to multiply the two different denominators to find a new one that works for both. This lets you add or subtract them fairly.

Property

1. Write fractions with common denominators.
2. Add or subtract numerators and whole numbers, regrouping if necessary.
3. Simplify the answer if possible.

Examples

• Add wholes and fractions: $3\frac{1}{2} + 1\frac{3}{4} \rightarrow 3\frac{2}{4} + 1\frac{3}{4} = 4\frac{5}{4} = 5\frac{1}{4}$.
• Subtract with regrouping: $3\frac{1}{2} - 1\frac{3}{4} \rightarrow 3\frac{2}{4} - 1\frac{3}{4} \rightarrow 2\frac{6}{4} - 1\frac{3}{4} = 1\frac{3}{4}$.
• A simple subtraction without regrouping: $5\frac{7}{8} - 2\frac{1}{4} \rightarrow 5\frac{7}{8} - 2\frac{2}{8} = 3\frac{5}{8}$.

Explanation

Treat mixed numbers like whole pizzas and extra slices. Add the wholes and fractions separately. For subtraction, if you're short on slices, you may need to 'borrow' a whole pizza and slice it up (regrouping). It’s just like breaking a one dollar bill to get quarters when you need to pay for something.