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Adding and subtracting fractions are probably the trickiest things you'll be asked to do in an introductory math class. The procedure isn't bad if you're careful to follow a few steps very precisely.

Warning - Adding the Wrong Way

A common mistake that I see is adding the numerator and the denominator. For example, students will try something like:

$$\frac{1}{2} + \frac{1}{2} = \frac{2}{4}$$

But take a closer look at that: 2/4 = 1/2. It isn't possible for a number added to itself to equal itself, that's like trying to say that 1 + 1 = 1 so clearly there's a problem with that method.

1. If the denominators are the same then skip to step five.
2. Find the least common multiple of the denominators. When we're working with fractions, this number is often called the least common denominator.
3. Divide the least common multiple by the denominator of the first fraction. Multiply the numerator and denominator of the first fraction by that number.
4. Divide the least common multiple by the denominator of the second fraction. Multiply the numerator and denominator of the second fraction by that number.
5. Make a new fraction whose numerator is the sum of the two new numerators from steps three and four and whose denominator is the common denominator.
6. Reduce the fraction.

# Example 1

Calculate 1/3 + 2/5.

 The denominators aren't the same so we can skip to step 2. The least common multiple of 3 and 5 is 15. lcm(3, 5) = 15 If we divide 15 by the denominator of the first fraction we get $\frac{15}{3} = 5$. If we multiply the numerator and denominator of the first fraction by 5 we get: $$\frac{1 \cdot 5}{3 \cdot 5} = \frac{5}{15}$$ Now we'll do the same thing with the second fraction. $\frac{15}{5} = 3$ so we'll multiply the numerator and denominator of the second fraction by three: $$\frac{2 \cdot 3}{5 \cdot 3} = \frac{6}{15}$$ Now we can add the two fractions we found in steps three and four together by adding their numerators and keeping their denomiantors the same. $$\frac{5}{15} + \frac{6}{15} = \frac{11}{15}$$ $\frac{11}{15}$ can't be reduced any further so that's our final answer.

We could summarize the whole process with this equation:

$$\frac{1}{3} + \frac{2}{5} = \frac{1 \cdot 5}{3 \cdot 5} + \frac{2 \cdot 3}{5 \cdot 3} = \frac{5}{15} + \frac{6}{15} = \frac{11}{15} = \frac{5 + 6}{15} = \frac{11}{15}$$

# Example 3

Calculate 4/3 + 5/3.

 In this example, the denominators are already the same so we can skip directly to step five. We can add the two fractions together by adding their numerators and keeping their denomiantors the same. $$\frac{4 + 5}{3} = \frac{9}{3}$$ We can reduce that by dividing both the numerator and denominator by three to get the final answer. $$\frac{9}{3} = \frac{9/3}{3/3} = \frac{3}{1} = 3$$

# Example 2

Simplify $\frac{5}{12} - \frac{4}{3}$.

Subtraction works with the same series of steps as addition.

 The denominators aren't the same so we can skip to step 2. The least common multiple of 3 and 12 is 12. lcm(3, 5) = 15 The denominator of the first fraction is already 12 so we can skip to step four. If we divide 12 by the denominator of the second fraction we get $\frac{12}{3} = 4$. If we multiply the numerator and denominator of the second fraction by 4 we get: $$\frac{4 \cdot 4}{3 \cdot 4} = \frac{16}{12}$$ Now we can subtract the two fractions we found in steps three and four together by subtracting their numerators and keeping their denomiantors the same. $\frac{5}{12} - \frac{16}{12} = -\frac{11}{12}$ $-\frac{11}{12}$ can't be reduced any further so that's our final answer.

We could summarize the whole process with this equation:

$$\frac{5}{12} - \frac{4}{3} = \frac{5}{12} - \frac{4 \cdot 4}{3 \cdot 4} = \frac{5}{12} - \frac{16}{12} = \frac{5 - 16}{12} = -\frac{11}{12}$$

# Example 4

Calculate 9/12 - 5/7.

Before we jump into our six steps, I'm going to add a "step zero" that you don't have to do to get the right answer but that can save you some work down the road:

 Reduce all the fractions in the original expression. In this example, 9/12 can be reduced by 3. $$\frac{9}{12} = \frac{9/3}{12/3} = \frac{3}{4}$$ The denominators aren't the same so we can skip to step 2. $$\frac{3}{4} - \frac{5}{7}$$ The least common multiple of 4 and 7 is 28. lcm(4, 7) = 28 If we divide 28 by the denominator of the first fraction we get $\frac{28}{3} = 7$. If we multiply the numerator and denominator of the first fraction by 7 we get: $$\frac{3 \cdot 7}{4 \cdot 7} = \frac{21}{28}$$ Now we'll do the same thing with the second fraction. $\frac{28}{7} = 4$ so we'll multiply the numerator and denominator of the second fraction by 4: $$\frac{5 \cdot 4}{7 \cdot 4} = \frac{20}{28}$$ Now we can subtract the two fractions we found in steps three and four together by subtracting their numerators and keeping their denomiantors the same. $$\frac{21}{28} - \frac{20}{28} = \frac{1}{28}$$ $\frac{1}{28}$ can't be reduced any further so that's our final answer.