Adding / Subtracting Fractions
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.
A common mistake that I see is adding the numerator and the denominator. For example, students will try something like:
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.
 If the denominators are the same then skip to step five.
 Find the least common multiple of the denominators. When we're working with fractions, this number is often called the least common denominator.
 Divide the least common multiple by the denominator of the first fraction. Multiply the numerator and denominator of the first fraction by that number.
 Divide the least common multiple by the denominator of the second fraction. Multiply the numerator and denominator of the second fraction by that number.
 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.
 Reduce the fraction.
Example 1
Calculate 1/3 + 2/5.



lcm(3, 5) = 15 

$$\frac{1 \cdot 5}{3 \cdot 5} = \frac{5}{15}$$ 

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

$$\frac{5}{15} + \frac{6}{15} = \frac{11}{15}$$ 

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.



$$\frac{4 + 5}{3} = \frac{9}{3}$$ 

$$\frac{9}{3} = \frac{9/3}{3/3} = \frac{3}{1} = 3$$ 
Example 2
Simplify $\frac{5}{12}  \frac{4}{3}$.
Subtraction works with the same series of steps as addition.



lcm(3, 5) = 15 



$$\frac{4 \cdot 4}{3 \cdot 4} = \frac{16}{12}$$ 

$\frac{5}{12}  \frac{16}{12} = \frac{11}{12}$ 

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:

$$\frac{9}{12} = \frac{9/3}{12/3} = \frac{3}{4}$$ 

$$\frac{3}{4}  \frac{5}{7}$$ 

lcm(4, 7) = 28 

$$\frac{3 \cdot 7}{4 \cdot 7} = \frac{21}{28}$$ 

$$\frac{5 \cdot 4}{7 \cdot 4} = \frac{20}{28}$$ 

$$\frac{21}{28}  \frac{20}{28} = \frac{1}{28}$$ 

Videos
Directions: This solution has 5 steps. To see a description of each step click on the boxes on the left side below. To see the calculations, click on the corresponding box on the right side. Try working out the solution yourself and use the descriptions if you need a hint and the calculations to check your solution.