Fractions, Decimals, and Percents Study Guide for the Math Basics

Operations with Fractions

Now that we have the basics down, we need to know how to combine and separate fractions using mathematical operations. For all operations with fractions, it’s helpful to turn all mixed numbers into improper fractions first.

Addition and Subtraction—When adding or subtracting fractions, the important thing to remember is that the two fractions must have a common denominator. If you remember that, you’ll be well on your way to victory.

To get common denominators, first find the Least Common Multiple (LCM) of the denominators (or Least Common Denominator = LCD).

For example, the LCD of \(\frac{3}{8}\) and \(\frac{5}{6}\) is \(24\).

Next, multiply the numerator and the denominator of the fraction (one at a time) by the factor which will turn the denominator into the LCD:

\(\dfrac{3}{8} =\dfrac{3 \cdot 3}{8 \cdot 3} = \dfrac{9}{24}\), and \(\dfrac{5}{6}=\dfrac{5 \cdot 4}{6 \cdot 4} = \dfrac{20}{24}\)

Next, let’s look at an example that you probably already know. What’s a quarter plus 2 quarters? Why, that’s three quarters, or 75 cents! This is a great example to remember because it will remind you of the basics: when adding, DON’T add the denominators.

\[\frac{1}{4} + \frac{2}{4} = \frac {1 + 2}{4} = \frac {3}{4}\]

To summarize, here are the steps to add or subtract fractions, along with an example for each.

1) Rewrite the fractions with common denominators.

\[\frac{3}{5} + \frac{4}{7} = \dfrac{3 \cdot 7}{5 \cdot 7} + \dfrac{4 \cdot 5}{7 \cdot 5} = \frac{21}{35} + \frac {20}{35}\]

2) Add the numerators, keep the denominators.

\[\frac{21}{35} + \frac {20}{35} = \frac{21+20}{35} = \frac{41}{35}\]

3) Reduce if necessary (also check to see which form is required).

41 and 35 have no common factors, so \(\frac{41}{35}\) is the answer.

If a mixed number is required, \(\frac{41}{35} = 1 \frac{6}{35}\)

Multiplication—Multiplication is the easiest operation when it comes to fractions. Simply multiply the numerators and the denominators. Simplify if necessary.

Example:

\[\frac{4}{5} \cdot \frac{3}{8} = \frac{4 \cdot 3}{5 \cdot 8} = \frac {12}{40}\]

Now reduce (simplify):

\[\dfrac{12}{40} = \dfrac{12 \div 4}{40 \div 4} = \dfrac {3}{10}\]

So, \(\dfrac{4}{5} \cdot \dfrac{3}{8} = \dfrac {3}{10}\)

Intuitively, if you like to bake, this should make sense. You might be using a recipe that calls for a half of a cup of flour. But maybe you only want to make a half of the recipe. So, what’s half of a half cup, or \(\frac{1}{2} \cdot \frac{1}{2}\) of a cup? You probably know the answer is a quarter cup. Mathematically, this is consistent with

\[\dfrac{1}{2} \cdot \dfrac{1}{2} = \dfrac {1 \cdot 1}{2 \cdot 2} = \dfrac{1}{4}\]

Division—To divide fractions, multiply the first by the reciprocal of the second. The reciprocal is the multiplicative inverse (\(\frac{a}{b}\) and \(\frac{b}{a}\) are reciprocals).

We’ve developed a helpful way to remember this. It’s one of your favorite fast food chains’ younger brother, KCF (not to be confused with the real one). KCF stands for Keep, Change, Flip. It tells you what to do with each item in a division of fractions problem.

Here’s how to use it. Start by writing the expression or equation in the top row of boxes, then do what it says to each item and write the result in the bottom row of boxes. Then carry out what your new figures say to do.

\[\begin{array}{|c|c|c|} \hline \quad\dfrac{3}{5}\quad & \div & \quad\dfrac{7}{8}\quad \\[2mm] \hline \quad\mathbf{Keep}\quad & \quad\mathbf{Change}\quad & \quad\mathbf{Flip}\quad \\[2mm] \hline \quad\dfrac{3}{5}\quad & \times & \quad\dfrac{8}{7}\quad \\[2mm] \hline \end{array}\]

To do this without actually drawing a chart, just start from the leftmost item, “keep, change, and flip.”

\[\frac{3}{5} \div \frac{7}{8}\] \[\frac{3}{5} \cdot \frac{8}{7}\] \[\frac{24}{35}\]