Fraction Review

To add or subtract fractions, we need to make them the “same kind” of fraction. We cannot just add/subtraction different kinds of fractions willy-nilly. This is similar to something like adding apples and oranges together and claiming that you have more apples as a result of adding some oranges to your pile of apples:

$$5\text{ apples}+3\text{ oranges}\neq8\text{ apples}$$

Similarly,

$$5\text{ apples}+3\text{ oranges}\neq8\text{ oranges}$$

However, if we can classify apples and oranges as some other thing, let’s say “fruits” we can conceptually add them together into the same pile of things.

$$5\text{ apple fruits}+3\text{ orange fruits}=8\text{ fruits}$$

Here, we found the common thing that relates apples or oranges: their aspect as fruits. When we add fractions, we have to find their “common denominator” (that number that will make them the same kind of object). Think of the denominator as that number which determined the “variety” of fraction we are dealing with (i.e. the kind of fruit in our fruit example). The numerator is how many of those kinds of fractions I have piled up together.

$$\frac{5}{\text{ apples}}+\frac{3}{\text{ oranges}}= ?$$

To add these fractions together, we need to find some aspect that apples and oranges share in common (i.e. a “common denominator”)

$$\frac{5\text{ fruits}}{\text{ apple fruits}}+\frac{3\text{ fruits}}{\text{ orange fruits}}=\frac{8\text{ fruits}}{\text{ fruits}}$$

This is a somewhat silly example, but it helps to explain why we cannot add different varieties of fractions together. Furthermore, you will notice that the denominator played no part in the ultimate addition/subtraction operation; that is, I did not add the denominators together in the fruit example: this is because they are not numbers but are rather the indicator of the type of fraction I am dealing with.

So, though in fraction addition/subtraction the denominator will be a number, don’t think of it quite like a normal number; it’s an identification of type.

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How to find Common Denominators:

So, let’s consider an actual fraction addition problem:

$$\frac{1}{2}+\frac{1}{3}=?$$

Well, these are not the same type of fraction: the first fraction is of the “half” variety while the second is of the “third” variety. So, we need to find a number that 2 and 3 share in common and adjust each fraction to be of that variety.

The most reliable and fast way of determining a common denominator is to multiply the two denominators together; this will tell you at least one possible denominator they share in common:

$$2(3)=6$$

Thus, if we can make each of these fractions of the “sixth” variety, we can add them together and sort them in the “sixth” pile.

Keep in mind that when we change the denominator, we have to change the numerator in the same way to keep the fraction fundamentally the same. Thinking back to our fruit example, I did not change the fact that each fraction was of apples and oranges; I merely appended “fruits” to the top and the bottom of the fraction, recognizing an aspect they share in common.

$$\frac{5\text{fruits}}{\text{apple fruits}}+\frac{3\text{fruits}}{\text{orange fruits}}=\frac{8\text{fruits}{\text{fruits}}$$

I could always remove the “fruits” label and return easily to what I had before if I needed to go backwards. I have not changed the fundamental nature of the objects (i.e. apples and oranges).

So, if I want to change my “halves” (i.e. $\frac{a}{2}$ fractions with $a$ just being how many halves we have) into “sixths” (i.e $\frac{b}{6}$ fractions with $b$ just being how many sixths we have as a result of this transformation), I need to multiply 2 in the denominator by 3; consequently, I also need to multiply the numerator by 3, just like I had to put “fruits” in the top and bottom of my fruit fractions.

$$\frac{1(3)}{2(3)}+\frac{1}{3}=?$$

Similarly, if I want to change my “thirds” (i.e. $\frac{c}{3}$ fractions with $c$ just being how many thirds we have) into “sixths” (i.e $\frac{b}{6}$ fractions with $b$ just being how many sixths we have as a result of this transformation), I need to multiply 3 in the denominator by 2; consequently, I also need to multiply the numerator by 2, just like I had to put “fruits” in the top and bottom of my fruit fractions.

$$\frac{1(3)}{2(3)}+\frac{1(2)}{3(2)}=\frac{3}{6}+\frac{2}{6}$$

Notice that if I wanted to simplify my new fractions, I would return to my original fractions, since these “sixth” fractions are fundamentally the same as the originals, although now seen as a common type (i.e. sixths).

$$\frac{3}{6}\rightarrow\frac{1}{2}$$

$$\frac{2}{6}\rightarrow\frac{1}{3}$$

Now, I can add these fractions together, since they are the same type of fraction:

$$\frac{3}{6}+\frac{2}{6}=\frac{5}{6}$$

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