08.05.2018

Episode #4 of the course Foundations of mathematics by John Robin

When you are presented with something like this, 57/168 + 17/199, what do you do? Thanks to what we learned yesterday, I’m going to show you today how to never panic again when you see this.

There is a quick and easy way to deal with any fraction addition.

If your fractions have this general form:

[Numerator left]/[Denominator left] + [Numerator right]/[Denominator right]

Then form the following:

[Numerator left]/[Denominator left] x [Denominator right]/[Denominator right]

+

[Numerator right]/[Denominator right] x [Denominator left]/[Denominator left]

Let’s break this into a step-by-step process and apply it to our above example:

0. Make sure both fractions are improper:

A mixed fraction looks like this: 1 1/2. Its improper form doesn’t have a whole number. 1 1/2 means two halves and another half, which is 3 halves. So 1 1/2 =3/2.

8 7/8 is a mixed fraction. In order to add, subtract, multiply, or divide it with another fraction, it must be in improper form (no whole number). Every time you see a mixed fraction, 1) multiply the whole number by the denominator (here, that’s 8×8 = 64), and 2) add that to the numerator (here, that’s 64 + 7 = 71), then eliminate the whole number and write the sum you calculated in the numerator and keep the denominator, i.e., 71/8.

In the case of our example, 57/168 + 17/199, both fractions are improper, so we’re good to go!

1. Form the following:

[Numerator left]/[Denominator left] x [Denominator right]/[Denominator right]

+

[Numerator right]/[Denominator right] x [Denominator left]/[Denominator left]

You’ll have:

57/168 + 17/199 =

57/168 x 199/199 + 17/199 x 168/168

2. Multiply together 57/168 x 199/199 and 17/199 x 168/168:

To do each of these, we follow the rules of fraction multiplication:

a. The numerator on the left is multiplied by numerator on the right.

b. The denominator on the left is multiplied by the denominator on the right.

So, our fractions become (57×199)/(168×199) and (17×168)/(199×168). Now we just need to follow the rules of multiplication of two numbers together. The two fractions now become 11343/33432 and 2856/33432.

So, putting everything together, we’ll have the following:

57/168 + 17/199 = 57/168 x 199/199 + 17/199 x 168/168 = 11343/33432 + 2856/33432

a. Make sure the fractions have the same denominator (which is why we did the above steps).

b. Add the two numerators, and combine into one fraction with the common denominator.

So, here: 11343/33432 + 2856/33432 = (11343 + 2856)/33432 = 14199/33432.

We (almost) have our answer: 57/168 + 17/199 = 14199/33432.

4. Simplify:

One step remains, and this is where our trick with prime numbers from yesterday is going to come in handy.

We must simplify 14199/33432 into lowest terms, which means we must factor the numerator and factor the denominator and divide out any common terms.

To do this, we can treat 14199 and 33,432 separately and put them through the drill from yesterday.

If we do that with 14199, we find: 14199 = 3(4733).

For 33432, we find: 33432 = 23(3)(7)(199).

We can now write our original fraction, 14199/33432, in terms of its prime factorization:

3(4733) / 23(3)(7)(199)

Notice that the numerator and denominator both have 3 in common. So, we can cancel that out to get:

4733 / 23(7)(199)

You can leave it in this form, or, multiply the denominator together to get one number again: 4733/11144.

We can be certain this number is in simplest terms, since we broke the denominator and numerator down into prime factors. We won’t have missed anything!

The beautiful thing about this all? The exact same rule works with subtracting two fractions. All the way up to Step 4, everything is identical. At Step 4, subtract the right numerator from the left. That’s all there is to it.

Dividing Fractions

One last thing about fractions before we close for the day: Division turns into multiplication. You just flip the denominator and numerator of the fraction on the right of the division sign, and the division turns into multiplication: 57/168 ÷ 17/199 = 57/168 x 199/17.

To finish this, we just need to follow the rules of fraction multiplication outlined above:

57/168 x 199/17

= (57×199)/(168×17)

= 11343/2856

= 3(19)(199) / 23(3)(7)(17)

= 3781/952

Of course, all that remains now is lots of practice for you to get used to these steps. But follow them all and you’ll never struggle with fractions again.

Tomorrow, we’ll explore the next important topic: square roots.