Maths for Diploma Studies

Ratios compare numbers proportionally. For example, if you had a mixed case of beer that had 4 IPAs and 8 lagers, they would be in a 4:8 ratio. We express ratios using the lowest number possible, so this ratio would actually be written as a 1:2 ratio. For every 1 bottle of IPA, you have 2 bottles of lager.

Fractions are made up of two elements: the denominator and the numerator, separated by a horizontal line like so:

The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. Let’s look at this in more detail to help understand this concept.

Imagine a slice of delicious chocolate cake. Before diving in with a fork in hand, let’s consider it in mathematical terms.

Figure: Fractions can be a piece of cake once you understand the basics

If we were to reassemble the entire cake, we would see that, as a whole, it was comprised of eight individual slices. So, using what we just learned about fractions, how would you represent this slice of cake as a fraction? To start, what is the value of the denominator?

It was said earlier that the denominator indicates how many parts make up a unit or a whole. Since the entire cake was comprised of eight equal slices, the denominator would be eight.

Figure: The entire cake representing the denominator

Moving on, what is the numerator that would represent our individual slice of cake? Remember, the numerator is the count or sum of how many of these parts we have before us. Knowing that the cake was originally cut into eight slices, and we only have one (sigh), the numerator would be the number one.

Figure: Our single slice as the fraction numerator

Putting this together, we have:

In this example, we are fortunate to have one-eighth of an amazing chocolate cake for dessert. What if, when everyone else went to bed, we snuck down to the kitchen and had two more slices for a delightful midnight snack? How would we represent how much of the cake we have consumed in fraction form?

In total, we have now eaten three slices out of the original whole cake that was cut into eight equal slices. This fraction would be three-eighths or three-parts out of eight:

After a fantastic chocolate-fuelled night’s sleep, you return to the kitchen and find that others in your household must have had a similar overnight snack, and now the entire cake is gone. With a bit of sadness, you turn to thinking about fractions to help improve your mood. In fraction form, how many slices of cake have been eaten in total?

Figure: The result of a tasty fractions lesson

Between you and your family, all eight out of the original eight slices were consumed. This new fraction would be represented by:

This gives us an opportunity to introduce a concept called simplifying fractions. In the example above, both the numerator and denominator are the same value. This fraction can be simplified – showing the fraction in its lowest terms – by dividing both numerator and denominator by eight:

If we do this, we are left with a fraction that looks like:

To continue familiarisation with this concept, let’s go through a few more examples.

How would you simplify the fraction?

We first think to ourselves, what value can be divided equally into both the numerator and denominator? In this instance, we have a seven in the numerator that divides nicely into the denominator. Since we know that:

and:

We get a simplified fraction of:

Let’s go through one more example of simplifying fractions. How would you simplify this one?

The same rules apply as we used before. We look for a common value to divide into both the numerator and the denominator. It should be fairly obvious that both 10 and 60 divide nicely by 10, so we have:

And:

This leaves us with the fraction:

You may be thinking to yourself that this fraction looks a bit odd, and you would be right. Typically fractions with a denominator of one are written simply as only the numerator:

All numbers can be written as a fraction with a one in the denominator. The reason we can make all numbers into fractions with 1 as the denominator is that anything divided by 1 is unchanged. Here are a few quick examples:

This is a very important concept and a tool we will use later in this lesson.