Math Lesson 4.1.1 - Definition of a Ratio

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Welcome to our Math lesson on Definition of a Ratio, this is the first lesson of our suite of math lessons covering the topic of Ratios, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Definition of a Ratio

There are many situations in practice when we need to compare two quantities. Basically, we use two methods for comparison of two quantities:

By subtraction; for example, when we calculate how much taller a child is compared to one year ago, so that we can calculate the increase in height of that child; and
By division; for example, when we calculate how much more fruit a large box has compared to a small box, so that we can calculate the number of boxes.

Often, it is not enough to simply find which quantity is greater but a quantitative comparison is needed (as seen in the example above), in order to know how much a quantity is bigger than another. In these scenarios we use the concept of a ratio - a method used to compare two quantities of the same type through division.

By definition, a ratio is a comparison of two or more numbers by means of division.

Thus, if we consider the second assumption about the wealth of the two brothers discussed in the introduction, it is obvious that they have almost the same wealth as 10,050,000 / 10,000,000 = 1.005. This means that your friend's brother is only 0.5 percent richer than your friend, so they can be considered as equally rich. This conclusion would be impossible to draw if you simply considered the difference between their wealth.

We know that rational numbers are all those numbers that can be expressed as fractions, it is evident that the term "rational number" derives from "ratio".

In other words, ratios are numbers of the "same type of quantity" written as a fraction or division. This means ratios compare by divisions only quantities of the same type, such as in the example discussed earlier where we compared through division two amounts of money. Hence, ratios have no unit as all units are simplified.

Another form of representing ratios is through the colon symbol (:). This is because the colon was the old symbol used for division. The colon is pariculalry suitable when representing ratios of more than two quantities.

Example 1

Two objects have the lengths: 40 cm and 60 cm respectively. What is the ratio of their lengths expressed in the simplest form?

Solution 1

We have to write the ratio as a fraction (or division) in the simplest terms, that is in the smallest numbers possible. Thus, since GCF of 40 and 60 is 20, we have

Ratio = 40 cm/60 cm
= 40 cm ÷ 20/60 cm ÷ 20
= 2/3 (or 2:3)

We can use the same approach as when calculating the part of a whole if we know the total and the ratio, and when the amounts represented by the ratio are required. In such situations, it is better to operate in terms of the greatest common factor, GCF. Thus, in the previous example, we could have expressed the greatest common factor (20) by k and the ratio therefore is expressed as

Ratio = 40 cm/60 cm
= 2 × 20 cm/3 × 20 cm
= 2k/3k
= 2/3

Let's see the utility of the greatest common factor k through another example.

Example 2

The female-male ratio in a company is 5:4 and the company has 63 employees in total. What is the number of employees from each gender?

Solution 2

First, we identify the GCF of the number of employees from each gender. We denote this GCF by k. Thus, from the ratio given in the clues, it is obvious that the number of female employees is 5k and that of males is 4k. Hence, the total number of employee is 5k + 4k = 9k, which corresponds to 63 employees. Hence,

9k = 63
k = 63/9 = 7

Now, we can find the exact number of each category by multiplying the corresponding part of ratio (share) by k. Hence,

Number of female employees = 5 × k = 5 × 7 = 35

and

Number of male employees = 4 × k = 4 × 7 = 28

The same procedure is also used when dealing with ratios containing more than two quantities. Let's see another example to clarify this point.

Example 3

Two knots are formed in a 24 m long rope producing three segments or rope a, b and c as shown in the figure, which have the ratio 3:4:5 to each other. Calculate the length of each segment.

$Math Tutorials: Ratios Example$

Solution 3

Given the ratio in the clues, we denote the pieces by 3k, 4k and 5k as usual. Thus, we have

a = 3k
b = 4k
c = 5k

In addition, since

a + b + c = 24 m

we can write the above clue in terms of k. Thus,

3k + 4k + 5k = 24

This allows us to find the GCF of the three pieces' lengths, here represented by k. We have

12k = 240 m
k = 240/12 = 20 m

Therefore, we obtain for the length of each piece:

a = 3k = 3 × 20 m = 60 m
b = 4k = 4 × 20 m = 80 m
c = 5k = 5 × 20 m = 100 m

The proof gives 60 m + 80 m + 100 m = 240 m, as the clues suggest.

More Ratios Lessons and Learning Resources

Ratio and Proportion Learning Material
Tutorial ID	Math Tutorial Title	Revision Notes	Revision Questions
4.1	Ratios
Lesson ID	Math Lesson Title	Lesson	Video Lesson
4.1.1	Definition of a Ratio
4.1.2	Ratio in a Number Line
4.1.3	Finding the Fraction of One Quantity Out of the Total
4.1.4	Part : Whole vs Part : Part Ratios
4.1.5	Finding the New Ratio When One Quantity Changes
4.1.6	Dividing a Given Ratio

Whats next?

Enjoy the "Definition of a Ratio" math lesson? People who liked the "Ratios lesson found the following resources useful:

Definition Feedback. Helps other - Leave a rating for this definition (see below)
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Ratio and Proportion Revision Notes: Ratios. Print the notes so you can revise the key points covered in the math tutorial for Ratios
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Continuing learning ratio and proportion - read our next math tutorial: Rates. Applications of Ratios and Rates in Practice

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