The objective of this lesson is show you how to write ratios using some situations or examples you encounter daily .

Things are not always the same size. Thus, a natural need arise to compare quantities using division to see how much bigger a quantity is when compared to another.

For example, looking at the two piles below made of red apples and green apples, you may not be satisfied just to know that there are more red apples.

3 green apples and 9 red apples

A comparison of red apples to green apples using division may help you to see how much more red apples there are.

There are 9 red apples and 3 green apples. We can do a ratio of red apples to green apples as shown below:

9 / 3
= 3

The quotient or answer to the ratio above is equal to 3 and we can quite interpret the answer.

It means that there are 3 times more red apples than green apples

Other examples:

Say for instance, you are in a classroom. In the classroom, there are 3 boys and six girls.

The ratio of boys to girls is 
3 / 6

3 / 6
is equal to  
1 / 2
or 0.5

3 / 6
is equal to  
1 / 2
or 0.5

It means that there are half as many boys as girls in the classroom.

However, the ratio of girls to boys is
6 / 3

6 / 3
 is equal to 2 and it means that there are two times as many girls as boys in the classroom.

You can also do the following ratios:

Ratio of girls to number of students in the classroom:
6 / 9

Ratio of boys to number of students in the classroom:
3 / 9

Ratio of number of students in the classroom to girls:
9 / 6

Ratio of number of students in the classroom to boys:
9 / 3

At this point you may have noticed that the order is important when defining a ratio. The number that comes after ' of ' is your numerator and the number that comes after ' to ' is your denominator.

Some formal definitions

A ratio is a comparison of two numbers using division.

The ratio of a to b is  
a / b
    with b ≠ 0

A ratio is an ordered pair of numbers, written a:b, with b ≠ 0

As you can see there are more than one way to express a ratio. For example, if you have 6 pencils and 2 pens all the followings are good ways to express the ratio of pens to pencils


Ratio of pens to pencils

2 / 6

The word "per" also means a ratio

For example, gas mileage such as 50 miles per 4 gallons means  
50 / 4

Wage such as 25 dollars per hour means 
25 / 1

Continued ratio: The ratio of three or more quantities is called continued ratio

The ratio of 4 to 8 to 12 is the continued ratio 4:8:12

We get the continued ratio above by combining 3 ratios

4:8, 8:12, and 4:12

When doing ratios, make sure that quantities are in the same units first

Ratio of 24 inches to 6 feet

Since 1 foot = 12 inches, 6 feet = 6 × 12 inches = 72 inches

Now, you can do the ratio of 24 inches to 72 inches

24 / 72

It may be useful to simplify a ratio sometimes such as the one immediately above

Just divide both numerator and the denominator by the greatest common factor

24 ÷ 24 / 72 ÷ 24

We get:  
1 / 3

A little word problem: A classroom has 50 students and the ratio of males to females is 2 to 3. How many students are females?

2 to 3 is the same thing as 20 to 30

20 + 30 = 50

Therefore, there are 30 females in this class

Ratios quiz

Take the quiz below to see how well you understand the lesson on this page.

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