I guess we all can agree that ratios are used in both mathematics and in professional environments. If not think about the time when your class teacher asked how many of you would like to participate in a particular event, or when you share a packet of sweets fairly among your friends. There are endless examples that we can take from our everyday life where we need to work out ratios and thus, it’s important to learn how to work these ratios out. However, before jumping into it, let’s quickly recall what a ratio is?
What is the ratio?
When we compare two quantities, we deal with ratios. For example, when we try to calculate how many girls in the class have red hair, or how many people in a town have dogs, we deal with two sets of numbers.
One number is the total number in the set, and the other is the one that we are trying to compare, as the total number of girls in the class maybe 30 and number of girls who have red hair maybe just 3. When we say 3 out of 30 girls have red hair, we are starting to talk about ratios.
In other words, ratio compares between two numbers and tells how many times one number is greater than the other using division.
Let’s try to understand this through an example, If there is a group of 30 people, with 10 Men and 20 Women, then the ratio of men to women in that group will be 10:20, and the ratio for women to men will be 20:10.
Ratios behave like fractions and can be simplified into smaller numbers. Simplifying ratios make it easier to solve complex ratio problems and hence it’s an important concept to understand for learning how to workout ratios. In the above example, if we try to simplify the ratio of men to women it will be 1:2.
Different ways to represent ratios
For example, if there are two oranges and three apples, their ratio can be represented as 2 to 3, 2 : 3, 2 / 3, where 2 is antecedent and 3 is consequent.
“Part-to-Part” and “Part-to-Whole” Ratios
Ratios represent comparison; this comparison can be a part to part comparison or a part-to-whole comparison. In the above example, we compare the number of men to the number of women that is a part-to-part comparison. If we have compared the number of men to the total number of people, it would have been a part-to-whole comparison.
Now let’s solve a word problem to understand how to workout ratios in part-to-part and part-to-whole form.
Word Problem: In a certain room, there are 24 women and 18 men. What is the ratio of men to women? What is the ratio of women to the total number of people?
Solution: In a certain room, there are 24 women and 18 men.
Ration of men to women= 18 : 24 or 3 : 4
Total people in the room = 18 + 24 = 42 people
The ratio of women to total number of people is
24 : 42 or 4 : 7
The ratio between women to total number of people is 4 : 7
Many numerical problems on how to workout ratios use the concept of scaling and hence it is important to get our hold on what it is and how it works. In general, ratio helps us in scaling i.e., to increase or decrease different amounts.
For example in maps, miles of distance are covered by just a few centimetres or in models gigantic buildings are replicated just a few inches tall. This is done by multiplying both antecedent and consequent by the same number.
Word problem : A map of the city is made using the scale 1 cm = 18 m. If the park in the city is 90 m long, what is its length on the map?
Scale 1 cm = 18 m
Length of the park in the city = 90 m
Length of the map = Length of the city park/ 18 = 90/18 = 5 cm
Word Problem : A scale model of a tower is made using the scale 1 in = 4 ft. If the height of the scale model is 20 in, what is the height of the tower?
Scale 1 in = 4 ft.
Height of the scale model = 20 in
Height of the tower =
The height of the scale model x scale unit per inch = 20 x 4 = 80 ft
Height of the tower = 80 ft.
What are equivalent ratios and proportions?
When two ratios or their simplified version is equal to each other they are called proportional or equivalent ratios. For example, If a car travels 40 km in 30 minutes and a truck travels 80 km in 60 minutes then the ratio of their distance to time ratio is equal to each other.
Let’s solve two simple questions to understand the concept of how to workout ratios and proportions better.
Question 1 – Are the ratios 3:4 and 6:8 said to be in Proportion?
Answer – 3:4= 3/4 = 0.75 and 6:8 = 6/8= 0.75
Since both the values are equal, they are said to be in proportion.
Question 2 – Are the two ratios 9:10 and 5:10 in proportion?
Answer – 9:10= 9/10= 0.9 and 5:10= 5/10= 0.5
Since both the values are not equal, they are not in proportion.
How to work out equivalent ratios?
You must have got some idea for finding the equivalent ratio and how to workout ratios, but the value may not always be this easy to simplify. Some other ways which can help you find the equivalent ratio are multiplication and division, as equivalent ratios have a multiplicative relationship.
Let’s try to understand this through an example.
Let’s take two ratios in their fractional form 32/24 and 16/12
To find whether these two are equivalent or not, we have to deduce them using multiplication or division.
Let’s divide 32/24 by their common factor 2, which will give us 16/12, equal to our other ratio; therefore, this is an equivalent ratio.
(You can also multiply 16/12 with 2 to get the answer)
Now let’s take another number, 24/16, with 32/24 and check whether they are equivalent or not.
24/16 divide by 8 will give 3/2, and 32/24 will give 4/3
This is not equal, hence this is not an equivalent ratio.
How to find unknown quantities from existing equivalent ratios?
One of the most popular types of questions asked is to find unknown quantities from existing ratios. Honestly, there are many ways to solve these types of problems, however, the most common and easiest one is cross-multiplication. Let’s understand this through an example.
Example – Jayden can read 4 sci-fi books in 10 hours. Assuming all of the books are of the same length, how many hours would it take him to read 50 sci-fi books?
Solution – Jayden can read 4 sci-fi books in 10 hours
So, Let’s take Jayden can read 50 sci-fi books in m hours
Step 1 – Putting value into proportions – 4 books / 10 hours = 50 books / m hours
Step 2 – On cross-multiplying, we will get – 4m = 500 => m = 500/4 = 125 hours
Answer = 125 hours
The ratio of girls to boys in a class is 4 to 5. If there are a total of 27 students in a class, how many boys are there in the class?
Types of Ratios
There are various types of question-based on how to workout ratios, these questions may involve different types of ratios, so before jumping to our practice worksheet let’s have a quick look at different types of ratios and how to solve them.
1. Compounded ratio: Compound or mixed Ratio is a product of antecedents (numerator) of the ratios and consequents (denominator) ratios. Let’s say a : b and m : n are two ratios; their compound ratio will be given by
(a x m) : (b x n) = am : bn
hence am : bn is the compound ratio of a:b and m : n
Question- Find the compound ratio of 3 : 5 and 8 : 15?
Solution- Using, (a x m) : (b x n) = am : bn
(3 x 8) : (5 x 15)
24 : 75
Question- Find the compounded ratio of 4: 9, 6: 11 and 11 :2?
Solution- (4 x 6 x 11) : (9 x 11 x 2) = 264:198
On simplifying = 4:3
2. Duplicate ratio: When the ratio a:b is compounded (or multiplied) by itself, it is called a Duplicate ratio.
(a : b ) x ( a : b ) = a2 : b2
For example, 16 : 25 is a duplicate ratio of 4 : 5
Question- Find the duplicate ratio of 3 : 4
Solution- The duplicate ratio of 3 : 4 = 3^2 : 4^2 = 9 : 16
Question- Find the duplicate ratio of 5 : 2
Solution- The duplicate ratio of 5 : 2 = 5^2 : 2^2 = 25 : 4
3. Triplicate ratio: When the ratio a/b is compounded with itself 3 times, it is called the triplicate ratio.
(a : b) x (a : b) x (a : b) = a3 : b3
For example – 33 : 43 = 27 : 64
Question- Find the Triplicate ratio of 2 : 3
Solution- The Triplicate ratio of 2 : 3 = 2^3 : 3^3 = 8 : 27
Question- Find the Triplicate ratio of 5 : 2
Solution- The Triplicate ratio of 4 : 5 = 4^3 : 5^3 = 64 : 125
4. Subduplicate ratio: The subduplicate ratio of a:b is √a : √b, so the subduplicate of a2 : b2 is a : b
For example: subduplicate of 16 : 49 = √16 : √49 = 4 : 7
Question- Find the subduplicate ratio of 9 : 16
Solution- The subduplicate ratio of 9:16 = √9 : √16 = 3 : 4
Question- Find the subduplicate ratio of 36 : 49
Solution- The subduplicate ratio of 36 : 49 = √36 : √49 = 6 : 7
5. Subtriplicate ratio: The subtriplicate ratio of a : b is 3√a : 3√b, so the subtriplicate of a3 : b3 is a:b. For example, subtriplicate of 8 : 27 = 3√8 : 3√27 = 2 : 3.
Question – Find the subtriplicate ratio of 64 : 27
Solution- The subtriplicate ratio of 64 : 27 = 3√64 : 3√27 = 4 : 3
Question – Find the subtriplicate ratio of 125 : 343
Solution- The subtriplicate ratio of 125 : 343 = 3√125 : 3√343= 5 : 7
6. Reciprocal ratio: The reciprocal ratio of the a : b, where a,b ≠ 0 is 1/a : 1/b = b : a. For example reciprocal ratio of 5 : 8 = 1 / 5 : 1 / 8 = 8 : 5
7. Ratio of equalities: If antecedent and consequent are equal in the ratio then it is called the ratio of equalities. For example, if the ratio is 7 : 7 then it is called the ratio of equality.
8. Ratio of inequalities: If antecedent and consequent are not equal in the ratio then it is called the ratio of inequalities. For example, if the ratio is 7 : 13 then it is called the ratio of inequalities.
9. Ratio of lesser inequalities: If the antecedent of a ratio is lesser than it’s consequent, it is called the ratio of lesser inequalities. For example, 9: 15 is a ratio of lesser inequalities.
10. Ratio of greater inequalities: If the antecedent of a ratio is greater than it’s consequent, it is called the ratio of greater inequalities. For example, 17: 15 is a ratio of lesser inequalities.
Common mistakes to avoid while working out ratios
Sometimes even if we know the correct way of solving the problem, we tend to make some simple mistakes, and when it comes to ratio, the chances of making these common mistakes increase. To make sure you don’t repeat the same while solving the practice worksheet, we have listed some common mistakes that you don’t want to repeat while figuring out how to work out ratios.
1. Correctly read the ratios
It may come out surprisingly, but one of the common mistakes children do while working out ratios is to mess up the order. For example, if there are 18 men and 12 women in a room and you are asked to calculate the ratio of women to people you may be tempted to write 12 : 18 but that would be incorrect because the question asks the ratio of women to people. Therefore, you need to calculate the total number of people ( 18 + 12 = 30 ) so the correct ratio will be, 12 : 30.
2. Be careful with the units
The ratio exists between the same unit quantities only; if the quantities are different, they need to be converted or expressed in the same unit before using them in ratios.
3. Read the order carefully
Another common mistake is the order, if it is asked the ratio of women to men, you need to write the number of women first (Left-hand side) and then the number of men (Right-hand side) i.e., 12 : 18.
Ratio and proportion worksheet
Download the Ratio and Proportion worksheet