Ratios and proportions are the topics that I used to love in school because they were the easiest topics of mathematics as compared to others (algebra, geometry, etc.). However, the deceptive nature of questions in exams made it hard for me and most of my classmates to figure out the solution. It was unsatisfying for me as a student – thought to be mastering a topic and then failing to
Today, I am going to explain both of these topics in detail and I will guide you on how to solve ratios and proportions by making the problems simple and straightforward.
By reading this post, you will get to learn how to find an unknown number in a proportion and a few examples for practical illustration. Let’s first understand both of the concepts by grasping their definitions.
What is a Ratio?
When we talk about the distance traveled by plane or a train in a specific time, we use kilometers per hour to calculate it. We express the measurement of distance over time using the concept of ratio. A ratio is a way to quantify two numbers by dividing them, such as meters per second, and compares meters and seconds.
A ratio can be expressed in three forms.
- a / b
- a : b
- a to b
All of the above ratios are read as "the ratio of a to b."
What is a proportion?
A proportion is a mathematical expression that states that two ratios are equal. For example, if two chocolate bars take 6 grams of cocoa powder to bake, it is the same as saying four chocolate bars take 12 grams of cocoa powder.
2/6 = 4/12
A proportion can be read as "a is to b as c is to d."
a / b = c / d
a : b = c : d
If one of the numbers in a proportion is not known, you can find it by using an online proportion calculator or by solving the proportion.
How to solve ratios and proportions? – The Simple Way
I will use simple but real-life examples to demonstrate the idea of finding proportional quantities. You can also use the proportion calculator mentioned above to verify the result when you are doing it on your own. Let’s go through my favorite cheese pizza example first.
If 50 grams of cheese is required to make 2 pizzas, how much cheese will be required to make 10 pizzas?
Here we have two ratios. One ratio is complete and the other one is missing one value. We can find that value using the idea of proportion.
- Write down both ratios.
Cheese / Pizzas = Cheese / Pizzas
- Pace the values in the equation and use a variable to represent the missing value. In this case, I will use “a.” We can solve this like every other equation if we put the unknown number in the numerator
50 : 2 = a : 10
We can write the above equation as:
50 / 2 = a / 10
- Multiply both sides of the equation by 10.
10 × 50 / 2 = 10 × a / 10
250 = a or
a = 250
We have found the missing value in the proportion. And we know that we will require 250 grams of cheese to make 10 pizzas if 50 grams is enough for 2 pizzas.
Solving Proportion using Cross Product
If the unknown number is in the denominator, another approach named cross product could be used to solve the proportion. We usually place missing value in the denominator in case of an indirect proportion.
An indirect proportion is a proportion in which a value increases if another value decreases or a value decreases if another value increases. On the other hand, direct proportion involves an increase in a value if another value increases or a decrease in a value if another value decreases.
In proportional problems, we can use cross-products to determine if two ratios are identical and form a proportion.
- To calculate the cross products of a proportion, multiply the outer values (extremes) and the middle terms (means).
- In the case of fractions, multiply the numerator of the first fraction with the denominator of the second fraction and the numerator of the second fraction with the denominator of the first fraction.
A solving proportion calculator can come in handy when you are dealing with proportions in fractional form.
In a toy manufacturing factory, 5 workers make 100 toys in a day. The manager of the factory wants to increase the production. He wants to know how many toys 10 workers will make in a day?
- First, we will make the proportion by using the ratios in form of fractions.
Workers / Toys = Workers / Toys
- Place the values in the above equation. Use a variable to represent the unknown value i.e., number of toys.
5 / 100 = 10 / a
- Apply the cross-multiplication as explained before.
5 × a = 10 × 100
5a = 1000
a = 1000/5
a = 200
So, 10 workers will make 200 toys in a day.
25 liters of paint is needed to paint 10 walls in a building complex. How many walls will be painted if there is 60 liters of paint available?
- Make the proportion by using the ratios as we did in the previous example.
Paint : Walls = Paint : Walls
- Substitute the values in the above equation. Place a variable for the unknown value.
25 : 10 = 60 : a
- Apply the cross multiplication to the proportion.
25 × a = 10 × 60
25a = 600
a = 600/25
a = 24
It means that with 60 liters of paint, we can paint 24 walls.
I hope this guide will clear out all of the ambiguities about ratios and proportions. You can practice your workbook questions by using these methods. Practicing will give you more confidence when you are sitting in an exam and staring at the tricky questions in the exam paper. It would be very fruitful for your studies if you work out some of the problems by yourself.