What do we know about the concept of ratios and proportions in general arithmetic? When we talk about the speed of a car and the distance it covers per km or maybe a recipe of a dish, we are talking about it in relation to ratios and proportions. But what actually are the concepts of ratio and proportions? What is the ratio and proportion formula? And why do we use the ratio and proportion formula? Why are they important to us? How do we use them in our day to day life? Let us elucidate all the clouds of questions one by one.
We always compare two or more things in our everyday life as per the need. This comparison between two or more quantities using the method of division is extremely efficient. So it will not be wrong if we say that ratio is actually, the comparison or simplification of two quantities of the same kind. This relation explains to us how many times is one quantity equal to the other quantity. In simple words, the ratio is the number that we use to express one quantity as a fraction of the other ones.
We can only compare two numbers in a ratio if they have the same unit and the sign that we use to denote a ratio is “:”. In a faction, it is writing using “/” and we also use “to” to represent a ratio.
Proportion validates if the two ratios are equivalent to each other. It judges the equality of two ratios. Now, for example, consider that two sets of numbers are given to you that are increasing or decreasing in the same ratio. So in proportion, we will say that the ratios are directly proportional to each other. Let us take another example, a train that is covering 100km per hour is actually equivalent to a train covering the distance of 500km for 5 hours because 100km/hr = 500km/5hrs = 100km/hr.
I think we are now very clear about the concepts of ratio and proportion. Ratios and proportions are actually the two faces of the same coin. If two ratios are equal then it is a proportion. Ratios and proportions are normally defined on the basis of fractions. When we define a fraction using “:” it becomes a ratio and when we represent two ratios using “::” it is a proportion. Both ratio and proportion are an important foundation that helps us to understand many crucial concepts in maths and science. The formula for ratio proportion is the foundation of many such concepts.
The ratio proportion formula is the key to solve any ratio and proportion problems. Using ratio proportion formula actually makes our work much easier and we save a lot of time. So, here are the ratio proportion formulas.
Ratio Formula
\[a : b \Rightarrow \frac{a}{b}\]
Proportion Formula
\[a : b : : c : d \Rightarrow \frac{a}{b} = \frac{c}{d}\]
Example 1) If person A and person B started a partnership business and decided to divide the profit between them in a ration of 2:4. If by the end of the financial year, the total profit would be RS 10,000, what will be their part of profit?
Solution 1) Their profit is to be divided into a ratio of 2:4
So we can find the profit of each one of them by:
A= 10,000 x 2/6 = 3333.33
B = 10,000 x 4/6 = 6666.67
Therefore their respective profit will be 3333.33 and 6666.67.
Example 2) Find a:b:c if the given ratios are as follows:
a:b = 2:3
b:c = 5:2
c:d = 1:4
Solution 2) If we multiply the first ratio by 5, the second ratio by 3 and the third ratio by 6, we will have:
a:b = 10:15
b:c = 15:6
c:d = 6:24
The ratios above have equal mean scores.
Therefore, a:b:c:d = 10:15:6:24
Example 3) In a handwriting competition, there are 5 boys and 3 girls. What will be the ratio between girls and boys?
Solution 3) The ratio between girls and boys will be 3 is to 5. We can also write it as 3/5 or 3:5.
Example 4) If sam in 2 hours covers a distance of 40 km. What distance will he cover in 8 hours?
Solution 4) Let the distance be x. With time the distance also increases,
therefore, 2:8 = 40:y
y = (40 x 8) / 2
= 160 km.
Sam can cover a distance of 160 in 8 hours.
Example 5) Find the numbers whose sum is 60 and they are in the ratio of 2:3.
Solution 5) Let the numbers be 2x and 3x respectively. According to the question, the sum of these two numbers is 60.
So 2x + 3x = 60
5x = 60
x = 12
Therefore, the two numbers are
2x = 2 x 12 = 24
3x = 3 x 12 = 36
24 and 36 are the two numbers.
Practise more and more solved examples for ratio and proportion formulas for better understanding of the topic. This will also help you to learn the formula for ratio proportion quickly.
Question 1) State the Important Properties of Proportion.
Answer 1) The important properties of proportion are:
Addendo: if a:b = c:d then a + c = b + d
Subtrendo: if a:b = c:d then a - c = b - d
Dividendo: if a:b = c:d then a - b:b = c - d:d
Componendo: if a:b = c:d then a + b:b = c + d:d
Alternendo: if a:b = c:d then a:c = b:d
Invertendo: if a:b = c:d then b:a = d:c
Componendo: if a:b = c:d then a + b:a - b = c + d:c - d
Question 2) What are the Differences Between Ratio and Proportion?
Answer 2) The most basic differences between ratio and proportion are:
Ratio | Proportion |
We use ratios to compare two things that share the same unit. | We use proportion because it allows us to check the equality of two ratios. |
We use a (:) or a (/) to represent a ratio. | We use a (::) double-colon or a (=) equal to sign to represent a proportion. |
A ratio is an expression. | A proportion is an equation. |
It offers to describe the quantitative relationship between two things. | It offers to describe the quantitative relationship between two things with a total. |
The keyword to identify a ration in a problem is “to every”. | The keyword to identify a proportion in a problem is “out of”. |