Ratio and Proportion: Definition, Formulas, Solved Examples

People must count and compare the quantity of various items in their daily lives in addition to understanding the relationship between them. For example, to determine how tall one person is in comparison to another, we must measure both people’s heights and compare the results. While simply measuring a person’s height provides us with an idea of how tall they are, ratios allow us to see how much taller one person is than the other.

Understanding Ratio and Proportion is useful for a variety of tasks, including deciphering food recipes, calculating relative distances on road journeys, making metal alloys, and combining multiple chemicals to make products, among others.

It is essential for students to learn important concepts like ratios and proportions which are used in daily life activities. Continue reading this article to learn everything there is to know about Ratios and Proportions, including basic vocabulary, properties, formulas, and examples.

Ratios and Proportions: Definition

Definition of Ratio: A ratio is a comparison of two or more numbers that indicates their quantities in relation to each other. A ratio of two different quantities is represented by using a dividend sign (:) between two numbers. For instance, if one were to represent the comparison between the number of boys and girls in a class, it can be done in the form of 30:45 as a ratio of boys to girls indicating the ratio of boys to girls in our hypothetical classroom is 2:3. While the number being divided (on the left ) is called an antecedent, the divisor (number on the right) is called a consequent.

Here are some common examples of expressing ratios in daily life:

• A car travelling at a speed of 60 Kilometres Per Hour (Km/pH).
• According to a number of total 80 cookies, there will be 2 cookies for every student.
• At a music concert, there were 3 times more children than adults.
• One out of 1000 people has a chance to win the lucky draw.

Definition of Proportion: Proportion is an equation that defines that the two given ratios are equivalent to each other. While ratios help us understand the relation between two different quantities, proportion helps understand the relationship between two ratios. Accordingly, when two ratios are equal in terms of value, they are said to be in proportion (or proportionate). When we have to denote proportions between two ratios, it can be done by using the double dividend (::) or equals (=) sign.

Here are some common examples of using proportion in daily life:

• Making 200 cupcakes for 100 people from a recipe for 10 cupcakes.
• Choosing one among two products based on discount deals on different quantities.
• According to the mileage and fuel capacity of a car, it can take a different quantity of fuel to top-up fuel for both cars across a trip.
• Calculating the price of a product being purchased from the USA and converting it for India (From USD to INR).

Ratios And Proportions: Properties

Properties of Ratios

A ratio remains the same if both antecedent and consequent are multiplied or divided by the same non-zero number,

• a/b = pa/pb = qa/qb , p, q ≠0
• a/b = (a/p)  /  (b/p) = (a/q)  / (b/q) , p, q ≠0

If two ratios a/b and c/d are equal

• a/b = c/d ⟹ b/a = d/c (Invertendo)
• a/b = c/d ⟹ a/c = b/d (Alternendo)
• a/b = c/d ⟹ (a+b)/b = (c+d)/d (Componendo)
• a/b = c/d ⟹ (a-b)/b = (c-d)/d (Dividendo)

Any two ratios in their fraction notation can be compared just as we compare real numbers like this:

• a/b = p/q ⟺ aq = bp
• a/b > p/q ⟺ aq > bp
• a/b < p/q ⟺ aq < bp

Properties of Proportions

If a:b = c:d is a proportion, then

• Product of extremes (numbers on extreme edges) = product of means i.e., ad = bc
• a, b, c, d,…. are in continued proportion means, a:b = b:c = c:d
• a:b = b:c then b is called mean proportional and b² = ac
• The third proportional of two numbers, a and b, is c, such that, a:b = b:c
• d is fourth proportional to numbers a, b, c if a:b = c:d

What is The Formula For Ratio And Proportion?

Formulas For Ratio

While comparing the two different quantities a and b, a dividend symbol can be used to denote the ratio.

a:b or a / b

Formulas For Proportion

To compare or equate two different ratios a:b and c:d, symbols double dividend (::) or equals can be used to represent equal proportion.

a:b::c:d OR a:b = c:d

How Do You Solve a Ratio Proportion And a Question?

Here are some of the common methods which can be used to solve ratios and proportions.

• If u/v = x/y, then uy = vx
• If u/v = x/y, then u/x = v/y
• If u/v = x/y, then v/u = y/x
• If u/v = x/y, then (u+v)/v = (x+y)/y
• If u/v = x/y, then (u-v)/v = (x-y)/y
• If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c ≠0, then a =b = c
• If u/v = x/y, then (u+v)/ (u-v) = (x+y)/(x-y) (also known as the componendo dividendo rule.

Ratios and Proportion: Solved Examples

Example 1: In a mixture of 60 litres, the ratio of milk and water 2:1. If this ratio is to be 1:2, then what amount of water should be added?

Quantity of milk: (60 x 2/3) liters = 40 liters

Quantity of water in it = (60 – 40) liters = 20 liters

If the new ratio is 1:2,

We have to let the quantity of water be x liters

Accordingly, if milk:water = (40 / 20 + x) = 1/2,

then 20 + x = 80

x = 60 liters.

Example 2: A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B’s share?

Solution: Let the shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively.

Then, 4x – 3x = 1000

x = 1000.

Hence, B’s share is 2x = Rs. (2 x 1000) = Rs. 2000

Example 3: The ratio of A’s salary to B’s was 4 : 5. A’s salary is increased by 10% and B’s by 20%, what is the ratio of their salaries now?

The current ratio is 4:5

If A’s salary is to be increased by 10% and B’s salary by 20%.

New ratio of salaries will be 4 × 1.1 :5 × 1.2 = 11 : 15.

Example 4: If x : y = 1 : 2, find the value of (2x + 3y) : (x + 4y)

Solution: x : y = 1 : 2 means x/y = 1/2

Now, (2x + 3y) : (x + 4y) = (2x + 3y)/(x + 4y)  [Dividing numerator and denominator by y]

= [(2x + 3y)/y]/[(x + 4y)/2] = [2(x/y) + 3]/[(x/y) + 4], put x/y = 1/2

We get = [2 (1/2) + 3)/(1/2 + 4)
= (1 + 3)/[(1 + 8)/2]
= 4/(9/2)
= 4/1 × 2/9
= 8/9
Accordingly, the value of (2x + 3y) : (x + 4y) = 8 : 9.

Related Concepts:

FAQs

Below are the frequently asked questions of Ratios and Proportion:

Q1: Where are proportions useful in daily life?
Ans: Proportions are useful in calculating material for recipes/formulas that include multiple elements in different quantities, calculating profit/expenses for business, calculating money required for fuel according to trip distance, and for many other types of computations.

Q2: How do you express ratios?
Ans: You can express ratios either as a/b or as a:b.

Q3: What is the concept of ratios and proportion?
Ans: The concept of ratio defines us to compare two quantities while proportion is an equation that shows that two ratios are equivalent.

Q4: What is a proportion in math?
Ans: A proportion is an equation that helps in identifying if two or more ratios are equal.

Q5: Where are ratios useful in daily life?
Ans: While Ratios are useful in changing values from one unit to another, expressing quantities in a mix (such as a number of boys/girls in a crowd or quantity of sugar in water), and expressing probability/chance.

We hope this article on Ratio and Proportion was helpful. Understanding ratios and proportions will help you in solving statistical problems and real-life challenges in a fraction of seconds. If you have any queries regarding this article, do let us know about it in the comment section below. We will get back to you at the earliest.

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