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# Detailed Notes: Proportion - CSAT Preparation - UPSC

## Introduction

• When two ratios are equal, the four quantities composing them are said to be proportional
• Thus if a/b = c/d, then a, b, c, d are proportional. This is expressed by saying that a is to b as c is to d, and the proportion is written as a:b::c:d or a:b = c:d. • The terms a and d are called the extremes while the terms b and c are called the means.

Thus, we can say:

Product of Extremes = Product of Means

Example: What is the value of x in the following expression?
5/8 =  x/12

Solution:

⇒ 5/8 =  x/12

⇒ x = 60/8 = 7.5

It can be calculated with the help of percentages also. In this question, the percentage increase in the denominator is 50%, so the numerator will also increase by 50%.

## Operations in Proportions

If four quantities a, b, c and d form a proportion, many other proportions may be deduced by the properties of fractions. The results of these operations are very useful. These operations are:

• Invertendo: If a/b = c/d then b/a = d/c
• Alternando: If a/b = c/d, then a/c = b/d
• Componendo: If a/b=c/d, then (a+b)/b=(c+d)/d
• Dividendo: If a/b=c/d, then (a-b)/b=(c-d)/d
Componendo and Dividendo: If a/b=c/d then (a + b)/(a – b) = (c + d)/(c – d)

## Variation

Two quantities A and B are said to vary with each other if there is some relationship between A and B such that the change in A and B is uniform and guided by some rule.

### ➢ Some Typical Examples of Variation

• Area (A) of a circle = π.R2, where R is the radius of the circle. Area of a circle depends upon the value of the radius of a circle, or, in other words, we can say that the area of a circle varies as the square of the radius of a circle.
• At a constant temperature, the pressure is inversely proportional to the volume.
• If the speed of any vehicle is constant, then the distance traversed is proportional to the time taken to cover the distance.

Essentially there are two kinds of proportions that two variables can be related by: 1. Direct Proportion

When it is said that A varies directly as B, you should understand the following implications:

(a) Logical Implication: When A increases B increases.

(b) Calculation Implication: If A increases by 10%, B will also increase by 10.

(c) Graphical Implications: The following graph is representative of this situation. 2. Inverse Proportion

When A varies inversely as B, the following implication arises.

(a) Logical Implication: When A increases B decreases.

(b) Calculation Implication: If A decreases by 9.09%, B will increase by 10%.

(c) Graphical Implications: The following graph is representative of this situation. (d) Equation Implication: The product A X B is constant.

Example 1: The height of a tree varies as the square root of its age (between 5 and 17 years). When the age of a tree is 9 years, its height is 4 feet. What will be the height of the tree at the age of 16?

Solution: Let us assume the height of the tree is H and its age is A years.

So, H ∝ √A, or, H = K x √A Now, 4 = K x √9 K = 4/3

So, height at the age of 16 years = H = K x √A = 4/3 x 4 = 16/3 = 5 feet 4 inches.

Question for Detailed Notes: Proportion
Try yourself:Total expenses at a hostel is partly fixed and partly variable. When the number of students is 20, total expense is Rs 15,000 and when the number of students is 30, total expense is Rs 20,000. What will be the expense when the number of students is 40?

Question for Detailed Notes: Proportion
Try yourself:Rs. 5783 is divided among Sherry, Berry, and Cherry in such a way that if Rs.28, Rs.37 and Rs.18 be deducted from their respective shares, they have money in the ratio 4:6:9. Find Sherry’s share.

Note: For problems based on this chapter we are always confronted with ratios and proportions between different number of variables. For the above problem we had three variables which were in the ratio of 4 : 6 : 9. When we have such a situation we normally assume the values in the same proportion, using one unknown ‘x’ only (in this example we could take the three values as 4x, 6x and 9x, respectively).Then, the total value is represented by the addition of the three giving rise to a linear equation, which on a solution, will result in the answer to the value of the unknown ‘x’.

• However, the student should realise that this unknown ‘x’ is not needed to solve the problem most of the time.
This is illustrated through the following alternative approach to solving the above problem:
Assume the three values as 4, 6 and 9. Then we have:
(4 + 28) + (6 + 37) + (9 + 18) = 5783
⇒ 19 = 5783–83 = 5700
⇒ 1 = 300
Hence, 4 + 28 = 1228.
While adopting this approach, the student should be careful in being able to distinguish the numbers in bold as pointing out the unknown variable.

Question for Detailed Notes: Proportion
Try yourself:If 10 persons can clean 10 floors by 10 mops in 10 days, in how many days can 8 persons clean 8 floors by 8 mops?

Question for Detailed Notes: Proportion
Try yourself:Three quantities A, B, C are such that AB = KC, where K is a constant. When A is kept constant, B var- ies directly as C; when B is kept constant, A varies directly C and when C is kept constant, A varies inversely as B.Initially, A was at 5 and A:B:C was 1:3:5. Find the value of A when B equals 9 at constant C.

Question for Detailed Notes: Proportion
Try yourself:The ratio of water and milk in a 30-litre mixture is 7 : 3. Find the quantity of water to be added to the mixture in order to make this ratio 6 :1.

This document is the final document of EduRev’s notes of Ratios & proportions chapter. In the next document we have tried to solve the different CAT previous year questions & curated some of the best tests for you to practice - as a CAT aspirant you should be well aware of types of questions and should have a good practice of questions. Hence we suggests you to practice all the questions in the upcoming documents & Tests.
The document Detailed Notes: Proportion | CSAT Preparation - UPSC is a part of the UPSC Course CSAT Preparation.
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## FAQs on Detailed Notes: Proportion - CSAT Preparation - UPSC

 1. What are operations in proportions? Ans. Operations in proportions refer to mathematical calculations performed on proportional relationships between two or more quantities. These operations include addition, subtraction, multiplication, and division.
 2. How are proportions useful in real-life situations? Ans. Proportions are useful in real-life situations as they help us compare and solve problems involving ratios and percentages. For example, they can be used to calculate discounts, determine ingredient quantities in recipes, or analyze financial data.
 3. What is variation in proportions? Ans. Variation in proportions refers to the change or fluctuation in the relationship between two or more quantities. It can be classified as direct variation, where the quantities increase or decrease together, or inverse variation, where one quantity increases while the other decreases.
 4. How can we solve problems involving proportions? Ans. To solve problems involving proportions, we can use the cross-multiplication method. This involves multiplying the numerator of one ratio by the denominator of the other ratio and vice versa. By equating the two products, we can find the value of the unknown quantity.
 5. Can proportions be used in scaling or resizing objects? Ans. Yes, proportions can be used in scaling or resizing objects. By maintaining the same ratio between the measurements of different sides or dimensions, we can accurately scale up or down the size of an object. This is commonly used in architecture, graphic design, and engineering.

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