CA Foundation Exam  >  CA Foundation Notes  >  Quantitative Aptitude for CA Foundation  >  Chapter Notes: Proportion

Proportion Chapter Notes | Quantitative Aptitude for CA Foundation PDF Download

Overview

Proportion Chapter Notes | Quantitative Aptitude for CA Foundation

Proportions

If a man's income increases in a specified ratio and the amount of increase is known, we can determine his new income using proportion principles. Similarly, if two men's ages are in a given ratio and the age of one man is known, we can calculate the age of the other man through proportion. An equality of two ratios is defined as a proportion. Four quantities a, b, c, and d are in proportion if a : b = c : d (also expressed as a : b :: c : d), meaning a/b = c/d or ad = bc. The quantities a, b, c, and d are referred to as the terms of the proportion, with a, b, c, and d identified as the first, second, third, and fourth terms, respectively. The first and fourth terms are known as extremes (or extreme terms), while the second and third terms are called means (or middle terms).

If a : b = c : d then d is called fourth proportional.

If a : b = c : d are in proportion then a/b = c/d i.e. ad = bc

i.e. product of extremes = product of means.

This is called cross product rule.

Three quantities a, b, c of the same kind (in same units) are said to be in continuous proportion

if a : b = b : c i.e. a/b = b/c i.e. b= ac

If a, b, c are in continuous proportion, then the middle term b is called the mean proportional between a and c, a is the first proportional and c is the third proportional.

Thus, if b is mean proportional between a and c, then b2 = ac i.e. b = √ac.

When three or more numbers are so related that the ratio of the first to the second, the ratio of the second to the third, third to the fourth etc. are all equal, the numbers are said to be in continued proportion. We write it as
x/y = y/z = z/w = w/p = p/q = ................................................. when
x, y, z, w, p and q are in continued proportion. If a ratio is equal to the reciprocal of the other, then either of them is in inverse (or reciprocal) proportion of the other. For example 5/4 is in inverse proportion of 4/5 and vice-versa. 

Note:

In a ratio a : b, both elements must be of the same type, whereas in a proportion a : b = c : d, the four elements do not all need to be of the same type. The first two elements should be of one kind, and the last two should be of another kind.

Applications: 
Illustration I:
₹ 6 : ₹ 8 = 12 toffees : 16 toffees are in a proportion.
Here 1st two quantities are of same kind and last two are of same kind. 

Example 1: The numbers 2.4, 3.2, 1.5, 2 are in proportion because these numbers satisfy the property the product of extremes = product of means.
Here 2.4 × 2 = 4.8 and 3.2 × 1.5 = 4.8 

Example 2: Find the value of x if 10/3 : x : : 5/2 : 5/4. 
Sol: 
10/3 : x = 5/2 : 5/4 Using cross product rule, x × 5/2 = (10/3) × 5/4 Or, x = (10/3) × (5/4) × (2/5) = 5/3

Example 3: Find the fourth proportional to 2/3, 3/7, 4. 
Sol: 
If the fourth proportional be x, then 2/3, 3/7, 4, x are in proportion. Using cross product rule, (2/3) × x = (3 × 4)/7 or, x = (3 × 4 × 3)/(7 × 2) = 18/7. 

Example 4: Find the third proportion to 2.4 kg, 9.6 kg. 
Sol:
Let the third proportion to 2.4 kg, 9.6 kg be x kg.
Then 2.4 kg, 9.6 kg and x kg are in continued proportion since b2 = ac
So, 2.4/9.6 = 9.6/x or, x = (9.6 × 9.6)/2.4 = 38.4
Hence the third proportional is 38.4 kg.

Example 5: Find the mean proportion between 1.25 and 1.8.
Sol:
Mean proportion between 1.25 and 1.8 is
Proportion Chapter Notes | Quantitative Aptitude for CA Foundation

Properties of Proportion

  1. If a : b = c : d, then ad = bc
    Proportion Chapter Notes | Quantitative Aptitude for CA Foundation
  2. If a : b = c : d, then b : a = d : c (Invertendo)
    Proportion Chapter Notes | Quantitative Aptitude for CA Foundation
    Hence, b : a = d : c.
  3.  If a : b = c : d, then a : c = b : d (Alternendo)
    Proportion Chapter Notes | Quantitative Aptitude for CA Foundation
    Dividing both sides by cd, we get
    Proportion Chapter Notes | Quantitative Aptitude for CA Foundation
  4.  If a : b = c : d, then a + b : b = c + d : d (Componendo)
    Proportion Chapter Notes | Quantitative Aptitude for CA Foundation
  5. If a : b = c : d, then a – b : b = c – d : d (Dividendo)
    Proportion Chapter Notes | Quantitative Aptitude for CA Foundation

  6.  If a : b = c : d, then a + b : a – b = c + d : c – d (Componendo and Dividendo)
    Proportion Chapter Notes | Quantitative Aptitude for CA Foundation
    Dividing (1) by (2) we get
    Proportion Chapter Notes | Quantitative Aptitude for CA Foundation

  7. If a : b = c : d = e : f = ………………..….., then each of these ratios (Addendo) is equal (a + c + e + ……..) : (b + d + f + …….)
    Proportion Chapter Notes | Quantitative Aptitude for CA Foundation
    ∴ a = bk, c = dk, e = fk, .............
    Proportion Chapter Notes | Quantitative Aptitude for CA Foundation
    Hence, (a + c + e + ……..) : (b + d + f + …….) is equal to each ratio

  8. Subtrahendo: If a : b = c : d = e : f = ………………..….., then each of these ratios is equal
    (a – c – e – ……..) : (b – d – f – …….)

Example 1: If a : b = c : d = 2.5 : 1.5, what are the values of ad : bc and a + c : b + d?
Sol: 
We have
Proportion Chapter Notes | Quantitative Aptitude for CA Foundation
From (1) ad = bc, or, ad/bc = 1 , i.e. ad : bc = 1 : 1
Proportion Chapter Notes | Quantitative Aptitude for CA Foundation (By addendo property)
Proportion Chapter Notes | Quantitative Aptitude for CA Foundation
Hence, the values of ad : bc and a + c : b + d are 1 : 1 and 5 : 3 respectively.

Example 2: Proportion Chapter Notes | Quantitative Aptitude for CA Foundation
Sol:

Proportion Chapter Notes | Quantitative Aptitude for CA Foundation (By addendo property)
Proportion Chapter Notes | Quantitative Aptitude for CA Foundation

Example 3: A dealer mixes tea costing ₹ 6.92 per kg. with tea costing ₹ 7.77 per kg and sells the mixture at ₹ 8.80 per kg and earns a profit of Proportion Chapter Notes | Quantitative Aptitude for CA Foundation on his sale price. In what proportion does he mix them?
Sol:
Let us first find the cost price (C.P.) of the mixture. If S.P. is ₹ 100, profit is
Proportion Chapter Notes | Quantitative Aptitude for CA Foundation
If S.P. is ₹ 8.80, C.P. is (165 × 8.80)/(2 × 100) = ₹ 7.26
 C.P. of the mixture per kg = ₹ 7.26
2nd difference = Profit by selling 1 kg. of 2nd kind @ ₹ 7.26
 = ₹ 7.77 – ₹ 7.26 = 51 Paise
1st difference = ₹ 7.26 – ₹ 6.92 = 34 Paise
We have to mix the two kinds in such a ratio that the amount of profit in the first case must balance the amount of loss in the second case.
Hence, the required ratio = (2nd diff) : (1st diff.) = 51 : 34 = 3 : 2.

The document Proportion Chapter Notes | Quantitative Aptitude for CA Foundation is a part of the CA Foundation Course Quantitative Aptitude for CA Foundation.
All you need of CA Foundation at this link: CA Foundation
114 videos|164 docs|98 tests

Top Courses for CA Foundation

FAQs on Proportion Chapter Notes - Quantitative Aptitude for CA Foundation

1. What are the basic concepts of proportions in mathematics?
Ans. Proportions are mathematical expressions that show the equality of two ratios. A ratio compares two quantities, and a proportion states that two ratios are equal. For example, if a:b = c:d, then a, b, c, and d are in proportion. This concept is fundamental in solving problems related to similar figures, scaling, and various applications in real-life situations.
2. How can I solve problems involving proportions in the CA Foundation exam?
Ans. To solve proportion problems in the CA Foundation exam, first identify the given ratios and set up an equation based on the property of proportions (cross-multiplication). For example, if a:b = c:d, then a × d = b × c. Simplify the equation to find the unknown variable. Practice a variety of problems to become familiar with different formats and applications.
3. What are some common types of proportion problems I should prepare for in the CA Foundation exam?
Ans. Common types of proportion problems include direct and inverse proportions, word problems involving mixtures or recipes, and problems related to speed, distance, and time. It’s also important to practice problems that involve scaling figures or converting measurements, as these are frequently tested in the exam.
4. Are there any shortcuts or tips for solving proportion questions quickly?
Ans. Yes, some helpful tips include memorizing the cross-multiplication method for quick calculations, recognizing familiar ratios, and simplifying fractions whenever possible. Additionally, practicing with timed quizzes can improve your speed and accuracy in identifying and solving proportion problems.
5. How important is understanding proportions for the CA Foundation exam overall?
Ans. Understanding proportions is crucial for the CA Foundation exam as they form the basis for various topics in mathematics, including algebra, geometry, and statistics. Proportions also appear in financial calculations and analysis, making them essential for a career in accounting and finance. Mastery of this concept can significantly enhance your problem-solving skills in the exam.
114 videos|164 docs|98 tests
Download as PDF
Explore Courses for CA Foundation exam

Top Courses for CA Foundation

Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

past year papers

,

MCQs

,

Previous Year Questions with Solutions

,

practice quizzes

,

Extra Questions

,

video lectures

,

Free

,

Important questions

,

pdf

,

Proportion Chapter Notes | Quantitative Aptitude for CA Foundation

,

Proportion Chapter Notes | Quantitative Aptitude for CA Foundation

,

Exam

,

shortcuts and tricks

,

Summary

,

mock tests for examination

,

Sample Paper

,

Semester Notes

,

Proportion Chapter Notes | Quantitative Aptitude for CA Foundation

,

Viva Questions

,

ppt

,

study material

,

Objective type Questions

;