Ratio and Proportion, Indices and Logarithm

Paper 4: Quantitative Aptitude Chapter 1 Part I: Ratio & Proportion Ms. Ritu Gupta, MA (Maths.)

Ratio

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Learning Objectives

How to compute

and compare two ratios

Effect of increase or decrease of a quantity

on the ratio

The concept and

application of different

kinds of ratio

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Ratio

A ratio is a comparison of the sizes of two or more quantities of the same kind (in same units) by methods of division.

If a and b are two quantities of the same kind then the fraction a/b is called the ratio of a to b.

It is written as a : b or a/b

The quantities a and b are called the terms of the ratio, a is called the first term or antecedent and b is called the second term or consequent.

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Points to Remember

1 • Both terms of a ratio can be multiplied or divided by the same (non –

zero) number. Usually a ratio is expressed in the lowest form or the simplest form. Example:- 10 : 15 = 10/15 = (5×2)/ (5×3) = 2/3 = 2:3

2 • Ratio exists only between quantities of the same kind. For Example:

There is no ratio between the height of a child and the salary of a teacher.

3 • The order of the terms in a ratio is important. For example - 4:5 ≠ 5:4

4 • If a quantity increases or decreases in the ratio a : b, then new quantity =

b of the original quantity / a

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Points to Remember - 2

Raju’s weight is 48.8 kg. If he reduces his weight in the ratio of 8:7, find his new weight.

Solution: Original weight of Raju = 48.8 kg

He reduces his weight in the ratio 8:7

His new weight = (7 × 48.8) / 8

= 42.7 kg

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5 • The fraction by which the original quantity is multiplied to

get a new quantity is called the factor multiplying ratio.

Points to Remember - 3

Example Ratio between 1 hour and 20 minutes = Ratio between (1x60) min. and 20 min. = 60 / 20 = 3/1 = 3:1

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6 • Quantities to be compared (by

division) must be in the same units.

Points to Remember - 4

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7 • To compare two ratios, convert them into equivalent like

fractions.

Different Kinds of Ratio- Inverse Ratio

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Different Kinds of Ratio- Ratio of Equality • A ratio a : b is said to be of greater equality if a > b, of less

equality if a<b and of equality if a = b. Example

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7 : 4 is a ratio of greater equality

5 : 9 is a ratio of less equality

5 : 5 is ratio of equality

Different Kinds of Ratio- Compounded Ratio

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Different Kinds of Ratio- Duplicate Ratio • A ratio compounded to itself is called its duplicate ratio.

Thus a² : b² is the duplicate ratio of a : b. Example Duplicate ratio of 5 : 7 is 52 : 72 = 25 : 49

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Different Kinds of Ratio- Triplicate Ratio • The compounded ratio of a ratio with its duplicate ratio is

called its triplicate ratio. Thus a³ : b³ is the triplicate ratio of a : b

Example Triplicate ratio of 2 : 3 is 23 : 33 = 8 : 27

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Different Kinds of Ratio- Sub – Duplicate Ratio

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Different Kinds of Ratio- Sub – Triplicate ratio

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Different Kinds of Ratio - Continued Ratio

• Continued Ratio is the relation (comparison) between the magnitudes of three or more quantities of the same kind. The continued ratio of three similar quantities a, b, c is written as a : b : c

Example The continued ratio of 200, 400 and 600 is 200 : 400 : 600 = 1 : 2 : 3

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Different Kinds of Ratio - Commensurable Ratio

• If the ratio of two similar quantities can be expressed as a ratio of two integers then the quantities are called commensurable e.g. 3:4

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Different Kinds of Ratio - Incommensurable Ratio

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Illustration 1

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Illustration 2 The ratio compounded of duplicate ratio of 4:5, triplicate ratio of 1:3. sub duplicate ratio of 81:256 and sub triplicate ratio of 125:512 is (a) 4 : 512 (b) 3 : 32 (c) 1 : 120 (d) None of these Solution The duplicate of ratio of 4 : 5 is 42 : 52 = 16 : 25 The triplicate ratio of 1 : 3 is 13 : 33 = 1 : 27

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Illustration 2- Continued

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Illustration 3

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Illustration- 4

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Illustration- 5

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Illustration- 5- Continued

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Illustration- 6

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Illustration- 6- Continued

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Illustration- 7

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Illustration- 8 A bag contains Rs. 187 in the form of 1 Rupee, 50 paise and 10 paise coins in the ratio of 3 : 4 : 5. Find the number of each type of coins. (a) 102, 136, 170 (b) 136, 102, 170 (c) 170, 102, 136 (d) None of these Solution Let the number of 1 Rupee, 50 paise and 10 paise coins be 3x, 4x and 5x respectively. Then,

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Illustration- 8- Continued

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Illustration- 9

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Illustration- 10

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Illustration - 11 Find in what ratio will the wages of the employees in a workshop be increased or decreased if there is a reduction in the number of employees in the ratio 7 : 4 and an increment in their wages in the ratio 16 : 21. (a) 2 : 7 (b) 4 : 3 (c) 4 : 1 (d) 7 : 3 Solution Let the original number of employees be x.

Therefore the number of employees after reduction will be 4x/7.

Let the (average) wages per worker be y

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Illustration – 11- Continued

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Illustration – 11- Continued

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Illustration - 12 The ratio of the number of boys to the number of girls in a dance school of 360 students is 3 : 5. If 15 new girls are admitted to the dance school, find how many new boys should be admitted so that the ratio of the number of boys to the number of girls becomes 4 : 5. (a) 75 (b) 57 (c) 55 (d) 45 Solution Let the number of boys and number of girls be 3x and 5x Therefore 3x+5x = 360

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Illustration – 12- Continued

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Illustration – 12- Continued

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Thank You Please see next part for e-Lecture on Proportion