# Class 8 – Arithmetic

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## Online Tests

Topic Sub Topic Online Practice Test
Arithmetic
• Properties of ratio
• Properties of proportion
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Arithmetic
• Simple interest
• Compound interest
• Discount
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Arithmetic
• Time and work
• Time and distance
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Arithmetic
• Averages
• Alligations and mixtures
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## Study Material

 Introduction: Arithmetic is the dynamic unit in math because of its usage in daily life. All conversions such as money, length, mass etc. uses the concept of arithmetic. Ratio: Ratio is the comparison of one quantity to another of the same kind. The ratio between two quantities a and b is written as a : b. Here a and b are called the terms of the ratio. Also ‘a’ is called the antecedent and b is called the consequent. Example: Ratio of 4 kg to 7 kg is . We divide a by b, to find what part or multiple is a of b. Thus, a ratio a : b is indicated as the fraction . Properties of ratio: i)      The value of a ratio remains unaltered when both the terms are multiplied or divided by the same non-zero constant. i.e.,   where n is constant. ii)     Two ratios can be compared by expressing them with a common denominator. e.g: To compare  and , we write them with a common denominator as  respectively. Here (i) , (ii) , (iii) Example: Compare  and  by considering   iii)   The ratio of two fractions can be expressed as the ratio of two numbers. i.e., Ratios are compounded by multiplying the fractions which denote them. i.e.,  can be compounded as

If the ratio between any two quantities, of the same kind and having the same unit, can be expressed exactly by the ratio between two integers; the quantities are said to be commensurable otherwise incommensurable.

e.g. (i) The ratio between  and  is

; which is the ratio between two integers 2 and 3.

Therefore,  and  are commensurable quantities.

(ii) The ratio between  and 5 is ; which can never be expressed as the ratio between two integers; therefore  and 5 are incommensurable quantities.

iv)  For the ratio  is called its duplicate ratio. Here the ratio is compounded by itself.

v)   For the ratio  is called its sub duplicate ratio.

vi)  For the ratio , the ratio  is called its triplicate ratio. Here the ratio is compounded thrice.

vii) For the ratio , the ratio  is called its sub triplicate ratio.

viii) In a ratio, when the antecedent is greater than the consequent, then it is called a ratio of greater inequality. A ratio of greater inequality is decreased when the same quantity is added to both the terms, i.e.,  if  is a ratio and a > b then, , where k is constant.

ix)   In a ratio, a : b, if a < b, then it is called a ratio of less inequality. A ratio of lesser inequality is increased when the same quantity is added to both the terms.
i.e., in the ratio a : b, if a < b then , where is a constant.

x)     A ratio of greater inequality is increased and a ratio of lesser inequality is decreased, when the same quantity is subtracted from both the terms.
i.e., if a > b then  and if a< b then , where is a constant.

xi)  Reciprocal of a ratio a : b is  or b : a.

xii)  To divide a quantity into two parts in a given ratio we multiply the quantity by .

, it is wrong to take  and y =3.

Let age of Ram is 24 years and the age of his elder brother Murthy is 32 years.

The ratio between the ages of Ram and Murthy = 24 years : 32 years = 3 : 4. It does not mean that Ram’s age is 3 years and Murthy’s age is 4 years.

Example: To divide ` 48 in the ratio 3 : 5,

We multiply by

` ` 18

` ` 30

This can be extended to any number of equal fractions.

Corollary. If  =

Example: Find the ratio between 80 cm and 1 m 20 cm in the simplest form.

Solution: Both the quantities to be compared should be in same units.

So1 m 20 cm = 120

80cm: 1 m 20 cm = 80 : 120 = 2 : 3

Example: Comparethe ratio 2: 5, 4 : 11 and 7 : 19. Write them in descending order.

Solution: The ratios are  The denominators are 5, 11 and 19. So the ratios can be written as

Descending order is

Theorem:  then each of these ratios is equal to  where p,q, r, n are any constants.

Proof:

Then a = bk, c = dk and e = fk

Example: Given that (x + 2) : (3x + 4) is the duplicate ratio of 3 : 5, find x.

Solution:

Example: Given that , find

Solution:

Example:

Solution:

Example:

Solution: