WBBSE Solutions For Class 8 Maths Arithmetic Chapter 3 Rational Number

Arithmetic Chapter 3 Rational Number

Natural Numbers: 1, 2, 3, 4, 5……., 125, ……. are counting numbers or natural numbers such that 1 is the first natural number and there is no last natural number.

⇒ The natural number is denoted by N and is written as N = (1, 2, 3, 4, . . . . . . . ., 125,…..)

Whole Numbers: The numbers 0, 1, 2, 3,….., and 125, are called whole numbers.

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⇒ The whole number is denoted by W and is written as W = (0, 1, 2, 3,…… 125, …)

Integers: The numbers …. -4, -3, -2, -1; 0, 1, 2, 3 ….. are called Integers.

⇒ The Integers is denoted by Z and is written as Z = (..,-3, -2, -1, 0, 1, 2, 3…..)

⇒ The integers greater than 0, i.e. 1, 2, 3 ….. are called positive Integers and the Integers less than 0, i,.e. -1, -2, -3,…. are called negative Integers.

⇒ 0 (zero) is an Integer that is neither positive nor negative.

Differece Between Rational And Irrational Numbers 

Rational Numbers: The numbers which can be expressed in the form of \(\frac{p}{q}\) where p and q are integers and q ≠ 0 are called Rational Numbers.

Example: 6, \(\frac{3}{9}\), 0, \(\frac{5}{6}\) etc. [All integers are Rational Numbers]

Irrational Numbers: The numbers which cannot be expressed in the form of \(\frac{p}{q}\) where p and q are integers and q ≠ 0 are called Irrational Numbers.

Example: √3, π etc.

Real Number:

1. Rational Number
2. Irrational Number

Some important points:

1. If two rational numbers x and y such that x < y there is a rational number. \(\frac{x+y}{2}\) i.e. x < \(\frac{x+y}{2}\) < y

2. If x and y are two rational numbers and x < y then n rational numbers between x and y are (x + d), (x + 2d), (x + 3d),…….. (x + nd), where d = \(\frac{y-x}{n+1}\)

3. If the rational numbers of the far \(\frac{p}{q}\) be expressed into decimals, it will be terminating into decimals it will be terminating decimal number, where q has prime factors 2 and 5 only.

4. If the rational numbers of the form be expressed into recurring decimals where has prime factors other than 2 and 5.

5. If rational numbers are expanded into decimals it will be terminating or recurring and the fraction whose decimal form is terminating or recurring will be rational.

6.

1. Sum of rational numbers are rational.
2. Difference of rational numbers are rational.
3. Product of rational numbers are rational.
4. Quotient of two rational numbers (not divided by 0) are rational.

7. If a, b, c are rational numbers.

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Associative Law:

⇒ a + (b + c) = (a + b) + c [Associative law of addition]

⇒ But a (b – c) (a – b) – c [Associative law does not exists for subtraction]

⇒ and a x (b x c) = (a x b) x c [Associative law for product]

Commutative law:

⇒ a + b = b + a (for addition)

⇒ a – b + b – a (not exist for subtraction)

⇒ a x b = b x a (for product)

⇒ \(\frac{a}{b}\) \(\frac{b}{a}\) (not exist for division)

Distributive law:

⇒ a x (b+c) = a x b + a x c

Arithmetic Chapter 3 Rational Number Examples

Example 1. Find the value of (2x + 5) when x = –\(\frac{3}{8}\)

Solution:

Given f(x) = 2x + 5 and x = –\(\frac{3}{8}\)

⇒  f(-\(\frac{3}{8}\)) = 2 × (-\(\frac{3}{8}\))+5

⇒  f(-\(\frac{3}{8}\)) = – (2 × (\(\frac{3}{8}\)) +5

⇒  f(-\(\frac{3}{8}\)) = \(\frac{3}{4}+5\)

⇒  f(-\(\frac{3}{8}\)) = \(\frac{-3+20}{4}\)

⇒  f(-\(\frac{3}{8}\)) = \(\frac{17}{4}\)

⇒  f(-\(\frac{3}{8}\)) = 4 \(\frac{1}{4}\)

∴ The Value of f(x) = 2x + 5 = 4 \(\frac{1}{4}\)

Example 2. Solve the following equations and express the roots in form (where q ≠ 0 and p, q are two integers)

1. 3x – 7 = 0
2. y = 15 + 10y

Solution:

Given That :

f(x) = 3x – 7 = 0

⇒ 3x = 7

⇒ x = \(\frac{7}{3}\)

∴ x = \(\frac{7}{3}\)

∴ The root of the equations is ( X – \(\frac{7}{3}\)) =0

f(y) ⇒ y = 15 + 10y

y = 15 + 10y

⇒ y – 10y = 15

⇒ – 9y = 15

⇒ y = –\(\frac{15}{9}\)

⇒ y = \(\frac{-5}{3}\)

∴ y = \(\frac{-5}{3}\)

∴ The root of the equations is ( y + \(\frac{5}{3}\) ) = 0

Example 3. Write the approximate number in the following boxes:

1. x \(\frac{-5}{3}\) = 1
2. (-\(\frac{6}{11}\)) + (\(\frac{7}{12}\)) = 

Solution :

1. x –\(\frac{5}{3}\) = 1 ⇒  = -1 x \(\frac{13}{5}\) = –\(\frac{13}{5}\)
2. (-\(\frac{6}{11}\)) + (\(\frac{7}{12}\)) = \(\frac{-72+77}{132}=\frac{5}{132}\)

Example 4. Write the product by multiplying \(\frac{4}{25}\) with the reciprocal of (-\(\frac{2}{15}\)).

Solution: The reciprocal of (-\(\frac{2}{15}\)) is (-\(\frac{15}{2}\))

\(\frac{4}{25} \times\left(-\frac{15}{2}\right)\)

 

= \(-\frac{6}{5}\)

Example 5. Find the value of the following with the help of Commutative law and Associative law.

1. \(\frac{7}{9} \times\left(-\frac{11}{25}\right) \times\left(-\frac{87}{42}\right) \times\left(\frac{5}{121}\right)\)
2. \(\frac{3}{4}+\left(-\frac{7}{10}\right)+\frac{5}{6}+\left(-\frac{12}{25}\right)\)

Solution:

1. \(\frac{7}{9} \times\left(-\frac{11}{25}\right) \times\left(-\frac{87}{42}\right) \times \frac{5}{121}\)

= \(\frac{7}{9} \times\left\{\left(-\frac{11}{25}\right) \times\left(-\frac{87}{42}\right)\right\} \times \frac{5}{121}\)

= \(\frac{7}{9} \times\left\{\left(-\frac{87}{42}\right) \times\left(-\frac{11}{25}\right)\right\} \times \frac{5}{121}\)

[I get with the help of Commutative Law and Associative Law]

2. \(\frac{3}{4}+\left(-\frac{7}{10}\right)+\frac{5}{6}+\left(-\frac{12}{25}\right)\)

= \(\frac{3}{4}+\left\{\left(-\frac{7}{10}\right)+\frac{5}{6}\right\}+\left(-\frac{12}{25}\right)=\frac{3}{4}+\left\{\frac{5}{6}+\left(-\frac{7}{10}\right)\right\}+\left(-\frac{12}{25}\right)\)

= \(\left(\frac{3}{4}+\frac{5}{6}\right)+\left\{\left(-\frac{7}{10}\right)+\left(-\frac{12}{25}\right)\right\}=\left(\frac{9+10}{12}\right)+\left\{-\left(\frac{35+24}{50}\right)\right\}\)

= \(\left(\frac{19}{12}\right)+\left(-\frac{59}{50}\right)\) [I get with the help of Commutative Law and Associative Law]

= \(\frac{475-354}{300}=\frac{121}{300}\)

Example 6. Write three rational numbers between (-5) and (-4).

Solution: 3 rational numbers are -4.1 or, –\(\frac{41}{10}\), -4.3 or, –\(\frac{43}{10}\), -4.5 or –\(\frac{45}{10}\) as –\(\frac{9}{2}\)

Example 7. Write 10 rational numbers between –\(\frac{3}{5}\) and \(\frac{1}{2}\)

Solution: –\(\frac{3}{5}\) = –\(\frac{6}{10}\) = –\(\frac{12}{20}\), \(\frac{1}{2}\) = \(\frac{5}{10}\) = –\(\frac{10}{20}\)

∴ 10 rational numbers are: –\(\frac{9}{20}\), –\(\frac{7}{20}\), –\(\frac{5}{20}\), –\(\frac{3}{20}\), –\(\frac{1}{20}\), \(\frac{1}{20}\), \(\frac{3}{20}\), \(\frac{5}{20}\), \(\frac{7}{20}\), \(\frac{9}{20}\)

Example 8. Plot the number \(\frac{1}{4}\) on number line.

Solution:

 

Example 9. Plot -2\(\frac{3}{5}\) on number line.

Solution :

 

Example 10. Write 5 rational number lying between \(\frac{3}{5}\) and \(\frac{4}{5}\).

Solution: Here x = \(\frac{3}{5}\), y = \(\frac{4}{5}\), x = 5

d = \(d=\frac{y-x}{x+1}=\frac{\frac{4}{5}-\frac{3}{5}}{5+1}=\frac{\frac{1}{5}}{6}=\frac{1}{30}\)

∴ 5 rational numbers are: \(\left(\frac{3}{5}+\frac{1}{30}\right),\left(\frac{3}{5}+\frac{2}{30}\right),\left(\frac{3}{5}+\frac{3}{30}\right),\left(\frac{3}{5}+\frac{4}{30}\right),\left(\frac{3}{5}+\frac{5}{30}\right)\)

i.e. \(\frac{19}{30}, \frac{20}{30}, \frac{21}{30}, \frac{22}{30}, \frac{23}{30}\)

Example 11. Write 6 rational numbers lying between 5 and 6.

Solution: Write the equivalent rational numbers of 5 and 6 which have (6 + 1) or 7 as denominator.

5 = \(\frac{35}{7}\), 6 = \(\frac{42}{7}\)

∴ 6 rational numbers are: \(\frac{36}{7}, \frac{37}{7}, \frac{38}{7}, \frac{39}{7}, \frac{40}{7}, \frac{41}{7}\)

Example 12. Choose the correct answer

1. √2 is a

1. Rational number
2. Irrational number
3. Natural number
4. Whole number

Solution: √2 = 1.414…..

∴ So √2 is irrational number.

∴ So the correct answer is 2. Irrational number

√2 is a Irrational number.

2. Product of \(\frac{7}{18}\) and reciprocal of (-\(\frac{5}{6}\)) is

1. –\(\frac{7}{15}\)
2. –\(\frac{15}{7}\)
3. \(\frac{7}{15}\)
4. \(\frac{7}{15}\)

Solution: The reciprocal of (-\(\frac{5}{6}\)) is (-\(\frac{6}{5}\))

∴ So the correct answer is 1. –\(\frac{7}{15}\)

Product of \(\frac{7}{18}\) and reciprocal of (-\(\frac{5}{6}\)) is –\(\frac{7}{15}\)

3. a x \(\frac{1}{a}\) =? [where a is rational number and a ≠ 0]

1. 1
2. a
3. \(\frac{1}{a}\)
4. None of these

Solution: a x \(\frac{1}{a}\)

∴ So correct answer is 1. a x \(\frac{1}{a}\)

a x \(\frac{1}{a}\) =1.

Example 13. Write ‘True’ or ‘False’:

1. Commutative law of subtraction does not exist for rational numbers.

Answer: True.

2. \(-\frac{21}{29} p-\left(\frac{21}{29}\right)=0\)

Answer: False

3. If x = –\(\frac{2}{5}\) then, \(\frac{1}{x}\) + \(\frac{x}{2}\) = –\(\frac{10}{27}\)

Solution: \(\frac{1}{x}+\frac{x}{2}=\frac{1}{-\frac{2}{5}}+\frac{-\frac{2}{5}}{2}\)

= \(-\frac{5}{2}-\frac{2}{5} \times \frac{1}{2}=-\left(\frac{5}{2}+\frac{1}{5}\right)=-\left(\frac{25+2}{10}\right)=-\frac{27}{10}\)

∴ So, the statement is false.

Example 14. Fill in the blanks

1. √47 is a ______ number.

Answer: Irrational number

√47 is a Irrational number

2. \(\frac{2}{9}\) + ____ = 0

Answer: –\(\frac{2}{9}\)

\(\frac{2}{9}\) + \(\frac{2}{9}\)= 0

3. (-\(\frac{2}{3}\)) + 0 = _____

Answer: Undefined

(-\(\frac{2}{3}\)) + 0 = Undefined