Rational Numbers - NCERT Class 8 Maths

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Question 1

Using appropriate properties find:

(i) \begin{align}\frac{{ - 2}}{3} \times \frac{3}{5} + \frac{5}{2} - \frac{3}{5} \times \frac{1}{6}\end{align}

(ii)\begin{align} \frac{2}{5} \times \left( {\frac{{ - 3}}{7}} \right) - \frac{1}{6} \times \frac{3}{2} + \frac{1}{{14}} \times \frac{2}{5}\end{align}

Solution

What is known?

Rational numbers with addition subtraction and multiplication.

What is unknown?

Result of addition, subtraction and multiplication of rational numbers.

Reasoning:

Steps:

(i)

\begin{align}\frac{{ - 2}}{3} \times \frac{3}{4} + \frac{5}{2} - \frac{3}{5} \times \frac{1}{6}\end{align}

\begin{align}= \frac{3}{5} \times \frac{{ - 2}}{3} + \frac{5}{2} - \frac{3}{5} \times \frac{1}{6}\end{align}

[By commutativity of multiplication]

\begin{align}= \frac{3}{5} \times \frac{{ - 2}}{3} - \frac{3}{5} \times \frac{1}{6} + \frac{5}{2}\end{align}

[Rearranging to take a common]

\begin{align} &= \frac{3}{5} \times \left( {\frac{{ - 2}}{3} - \frac{1}{6}} \right) + \frac{5}{2}\\ &= \frac{3}{5} \times \left( {\frac{{ - 4 - 1}}{6}} \right) + \frac{5}{2}\\& = \frac{3}{5} \times \frac{{ - 5}}{6} + \frac{5}{2}\\ &= - \frac{1}{2} + \frac{5}{2}\\ &= \frac{{ - 1 + 5}}{2}\\& = \frac{4}{2}\\&= 2\end{align}

Answer is $$2$$

(ii)

\begin{align} \frac{2}{5} \times \left( {\frac{{ - 3}}{7}} \right) - \frac{1}{6} \times \frac{3}{2} + \frac{1}{{14}} \times \frac{2}{5}\end{align}

Rearranging

\begin{align}= \frac{2}{5} \times \left( {\frac{{ - 3}}{7}} \right) + \frac{2}{5} \times \frac{1}{{14}} - \frac{1}{6} \times \frac{3}{2}\end{align}

Taking \begin{align}\frac{2}{5}\end{align} common

$= \frac{2}{5} \times \left[ {\left( {\frac{{ - 3}}{7}} \right) + \frac{1}{{14}}} \right] - \frac{1}{6} \times \frac{3}{2}$

[By distributivity]

\begin{align}&= \frac{2}{5} \times \left( {\frac{{ - 3 \times 2 + 1}}{{14}}} \right) - \frac{1}{6} \times \frac{3}{2}\\& = \frac{2}{5} \times \left( {\frac{{ - 6 + 1}}{{14}}} \right) - \frac{1}{4}\\&= \frac{2}{5} \times \frac{{ - 5}}{{14}} - \frac{1}{4}\\&= \frac{{ - 1}}{7} - \frac{1}{4}\\ &= \frac{{( - 1 \times 4)}}{{(7 \times 4)}} - \frac{{(1 \times 7)}}{{(4 \times 7)}}\\ &= - \frac{4}{{28}} - \frac{7}{{28}}\\& = \frac{{ - 4 - 7}}{{28}}\\& = - \frac{{11}}{{28}}\end{align}

Answer is \begin{align}-\frac{11}{28}\end{align}

Question 2

Write the additive inverse of each of the following

(i)  \begin{align}\frac{2}{8}\end{align}

(ii)\begin{align}\frac{{ - 5}}{9}\end{align}

(iii)\begin{align}\frac{{ - 6}}{{ - 5}}\end{align}

(iv)\begin{align}\frac{2}{{ - 9}}\end{align}

(v) \begin{align}\frac{{19}}{{ - 6}}\end{align}

Solution

What is known?

Rational numbers

What is unknown?

Reasoning:

The negative of a rational number is called additive inverse.

Steps:

(i) \begin{align}\quad\frac{2}{8}\end{align}

Additive inverse of \begin{align}\frac{2}{8}\end{align} is

\begin{align} - \left( {\frac{2}{8}} \right) = - \frac{2}{8}\end{align}

(ii) \begin{align}\quad\frac{{ - 5}}{9}\end{align}

Additive inverse of \begin{align}\frac{{ - 5}}{9}\end{align} is

\begin{align} - \left( {\frac{{ - 5}}{9}} \right) = \frac{5}{9}\end{align}

(iii) \begin{align}\quad\frac{{ - 6}}{{ - 5}}\end{align}

The rational number is \begin{align}\frac{{ - 6}}{{ - 5}} = \frac{6}{5}\end{align}

Additive inverse of \begin{align}\frac{{ - 6}}{{ - 5}}\end{align} is

\begin{align} - \left( {\frac{6}{5}} \right) = - \frac{6}{5}\end{align}

(iv) \begin{align}\;\frac{2}{{ - 9}}\end{align}

Additive inverse of \begin{align}\;\frac{2}{{ - 9}}\end{align} is

\begin{align} - \left( {\frac{2}{{ - 9}}} \right) = \frac{2}{9}\end{align}

(v) \begin{align}\,\frac{{19}}{{ - 6}}\end{align}

Additive inverse of \begin{align}\frac{{19}}{{ - 6}}\end{align} is

\begin{align} - \left( {\frac{{19}}{{ - 6}}} \right) = \frac{{19}}{6}\end{align}

Question 3

Verify that  \begin{align}- \left( { - x} \right) = x\end{align} for

(i) \begin{align} x = \frac{{11}}{{15}}\end{align}

(ii) \begin{align} x = - \frac{{13}}{7}\end{align}

Solution

What is known?

Rational number.

What is unknown?

The negative of the negative of a rational number.

Reasoning:

The negative of the negative of a rational number is that rational number of Self.

Steps:

(i) \begin{align} \quad x = \frac{{11}}{{15}}\end{align}

\begin{align}- ( - x)&= - \left( { - \frac{{11}}{{15}}} \right)\\&= \frac{{11}}{{15}}\\&= x\end{align}

Hence Proved.

(ii) \begin{align}\quad x = - \frac{{13}}{7}\end{align}

\begin{align}- ( - x) &= - \left[ { - \left( { - \frac{{13}}{{17}}} \right)} \right]\\&=- \left[ {\frac{{13}}{{17}}} \right]\\&= - \frac{{13}}{{17}}\\&= x\end{align}

Hence Proved.

Question 4

Find the multiplicative inverse of the following.

(i) \begin{align} \; - 13 \end{align}

(ii) \begin{align} \; \frac{{ - 13}}{{19}}\end{align}

(iii) \begin{align} \;\frac{1}{5}\end{align}

(iv) \begin{align} \; \frac{{ - 5}}{8} \times \frac{{ - 3}}{7} \end{align}

(v) \begin{align} \; - 1 \times \frac{{ - 2}}{5} \end{align}

(vi) \begin{align} \; - 1\end{align}

Solution

What is known?

Rational number

What is unknown?

The multiplicative inverse.

Reasoning:

The reciprocal of the given rational number is the multiplicative inverse. [The product of the rational number and its multiplicative inverse is $$1$$]

Steps:

(i) \begin{align} \; - 13 \end{align}

The Multiplicative inverse of \begin{align} - 13 \end{align} is \begin{align} \frac{{ - 1}}{{13}} \end{align}

\begin{align}\left[ { - 13 \times \frac{{ - 1}}{{13}} = 1} \right]\end{align}

(ii) \begin{align} \; \frac{{ - 13}}{{19}}\end{align}

The Multiplicative inverse of \begin{align}\frac{{ - 13}}{{19}}\end{align} is \begin{align}\frac{{19}}{{ - 13}}\end{align}

\begin{align}\left[ {\frac{{ - 13}}{{19}} \times \frac{{19}}{{ - 13}} = 1} \right]\end{align}

(iii) \begin{align} \; \frac{1}{5}\end{align}

The Multiplicative inverse of \begin{align}\frac{1}{5}\end{align} is \begin{align}\frac{5}{1}\end{align}

\begin{align}\left[ {\frac{1}{5} \times \frac{5}{1} = 1} \right]\end{align}

(iv) \begin{align} \; \frac{{ - 5}}{8} \times \frac{{ - 3}}{7}=\frac{{15}}{56} \end{align}

The Multiplicative inverse of \begin{align}\frac{{15}}{{56}}\end{align} is \begin{align}\frac{{56}}{{15}}\end{align}

\begin{align}\left[ {\frac{{15}}{{56}} \times \frac{{56}}{{15}} = 1} \right]\end{align}

(v) \begin{align} \; - 1 \times \frac{{ - 2}}{5} \end{align}

This can be simplified as:

\begin{align} - 1 \times \frac{{ - 2}}{5} &= \frac{{( - 1) \times ( - 2)}}{5}\\ &= \frac{2}{5}\end{align}

The multiplicative inverse of \begin{align}\frac{2}{5} \end{align} is \begin{align}\frac{5}{2} \end{align}

\begin{align}\left[ {\frac{2}{5} \times \frac{5}{2} = 1} \right]\end{align}

(vi) \begin{align} \; - 1\end{align}

The multiplicative inverse of \begin{align} - 1\end{align} is \begin{align} - 1\end{align}.

\begin{align}( - 1) \times ( - 1) = 1\end{align}

Question 5

Name the property under multiplication used in each of the following:

(i) \begin{align}\;\frac{{ - 4}}{5} \times 1 = 1 \times \frac{{ - 4}}{5} = \frac{{ - 4}}{5}\end{align}

(ii) \begin{align}\;\frac{{ - 13}}{{17}} \times \frac{{ - 2}}{7} = \frac{{ - 2}}{7} \times \frac{{ - 13}}{{17}}\end{align}

(iii) \begin{align}\;\frac{{ - 19}}{{29}} \times \frac{{29}}{{ - 19}} = 1\end{align}

Solution

What is known?

Rational number.

What is unknown?

Name of the property.

(i) \begin{align}\;\frac{{ - 4}}{5} \times 1 = 1 \times \frac{{ - 4}}{5} = \frac{{ - 4}}{5}\end{align}

Reasoning:

So, $$1$$ is the multiplicative identity.

Steps:

\begin{align}\;\frac{{ - 4}}{5} \times 1 = 1 \times \frac{{ - 4}}{5} = \frac{{ - 4}}{5}\end{align}

$$\therefore 1$$ is the multiplicative identity and here, property of multiplicative identity is used.

(ii) \begin{align}\;\frac{{ - 13}}{{17}} \times \frac{{ - 2}}{7} = \frac{{ - 2}}{7} \times \frac{{ - 13}}{{17}}\end{align}

Reasoning:

In general, $$a \times b = b \times a$$ for any two rational numbers. This is called commutativity of multiplication.

Steps:

\begin{align}\frac{{ - 13}}{{17}} \times \frac{{ - 2}}{7} &= \frac{{ - 2}}{7} \times \frac{{ - 13}}{{17}}\\ [a \times b] &= [b \times a]\end{align}

Commutativity of multiplication of rational numbers is used here.

(iii) \begin{align}\;\frac{{ - 19}}{{29}} \times \frac{{29}}{{ - 19}} = 1\end{align}

Reasoning:

For a rational number \begin{align}\frac{a}{b}\end{align} the multiplicative inverse is the reciprocal of that number that is \begin{align}\frac{b}{a}\end{align} . So that the product of the rational number and its multiplicative inverse is $$1.$$

Steps:

\begin{align}\frac{{ - 19}}{{29}} \times \frac{{29}}{{ - 19}} &= 1\\\left[ {\left( {\frac{a}{b}} \right) \times \left( {\frac{b}{a}} \right)} \right] &= 1\end{align}

Multiplicative Inverse.

Question 6

Multiply \begin{align}\frac{6}{{13}}\end{align} by the reciprocal of \begin{align}\frac{{ - 7}}{{16}}\end{align}

Solution

What is known?

Rational numbers.

What is unknown?

Product of the rational numbers.

Reasoning:

Reciprocal of a rational number is its multiplicative inverse.

Steps:

Reciprocal of \begin{align}\frac{-7}{16}\end{align}is \begin{align}\frac{16}{-7}\end{align}

\begin{align}\frac{6}{{13}} \times \end{align} Reciprocal of \begin{align}\frac{7}{{16}}\end{align}

\begin{align}&= \frac{6}{{13}} \times \frac{{16}}{{ - 7}}\\ &= \frac{{6 \times 16}}{{13 \times ( - 7)}}\\& = \frac{{96}}{{ - 91}}\end{align}

Thus answer is \begin{align}\frac{{ - 96}}{{91}}\end{align}

Question 7

Tell what property allow you to compute \begin{align}\frac{1}{3} \times \left( {6 \times \frac{4}{5}} \right)\end{align} as \begin{align}\left( {\frac{1}{3} \times 6} \right) \times \frac{4}{3}\end{align}

Solution

What is known?

Rational numbers.

What is unknown?

Property.

Reasoning:

Multiplication is associative for rational numbers. For any rational numbers $$a,\, b,\, c$$

\begin{align}a \times (b \times c) = (a \times b) \times c \end{align}

Steps:

\begin{align}\,&[a \times (b \times c) = (a \times b) \times c] \\\\&\frac{1}{3} \times \left( {6 \times \frac{4}{3}} \right)\,\,\,{\rm{as}}\,\,\,\left( {\frac{1}{3} \times 6} \right)\frac{4}{3}\end{align}

Associativity of multiplication of rational numbers is used here.

Question 8

Is \begin{align}\frac{8}{9}\end{align} the multiplicative inverse of \begin{align}- 1\frac{1}{8}\end{align}?

Why or why not?

Solution

What is known?

Rational numbers.

What is unknown?

Multiplicative or not

Reasoning:

The product of a rational number with its multiplicative inverse is $$1.$$

Steps:

\begin{align} - 1\frac{1}{8} = - \frac{9}{8}\end{align}

Now: \begin{align} = \frac{8}{9} \times - \frac{9}{8} = - 1 \ne 1\end{align}

So, \begin{align}\frac{8}{9} \end{align} is not the multiplicative inverse of \begin{align}- 1\frac{1}{8}.\end{align}

\begin{align}\frac{8}{9}\end{align} is not the multiplicative inverse of \begin{align} - 1\frac{1}{8} \end{align} because the product of \begin{align}\frac {{8}}{9}\end{align} and \begin{align}- 1\frac{1}{8}\end{align} is $$-1$$ , and it should be $$1$$ to be a multiplicative inverse.

Question 9

Is $$0.3$$ the multiplicative inverse of \begin{align}3\frac{1}{3}\end{align}?

Why or why not?

Solution

What is known?

Rational number.

What is unknown?

Multiplicative inverse or not?

Reasoning:

The product of the rational number and its multiplicative inverse is $$1.$$

Steps:

$$0.3$$ can be written as \begin{align} \frac{3}{10} \end{align}

Given rational number \begin{align}3\frac{1}{3}\end{align} can be written as \begin{align}\frac{{10}}{3}\end{align}

So, \begin{align}\frac{3}{{10}} \times \frac{{10}}{3} = 1\end{align}

The answer is $$8$$

Yes, $$0.3$$ is the multiplicative inverse of \begin{align}3\frac{1}{3}\end{align} because their product is $$1.$$

Question 10

Write:

(i) The rational number that does not have a reciprocal.

(ii) The rational numbers that are equal to their reciprocals.

(iii) The rational number that is equal to its negative.

Solution

Steps:

(i) Rational number does not have a reciprocal

$$0$$ (zero) is the rational number which does not have a reciprocal.

(ii) The rational number that is equal to its reciprocals.

The rational numbers $$1$$ and ($$–1$$) are equal to their own reciprocals.

(iii) The rational number that is equal to its negative.

Rational number $$0$$ is equal to its negative.

Question 11

Fill in the blanks.

(i) Zero has ________ reciprocal.

(ii) The numbers ________ and ________ are their own reciprocals

(iii) The reciprocal of $$– 5$$ is ________.

(iv) Reciprocal of \begin{align}\frac{1}{x},\end{align} where \begin{align}x \ne 0\end{align} is________.

(v) The product of two rational numbers is always a _______.

(vi) The reciprocal of a positive rational number is ________.

Solution

Steps:

(i) Zero has     no    reciprocal

(ii) The numbers $$\underline{\,\,\,1\,\,\,}$$   and $$\underline{\;\;(–1)\;\;}$$ are their own reciprocals.

(iii) The reciprocal of $$(–5)$$ is \begin{align}\underline{\quad\frac{1}{{ - 5}}\quad}\end{align}

(iv) Reciprocal of \begin{align}\frac{1}{x}\end{align} where \begin{align}x \ne 0 \end{align} is\begin{align}\underline{\quad x.\quad}\end{align}

(v) The product of two rational numbers is always a rational number.

(vi) The reciprocal of a positive rational number is positive.

The chapter 1 begins with an introduction to Rational Numbers by recalling the concepts of Natural numbers, whole numbers and integers.Then various properties of Rational Numbers such as Closure Property ,Commutative Property, Associative Property and Distributive Property is explained. All of these are discussed separately for Whole numbers, Integers and Rational Numbers. Under rational numbers, it is dealt individually for Addition, Subtraction, Multiplication and Division.Negative and reciprocal of a rational number is dealt subsequently.The next section of the chapter deals with representation of rational numbers on the number line and lastly, we have  rational numbers between two rational numbers.