# Excercise 1.1 Rational Numbers- NCERT Solutions Class 8

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## Chapter 1 Ex.1.1 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}

## Chapter 1 Ex.1.1 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}

## Chapter 1 Ex.1.1 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.

## Chapter 1 Ex.1.1 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}

## Chapter 1 Ex.1.1 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.

## Chapter 1 Ex.1.1 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}

## Chapter 1 Ex.1.1 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.

## Chapter 1 Ex.1.1 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.

## Chapter 1 Ex.1.1 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 $$1.$$

## Chapter 1 Ex.1.1 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.

## Chapter 1 Ex.1.1 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.

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