Ex.1.1 Q1 Rational Numbers Solution - NCERT Maths Class 8

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Question

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

Text Solution

What is known?

Rational numbers with addition subtraction and multiplication.

What is unknown?

Result of addition, subtraction and multiplication of rational numbers.

Reasoning:

By using commutativity of multiplication and addition getting the answer.

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]

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

[Commutativity of addition]

[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 \(\begin{align}-\frac{11}{28}\end{align}\)