Ex.9.1 Q2 Rational-Numbers Solution - NCERT Maths Class 7

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Question

Write four more rational numbers in each of the following patterns:

\(\begin{align}{\rm{(i)}}\,\,\,\,\frac{{ - 3}}{5},\frac{{ - 6}}{5},\frac{{ - 9}}{5},\frac{{ - 12}}{5}\end{align}\)

\(\begin{align}{\rm{(ii)}}\,\,\,\,\frac{{ - 1}}{4},\frac{{ - 2}}{8},\frac{{ - 3}}{{12}}\end{align}\)

\(\begin{align}{\rm{(iii)}}\frac{{ - 1}}{6},\frac{2}{{ - 12}},\frac{3}{{ - 18}},\frac{4}{{ - 24}}\end{align}\)

\(\begin{align}{\rm{(iv)}}\frac{{ - 2}}{3},\frac{2}{{ - 3}},\frac{4}{{ - 6}},\frac{6}{{ - 9}}\end{align}\)

Text Solution

What is known?

Patterns of the numbers.

What is unknown?

Four more rational numbers in each of the given patterns.

Reasoning:

This question is based on a definite pattern, while solving such type of questions observe the numerator and denominator carefully. Here both numerator and denominator are following a definite pattern, observe this pattern and follow the same pattern, you can easily find out the next four rational numbers.

Steps:

\(\begin{align}{\rm{i)}}\quad \frac{{ - 3}}{5},\frac{{ - 6}}{{10}},\frac{{ - 9}}{{15}},\frac{{ - 12}}{{20}}\end{align}\)

\[\begin{align}\frac{{ - 3 \times 1}}{{5 \times 1}},\frac{{ - 3 \times 2}}{{5 \times 2}},\frac{{ - 3 \times 3}}{{5 \times 3}},\frac{{ - 3 \times 4}}{{5 \times 4}} \ldots \ldots\end{align} \]

Next four rational numbers in the same pattern,

\[\begin{align}\frac{{ - 3 \times 5}}{{5 \times 5}},\frac{{ - 3 \times 6}}{{5 \times 6}},\frac{{ - 3 \times 7}}{{5 \times 7}},\frac{{ - 3 \times 8}}{{5 \times 8}}\end{align}\]

Therefore, the numbers are \(\begin{align}\frac{{ - 15}}{{25}},\frac{{ - 18}}{{30}},\frac{{ - 21}}{{35}},\;\frac{{ - 24}}{{40}}\end{align}\)

\(\begin{align} {\rm{(ii)}} & \frac{{ - 1}}{4},\frac{{ - 2}}{8},\frac{{ - 3}}{{12}}\\ & \frac{{ - 1 \times 1}}{{4 \times 1}},\frac{{ - 1 \times 2}}{{4 \times 2}},\frac{{ - 1 \times 3}}{{4 \times 3}} \ldots \ldots  \end{align}\)

Next four rational numbers in the same pattern,

\[\begin{align}\;\frac{{ - 1\; \times \;4}}{{4\; \times \;4}},\frac{{ - 1\; \times \;5}}{{4\; \times \;5}},\frac{{ - 1\; \times \;6}}{{4\; \times \;6}},\frac{{ - 1\; \times \;7}}{{4\; \times \;7}}\end{align}\]

Therefore, the numbers are \(\begin{align}\frac{{ - 4}}{{16}},\frac{{ - 5}}{{20}},\frac{{ - 6}}{{24}},\frac{{ - 7}}{{28}} \cdots\end{align}\)

\(\begin{align} {\rm{(iii)}} & \frac{{ - 1}}{6},\frac{2}{{ - 12}},\frac{3}{{ - 18}},\frac{4}{{ - 24}}\\ & \frac{{1 \times 1}}{{6{\times1}}},\frac{{1\times2}}{{6{\times 2}}},\frac{{1{\times3}}}{{6\times 3}},\frac{{1{\times4}}}{{6{\times 4}}} \ldots \end{align}\)

Next four rational numbers in the same pattern,

\[\begin{align}\frac{{1 \times 5}}{{-6 \times5}},\frac{{1 \times 6}}{{-6 \times 6}},\frac{{1 \times 7}}{{-6 \times7}},\frac{{1 \times 8}}{{-6 \times8}}\end{align}\]

Therefore, the numbers are\(\begin{align}\frac{5}{{ - 30}},\frac{6}{{ - 36}},\frac{7}{{ - 42}},\frac{8}{{ - 48}} \ldots \ldots\end{align}\)

\(\begin{align} {\rm{(iv)}} & \frac{{ - 2}}{3},\frac{2}{{ - 3}},\frac{4}{{ - 6}},\frac{6}{{ - 9}}\\ & \frac{{-2 \times1}}{{3 \times 1}},\frac{{2 \times 1}}{{-3 \times1}},\frac{{2 \times 2}}{{-3 \times 2}},\frac{{2 \times 3}}{{-3 \times 3}} \ldots  \end{align}\)

Next four rational numbers in the same pattern,

\[\begin{align}\frac{{2 \times 4}}{{-3 \times4}},\frac{{2 \times 5}}{{-3 \times5}},\frac{{2 \times 6}}{{-3 \times6}},\frac{{2 \times 7}}{{-3 \times7}}\end{align}\]

Therefore, the numbers are\(\begin{align}\frac{8}{{ - 12}},\frac{{10}}{{ - 15}},\frac{{12}}{{ - 18}},\;\frac{{14}}{{ - 21}}\end{align}\)

  
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