DAV Class 7 Maths Chapter 1 Worksheet 2
Rational Numbers Worksheet 2
1. In each of the following cases, show that the rational numbers are equivalent.
(i) \( \displaystyle \frac{4}{9} \) and \( \displaystyle \frac{44}{99} \)
Answer
\[ \begin{align*} 4 \times 99 & = 396 \\ 9 \times 44 & = 396 \\ \end{align*} \] \(\displaystyle \frac{4}{9} \) and \( \displaystyle \frac{44}{99} \) are equivalent.
(ii) \( \displaystyle \frac{7}{-3} \) and \( \displaystyle \frac{35}{-15} \)
Answer
\[ \begin{align*} 7 \times (-15) & = -105 \\ (-3) \times 35 & = -105 \\ \end{align*} \] \( \displaystyle \frac{7}{-3} \) and \( \displaystyle \frac{35}{-15} \) are equivalent rational numbers.
(iii) \( \displaystyle \frac{-3}{5} \) and \( \displaystyle \frac{-12}{20} \)
Answer
\[ \begin{align*} (-3) \times 20 & = -60 \\ 5 \times (-12) & = -60 \\ \end{align*} \] \( \displaystyle \frac{-3}{5} \) and \( \displaystyle \frac{-12}{20} \) are equivalent rational numbers.
2. In each of the following cases, show that rational numbers are not equivalent.
(i) \( \displaystyle \frac{4}{9} \) and \( \displaystyle \frac{16}{27} \)
Answer
\[ \begin{align*} 4 \times 27 & = 108 \\ 9 \times 16 & = 144 \\ \end{align*} \] \( \displaystyle \frac{4}{9} \) and \( \displaystyle \frac{16}{27} \) are not equivalent rational numbers.
(ii) \( \displaystyle \frac{-100}{3} \) and \( \displaystyle \frac{300}{9} \)
Answer
\[ \begin{align*} (-100) \times 9 & = -900 \\ 3 \times 300 & = 900 \\ \end{align*} \] \( \displaystyle \frac{-100}{3} \) and \( \displaystyle \frac{300}{9} \) are not equivalent rational numbers.
(iii) \( \displaystyle \frac{3}{-17} \) and \( \displaystyle \frac{8}{-51} \)
Answer
\[ \begin{align*} 3 \times (-51) & = -153 \\ (-17) \times 8 & = -136 \\ \end{align*} \] \( \displaystyle \frac{3}{-17} \) and \( \displaystyle \frac{8}{-51} \) are not equivalent rational numbers.
3. Write three rational numbers, equivalent to each of the following:
(i) \( \displaystyle \frac{4}{7} \)
Answer
\[ \begin{align*} \frac{4 \times {\color{green} 2}}{7 \times { \color{green} 2}} &= \frac{8}{14}\\ \\ \frac{4 \times {\color{green} 3}}{7 \times { \color{green} 3}} &= \frac{12}{21}\\ \\ \frac{4 \times {\color{green} 4}}{7 \times { \color{green} 4}} &= \frac{16}{28} \\ \\ \text{Equivalent rational numbers} & = \displaystyle \color{red} \frac{8}{14} , \frac{12}{21} , \frac{16}{28} \end{align*} \]
(ii) \( \displaystyle \frac{36}{108} \)
Answer
\[ \begin{align*} \frac{36 \div { \color{green} 2}}{108 \div {\color{green} 2}} &= \frac{18}{54}\\ \\ \frac{36 \div { \color{green} 3}}{108 \div {\color{green} 3}} &= \frac{12}{36}\\ \\ \frac{36 \div { \color{green} 4}}{108 \div {\color{green} 4}} &= \frac{9}{27} \\ \\ \text{Equivalent rational numbers} & = \displaystyle \color{red} \frac{18}{54}, \frac{12}{36}, \frac{9}{27} \end{align*} \]
(iii) \( \displaystyle \frac{-5}{-7} \)
Answer
\[ \begin{align*} \frac{-5 \times {\color{green} (-2)}}{-7 \times {\color{green} (-2)}} &= \frac{10}{14}\\ \\ \frac{-5 \times {\color{green} (-3)}}{-7 \times {\color{green} (-3)}} &= \frac{15}{21}\\ \\ \frac{-5 \times {\color{green} (-4)}}{-7 \times {\color{green} (-4)}} &= \frac{20}{28} \\ \\ \text{Equivalent rational numbers} & = \displaystyle \color{red} \frac{10}{14}, \frac{15}{21}, \frac{20}{28} \end{align*} \]
(iv) \( \displaystyle \frac{-72}{180} \)
Answer
\[ \begin{align*} \frac{-72 \div {\color{green} 2}}{180 \div {\color{green} 2}} &= \frac{-36}{90} \\ \\ \frac{-72 \div {\color{green} 3}}{180 \div {\color{green} 3}} &= \frac{-24}{60} \\ \\ \frac{-72 \div {\color{green} 4}}{180 \div {\color{green} 4}} &= \frac{-18}{45} \\ \\ \text{Equivalent rational numbers} & = \displaystyle \color{red} \frac{-36}{90}, \frac{-24}{60}, \frac{-18}{45} \end{align*} \]
4. Express \( \displaystyle \frac{3}{5} \) as rational number with numerator,
(i) \( \displaystyle \frac{3}{5} \) with numerator -21
Answer
\[ \begin{align*} \frac{3}{5} &= \frac{-21}{\boxed{?}}\\ \\ \text{Multiply numerator} & \text{ and denominator by } \color{green} (-7) \\ \frac{3 \times {\color{green} (-7)}}{5 \times {\color{green} (-7)}} &= \frac{-21}{-35}\\ \\ \frac{3}{5} &= \color{red}\frac{-21}{-35} \end{align*} \]
(ii) \( \displaystyle \frac{3}{5} \) with numerator 150
Answer
\[ \begin{align*} \frac{3}{5} &= \frac{150}{\boxed{?}}\\ \\ \text{Multiply numerator} & \text{ and denominator by } \color{green} 50 \\ \frac{3 \times {\color{green}50}}{5 \times {\color{green} 50}} &= \frac{150}{250}\\ \\ \frac{3}{5} &= \color{red} \frac{150}{250} \end{align*} \]
5. Express \( \displaystyle \frac{4}{-7} \) as rational number with denominator,
(i) \( \displaystyle \frac{4}{-7} \) with denominator 84
Answer
\[ \begin{align*} \frac{4}{-7} &= \frac{\boxed{?}}{84}\\ \\ \text{Multiply numerator} & \text{ and denominator by } \color{green} (-12) \\ \frac{4 \times {\color{green} (-12)}}{-7 \times {\color{green} (-12)}} &= \frac{-48}{84}\\ \\ \frac{4}{-7} &= \color{red} \frac{-48}{84} \end{align*} \]
(ii) \( \displaystyle \frac{4}{-7} \) with denominator -28
Answer
\[ \begin{align*} \frac{4}{-7} &= \frac{\boxed{?}}{-28}\\ \\ \text{Multiply numerator} & \text{ and denominator by } \color{green} 4 \\ \frac{4 \times {\color{green} 4}}{-7 \times {\color{green} 4}} &= \frac{16}{-28}\\ \\ \frac{4}{-7} &= \color{red} \frac{16}{-28} \end{align*} \]
6. Express \( \displaystyle \frac{90}{216} \) as rational number with numerator 5.
Answer
\[ \begin{align*} \frac{90}{216} &= \frac{5}{\boxed{?}}\\ \\ \text{Divide numerator} & \text{ and denominator by } \color{green} 18 \\ \frac{90 \div {\color{green} 18}}{216 \div {\color{green} 18}} &= \frac{5}{\boxed{12}}\\ \\ \frac{90}{216} &= \color{red} \frac{5}{12} \end{align*} \]
7. Express \( \displaystyle \frac{-64}{256} \) as rational number with denominator 8
Answer
\[ \begin{align*} \frac{-64}{256} &= \frac{\boxed{?}}{8}\\ \\ \text{Divide numerator} & \text{ and denominator by } \color{green} 32 \\ \frac{-64 \div {\color{green} 32}}{256 \div {\color{green} 32}} &= \frac{\boxed{-2}}{8}\\ \\ \frac{-64}{256} &=\color{red} \frac{-2}{8} \end{align*} \]
8. Find equivalent forms of the rational numbers having a common denominator in each of the following collections of rational numbers.
(i) \( \displaystyle \frac{2}{5} , \frac{6}{13} \)
Solution
\[ \begin{array}{c|c} 5 & 5 , 13 \\ \hline 13 & 1, 13 \\ \hline & 1, 1 \\ \end{array} \] \[ \begin{align*} \text{LCM} = 5 \times 13 &\implies 65 \\ \\ \frac{2 \times {\color{green} 13}}{5 \times {\color{green} 13}} &= \frac{26}{65} \\ \\ \frac{6 \times {\color{green} 5 }}{13 \times {\color{green} 5 }} &= \frac{30}{65} \\ \\ \end{align*} \]
Answer \( \color{red} \displaystyle \frac{26}{65} , \frac{30}{65} \)
(ii) \( \displaystyle \frac{1}{7} , \frac{2}{8} , \frac{3}{14} \)
Solution
\[ \begin{array}{c|c} 7 & 7 , 8, 14 \\ \hline 2 & 1 , 8, 2 \\ \hline 2 & 1 , 4, 1 \\ \hline 2 & 1 , 2, 1 \\ \hline & 1 , 1, 1 \\ \end{array} \] \[ \begin{align*} \text{LCM} &= 7 \times 2 \times 2 \times 2 \\ &= 56 \\ \\ \frac{1 \times{\color{green} 8}}{7 \times {\color{green} 8}} &= \frac{8}{56} \\ \\ \frac{2 \times{\color{green} 7}}{8 \times {\color{green} 7}} &= \frac{14}{56} \\ \\ \frac{3 \times{\color{green} 4}}{14 \times {\color{green} 4}} &= \frac{12}{56} \\ \\ \end{align*} \]
Answer \( \color{red} \displaystyle \frac{8}{56} , \frac{14}{56} , \frac{12}{56} \)
(iii) \( \displaystyle \frac{5}{12} , \frac{7}{4} , \frac{9}{60} , \frac{11}{3} \)
Solution
\[ \begin{array}{c|c} 2 & 12, 4, 60, 3 \\ \hline 2 & 6, 2, 30, 3 \\ \hline 3 & 3, 1, 15, 3 \\ \hline 5 & 1, 1, 5, 1 \\ \hline & 1, 1, 1, 1 \\ \end{array} \] \[ \begin{align*} \text{LCM} &= 2 \times 2 \times 3 \times 5 \\ &= 60 \\ \\ \frac{5 \times {\color{green} 5}}{12 \times {\color{green} 5}} &= \frac{25}{60} \\ \\ \frac{7 \times {\color{green} 15}}{4 \times {\color{green} 15}} &= \frac{105}{60} \\ \\ \frac{9 \times {\color{green} 1}}{60 \times {\color{green} 1}} &= \frac{9}{60} \\ \\ \frac{11 \times {\color{green} 20}}{3 \times {\color{green} 20}} &= \frac{220}{60} \\ \\ \end{align*} \]
Answer \( \displaystyle \color{red} \frac{25}{60} , \frac{105}{60} , \frac{9}{60} , \frac{220}{60} \)