RD Sharma solution class 7 chapter 4 Rational numbers Exercise 4.2

Exercise 4.2

Page-4.8

Question 1:

Express each of the following as a rational number with positive denominator:
(i) $\frac{-15}{-28}$
(ii) $\frac{6}{-9}$
(iii) $\frac{-28}{-11}$
(iv) $\frac{19}{-7}$

Answer 1:

Rational number with positive denominators:

(i) Multiplying the number by $-$1, we get:

$\frac{-15}{-28}=\frac{-15\times -1}{-28\times -1}=\frac{15}{28}$

(ii) Multiplying the number by $-$1, we get: 

$\frac{6}{-9}=\frac{6\times -1}{-9\times -1}=\frac{-6}{9}$

(iii) Multiplying the number by $-$1, we get:

$\frac{-28}{-11}=\frac{-28\times -1}{-11\times -1}=\frac{28}{11}$

(iv) Multiplying the number by $-$1, we get:

$\frac{19}{-7}=\frac{19\times -1}{-7\times -1}=\frac{-19}{7}$

Question 2:

Express $\frac{3}{5}$ as a rational number with numerator:
(i) 6
(ii) −15
(iii) 21
(iv) −27

Answer 2:

Rational number with numerator:

(i) 6 is:

$\frac{3\times 2}{5\times 2} = \frac{6}{10} (\mathrm{multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator}\text{ by} 2)$

(ii)

$$\begin{gathered} -15 \mathrm{is}: \\ \frac{3\times -5}{5\times -5} = \frac{-15}{-25} (\mathrm{multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} -5) \end{gathered}$$

(iii)

$$\begin{gathered} 21 \mathrm{is}: \\ \frac{3\times 7}{5\times 7} = \frac{21}{35} (\mathrm{multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} 7) \end{gathered}$$

(iv)

$$\begin{gathered} -27 \mathrm{is}: \\ \frac{3\times -9}{5\times -9} = \frac{-27}{-45} (\mathrm{multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} -9) \end{gathered}$$

Question 3:

Express $\frac{5}{7}$ as a rational number with denominator:
(i) −14
(ii) 70
(iii) −28
(iv) −84

Answer 3:

$\frac{5}{7}$ as a rational number with denominator:
(i) −14 is:

$\frac{5\times -2}{7\times -2 } = \frac{-10}{-14} (\mathrm{Multiplyin}\text{g} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} -2)$
(ii) 70 is:

$\frac{5\times 10}{7\times 10} = \frac{50}{70} (\mathrm{Multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} 10)$

(iii) −28 is:

$\frac{5\times -4}{7\times -4} = \frac{-20}{-28} (\mathrm{Multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} -4)$

(iv) −84 is:

$\frac{5\times -12}{7\times -12} = \frac{-60}{-84} (\mathrm{Multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} -12)$

Page-4.9

Question 4:

Express $\frac{3}{4}$ as a rational number with denominator:
(i) 20
(ii) 36
(iii) 44
(iv) −80

Answer 4:

3/4 as rational number with denominator:

(i)
$$\begin{gathered} 20 \mathrm{is}: \\ \frac{3\times 5}{4\times 5}=\frac{15}{20} (\mathrm{multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} 5) \end{gathered}$$


(ii)
$$\begin{gathered} 36 \mathrm{is}: \\ \frac{3\times 9}{4\times 9} = \frac{27}{36} (\mathrm{multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} 9) \end{gathered}$$


(iii)
$$\begin{gathered} 44 \mathrm{is}: \\ \frac{3\times 11}{4\times 11} = \frac{33}{44} (\mathrm{multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} 11) \end{gathered}$$

(iv)
$$\begin{gathered} -80 \mathrm{is}: \\ \frac{3\times -20}{4\times -20} = \frac{-60}{-80} (\mathrm{multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} -20) \end{gathered}$$

Question 5:

Express $\frac{2}{5}$ as a rational number with numerator:
(i) −56
(ii) 154
(iii) −750
(iv) 500

Answer 5:

2/5 as a rational number with numerator:

(i)
$$\begin{gathered} -56 \mathrm{is}: \\ \frac{2\times 28}{5\times -28} =\frac{-56}{-140} (\mathrm{multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} -28) \end{gathered}$$

(ii)
$$\begin{gathered} 154 \mathrm{is}: \\ \frac{2\times 77}{5\times 77} = \frac{154}{385} (\mathrm{multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator}\text{ by} 77) \end{gathered}$$

(iii)
$$\begin{gathered} -750 \mathrm{is}: \\ \frac{2\times -375}{5\times -375} = \frac{-750}{-1875} (\mathrm{multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} -375) \end{gathered}$$

(iv)
$$\begin{gathered} 500 \mathrm{is}: \\ \frac{2\times 250}{5\times 250} = \frac{500}{1250} (\mathrm{multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} 250) \end{gathered}$$

Question 6:

Express $\frac{-192}{108}$ as a rational number with numerator:
(i) 64
(ii) −16
(iii) 32
(iv) −48

Answer 6:

Rational number with numerator:

$$\begin{gathered} \left(\mathrm{i}\right) 64 \mathrm{as} \mathrm{numerator}: \frac{-192/-3}{108/-3} =\frac{64}{-36} (\mathrm{Dividing} \mathrm{the} \mathrm{numerator} \mathrm{and} \mathrm{denomintor} \mathrm{by} -3) \\ \left(\mathrm{ii}\right) -16 \mathrm{as} \mathrm{numerator}: \frac{-192/12}{108/12} = \frac{-16}{9} (\mathrm{Dividing} \mathrm{the} \mathrm{numerator} \mathrm{and} \mathrm{denomintor} \mathrm{by} 12) \\ \left(\mathrm{iii}\right) 32 \mathrm{as} \mathrm{numerator}: \frac{-192/-6}{108/-6} = \frac{32}{-18} (\mathrm{Dividing} \mathrm{the} \mathrm{numerator} \mathrm{and} \mathrm{denomintor} \mathrm{by} -6) \\ \left(\mathrm{iv}\right) -48 \mathrm{as} \mathrm{numerator}: \frac{-192/4}{108/4} = \frac{-48}{27} (\mathrm{Dividing} \mathrm{the} \mathrm{numerator} \mathrm{and} \mathrm{denomintor} \mathrm{by} 4) \end{gathered}$$

Question 7:

Express $\frac{168}{-294}$ as a rational number with denominator:
(i) 14
(ii) −7
(iii) −49
(iv) 1470

Answer 7:

Rational number with denominator:

$$\begin{gathered} \left(\mathrm{i}\right) 14 \mathrm{as} \mathrm{denominator}: \frac{168/-21}{-294/-21} = \frac{-8}{14} (\mathrm{Dividing} \mathrm{the} \mathrm{numerator} \mathrm{and} \mathrm{denomintor} \mathrm{by} -21) \\ \left(\mathrm{ii}\right) -7 \mathrm{as} \mathrm{denominator}: \frac{168/42}{-294/42} = \frac{4}{-7} (\mathrm{Dividing} \mathrm{the} \mathrm{numerator} \mathrm{and} \mathrm{denomintor} \mathrm{by} 42) \\ \left(\mathrm{iii}\right) -49 \mathrm{as} \mathrm{denominator}: \frac{168/6}{-294/6} = \frac{28}{-49} (\mathrm{Dividing} \mathrm{the} \mathrm{numerator} \mathrm{and} \mathrm{denomintor} \mathrm{by} 6) \\ \left(\mathrm{iv}\right) 1470 \mathrm{as} \mathrm{denominator}: \frac{168\times -5}{-294\times -5} = \frac{-840}{1470} (\mathrm{Multiplying} \mathrm{the} \mathrm{numerator} \mathrm{and} \mathrm{denomintor} \mathrm{by} -5) \end{gathered}$$

Question 8:

Write $\frac{-14}{42}$ in a form so that the numerator is equal to:
(i) −2
(ii) 7
(iii) 42
(iv) −70

Answer 8:

Rational number with numerator:

$\left(i\right) -2 \mathrm{is}: \frac{-14/7}{42/7}= \frac{-2}{6} ( \mathrm{Dividing} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} 7)$
$\left(ii\right) 7 \mathrm{is}: \frac{-14/-2}{42/-2} = \frac{7}{-21} ( \mathrm{Dividing} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} -2)$
$\left(iii\right) 42 \mathrm{is}: \frac{-14\times -3}{42\times -3} = \frac{42}{-126} ( \text{M}\mathrm{ultiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator}\text{ by} -3)$
$\left(iv\right) -70 \mathrm{is}: \frac{-14\times 5}{42\times 5} = \frac{-70}{210} ( \mathrm{Multiplying} \mathrm{numerator} \mathrm{and} \mathrm{denominator} \text{by} 5)$

Question 9:

Select those rational numbers which can be written as a rational number with numerator 6:

$\frac{1}{22}, \frac{2}{3}, \frac{3}{4}, \frac{4}{-5}, \frac{5}{6}, \frac{-6}{7}, \frac{-7}{8}$

Answer 9:

Given rational numbers that can be written as a rational number with numerator 6 are:

$$\begin{gathered} \frac{1}{22} (\mathrm{On} \mathrm{multiplying} \mathrm{by} 6) = \frac{6}{132} \\ \frac{2}{3} (\mathrm{On} \mathrm{multiplying} \mathrm{by} 3) = \frac{6}{9} \\ \frac{3}{4} (\mathrm{On} \mathrm{multiplying} \mathrm{by} 2) = \frac{6}{8} \\ \frac{-6}{7} (\mathrm{On} \mathrm{multiplying} \mathrm{by} -1) = \frac{6}{-7} \end{gathered}$$

Question 10:

Select those rational numbers which can be written as a rational number with denominator 4:

$\frac{7}{8}, \frac{64}{16}, \frac{36}{-12}, \frac{-16}{17}, \frac{5}{-4}, \frac{140}{28}$

Answer 10:

Given rational numbers that can be written as a rational number with denominator 4 are:

$$\begin{gathered} \frac{7}{8} (\mathrm{On} \mathrm{dividing} \mathrm{by} 2) = \frac{3.5}{4} \\ \frac{64}{16} (\mathrm{On} \mathrm{dividing} \mathrm{by} 4) =\frac{16}{4} \\ \frac{36}{-12}(\mathrm{On} \mathrm{dividing} \mathrm{by} 3) =\frac{12}{-4} = \frac{-12}{4} \\ \frac{-16}{17} \mathrm{can}\text{'}\mathrm{t} \mathrm{be} \mathrm{expressed} \mathrm{with} \mathrm{a} \mathrm{denominator} 4. \\ \frac{5}{-4}(\mathrm{On} \mathrm{multiplying} \mathrm{by} -1) =\frac{-5}{4} \\ \frac{140}{28}(\mathrm{On} \mathrm{dividing} \mathrm{by} 7) =\frac{20}{4} \end{gathered}$$
122 (On multiplying by 6) = 613223 (On multiplying by 3) = 6934 (On multiplying by 2) = 68−67 (On multiplying by −1) = 6−7

Question 11:

In each of the following, find an equivalent form of the rational number having a common denominator:
(i) $\frac{3}{4} \mathrm{and} \frac{5}{12}$
(ii) $\frac{2}{3}, \frac{7}{6} \mathrm{and} \frac{11}{12}$
(iii) $\frac{5}{7}, \frac{3}{8}, \frac{9}{14} \mathrm{and} \frac{20}{21}$

Answer 11:

Equivalent forms of the rational number having common denominator are:

(i)  $\frac{3}{4}=\frac{3\times 3}{4\times 3}=\frac{9}{12} \mathrm{and} \frac{5}{12}$.

(ii)
$$\begin{gathered} \frac{2}{3}=\frac{2\times 4}{3\times 4}=\frac{8}{12} \mathrm{and} \frac{7}{6}=\frac{7\times 2}{6\times 2}=\frac{14}{12} and \frac{11}{12} \\ \mathrm{Forms} \mathrm{are} \frac{8}{12}, \frac{14}{12} \mathrm{and} \frac{11}{12} \end{gathered}$$


(iii)
$$\begin{gathered} \frac{5}{7}=\frac{5\times 24}{7\times 24}=\frac{120}{168} , \frac{3}{8}=\frac{3\times 21}{8\times 21}=\frac{63}{168} , \frac{9}{14}=\frac{9\times 12}{14\times 12}=\frac{108}{168} and \frac{20}{21}=\frac{20\times 8}{21\times 8}=\frac{160}{168} \\ \mathrm{Forms} \mathrm{are} \frac{120}{168}, \frac{63}{168},\frac{108}{168} \mathrm{and} \frac{160}{168} \end{gathered}$$