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Chapter 4

Rational Numbers - Exercise 4(A)

Class - 7 RS Aggarwal Mathematics Solutions

Exercise 4(A)

Question 1

What are rational numbers? Give four examples of each of positive rationals and negative rationals. Give an example of a rational number which is neither positive nor negative.

Answer

A rational number is a number that can be expressed in the form pq\dfrac{p}{q}, where p and q are integers and q ≠ 0. In pq\dfrac{p}{q}, we have numerator = p and denominator = q.

Examples of Positive Rational Numbers:

A rational number is positive if its numerator and denominator are both of the same sign (both positive or both negative).

(1) 34\dfrac{3}{4}

(2) 1120\dfrac{11}{20}

(3) 5 (which is 51\dfrac{5}{1})

(4) −7−9\dfrac{-7}{-9}

Examples of Negative Rational Numbers:

A rational number is negative if its numerator and denominator are of opposite signs.

(1) −12-\dfrac{1}{2}

(2) −58\dfrac{-5}{8}

(3) 4−7\dfrac{4}{-7}

(4)-10 (which is −101\dfrac{-10}{1})

Example of a rational number which is neither positive nor negative is 0.

Question 2

Which of the following are rational numbers?

(i) 515\dfrac{5}{15}

(ii) −623\dfrac{-6}{23}

(iii) 17

(iv) -25

(v) 0

(vi) 80\dfrac{8}{0}

(vii) 00\dfrac{0}{0}

(viii) 08\dfrac{0}{8}

(ix) −23−37\dfrac{-23}{-37}

(x) −17\dfrac{-1}{7}

Answer

(i) 515\dfrac{5}{15}

⇒ It is in pq\dfrac{p}{q} form where both 5 and 15 are integers and q ≠ 0.

Hence, it is a rational number.

(ii) −623\dfrac{-6}{23}

⇒ Both -6 and 23 are integers and the denominator is not zero.

Hence, it is a rational number.

(iii) 17

⇒ An integer that can be written as 171\dfrac{17}{1}.

Hence, it is a rational number.

(iv) -25

⇒ A negative integer that can be written as −251\dfrac{-25}{1}.

Hence, it is a rational number.

(v) 0

⇒ Zero is a rational number because it can be written as 01\dfrac{0}{1}.

Hence, it is a rational number.

(vi) 80\dfrac{8}{0}

⇒ Division by zero is undefined; the denominator q cannot be 0.

Hence, it is not a rational number.

(vii) 00\dfrac{0}{0}

⇒ The denominator is zero, which is not allowed in the definition of a rational number.

Hence, it is not a rational number.

(viii) 08\dfrac{0}{8}

⇒ The numerator can be 0 as long as the denominator is a non-zero integer.

Hence, it is a rational number.

(ix) −23−37\dfrac{-23}{-37}

⇒ Both are integers and the denominator is not zero (this simplifies to 2337\dfrac{23}{37}).

Hence, it is a rational number.

(x) −17\dfrac{-1}{7}

⇒ Both are integers and the denominator is not zero.

Hence, it is a rational number.

Question 3

Write down the numerator and the denominator of each of the following rational numbers :

(i) 1217\dfrac{12}{17}

(ii) 6−23\dfrac{6}{-23}

(iii) −215\dfrac{-21}{5}

(iv) 7

(v) -8

Answer

(i) We have:

1217\dfrac{12}{17}

Numerator = 12, Denominator = 17

(ii) We have:

6−23\dfrac{6}{-23}

Numerator = 6, Denominator = -23

(iii) We have:

−215\dfrac{-21}{5}

Numerator = -21, Denominator = 5

(iv) We have:

7

It can be written as 71\dfrac{7}{1}

Numerator = 7, Denominator = 1

(v) We have:

-8

It can be written as −81\dfrac{-8}{1}

Numerator = -8, Denominator = 1

Question 4

Which of the following are positive rational numbers?

(i) −78\dfrac{-7}{8}

(ii) −1317\dfrac{-13}{17}

(iii) −8−11\dfrac{-8}{-11}

(iv) 08\dfrac{0}{8}

(v) 0−7\dfrac{0}{-7}

Answer

(i) −78\dfrac{-7}{8}

⇒ The numerator is negative and the denominator is positive (opposite signs), making it a negative rational number.

Hence, it is not a positive rational number.

(ii) −1317\dfrac{-13}{17}

⇒ The signs are opposite, so it is a negative rational number.

Hence, it is not a positive rational number.

(iii) −8−11\dfrac{-8}{-11}

⇒ Both the numerator and denominator are negative, the signs cancel out (−8−11=811\dfrac{-8}{-11} = \dfrac{8}{11}), making the value positive.

Hence, it is a positive rational number.

(iv) 08\dfrac{0}{8}

⇒ The value is 0, and zero is neither positive nor negative.

Hence, it is not a positive rational number.

(v) 0−7\dfrac{0}{-7}

⇒ The value is 0, and zero is neither positive nor negative.

Hence, it is not a positive rational number.

Question 5

Which of the following are negative rational numbers?

(i) −165\dfrac{-16}{5}

(ii) −10−11\dfrac{-10}{-11}

(iii) -21

(iv) 0−3\dfrac{0}{-3}

(v) 17

Answer

(i) −165\dfrac{-16}{5}

⇒ The numerator is negative and the denominator is positive (opposite signs).

Hence, it is a negative rational number.

(ii) −10−11\dfrac{-10}{-11}

⇒ Both are negative, so the signs cancel out (1011\dfrac{10}{11}), making it positive.

Hence, it is not a negative rational number.

(iii) -21

⇒ It can be written as −211\dfrac{-21}{1}. The signs are opposite.

Hence, it is a negative rational number.

(iv) 0−3\dfrac{0}{-3}

⇒ The value is 0, which is neutral (neither positive nor negative).

Hence, it is not a negative rational number.

(v) 17

⇒ It is a positive integer (can be written as 171\dfrac{17}{1}).

Hence, it is not a negative rational number.

Question 6

Find four rational numbers equivalent to each of the following :

(i) 310\dfrac{3}{10}

(ii) −59\dfrac{-5}{9}

(iii) 6−13\dfrac{6}{-13}

(iv) 9

(v) -1

Answer

(i) 310\dfrac{3}{10}

We have:

310=3×210×2=3×310×3=3×410×4=3×510×5\dfrac{3}{10} = \dfrac{3 \times 2}{10 \times 2} = \dfrac{3 \times 3}{10 \times 3} = \dfrac{3 \times 4}{10 \times 4} = \dfrac{3 \times 5}{10 \times 5} [Multiply the numerator and denominator by 2, 3, 4, and 5]

⇒310=620=930=1240=1550\Rightarrow \dfrac{3}{10} = \dfrac{6}{20} = \dfrac{9}{30} = \dfrac{12}{40} = \dfrac{15}{50}

∴ Four rational numbers equivalent to 310\dfrac{3}{10} are:

620,930,1240,1550\dfrac{6}{20}, \dfrac{9}{30}, \dfrac{12}{40} , \dfrac{15}{50}

(ii) −59\dfrac{-5}{9}

We have:

−59=(−5)×29×2=(−5)×39×3=(−5)×49×4=(−5)×59×5\dfrac{-5}{9} = \dfrac{(-5) \times 2}{9 \times 2} = \dfrac{(-5) \times 3}{9 \times 3} = \dfrac{(-5) \times 4}{9 \times 4} = \dfrac{(-5) \times 5}{9 \times 5} [Multiply the numerator and denominator by 2, 3, 4, and 5]

⇒−59=−1018=−1527=−2036=−2545\Rightarrow \dfrac{-5}{9} = \dfrac{-10}{18} = \dfrac{-15}{27} = \dfrac{-20}{36} = \dfrac{-25}{45}

∴ Four rational numbers equivalent to −59\dfrac{-5}{9} are:

−1018,−1527,−2036,−2545\dfrac{-10}{18}, \dfrac{-15}{27}, \dfrac{-20}{36} , \dfrac{-25}{45}

(iii) 6−13\dfrac{6}{-13}

We have:

6−13=6×2(−13)×2=6×3(−13)×3=6×4(−13)×4=6×5(−13)×5\dfrac{6}{-13} = \dfrac{6 \times 2}{(-13) \times 2} = \dfrac{6 \times 3}{(-13) \times 3} = \dfrac{6 \times 4}{(-13) \times 4} = \dfrac{6 \times 5}{(-13) \times 5} [Multiply the numerator and denominator by 2, 3, 4, and 5]

⇒6−13=12−26=18−39=24−52=30−65\Rightarrow \dfrac{6}{-13} = \dfrac{12}{-26} = \dfrac{18}{-39} = \dfrac{24}{-52} = \dfrac{30}{-65}

∴ Four rational numbers equivalent to 6−13\dfrac{6}{-13} are:

12−26,18−39,24−52,30−65\dfrac{12}{-26}, \dfrac{18}{-39}, \dfrac{24}{-52} , \dfrac{30}{-65}

(iv) 9

Since 9=919 = \dfrac{9}{1}, we have:

91=9×21×2=9×31×3=9×41×4=9×51×5\dfrac{9}{1} = \dfrac{9 \times 2}{1 \times 2} = \dfrac{9 \times 3}{1 \times 3} = \dfrac{9 \times 4}{1 \times 4} = \dfrac{9 \times 5}{1 \times 5} [Multiply the numerator and denominator by 2, 3, 4, and 5]

⇒9=182=273=364=455\Rightarrow 9 = \dfrac{18}{2} = \dfrac{27}{3} = \dfrac{36}{4} = \dfrac{45}{5}

∴ Four rational numbers equivalent to 9 are:

182,273,364,455\dfrac{18}{2}, \dfrac{27}{3}, \dfrac{36}{4} , \dfrac{45}{5}

(v) -1

Since −1=−11-1 = \dfrac{-1}{1}, we have

−11=(−1)×21×2=(−1)×31×3=(−1)×41×4=(−1)×51×5\dfrac{-1}{1} = \dfrac{(-1) \times 2}{1 \times 2} = \dfrac{(-1) \times 3}{1 \times 3} = \dfrac{(-1) \times 4}{1 \times 4} = \dfrac{(-1) \times 5}{1 \times 5} [Multiply the numerator and denominator by 2, 3, 4, and 5]

⇒−1=−22=−33=−44=−55\Rightarrow -1 = \dfrac{-2}{2} = \dfrac{-3}{3} = \dfrac{-4}{4} = \dfrac{-5}{5}

∴ Four rational numbers equivalent to -1 are:

−22,−33,−44,−55\dfrac{-2}{2}, \dfrac{-3}{3}, \dfrac{-4}{4}, \dfrac{-5}{5}

Question 7

Write each of the following rational numbers with positive denominator :

(i) 16−21\dfrac{16}{-21}

(ii) 1−5\dfrac{1}{-5}

(iii) −7−12\dfrac{-7}{-12}

(iv) 5−1\dfrac{5}{-1}

(v) −6−1\dfrac{-6}{-1}

Answer

(i) We have:

16−21=16×(−1)−21×(−1)=−1621\dfrac{16}{-21} = \dfrac{16 \times (-1)}{-21 \times (-1)} = \dfrac{-16}{21}

∴ The rational number with a positive denominator is −1621\dfrac{-16}{21}.

(ii) We have:

1−5=1×(−1)−5×(−1)=−15\dfrac{1}{-5} = \dfrac{1 \times (-1)}{-5 \times (-1)} = \dfrac{-1}{5}

∴ The rational number with a positive denominator is −15\dfrac{-1}{5}.

(iii) We have:

−7−12=(−7)×(−1)(−12)×(−1)=712\dfrac{-7}{-12} = \dfrac{(-7) \times (-1)}{(-12) \times (-1)} = \dfrac{7}{12}

∴ The rational number with a positive denominator is 712\dfrac{7}{12}.

(iv) We have:

5−1=5×(−1)−1×(−1)=−51\dfrac{5}{-1} = \dfrac{5 \times (-1)}{-1 \times (-1)} = \dfrac{-5}{1}

∴ The rational number with a positive denominator is −51\dfrac{-5}{1}.

(v) We have:

−6−1=(−6)×(−1)−1×(−1)=61\dfrac{-6}{-1} = \dfrac{(-6) \times (-1)}{-1 \times (-1)} = \dfrac{6}{1}

∴ The rational number with a positive denominator is 61\dfrac{6}{1}.

Question 8

Express 49\dfrac{4}{9} as a rational number with numerator

(i) 24

(ii) -20

Answer

(i) Numerator 24

49=24x\dfrac{4}{9} = \dfrac{24}{x}

4 × x = 24 × 9 \quad [cross multiplication]

x = 246×941\dfrac{{\overset{6}{\cancel{24}}} × 9}{\underset{1}{\cancel{4}}}

x = 6×91\dfrac{6 × 9}{1}

x = 54

∴ The rational number with numerator 24 is 2454\dfrac{24}{54}.

(ii) Numerator -20

49=−20x\dfrac{4}{9} = \dfrac{-20}{x}

4 × x = -20 × 9 \quad [cross multiplication]

x = −20−5×941\dfrac{{\overset{-5}{\cancel{-20}}} × 9}{\underset{1}{\cancel{4}}}

x = −5×91\dfrac{-5 × 9}{1}

x = -45

∴ The rational number with numerator -20 is −20−45\dfrac{-20}{-45}.

Question 9

Express 38\dfrac{3}{8} as a rational number with denominator

(i) 48

(ii) -32

Answer

(i) Denominator 48

38=x48\dfrac{3}{8} = \dfrac{x}{48}

8 × x = 3 × 48 \quad [cross multiplication]

x = 3×48681\dfrac{3 × {\overset{6}{\cancel{48}}}}{{\underset{1}{\cancel{8}}}}

x = 3×61\dfrac{3 × 6}{1}

x = 18

∴ The rational number with denominator 48 is 1848\dfrac{18}{48}.

(ii) Denominator -32

38=x−32\dfrac{3}{8} = \dfrac{x}{-32}

8 × x = 3 × -32 \quad [cross multiplication]

x = 3×−32−481\dfrac{3 × {\overset{-4}{\cancel{-32}}}}{{\underset{1}{\cancel{8}}}}

x = 3×−41\dfrac{3 × -4}{1}

x = -12

∴ The rational number with denominator -32 is −12−32\dfrac{-12}{-32}.

Question 10

Express −611\dfrac{-6}{11} as a rational number with numerator

(i) -36

(ii) 42

Answer

(i) Numerator -36

−611=−36x\dfrac{-6}{11} = \dfrac{-36}{x}

-6 × x = -36 × 11 \quad [cross multiplication]

x = −366×11−61\dfrac{{\overset{6}{\cancel{-36}}} × 11}{\underset{1}{\cancel{-6}}}

x = 6×111\dfrac{6 × 11}{1}

x = 66

∴ The rational number with numerator -36 is −3666\dfrac{-36}{66}.

(ii) Numerator 42

−611=42x\dfrac{-6}{11} = \dfrac{42}{x}

-6 × x = 42 × 11 \quad [cross multiplication]

x = 427×11−6−1\dfrac{{\overset{7}{\cancel{42}}} × 11}{\underset{-1}{\cancel{-6}}}

x = 7×11−1\dfrac{7 × 11}{-1}

x = -77

∴ The rational number with numerator 42 is 42−77\dfrac{42}{-77}.

Question 11

Express 2−7\dfrac{2}{-7} as a rational number with denominator

(i) 42

(ii) -28

Answer

(i) Denominator 42

2−7=x42\dfrac{2}{-7} = \dfrac{x}{42}

-7 × x = 2 × 42 \quad [cross multiplication]

x = 2×426−7−1\dfrac{2 × {\overset{6}{\cancel{42}}}}{{\underset{-1}{\cancel{-7}}}}

x = 2×6−1\dfrac{2 × 6}{-1}

x = -12

∴ The rational number with denominator 42 is −1242\dfrac{-12}{42}.

(ii) Denominator -28

2−7=x−28\dfrac{2}{-7} = \dfrac{x}{-28}

-7 × x = 2 × -28 \quad [cross multiplication]

x = 2×−284−71\dfrac{2 × {\overset{4}{\cancel{-28}}}}{{\underset{1}{\cancel{-7}}}}

x = 2×41\dfrac{2 × 4}{1}

x = 8

∴ The rational number with denominator -28 is 8−28\dfrac{8}{-28}.

Question 12

Express −4836\dfrac{-48}{36} as a rational number with numerator

(i) -4

(ii) 8

Answer

(i) Numerator -4

−4836=−4x\dfrac{-48}{36} = \dfrac{-4}{x}

-48 × x = -4 × 36 \quad [cross multiplication]

x = −4×36−48\dfrac{-4 × 36}{-48}

x = −41×36−4812\dfrac{{\overset{1}{\cancel{-4}}} × 36}{\underset{12}{\cancel{-48}}}

x = 1×363121\dfrac{1 × {\overset{3}{\cancel{36}}}}{\underset{1}{\cancel{12}}}

x = 1×31\dfrac{1 × 3}{1}

x = 3

∴ The rational number with numerator -4 is −43\dfrac{-4}{3}.

(ii) Numerator 8

−4836=8x\dfrac{-48}{36} = \dfrac{8}{x}

-48 × x = 8 × 36 \quad [cross multiplication]

x = 8×36−48\dfrac{8 × 36}{-48}

x = 81×36−48−6\dfrac{{\overset{1}{\cancel{8}}} × 36}{\underset{-6}{\cancel{-48}}}

x = 1×366−6−1\dfrac{1 × {\overset{6}{\cancel{36}}}}{{\underset{-1}{\cancel{-6}}}}

x = 1×6−1\dfrac{1 × 6}{-1}

x = -6

∴ The rational number with numerator 8 is 8−6\dfrac{8}{-6}.

Question 13

Express 78−117\dfrac{78}{-117} as a rational number with numerator

(i) -6

(ii) 2

Answer

(i) Numerator -6

78−117=−6x\dfrac{78}{-117} = \dfrac{-6}{x}

78 × x = -6 × (-117) \quad [cross multiplication]

x = −6×(−117)78\dfrac{-6 × (-117)}{78}

x = −6−1×−1177813\dfrac{{\overset{-1}{\cancel{-6}}} × -117}{\underset{13}{\cancel{78}}}

x = −1×−117−9131\dfrac{-1 × {\overset{-9}{\cancel{-117}}}}{{\underset{1}{\cancel{13}}}}

x = −1×−91\dfrac{-1 × -9}{1}

x = 9

∴ The rational number with numerator -6 is −69\dfrac{-6}{9}.

(ii) Numerator 2

78−117=2x\dfrac{78}{-117} = \dfrac{2}{x}

78 × x = 2 × (-117) \quad [cross multiplication]

x = 2×(−117)78\dfrac{2 × (-117)}{78}

x = 21×−1177839\dfrac{{\overset{1}{\cancel{2}}} × {-117}}{\underset{39}{\cancel{78}}}

x = 1×−117−3391\dfrac{1 × {\overset{-3}{\cancel{-117}}}}{{\underset{1}{\cancel{39}}}}

x = 1×−31\dfrac{1 × -3}{1}

x = -3

∴ The rational number with numerator 2 is 2−3\dfrac{2}{-3}.

Question 14

Write each of the following rational numbers in standard form :

(i) 5632\dfrac{56}{32}

(ii) 16−40\dfrac{16}{-40}

(iii) −3654\dfrac{-36}{54}

(iv) −22−77\dfrac{-22}{-77}

(v) 78−65\dfrac{78}{-65}

(vi) −95114\dfrac{-95}{114}

(vii) −69115\dfrac{-69}{115}

(viii) 155−217\dfrac{155}{-217}

Answer

(i) 5632\dfrac{56}{32}

The HCF of 56 and 32 is 8.

Divide the numerator and denominator by 8:

5632=56÷832÷8=74\dfrac{56}{32} = \dfrac{56 \div 8}{32 \div 8} = \dfrac{7}{4}

∴ The standard form is 74\dfrac{7}{4}.

(ii) 16−40\dfrac{16}{-40}

First, multiply the numerator and denominator by -1 to make the denominator positive:

16×(−1)−40×(−1)=−1640\dfrac{16 \times (-1)}{-40 \times (-1)} = \dfrac{-16}{40}

The HCF of 16 and 40 is 8.

Divide the numerator and denominator by 8:

−16÷840÷8=−25\dfrac{-16 \div 8}{40 \div 8} = \dfrac{-2}{5}

∴ The standard form is −25\dfrac{-2}{5}.

(iii) −3654\dfrac{-36}{54}

The HCF of 36 and 54 is 18.

Divide the numerator and denominator by 18:

−36÷1854÷18=−23\dfrac{-36 \div 18}{54 \div 18} = \dfrac{-2}{3}

∴ The standard form is −23\dfrac{-2}{3}.

(iv) −22−77\dfrac{-22}{-77}

First, multiply the numerator and denominator by -1 to make the denominator positive:

−22×(−1)−77×(−1)=2277\dfrac{-22 \times (-1)}{-77 \times (-1)} = \dfrac{22}{77}

The HCF of 22 and 77 is 11.

22÷1177÷11=27\dfrac{22 \div 11}{77 \div 11} = \dfrac{2}{7}

∴ The standard form is 27\dfrac{2}{7}.

(v) 78−65\dfrac{78}{-65}

First, multiply the numerator and denominator by -1 to make the denominator positive:

78×(−1)−65×(−1)\dfrac{78 \times (-1)}{-65 \times (-1)} = −7865\dfrac{-78}{65}

The HCF of 78 and 65 is 13.

−78÷1365÷13=−65\dfrac{-78 \div 13}{65 \div 13} = \dfrac{-6}{5}

∴ The standard form is −65\dfrac{-6}{5}.

(vi) −95114\dfrac{-95}{114}

The HCF of 95 and 114 is 19.

−95÷19114÷19=−56\dfrac{-95 \div 19}{114 \div 19} = \dfrac{-5}{6}

∴ The standard form is −56\dfrac{-5}{6}.

(vii) −69115\dfrac{-69}{115}

The HCF of 69 and 115 is 23.

−69÷23115÷23=−35\dfrac{-69 \div 23}{115 \div 23} = \dfrac{-3}{5}

∴ The standard form is −35\dfrac{-3}{5}.

(viii) 155−217\dfrac{155}{-217}

First, multiply the numerator and denominator by -1 to make the denominator positive:

155×(−1)−217×(−1)\dfrac{155 \times (-1)}{-217 \times (-1)} = −155217\dfrac{-155}{217}

The HCF of 155 and 217 is 31.

−155÷31217÷31=−57\dfrac{-155 \div 31}{217 \div 31} = \dfrac{-5}{7}

∴ The standard form is −57\dfrac{-5}{7}.

Question 15

Find the value of x such that :

(i) −23=14x\dfrac{-2}{3} = \dfrac{14}{x}

(ii) 8−3=x6\dfrac{8}{-3} = \dfrac{x}{6}

(iii) 59=x−27\dfrac{5}{9} = \dfrac{x}{-27}

(iv) 116=−55x\dfrac{11}{6} = \dfrac{-55}{x}

(v) 15x=−3\dfrac{15}{x} = -3

(vi) −36x=2\dfrac{-36}{x} = 2

Answer

(i) −23=14x\dfrac{-2}{3} = \dfrac{14}{x}

We have,

−23=14x⇒(−2)×x=3×14[cross multiplication]⇒x=3×14−2⇒x=3×(−7)=−21\dfrac{-2}{3} = \dfrac{14}{x} \\[1em] \Rightarrow (-2) \times x = 3 \times 14 \quad \text{[cross multiplication]} \\[1em] \Rightarrow x = \dfrac{3 \times 14}{-2} \\[1em] \Rightarrow x = 3 \times (-7) = -21

Hence, x = -21.

(ii) 8−3=x6\dfrac{8}{-3} = \dfrac{x}{6}

We have,

8−3=x6⇒(−3)×x=8×6[cross multiplication]⇒x=8×6−3⇒x=8×(−2)=−16\dfrac{8}{-3} = \dfrac{x}{6} \\[1em] \Rightarrow (-3) \times x = 8 \times 6 \quad \text{[cross multiplication]} \\[1em] \Rightarrow x = \dfrac{8 \times 6}{-3} \\[1em] \Rightarrow x = 8 \times (-2) = -16

Hence, x = -16.

(iii) 59=x−27\dfrac{5}{9} = \dfrac{x}{-27}

We have,

59=x−27⇒9×x=5×(−27)[cross multiplication]⇒x=5×(−27)9⇒x=5×(−3)=−15\dfrac{5}{9} = \dfrac{x}{-27} \\[1em] \Rightarrow 9 \times x = 5 \times (-27) \quad \text{[cross multiplication]} \\[1em] \Rightarrow x = \dfrac{5 \times (-27)}{9} \\[1em] \Rightarrow x = 5 \times (-3) = -15

Hence, x = -15.

(iv) 116=−55x\dfrac{11}{6} = \dfrac{-55}{x}

We have,

116=−55x⇒11×x=6×(−55)[cross multiplication]⇒x=6×(−55)11⇒x=6×(−5)=−30\dfrac{11}{6} = \dfrac{-55}{x} \\[1em] \Rightarrow 11 \times x = 6 \times (-55) \quad \text{[cross multiplication]} \\[1em] \Rightarrow x = \dfrac{6 \times (-55)}{11} \\[1em] \Rightarrow x = 6 \times (-5) = -30

Hence, x = -30.

(v) 15x=−3\dfrac{15}{x} = -3

Since −3=−31-3 = \dfrac{-3}{1}, we have

15x=−31⇒(−3)×x=15×1[cross multiplication]⇒x=15−3=−5\dfrac{15}{x} = \dfrac{-3}{1} \\[1em] \Rightarrow (-3) \times x = 15 \times 1 \quad \text{[cross multiplication]} \\[1em] \Rightarrow x = \dfrac{15}{-3} = -5

Hence, x = -5.

(vi) −36x=2\dfrac{-36}{x} = 2

Since 2=212 = \dfrac{2}{1}, we have

−36x=21⇒2×x=(−36)×1[cross multiplication]⇒x=−362=−18\dfrac{-36}{x} = \dfrac{2}{1} \\[1em] \Rightarrow 2 \times x = (-36) \times 1 \quad \text{[cross multiplication]} \\[1em] \Rightarrow x = \dfrac{-36}{2} = -18 \\[1em]

Hence, x = -18.

Question 16

State whether the given statement is true or false :

(i) The quotient of two integers is always a rational number.

(ii) Every rational number is a fraction.

(iii) Zero is the smallest rational number.

(iv) Every fraction is a rational number.

Answer

(i) False
Reason — While a rational number is defined as the quotient of two integers pq\dfrac{p}{q}, the denominator q cannot be zero. If the divisor (the second integer) is zero, the quotient is undefined and is not a rational number.

(ii) False
Reason — Fractions typically refer to a part of a whole and are usually represented as a ratio of two natural numbers. Rational numbers include negative values (like −23-\dfrac{2}{3}) and integers (like -5), which are not traditionally considered fractions in their standard form.

(iii) False
Reason — Rational numbers include negative values. Therefore, any negative rational number (such as -1 or −12-\dfrac{1}{2}) is smaller than zero. Because the set of rational numbers extends infinitely in the negative direction, there is no "smallest" rational number.

(iv) True
Reason — A fraction is represented as ab\dfrac{a}{b} where a and b are whole numbers and b ≠ 0. Since whole numbers are also integers, every fraction fits the definition of a rational number.

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