NCERT solution class 7 chapter 9 Rational numbers exercise 9.1 mathematics

Exercise 9.1


QUESTION 1

List five rational numbers between:

(i) -1 and 0

Sol :

Let us write — 1 and 0 as rational numbers with denominator 6

Here we take 6 because we want five rational number

-1 = \dfrac{-6}{\phantom{-}6} and 0=\dfrac{0}{6}

\dfrac{-6}{\phantom{-}6}<\dfrac{-5}{\phantom{-}6}<\dfrac{-4}{\phantom{-}6}<\dfrac{-3}{\phantom{-}6}<\dfrac{-2}{\phantom{-}6}<\dfrac{-1}{\phantom{-}6}<\dfrac{0}{6}

-1<\dfrac{-5}{\phantom{-}6}<\dfrac{-4}{\phantom{-}6}<\dfrac{-3}{\phantom{-}6}<\dfrac{-2}{\phantom{-}6}<\dfrac{-1}{\phantom{-}6}<0


(ii) -2 and -1

Sol :

Let us write —2 and — 1 as rational numbers with denominator 6.

-2 = \dfrac{-12}{\phantom{-}6} and -1=\dfrac{-6}{\phantom{-}6}

\dfrac{-12}{\phantom{-}6}<\dfrac{-11}{\phantom{-}6}<\dfrac{-10}{\phantom{-}6}<\dfrac{-9}{\phantom{-}6}<\dfrac{-8}{\phantom{-}6}<\dfrac{-7}{\phantom{-}6}<\dfrac{-6}{\phantom{-}6}

-2<\dfrac{-11}{\phantom{-}6}<\dfrac{-10}{\phantom{-}6}<\dfrac{-9}{\phantom{-}6}<\dfrac{-8}{\phantom{-}6}<\dfrac{-7}{\phantom{-}6}<-1


(iii) \dfrac{-4}{\phantom{-}5} \text{ and } \dfrac{-2}{\phantom{-}3}

Sol :

Lets make the denominator same

On taking H.C.F of 3 and 5 is 15 . So , denominator have to be 15 and to make denominator 15 we multiply \dfrac{3}{3} on L.H.S and by \dfrac{5}{5} on R.H.S

\dfrac{-4}{\phantom{-}5}\times \dfrac{3}{3}=\dfrac{-12}{\phantom{-}15} and \dfrac{-2}{\phantom{-}3}\times \dfrac{5}{5}=\dfrac{-10}{\phantom{-}15}

\dfrac{-12}{\phantom{-}15}\times \dfrac{3}{3} and \dfrac{-10}{\phantom{-}15}\times \dfrac{3}{3} [Multiplying both sides by \dfrac{3}{3} to get five rational numbers ]

\dfrac{-36}{\phantom{-}45} and \dfrac{-31}{\phantom{-}45}

Five rational numbers are

\dfrac{-35}{\phantom{-}45},\dfrac{-34}{\phantom{-}45},\dfrac{-33}{\phantom{-}45},\dfrac{-32}{\phantom{-}45},\dfrac{-31}{\phantom{-}45}

 

(iv) \dfrac{1}{2} \text{ and } \dfrac{2}{3}

Sol :

H.C.F of 2 and 3 is 6 . So , the denominator have to be 6

\dfrac{1}{2}\times \dfrac{3}{3}=\dfrac{3}{6}  and \dfrac{2}{3}\times\dfrac{2}{2}=\dfrac{4}{6}

\dfrac{3}{6}\times \dfrac{6}{6}=\dfrac{18}{36} and \dfrac{4}{6}\times \dfrac{6}{6}=\dfrac{24}{36}

\dfrac{19}{36},\dfrac{20}{36} , \dfrac{21}{36} , \dfrac{22}{36}, \dfrac{23}{36}

 


QUESTION 2

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

(i) \dfrac{-3}{\phantom{-}5},\dfrac{-6}{\phantom{-}10},\dfrac{-9}{\phantom{-}15},\dfrac{-12}{\phantom{-}20}\dots

Sol : \dfrac{-3}{\phantom{-}5}\times \dfrac{1}{1},\dfrac{-3}{\phantom{-}5}\times \dfrac{2}{2},\dfrac{-3}{\phantom{-}5}\times \dfrac{3}{3},\dfrac{-3}{\phantom{-}5}\times \dfrac{4}{4}\dots

It can be observed that the numerator is a multiple of 3 while the denominator is a multiple of 5 and as we increase them further, these multiples are increasing. Therefore, the next four rational numbers in this pattern are

\dfrac{-3}{\phantom{-}5}\times \dfrac{5}{5},\dfrac{-3}{\phantom{-}5}\times \dfrac{6}{6},\dfrac{-3}{\phantom{-}5}\times \dfrac{7}{7},\dfrac{-3}{\phantom{-}5}\times \dfrac{8}{8}\dots

\dfrac{-15}{\phantom{-}25},\dfrac{-18}{\phantom{-}30},\dfrac{-21}{\phantom{-}35},\dfrac{-24}{\phantom{-}40}\dots

 

(ii) \dfrac{-1}{\phantom{-}4},\dfrac{-2}{\phantom{-}8},\dfrac{-3}{\phantom{-}12}\dots

Sol :

\dfrac{-1}{\phantom{-}4}\times \dfrac{1}{1}, \dfrac{-1}{\phantom{-}4} \times \dfrac{2}{2}, \dfrac{-1}{\phantom{-}4} \times \dfrac{3}{3} ,\dots

The next four rational numbers in this patterns are

\dfrac{-1}{\phantom{-}4} \times \dfrac{4}{4} , \dfrac{-1}{\phantom{-}4} \times \dfrac{5}{5} , \dfrac{-1}{\phantom{-}4} \times \dfrac{6}{6} , \dfrac{-1}{\phantom{-}4} \times \dfrac{7}{7}

\dfrac{-4}{\phantom{-}16} , \dfrac{-5}{\phantom{-}20} , \dfrac{-6}{\phantom{-}24} , \dfrac{-7}{\phantom{-}28} \dots

 

(iii) \dfrac{-1}{\phantom{-}6},\dfrac{2}{-12},\dfrac{3}{-18},\dfrac{4}{-24}\dots

Sol :

\dfrac{-1}{\phantom{-}6}\times \dfrac{1}{1} , \dfrac{-1}{\phantom{-}6}\times \dfrac{2}{2} , \dfrac{-1}{\phantom{-}6}\times \dfrac{3}{3} , \dfrac{-1}{\phantom{-}6}\times \dfrac{4}{4}\dots

The next four rational numbers in this pattern are

\dfrac{-1}{\phantom{-}6}\times \dfrac{5}{5} , \dfrac{-1}{\phantom{-}6}\times \dfrac{6}{6}  , \dfrac{-1}{\phantom{-}6}\times \dfrac{7}{7} , \dfrac{-1}{\phantom{-}6}\times \dfrac{8}{8}\dots

\dfrac{5}{-30} , \dfrac{6}{-36} , \dfrac{7}{-42} , \dfrac{8}{-48}\dots

 

(iv) \dfrac{-2}{\phantom{-}3},\dfrac{2}{-3},\dfrac{4}{-6},\dfrac{6}{-9}\dots

Sol :

\dfrac{-2}{\phantom{-}3} , \dfrac{2}{-3} , \dfrac{2}{-3}\times \dfrac{2}{2} , \dfrac{-2}{\phantom{-}3} \times \dfrac{3}{3}\dots

The next four rational numbers in this pattern are

\dfrac{2}{-3}\times \dfrac{4}{4} , \dfrac{2}{-3}\times \dfrac{5}{5} , \dfrac{2}{-3}\times \dfrac{6}{6}  , \dfrac{2}{-3}\times \dfrac{7}{7} \dots

\dfrac{8}{-12} , \dfrac{10}{-15} , \dfrac{12}{-18} , \dfrac{14}{-21} \dots

 


QUESTION 3

Give four rational numbers equivalent to:

(i) \dfrac{-2}{7}

Sol :

\dfrac{-2}{7}\times \dfrac{2}{2} , \dfrac{-2}{7} \times \dfrac{3}{3} , \dfrac{-2}{7} \times \dfrac{4}{4} , \dfrac{-2}{7} \times \dfrac{5}{5}

\dfrac{-4}{\phantom{-}14} , \dfrac{-6}{\phantom{-}21} , \dfrac{-8}{\phantom{-}28} , \dfrac{-10}{\phantom{-}35}

 

(ii) \dfrac{5}{-3}

Sol :

\dfrac{5}{-3}\times \dfrac{2}{2} , \dfrac{5}{-3} \times \dfrac{3}{3} , \dfrac{5}{-3} \times \dfrac{4}{4} , \dfrac{5}{-3} \times \dfrac{5}{5}

\dfrac{10}{-6} , \dfrac{15}{-9} , \dfrac{20}{-12} , \dfrac{25}{-15}

 

(iii) \dfrac{4}{9}

Sol :

\dfrac{4}{9}\times \dfrac{2}{2} , \dfrac{4}{9} \times \dfrac{3}{3} , \dfrac{4}{9} \times \dfrac{4}{4} , \dfrac{4}{9} \times \dfrac{5}{5}


QUESTION 4

Draw the number line and represent the following rational numbers on it:

(i) \dfrac{3}{4}

Sol :

This fraction represents 3 parts out of 4 equal parts. Therefore, each space between two integers on number line must be divided into 4 equal parts.

 can be represented as


(ii) \dfrac{-5}{\phantom{-}8}

Sol :

This fraction represents 5 parts out of 8 equal parts. Negative sign represents that it is on the negative side of number line. Therefore, each space between two integers on number line must be divided into 8 equal parts.

 can be represented as


(iii) \dfrac{-7}{\phantom{-}4}

Sol :

This fraction represents 1 full part and 3 parts out of 4 equal parts. Negative sign represents that it is on the negative side of number line. Therefore, each space between two integers on number line must be divided into 4 equal parts.

 can be represented as


(iv) \dfrac{7}{8}

Sol :

This fraction represents 7 parts out of 8 equal parts. Therefore, each space between two integers on number line must be divided into 8 equal parts.

 can be represented as


QUESTION 5

The points P, Q, R, S, T, U, A and B on the number line are such that,

TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by P, Q, R and S.

Sol :

Distance between U and T = 1 unit

It is divided into 3 equal parts.

TR = RS = SU =

R = 

S = 

Similarly,

AB = 1 unit

It is divided into 3 equal parts.

P = 

Q = 


QUESTION 6

Which of the following pairs represent the same rational number?

(i)  (ii)  (iii) 

(iv)  (v)  (vi) 

(vii) 

Sol:

(i) 

As, therefore, it does not represent same rational numbers.

(ii) 

Therefore, it represents same rational numbers.

(iii) 

Therefore, it represents same rational numbers.

(iv) 

Therefore, it represents same rational numbers.

(v) 

Therefore, it represents same rational numbers.

(vi) 

As, therefore, it does not represent same rational numbers.

(vii) 


QUESTION 7

Rewrite the following rational numbers in the simplest form:

(i)  (ii) 

(iii)  (iv) 

Answer:

(i) 

(ii) 

(iii) 

(iv) 


QUESTION 8

Fill in the boxes with the correct symbol out of >, <, and =

(i)  (ii)  (iii) 

(iv)  (v)  (vi) 

(vii) 

Sol :

(i)

As −15 < 14,

Therefore,

(ii)

As −28 < −25

Therefore, 

(iii) Here, 

Therefore, 

(iv)

As −32 > −35,

Therefore, 

(v)

As −4 < −3,

Therefore, 

(vi) 

(vii) 


Page No 184:

QUESTION 9

Which is greater in each of the following?

(i)  (ii)  (iii) 

(iv)  (v) 

Answer:

(i) 

By converting these into like fractions,

As 15 > 4, therefore, is greater.

(ii) 

(iii)

By converting these into like fractions,

(iv) 

(v) 

By converting these into like fractions,


QUESTION 10

Write the following rational numbers in ascending order:

(i)  (ii)  (iii) 

Answer:

(i) 

As −3 < −2 < −1,

(ii) 

By converting these into like fractions,

As −12 < −3 < −2,

(iii) 

By converting these into like fractions,

As −42 < −21 < −12,


 

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