Wednesday, May 27, 2020

NCERT solution class 7 chapter 9 Rational numbers exercise 9.1 mathematics

Exercise 9.1



"Rational Numbers" Chapter 9 - Introduction - NCERT Class 7th Maths Solutions


Q 1, Ex 9.1 - Rational Numbers - Chapter 9 - Maths Class 7th - NCERT

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 $\dfrac{4}{6}\times \dfrac{6}{6}=\dfrac{24}{36}$
⇒ $\dfrac{19}{36},\dfrac{20}{36} , \dfrac{21}{36} , \dfrac{22}{36}, \dfrac{23}{36}$



Q 2, Ex 9.1 - Rational Numbers - Chapter 9 - Maths Class 7th - NCERT



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
$\dfrac{2}{-3}\times \dfrac{4}{4} , \dfrac{2}{-3}\times \dfrac{5}{5} , \dfrac{2}{-3}\times \dfrac{6}{6}  , \dfrac{2}{-3}\times \dfrac{7}{7} \dots$
$\dfrac{8}{-12} , \dfrac{10}{-15} , \dfrac{12}{-18} , \dfrac{14}{-21} \dots$



Q 3, Ex 9.1 - Rational Numbers - Chapter 9 - Maths Class 7th - NCERT



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


Q 4, Ex 9.1 - Rational Numbers - Chapter 9 - Maths Class 7th - NCERT



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 = 


Q 6, Ex 9.1 - Rational Numbers - Chapter 9 - Maths Class 7th - NCERT



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) 


Q 7, Ex 9.1 - Rational Numbers - Chapter 9 - Maths Class 7th - NCERT



QUESTION 7
Rewrite the following rational numbers in the simplest form:
(i)  (ii) 
(iii)  (iv) 

Answer:

(i) 
(ii) 
(iii) 
(iv) 


Q 8, Ex 9.1 - Rational Numbers - Chapter 9 - Maths Class 7th - NCERT



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:

Q 9, Ex 9.1 - Rational Numbers - Chapter 9 - Maths Class 7th - NCERT

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,

Q 10, Ex 9.1 - Rational Numbers - Chapter 9 - Maths Class 7th - NCERT


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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