GSEB Solutions Class 7 Maths Chapter 9 Rational Numbers Ex 9.1

Gujarat Board GSEB Textbook Solutions Class 7 Maths Chapter 9 Rational Numbers Ex 9.1 Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 7 Maths Chapter 9 Rational Numbers Ex 9.1

Question 1.
List five rational numbers between:
(i) -1 and 0
(ii) -2 and -1
(iii) $\frac { -4 }{ 5 }$ and $\frac { -2 }{ 3 }$
(iv) $\frac { 1 }{ 2 }$ and $\frac { 2 }{ 3 }$
Solution:
(i) -1 and 0
Since – 1 = $\frac { -1 }{ 1 }$ = $\frac { (-1)×10 }{ 1×10 }$ = $\frac { -10 }{ 10 }$
and 0 = $\frac { 0 }{ 1 }$ = $\frac { 0×10 }{ 1×10 }$ = $\frac { 0 }{ 10 }$
Also, $\frac { -10 }{ 10 }$ < $\frac { -9 }{ 10 }$ < $\frac { -8 }{ 10 }$ < $\frac { -7 }{ 10 }$ < $\frac { -6 }{ 10 }$ < $\frac { -5 }{ 10 }$ < $\frac { 0 }{ 10 }$
i.e $\frac { -9 }{ 10 }$, $\frac { -8 }{ 10 }$, $\frac { -7 }{ 10 }$, $\frac { -6 }{ 10 }$ and $\frac { -5 }{ 10 }$ are five rational numbers between $\frac { -10 }{ 10 }$ and $\frac { 0 }{ 10 }$(i.e. between -1 and 0)
Thus, the five rational numbers between -1 and 0 are $\frac { -9 }{ 10 }$, $\frac { -8 }{ 10 }$, $\frac { -7 }{ 10 }$, $\frac { -6 }{ 10 }$ and $\frac { -5 }{ 10 }$
or $\frac { -9 }{ 10 }$, $\frac { -4 }{ 5 }$, $\frac { -7 }{ 10 }$, $\frac { -3 }{ 5 }$, $\frac { -1 }{ 2 }$

(ii) – 2 and -1
Since – 2= $\frac { -2}{ 1 }$ = $\frac { (-2)×10 }{ 1×10 }$ = $\frac { -20 }{ 10 }$
– 1 = $\frac { -1 }{ 1 }$ = $\frac { (-1)×10 }{ 1×10 }$ = $\frac { -10 }{ 10 }$
Since, $\frac { -20 }{ 10 }$ < $\frac { -19 }{ 10 }$ < $\frac { -18 }{ 10 }$ < $\frac { -17 }{ 10 }$ < $\frac { -16 }{ 10 }$ < $\frac { -15 }{ 10 }$ < $\frac { -10 }{ 10 }$
or – 2 < $\frac { -19 }{ 10 }$ < $\frac { -9 }{ 5 }$ < $\frac { -17 }{ 10 }$ < $\frac { -8 }{ 5 }$ and $\frac { -3 }{ 2 }$ < – 1
Thus, the five rational numbers between – 2 and – 1 are $\frac { -19 }{ 10 }$, $\frac { -9 }{ 10 }$, $\frac { -17 }{ 10 }$, $\frac { -8 }{ 5 }$ and $\frac { -3 }{ 2 }$

(iii) $\frac { -4 }{ 5 }$ and $\frac { -2 }{ 3 }$

Thus, the five rational numbers between $\frac { -4 }{ 5 }$ and $\frac { -2 }{ 5 }$ are $\frac { -47 }{ 60 }$, $\frac { -23 }{ 30 }$, $\frac { -3 }{ 4 }$, $\frac { -11 }{ 15 }$ and $\frac { -43 }{ 60 }$

(iv) $\frac { 1 }{ 2 }$ and $\frac { 2 }{ 3 }$

Question 2.
Write four more rational numbers in each of the following patterns:
(i) $\frac { -3 }{ 5 }$, $\frac { -6 }{ 10 }$, $\frac { -9 }{ 15 }$, $\frac { -12 }{ 20 }$, ….
(ii) $\frac { -1 }{ 4 }$, $\frac { -2 }{ 8 }$, $\frac { -3 }{ 12 }$, …..
(iii) $\frac { -1 }{ 6 }$, $\frac { 2 }{ -12 }$, $\frac { 3 }{ -18 }$, $\frac { 4 }{ -24 }$, ….
(iv) $\frac { -2 }{ 3 }$, $\frac { 2 }{ -3 }$, $\frac { 4 }{ -6 }$, $\frac { 6 }{ -9 }$, ….
Solution:

∴ We have a pattern in these numbers. Obviously, the next four rational numbers would be:
$\frac { (-3)×5 }{ 5×5 }$ = $\frac { -15 }{ 25 }$
$\frac { (-3)×6 }{ 5×6 }$ = $\frac { -18 }{ 30 }$
$\frac { (-3)×7 }{ 5×7 }$ = $\frac { -21 }{ 35 }$
$\frac { (-3)×8 }{ 5×8 }$ = $\frac { -24 }{ 40 }$
∴ The next four required rational numbers are $\frac { -15 }{ 25 }$, $\frac { -18 }{ 30 }$, $\frac { -21 }{ 35 }$, $\frac { -24 }{ 40 }$.

(ii) $\frac { -1 }{ 4 }$, $\frac { -2 }{ 8 }$, $\frac { -3 }{ 12 }$, …..
∵ $\frac { -1 }{ 4 }$ = $\frac { (-1)×1 }{ 4×1 }$
$\frac { -2 }{ 8 }$ = $\frac { (-1)×2 }{ 4×2 }$
$\frac { -3 }{ 12 }$ = $\frac { (-1)×3 }{ 4×3 }$
i.e We have a pattern in these numbers.
∴ Next four rational numbers would be:
$\frac { (-1)×4 }{ 4×4 }$ = $\frac { -4 }{ 16 }$
$\frac { (-1)×5 }{ 4×5 }$ = $\frac { -5 }{ 20 }$
$\frac { (-1)×6 }{ 4×6 }$ = $\frac { -6 }{ 24 }$
$\frac { (-1)×7 }{ 4×7 }$ = $\frac { -7 }{ 28 }$
∴ The next four required rational numbers are $\frac { -15 }{ 25 }$, $\frac { -18 }{ 30 }$, $\frac { -21 }{ 35 }$, $\frac { -24 }{ 40 }$.

(iii) $\frac { -1 }{ 6 }$, $\frac { 2 }{ -12 }$, $\frac { 3 }{ -18 }$, $\frac { 4 }{ -24 }$, ….

Thus, the next four required rational numbers are $\frac { -5 }{ 30 }$, $\frac { -6 }{ 36 }$, $\frac { -7 }{ 42 }$, $\frac { -8 }{ 48 }$.

Thus, the next four required rational numbers are $\frac { 8 }{ -12 }$, $\frac { 10 }{ -15 }$, $\frac { 12 }{ -18 }$, $\frac { 14 }{ -21 }$.

Question 3.
Give four rational numbers equivalent to:
(i) $\frac { -2 }{ 7 }$
(ii) $\frac { 5 }{ -3 }$
(iii) $\frac { 4 }{ 9 }$
Solution:

∴ Four required rational numbers equivalent to

Thus, the four required rational numbers equivalent to

Thus, the four required rational numbers equivalent to
$\frac { 4 }{ 9 }$ are $\frac { 8 }{ 18 }$, $\frac { 12 }{ 27 }$, $\frac { 16 }{ 36 }$ and
$\frac { 20 }{ 45 }$.

Question 4.
Draw the number line and represent the following rational numbers on it:
(i) $\frac { 3 }{ 4 }$
(ii) $\frac { -5 }{ 8 }$
(iii) $\frac { -7 }{ 4 }$
(iv) $\frac { 7 }{ 8 }$
Solution:
(i) $\frac { 3 }{ 4 }$

(ii) $\frac { -5 }{ 8 }$

(iii) $\frac { -7 }{ 4 }$

(iv) $\frac { 7 }{ 8 }$

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.

Solution:
Since AP = PQ = QB
∴ Distance between 2 and 3 is divided into 3 equal parts.
Similarly, distance between -2 and -1 is also divided into three equal parts.

Question 6.
Which of the following pairs represent the same rational number?
(i) $\frac { -7 }{ 21 }$ and $\frac { 3 }{ 9 }$
(ii) $\frac { -16 }{ 20 }$ and $\frac { 20 }{ -25 }$
(iii) $\frac { -2 }{ -3 }$ and $\frac { 2 }{ 3 }$
(iv) $\frac { -3 }{ 5 }$ and $\frac { -12 }{ 20 }$
(v) $\frac { 8 }{ -5 }$ and $\frac { -24 }{ 15 }$
(vi) $\frac { 1 }{ 3 }$ and $\frac { -1 }{ 9 }$
(vii) $\frac { -5 }{ -9 }$ and $\frac { 5 }{ -9 }$
Solution:
(i) $\frac { -7 }{ 21 }$ and $\frac { 3 }{ 9 }$
Here, $\frac { -7 }{ 21 }$ is a negative rational number and $\frac { 3 }{ 9 }$ is a positive rational number.
∴ $\frac { -7 }{ 21 }$ ≠ $\frac { 3 }{ 9 }$

(ii) $\frac { -16 }{ 20 }$ and $\frac { 20 }{ -25 }$
We have
$\frac { -16 }{ 20 }$ = $\frac { (-16)÷4 }{ 20÷4 }$ = $\frac { -4 }{ 5 }$
= –$\frac { 4 }{ 5 }$
and $\frac { 20 }{ -25 }$ = $\frac { 20÷5 }{ (-25)÷5 }$ = $\frac { 4 }{ -5 }$
= –$\frac { 4 }{ 5 }$
∴ $\frac { -16 }{ 20 }$ and $\frac { 20 }{ -25 }$ represent the same rational number.

(iii) $\frac { -2 }{ -3 }$ and $\frac { 2 }{ 3 }$
We have
$\frac { -2 }{ -3 }$ = $\frac { (-2)÷(-1) }{ (-3)÷(-1) }$ = $\frac { 2 }{ 3 }$
∴ $\frac { -2 }{ -3 }$ = $\frac { 2 }{ 3 }$
Thus, $\frac { -2 }{ -3 }$ and $\frac { 2 }{ 3 }$ represent the same rational number.

(iv) $\frac { -3 }{ 5 }$ and $\frac { -12 }{ 20 }$
We have
$\frac { -3 }{ 5 }$ = $\frac { (-3)×4 }{ 5×4) }$ = $\frac { -12 }{ 20 }$
∴ $\frac { -3 }{ 5 }$ = $\frac { -12 }{ 20 }$
Thus, $\frac { -3 }{ 5 }$ and $\frac { -12 }{ 20 }$ represent the same rational number.

(v) $\frac { 8 }{ -5 }$ and $\frac { -24 }{ 15 }$
We have
$\frac { 8 }{- 5 }$ = $\frac { 8×3 }{ ((-5)×3) }$ = $\frac { 24 }{ -15 }$ = $\frac { -24 }{ 15 }$
∴ $\frac { 8 }{ -5 }$ = $\frac { -24 }{ 15 }$
Thus, $\frac { 8 }{ -5 }$ and $\frac { -24 }{ 15 }$ represent the same rational number.

(vi) $\frac { 1 }{ 3 }$ and $\frac { -1 }{ 9 }$
Here, $\frac { 1 }{ 3 }$ is a positive integer and $\frac { -1 }{ 9 }$ is a negative integer.
∴ $\frac { 1 }{ 3 }$ ≠ $\frac { -1 }{ 9 }$

(vii) $\frac { -5 }{ -9 }$ and $\frac { 5 }{ -9 }$
Since $\frac { -5 }{ -9 }$ is a positive integer $\frac { 5 }{ -9 }$ is a negative integer.
∴ $\frac { -5 }{ -9 }$ ≠ $\frac { 5 }{ -9 }$

Question 7.
Rewrite the following rational numbers in the simplest form:
(i) $\frac { -8 }{ 6 }$
(ii) $\frac { 25 }{ 45 }$
(iii) $\frac { -44 }{ 72 }$
(iv) $\frac { -8 }{ 10 }$
Solution:
(i) $\frac { -8 }{ 6 }$
∵ HCF of 8 and 6 is 2.
∴ $\frac { -8 }{ 6 }$ = $\frac { (-8)÷2 }{ 6÷2 }$ = $\frac { -4 }{ 3 }$
The simplest form of $\frac { -8 }{ 6 }$ is $\frac { -4 }{ 3 }$.

(ii) $\frac { 25 }{ 45 }$
∵ HCF of 25 and 45 is 5.
∴ $\frac { 25 }{ 45 }$ = $\frac { 25÷5 }{ 45÷5 }$
= $\frac { 5 }{ 9 }$
Thus, the simplest form of $\frac { 25 }{ 45 }$ is $\frac { 5 }{ 9 }$ .

(iii) $\frac { -44 }{ 72 }$
∵ HCF of 44 and 72 is 4.
∴ $\frac { -44 }{ 72 }$ = $\frac { (-44)÷4 }{ 72÷4 }$
= $\frac { -11 }{ 18 }$
Thus, the simplest form of $\frac { -44 }{ 72 }$ is $\frac { -11 }{ 18 }$ .

(iv) $\frac { -8 }{ 10 }$
∵ HCF of 25 and 45 is 5.
∴ $\frac { -8 }{ 10 }$ = $\frac { (-8)÷2 }{ 10÷2 }$
= $\frac { -4 }{ 5 }$
Thus, the simplest form of $\frac { -8 }{ 10 }$ is $\frac { -4 }{ 5 }$ .

Question 8.
Fill in the boxes with the correct symbol out of >, < and =.

Solution:

Question 9.
Which is greater in each of the following:
(i) $\frac { 2 }{ 3 }$ and $\frac { 5 }{ 2 }$
(ii) $\frac { -5 }{ 6 }$ and $\frac { -4 }{ 3 }$
(iii) $\frac { -3 }{ 4 }$ and $\frac { 2 }{ -3 }$
(iv) $\frac { -1 }{ 4 }$ and $\frac { 1 }{ 4 }$
(v) -3$\frac { 2 }{ 7 }$, -3$\frac { 4 }{ 5 }$
Solution:
(i) $\frac { 2 }{ 3 }$ and $\frac { 5 }{ 2 }$

Thus, $\frac { 5 }{ 2 }$ is greater rational number.

(ii) $\frac { -5 }{ 6 }$ and $\frac { -4 }{ 3 }$

Thus, $\frac { -5 }{ 6 }$ is greater rational number.

(iii) $\frac { -3 }{ 4 }$ and $\frac { 2 }{ -3 }$

Thus, the rational number $\frac { 2 }{ -3 }$ is greater.

(iv) $\frac { -1 }{ 4 }$ and $\frac { 1 }{ 4 }$
Since a positive rational number is always greater than a negative rational number.
∴ $\frac { 1 }{ 4 }$ and $\frac { -1 }{ 4 }$
i.e The greater rational number is $\frac { 1 }{ 4 }$.

(v) -3$\frac { 2 }{ 7 }$, -3$\frac { 4 }{ 5 }$

Thus, the rational number -3$\frac { 2 }{ 7 }$ is greater.

Question 10.
Write the following rational numbers in ascending order:
(i) $\frac { -3 }{ 5 }$, $\frac { -2 }{ 5 }$, $\frac { -1 }{ 5 }$
(ii) $\frac { -1 }{ 3 }$, $\frac { -2 }{ 9 }$, $\frac { -4 }{ 3 }$
(iii) $\frac { -3 }{ 7 }$, $\frac { -3 }{ 2 }$, $\frac { -3 }{ 4 }$
Solution:
(i) $\frac { -3 }{ 5 }$, $\frac { -2 }{ 5 }$, $\frac { -1 }{ 5 }$
Since (-3) < (-2) < (-1)
∴$\frac { -3 }{ 5 }$ < $\frac { -2 }{ 5 }$ < $\frac { -1 }{ 5 }$
∴ The ascending order of the given rational numbers is $\frac { -3 }{ 5 }$ < $\frac { -2 }{ 5 }$ < $\frac { -1 }{ 5 }$.

(ii) $\frac { -1 }{ 3 }$, $\frac { -2 }{ 9 }$, $\frac { -4 }{ 3 }$
Since, LCM of 3 and 9 is 9.

Thus The ascending order of the given rational numbers is $\frac { -4 }{ 3 }$, $\frac { -1 }{ 3 }$, $\frac { -2 }{ 9 }$.

(iii) $\frac { -3 }{ 7 }$, $\frac { -3 }{ 2 }$, $\frac { -3 }{ 4 }$
∵ LCM of 7, 2 and 4 is 28.

Thus The ascending order of the given rational numbers is $\frac { -3 }{ 2 }$, $\frac { -3 }{ 4 }$, $\frac { -3 }{ 7 }$.