# 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 $$\frac { -3 }{ 2 }$$, $$\frac { -3 }{ 4 }$$, $$\frac { -3 }{ 7 }$$.