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 }).