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 \(\frac { -3 }{ 2 }\), \(\frac { -3 }{ 4 }\), \(\frac { -3 }{ 7 }\).