NCERT Exercise 9.1 Question 1
Rational Numbers 7th Math
NCERT Exercise 9.1 Question 1
In this section we will solve the questions given in the NCERT Exercise 9.1, class seven, mathematics. We will learn to find the rational numbers between given two rational numbers. Solution of the NCERT Exercise 9.1 questions 1(i), 1(ii), 1(iii), and 1(iv).
NCERT Exercise 9.1 Solution of questions 1(i) to question 1(iv)
Question (1) List five rational numbers between:
Question (1)(i) – 1 and 0
Solution
Given, rational numbers are – 1 and 0
Thus, find 5 rational numbers between given rational numbers.
Now, – 1 = – 1/1
And, 0 = 0/1
Here, we have to find 5 rational numbers between the given rational numbers –1 and 0.
Thus, after multiplying the numerators and denominators of the given rational numbers by 5 + 1 = 6, we get
– 1/1 = – 1 × 6/1 × 6 = – 6/6
And, 0/1 = 0 × 6/1 × 6 = 0/6
Now, we have two rational numbers, – 6/6 and 0/6
Thus, five rational numbers between – 6/6 and 0/6 are
– 5/6, – 4 2/6 3, – 3/6 2, – 2/6 3 and – 1/6
= – 5/6, – 2/3, – 1/2, – 1/3 and – 1/6
Thus five rational numbers between given rational numbers – 1 and 0 are
– 5/6, – 2/3, – 1/2, – 1/3 and – 1/6 Answer
Alternate method to find the five rational numbers between given rational number – 1 and 0 using calculation of average between two rational numbers
We can find a rational number between two rational numbers by calculating their average.
1 and 0.= –1 + 0/2 = – 1/2
2nd rational number between – 1 and 0
– 1 + (– 1)/2/2
= – 1 – 1/2/2
= – 2 – 1/2/2
= – 3/2/2
= – 3/2 × 2
= – 3/4
∴ 2nd rational number between – 1 and 0 = – 3/4
= Average of –1 and –3/4
= – 1 + (–3)/4/2
= – 1 – 3/4/2
= – 4 –3/4/2
= – 7/4/2
= – 7/4 × 2
= –7/8
∴ 3rd number between –1 and 0 = – 7/8
4th rational number between –1 and 0
Average of – 1 and – 7/8
= – 1 + (– 7)/8/2
= – 1 – 7/8/2
= – 8 –7/8/2
= –15/8/2
= – 15/ 8 × 2
= – 15/16
Thus 4th rational number between – 1 and 0 = – 15/16
5th rational number between – 1 and 0
Average of – 1 and – 15/16
= – 1 + (– 15)/16/2
= – 1 – 15/16/2
= – 16 – 15/16/2
= – 31/16/2
= – 31/16 × 2
= – 31/32
∴ 5th rational number between –1 and 0 will be
– 1/2, – 3/4, – 7/8, – 15/16 and – 31/32 Answer
Question (1) (ii) – 2 and – 1
Solution:
Given, –2 and –1
Thus, 5 rational numbers between given rational numbers = ?
Now, given rational numbers can be written in the form of p/q as follows
–2 = – 2/1 and – 1 = – 1/1
Now, since we have to find 5 rational numbers between given rational numbers, thus after multiplying the numerator and denominators of given rational numbers with 5 + 1= 6, we get
– 2/1 = – 2 × 6/1 × 6 = – 12/6
And, – 1/1 = – 1 × 6/1 × 6 = – 6/6
Now, clearly, the 5 rational numbers between given rational numbers are
– 11/6, – 10 5 /6 3, – 9 3/6 2, – 8 4/ 6 3 and – 7/6 Answer
Thus five rational numbers between – 2 and – 1 are
– 11/6, – 5/3, – 3/2, – 4/3, and – 7/6 Answer
Alternate method to find the five rational numbers between given rational numbers – 2 and – 1 using calculation of average between two rational numbers
Solution
Given, –2 and –1
To find five rational number between them
we can find the rational numbers between two numbers by finding their average
∴ Rational number between
–2 and –1
= – 2 + (– 1)/2
= – 2 – 1/2
= – 3/2 ---(i)
2nd rational number between – 2 and – 3/2
= – 2 + (– 3/2)/2
= – 2 – 3/2/2
= – 4 – 3/2/2
= – 7/2/2
= – 7/2 × 2
= – 7/4 - - - - (ii)
3rd rational number between – 2 and – 7/4
= – 2 + (– 7/4)/2
= – 2 – 7/4/2
= – 8 – 7/4/2
= – 15/4/2
= – 15/4 × 2
= – 15/8 ----(iii)
4th rational number between – 2 and – 15/8
= – 2 + (– 15)/8/2
= – 2 – 15/8/2
= – 16 – 15/8/2
= – 31/8/2
= – 31/8 × 2
= – 31/16 ----(iv)
5th rational number between – 2 and – 31/16
= – 2 + (– 31)/16/2
= – 2 – 31/16/2
= – 32 – 31/16/2
= – 63/16/2
= – 63/16 × 2
= – 63/32 ----(v)
Thus five rational number between – 2 and – 1 are
– 3/2, – 7/4, – 15/8, – 31/16, and – 63/32 Answer
(iii) – 4/5 and – 2/3
Solution :
Given, rational numbers are – 4/5 and – 2/3
Thus, 5 rational numbers between given rational numbers = ?
The LCM of denominators 5 and 3 of given rational number = 5 × 3 = 15
Thus, in order to finding the 5 rational numbers between given rational number, to make the denominators of both of the rational numbers equal, i.e. = LCM × 4 = 15 × 4 = 60
After multiplying the numerators and denominators of the first rational number by 12 and the second rational number by 20, we get
– 4/5 = – 4 × 12/5 × 12 = – 48/60
And, – 2/3 = – 2 × 20/3 × 20 = – 40/60
Now, clearly, the some of the rational numbers between obtained rational numbers, – 48/60 and – 40/60 are
– 47/60, – 46 23/ 60 30, – 45 3/60 4, – 44 11/ 60 15, – 43/60, – 42 7/ 60 10 , and – 41/60
= – 47/60, – 23/30, – 3/4, – 11/15, – 43/60, – 7/10, and – 41/60
Thus, by taking any five, we have the 5 rational numbers between given rational numbers are
= – 47/60, – 23/30, – 11/15, – 43/60, and – 41/60 Answer
Alternate Method: Finding the five rational numbers between – 4/5 and – 2/3 using calculation of average of two rational numbers.
Given, rational numbers are – 4/5 and – 2/3
Thus, 5 rational numbers between given rational numbers = ?
The LCM of denominators 5 and 3 of given rational number = 5 × 3 = 15
Thus, in order to find the rational numbers between given rational number, to make the denominators of both of the rational numbers equal,
After multiplying the numerators and denominators of the first rational number by 3 and the second rational number by 5, we get
– 4/5 = – 4 × 3/5 × 3 = – 12/15
And, – 2/3 = – 2 × 5/3 × 5 = – 10/15
Thus, rational number between
– 12/15 and – 10/15 = – 11/15
Now, to find the rational number between – 11/15 and – 10/15 we will calculate the average between them.
= – 11/15 + – 10/15/2
= – 11 + (– 10)/15/2
= – 11 – 10/15 × 2
= – 21/30
Now, to another rational numbers between – 21/30 and – 10/15 are clearly
– 20/30 , – 19/30 , – 18/30 , – 17/30 , – 16/30 , – 15/30, etc.
Thus, taking any five
– 11/15 , – 20 2/ 30 3 , – 19/30 , – 18 3/ 30 5 and – 17/30
Thus, taking any five we have 5 rational numbers between – 4/5 and – 2/3 are
= – 11/15, – 2/3, – 19/30, – 3/5, and – 17/30 Answer
Question (1) (iv) – 1/2 and 2/3
Solution :
Given, rational numbers are – 1/2 and 2/3
Thus, 5 rational numbers between given rational numbers = ?
The LCM of denominators of given rational numbers = 2 × 3 = 6
Now, to find 5 rational numbers between given numbers make the denominators of given rational numbers equal to LCM of their denominator = 6
Thus, – 1/2 = – 1 × 3/2 × 6 = – 3/6
And, 2/3 = 2 × 2/3 × 2 = 4/6
Thus, rational numbers between obtained rational numbers are clearly
– 2 1/ 6 3 , – 1/6 , 0/6 , 1/6 , 2 1/ 6 3 and 3 1/ 6 2
Thus, by taking any five the rational numbers between – 1/2 and 2/3 are
– 1/3 , – 1/6 , 1/6 , 1/3 and 1/2 Answer
Alternate method: To find the five rational numbers between – 1/2 and 2/3 using calculation of the average of two rational numbers
Solution :
Given, rational numbers are – 1/2 and 2/3
Thus, 5 rational numbers between given rational numbers = ?
Two rational numbers between given rational numbers are clearly
– 0/2 and 1/3
3rd rational number between – 1/2 and 2/3 can be obtained by getting average between them
= – 1/2 + 2/3/2
= – 3 + 4/6/2
= – 1/6 × 2 = – 1/12
4th rational number between given rational numbers can be calculated by getting average between 1/3 and 2/3
= 1/3 + 2/3/2
= 1 + 2/3/2
= 3/3/2 = 3/3 × 2
= 3/6
5th rational number between 3/6 and 2/3 can be obtained by calculating average between them
= 3/6 + 2/3/2
= 3 + (2 × 2)/6/2
= 3 + 4/6 × 2
= 7/12
Thus, five rational numbers between given rational numbers – 1/2 and 2/3 are
– 1/12 , – 0/2 , 1/3 , 3/6 and 7/12 Answer
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