1. Chapter 1. Number Systems NIOS Class 10 Mathematics Textbook Solutions
Chapter 1. Number Systems NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation
Chapter 1: Number Systems
CHECK YOUR PROGRESS 1.1 for Number Systems
1. Identify rational numbers and integers from the following:
Solution: Rational numbers: 
Integers: – 36 , – 6 , 4
[ Note: A rational number is any number that can be expressed as a fraction 
where p and q are integers and
. Integers are whole numbers (positive, negative, or zero) with no fractional or decimal part. ]
2. From the following identify those which are not :
(i) natural numbers (ii) whole numbers (iii) integers (iv) rational numbers
Solution:
|
Category |
Numbers that are NOT in it |
|
Natural numbers |
|
|
Whole numbers |
|
|
Integers |
|
|
Rational numbers |
None |
3. By making the following rational numbers with same denominator, simplify the following and specify whether the result in each case is a natural number, whole number, integer or a rational number:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
Solution:
|
Expression |
Simplified Result |
Classification |
|
|
(i) |
|
Rational number |
|
|
(ii) |
|
Rational number |
|
|
(iii) |
– 8 − 10 = – 18 |
Integer |
|
|
(iv) |
12 – 12 = 0 |
Whole number |
|
|
(v) |
|
Natural number |
|
|
(vi) |
|
Rational number |
|
|
(vii) |
|
Rational number |
|
4. Use the number line to add the following:
(i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
5. Which of the following are rational numbers in lowest term?
Solution: We have, 
is the lowest form of the rational number 
.
is the lowest form of the rational number 
.
is the lowest form of the rational number 
.
is the lowest form of the rational number 
.
is the lowest form of the rational number .
is the lowest form of the rational number 
.
6. Which of the following rational numbers are integers?
Solution: We have,
The rational numbers that are integers are: 
7. Write 3 rational numbers equivalent to given rational numbers:
Solution: We have,
and 
8. Represent the following rational numbers on the number line.
Solution: (a) The rational number 
is represented on the number line.
(b) The rational number 
is represented on the number line.
(c) The rational number 
is represented on the number line.
9. Compare the following rational numbers by (i) changing them to rational numbers in equivalent forms (ii) using number line:
(a) 
(b) 
(c) 
(d) 
(e) 
Solution: We have,
(a)
(b)
(c)
(d)
(e)
CHECK YOUR PROGRESS 1.2 for Number Systems
1. Add the following rational numbers:
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
(iv) We have,
2. Add the following rational numbers: (i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
3. Perform the indicated operations:
(i) 
(ii) 
Solution: (i) We have, 
(ii) We have, 
4. Subtract: (i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
5. Simplify: (i) 
(ii) 
Solution: (i) We have, 
(ii) We have,
6. Multiply : (i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have,
(iii) We have, 
7. Divide :
(i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
8. Simplify the following:
(i) 
(ii) 
Solution: (i) We have, 
(ii) We have , 
9. Divide the sum of 
by their difference.
Solution: We have, 
and 
Now, 
10. A number when multiplied by 
. Find the number.
Solution: We have, 
CHECK YOUR PROGRESS 1.3 for Number Systems
1. Represent the following rational numbers in the decimal form:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
(iv) We have, 
(v) We have, 
2. Represent the following rational numbers in the decimal form:
(i) 
(ii) 
(iii) 
Solution: (i)
Now,
(ii)
Now ,
(iii)
Now,
3. Represent the following decimals in the form :
(a) (i) 2.3 (ii) – 3.12 (iii) –0.715 (iv) 8.146
(b) (i) 333.0 (ii) 42.3 (iii) – 0.315315315...
Solution: (a) (i) We have, 
(ii) We have, 
(iii) We have, 
(iv) We have, 
(b) (i) We have, 
(ii) We have, 
(iii) We have, 
CHECK YOUR PROGRESS 1.4 for Number Systems
1. Find a rational number between the following rational numbers:
(i) 
(ii) 5 and 6 (iii) 
Solution: (i) 
is a rational number between the rational numbers 
(ii) 5 and 6
is a rational number between the rational numbers 5 and 6 .
(iii)
is a rational number between the rational numbers 
2. Find two rational numbers between the following rational numbers:
(i) 
(ii) 
Solution: (i) 
Therefore, the two rational numbers between 
(ii) 
Therefore, the two rational numbers between 
3. Find 5 rational numbers between the following rational numbers:
(i) 0.27 and 0.30 (ii) 7.31 and 7.35 (iii) 20.75 and 26.80 (iv) 1.001 and 1.002
Solution: (i) 0.27 and 0.30
Therefore, the 5 rational numbers between 0.27 and 0.30 are : 
(ii) 7.31 and 7.35
Therefore, the 5 rational numbers between 0.27 and 0.30 are : 
(iii) 20.75 and 26.80
Therefore, the 5 rational numbers between 20.75 and 26.80 are : 
(iv) 1.001 and 1.002
Therefore, the 5 rational numbers between 1.001 and 1.002 are : 
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