Chapter 1: Rational Numbers (NCERT Solutions)
Exercise 1.1
(i) -2/3 × 3/5 + 5/2 - 3/5 × 1/6
(ii) 2/5 × (-3/7) - 1/6 × 3/2 + 1/14 × 2/5
(i) (-2/3 × 3/5) + 5/2 - (3/5 × 1/6)
Using commutative property of addition, rearrange terms:
= (-2/3 × 3/5) - (3/5 × 1/6) + 5/2
Using distributive property [a×b - a×c = a×(b - c)] with a = 3/5:
= 3/5 × (-2/3 - 1/6) + 5/2
= 3/5 × [(-4 - 1) / 6] + 5/2
= 3/5 × (-5/6) + 5/2
= -15/30 + 5/2
= -1/2 + 5/2
= (-1 + 5) / 2 = 4 / 2 = 2.
(ii) 2/5 × (-3/7) - 1/6 × 3/2 + 1/14 × 2/5
Rearrange terms using commutative property:
= 2/5 × (-3/7) + 1/14 × 2/5 - 1/6 × 3/2
Using distributive property on the first two terms with 2/5 common:
= 2/5 × (-3/7 + 1/14) - (1/6 × 3/2)
= 2/5 × [(-6 + 1) / 14] - (3/12)
= 2/5 × (-5/14) - 1/4
= -10/70 - 1/4
= -1/7 - 1/4
= (-4 - 7) / 28 = -11/28.
(i) 2/8 (ii) -5/9 (iii) -6/-5 (iv) 2/-9 (v) 19/-6
The additive inverse of a rational number a/b is -a/b, such that (a/b) + (-a/b) = 0.
(i) Additive inverse of 2/8 is -2/8.
(ii) Additive inverse of -5/9 is -(-5/9) = 5/9.
(iii) -6/-5 is equivalent to 6/5. Its additive inverse is -6/5.
(iv) 2/-9 is equivalent to -2/9. Its additive inverse is 2/9.
(v) 19/-6 is equivalent to -19/6. Its additive inverse is 19/6.
(i) x = 11/15
(ii) x = -13/17
(i) For x = 11/15:
-x = -(11/15) = -11/15
-(-x) = -(-11/15) = 11/15 = x. Verified.
(ii) For x = -13/17:
-x = -(-13/17) = 13/17
-(-x) = -(13/17) = -13/17 = x. Verified.
(i) -13 (ii) -13/19 (iii) 1/5 (iv) -5/8 × -3/7 (v) -1 × -2/5 (vi) -1
Multiplicative inverse (reciprocal) of a/b is b/a.
(i) Multiplicative inverse of -13 (-13/1) is -1/13.
(ii) Multiplicative inverse of -13/19 is 19/-13 or -19/13.
(iii) Multiplicative inverse of 1/5 is 5/1 = 5.
(iv) First solve: -5/8 × -3/7 = 15/56. Its multiplicative inverse is 56/15.
(v) First solve: -1 × -2/5 = 2/5. Its multiplicative inverse is 5/2.
(vi) Multiplicative inverse of -1 (-1/1) is 1/-1 = -1.
(i) -4/5 × 1 = 1 × -4/5 = -4/5
(ii) -13/17 × -2/7 = -2/7 × -13/17
(iii) -19/29 × 29/-19 = 1
(i) Multiplicative Identity Property (1 is the multiplicative identity).
(ii) Commutative Property of Multiplication (a × b = b × a).
(iii) Multiplicative Inverse Property (a/b × b/a = 1).
The reciprocal of -7/16 is 16/-7 (or -16/7).
6/13 × (-16/7) = (6 × -16) / (13 × 7)
= -96/91.
The property is Associativity of Multiplication [a × (b × c) = (a × b) × c].
Convert mixed fraction to improper fraction: -1 1⁄8 = -9/8.
For two numbers to be multiplicative inverses, their product must be 1.
Product = 8/9 × (-9/8) = -72 / 72 = -1.
Since the product is -1 (not 1), No, 8/9 is not the multiplicative inverse of
-1 1⁄8.
Convert 0.3 to fraction: 0.3 = 3/10.
Convert mixed fraction to improper fraction: 3 1⁄3 = (3×3 +
1)/3 = 10/3.
Product = (3/10) × (10/3) = 30 / 30 = 1.
Since the product is 1, Yes, 0.3 is the multiplicative inverse of 3
1⁄3.
(i) The rational number that does not have a reciprocal.
(ii) The rational numbers that are equal to their reciprocals.
(iii) The rational number that is equal to its negative.
(i) 0. (Because 1/0 is not defined or infinite).
(ii) 1 and -1. (Because reciprocal of 1 is 1/1 = 1, and reciprocal of -1 is 1/-1 = -1).
(iii) 0. (Because 0 = -0).
(i) Zero has ________ reciprocal.
(ii) The numbers ________ and ________ are their own reciprocals.
(iii) The reciprocal of -5 is ________.
(iv) Reciprocal of 1/x, where x ≠ 0 is ________.
(v) The product of two rational numbers is always a ________.
(vi) The reciprocal of a positive rational number is ________.
(i) Zero has no reciprocal.
(ii) The numbers 1 and -1 are their own reciprocals.
(iii) The reciprocal of -5 is -1/5.
(iv) Reciprocal of 1/x, where x ≠ 0 is x.
(v) The product of two rational numbers is always a rational number (Closure property).
(vi) The reciprocal of a positive rational number is positive.
Exercise 1.2
(i) 7/4: Convert to mixed fraction: 1 3⁄4. This
number lies between 1 and 2.
Draw a number line. Mark 0, 1, 2. Divide the distance between 0 and 1, and between 1 and 2 into
4 equal parts each.
The parts from 0 are: 1/4, 2/4, 3/4, 4/4 (which is 1), 5/4, 6/4, 7/4, 8/4
(which is 2).
Mark the point corresponding to 7/4.
(ii) -5/6: This number lies between 0 and -1.
Draw a number line. Divide the distance between 0 and -1 into 6 equal parts.
The parts from 0 towards the left are: -1/6, -2/6, -3/6, -4/6, -5/6, -6/6
(which is -1).
Mark the point corresponding to -5/6.
All these numbers have the same denominator (11) and are between 0 and -1.
Draw a number line with 0 on the right and -1 on the left.
Divide the distance between 0 and -1 into 11 equal parts.
Starting from 0 to the left, the points will be -1/11, -2/11, -3/11, -4/11,
-5/11, -6/11, -7/11, -8/11, -9/11, -10/11, -11/11 (which is
-1).
Mark the 2nd, 5th, and 9th parts.
Any negative rational number, 0, or any positive number less than 2.
Examples: 1, 1/2, 0, -1, -2.
First, convert them to have the same denominator. LCM of 5 and 2 is 10.
-2/5 = -4/10
1/2 = 5/10
There are 8 rational numbers between -4/10 and 5/10 (-3/10, -2/10, ..., 4/10). We need 10.
So, let's multiply numerator and denominator by 2 to expand the range.
-4/10 × 2/2 = -8/20
5/10 × 2/2 = 10/20
Now we can pick any 10 numbers between -8/20 and 10/20.
For instance: -7/20, -6/20, -5/20, -4/20, -3/20, -2/20, -1/20, 0/20, 1/20,
2/20.
(i) 2/3 and 4/5 (ii) -3/2 and 5/3 (iii) 1/4 and 1/2
(i) 2/3 and 4/5:
LCM of 3 and 5 is 15. 2/3 = 10/15, 4/5 = 12/15.
Only 11/15 is between them. Multiply by 6 to get a wider range.
10/15 × 6/6 = 60/90
12/15 × 6/6 = 72/90
Five numbers: 61/90, 62/90, 63/90, 64/90, 65/90.
(ii) -3/2 and 5/3:
LCM of 2 and 3 is 6. -3/2 = -9/6, 5/3 = 10/6.
There are many numbers between them. Pick any five:
-8/6, -7/6, 0, 1/6, 2/6.
(iii) 1/4 and 1/2:
1/2 can be written as 2/4. We need numbers between 1/4 and 2/4.
Multiply by 6: 1/4 × 6/6 = 6/24; 2/4 × 6/6 = 12/24.
Five numbers: 7/24, 8/24, 9/24, 10/24, 11/24.
There are infinitely many rational numbers greater than -2.
Five examples are: -1, 0, 1, 2, 3.
LCM of 5 and 4 is 20.
3/5 = 12/20
3/4 = 15/20
To find 10 numbers, multiply numerator and denominator by 5 or more (let's say we multiply by 10
for simplicity):
12/20 × 10/10 = 120/200
15/20 × 10/10 = 150/200
Ten numbers between them can be:
121/200, 122/200, 123/200, 124/200, 125/200, 126/200, 127/200, 128/200, 129/200,
130/200.
Chapter 1: Rational Numbers (Practice Questions)
RD Sharma / Extra Practice Questions
LCM of 12 and 8 is 24.
5/12 = (5 × 2) / (12 × 2) = 10/24
3/8 = (3 × 3) / (8 × 3) = 9/24
Sum = (10/24) + (9/24) = (10 + 9) / 24 = 19/24.
(2/5) - (-3/7) = (2/5) + (3/7)
LCM of 5 and 7 is 35.
= [(2 × 7) + (3 × 5)] / 35
= (14 + 15) / 35 = 29/35.
Product = (5 / 8) × (-3 / 4)
= (5 × -3) / (8 × 4)
= -15 / 32.
(-8 / 15) ÷ (4 / 5) = (-8 / 15) × (5 / 4)
= (-8 × 5) / (15 × 4)
= -40 / 60
Dividing numerator and denominator by 20:
= -2/3.
Let the number be x.
(-5/8) + x = 3/2
x = (3/2) - (-5/8)
x = (3/2) + (5/8)
LCM of 2 and 8 is 8.
x = [(3 × 4) + (5 × 1)] / 8
x = (12 + 5) / 8 = 17/8.
Let the number be x.
(-2/3) - x = -1/6
x = (-2/3) - (-1/6)
x = (-2/3) + (1/6)
LCM of 3 and 6 is 6.
x = [-4/6] + [1/6] = (-4 + 1) / 6 = -3 / 6 = -1/2.
The multiplicative inverse (or reciprocal) of a/b is b/a.
The multiplicative inverse of -3/5 is 5 / -3, which is written as -5/3.
7/-12 is the same as -7/12.
The additive inverse of a number x is -x.
So, additive inverse of -7/12 is -(-7/12) = 7/12.
Let the other number be x.
(7/9) × x = -14/27
x = (-14/27) ÷ (7/9)
x = (-14/27) × (9/7)
x = (-14 × 9) / (27 × 7)
x = (-2 × 1) / (3 × 1) = -2/3.
Let the number be x.
(-8/39) × x = 1/26
x = (1/26) ÷ (-8/39)
x = (1/26) × (39 / -8)
Both 26 and 39 are divisible by 13.
x = (1/2) × (3 / -8) = 3 / -16 = -3/16.
Using the Distributive Property [ a × b + a × c = a × (b + c) ]:
Here a = 2/3, b = 5/7, c = 2/7.
= (2/3) × (5/7 + 2/7)
= (2/3) × [(5 + 2) / 7]
= (2/3) × (7/7)
= (2/3) × 1 = 2/3.
First term: (-3/2) × (4/5) = (-3 × 2) / 5 = -6/5.
Second term: (9/5) × (-10/3) = (3 × -2) / 1 = -6.
Third term: (1/2) × (3/4) = 3/8.
Now combine: (-6/5) + (-6/1) - (3/8)
LCM of 5, 1, and 8 is 40.
= [-(6×8) - (6×40) - (3×5)] / 40
= [-48 - 240 - 15] / 40
= -303 / 40. (-303/40)
Method 1: Make denominators equal.
LCM of 3 and 2 is 6.
1/3 = 2/6 and 1/2 = 3/6.
To find 3 numbers, multiply numerator and denominator by 4.
2/6 = 8/24 and 3/6 = 12/24.
The rational numbers between 8/24 and 12/24 are:
9/24, 10/24, and 11/24 (which can be simplified to 3/8, 5/12, 11/24).
LCM of 5 and 2 is 10.
-2/5 = -4/10
1/2 = 5/10
Rational numbers between -4/10 and 5/10 are:
-3/10, -2/10, -1/10, 0/10, 1/10, 2/10...
Any five can be chosen, e.g., -3/10, -1/5, -1/10, 0, 1/10.
There are infinitely many rational numbers greater than -2. Some simple examples are:
-1, 0, 1/2, 1, 3/2.
-4/7 lies between 0 and -1.
Divide the distance between 0 and -1 into 7 equal parts.
The points will represent -1/7, -2/7, -3/7, -4/7, -5/7, -6/7.
Mark the 4th point to the left of 0. That point represents -4/7.
Given x = -5/11.
First, find -x: -x = -(-5/11) = 5/11.
Then, find -(-x): -(-x) = -(5/11) = -5/11.
Comparing this with x, we get -(-x) = x.
Hence verified.
This is of the form a × (b × c) = (a × b) × c.
This is the Associativity Property of Multiplication.
True. Let a/b be a positive rational number (meaning a and b have the same sign). Its reciprocal is b/a. Since a and b have the same sign, b/a will also be positive.
The rational number exactly halfway between a and b is their mean: (a + b) / 2.
Let a = 1/4, b = 1/2 = 2/4.
Mean = (1/4 + 2/4) / 2 = (3/4) / 2 = 3/4 × 1/2 = 3/8.
Chapter 1: Rational Numbers (Concepts & Formulas)
1. Introduction
- Natural Numbers: Counting numbers (1, 2, 3, ...)
- Whole Numbers: Natural numbers along with zero (0, 1, 2, 3, ...)
- Integers: Whole numbers along with negative numbers (..., -3, -2, -1, 0, 1, 2, 3, ...)
- Rational Numbers: Any number that can be expressed
in the form p/q, where p and q are integers and q ≠
0.
Examples: 2/3, -5/7, 0, 4 (since 4 = 4/1).
2. Properties of Rational Numbers
- Closure Property:
Rational numbers are closed under addition, subtraction, and multiplication.
For any two rational numbers a and b, (a + b), (a - b), and (a × b) are also rational numbers.
Note: They are not closed under division because division by zero is not defined. - Commutative Property:
Addition: a + b = b + a
Multiplication: a × b = b × a
Note: Subtraction and division are NOT commutative for rational numbers. - Associative Property:
Addition: a + (b + c) = (a + b) + c
Multiplication: a × (b × c) = (a × b) × c
Note: Subtraction and division are NOT associative for rational numbers. - Distributive Property:
Multiplication over addition: a × (b + c) = (a × b) + (a × c)
Multiplication over subtraction: a × (b - c) = (a × b) - (a × c)
3. Identity and Inverse
- Additive Identity (Zero):
0 is the additive identity for rational numbers.
a + 0 = 0 + a = a - Multiplicative Identity (One):
1 is the multiplicative identity for rational numbers.
a × 1 = 1 × a = a - Additive Inverse (Negative of a number):
The additive inverse of a/b is -a/b, and vice-versa.
(a/b) + (-a/b) = 0 - Multiplicative Inverse (Reciprocal):
The multiplicative inverse of a/b is b/a (where a ≠ 0).
(a/b) × (b/a) = 1
4. Rational Numbers on the Number Line
- Rational numbers can be represented on a number line, just like integers.
- Positive rational numbers lie to the right of 0, and negative rational numbers lie to the left of 0.
- Finding rational numbers between two rational
numbers:
There are countless (infinitely many) rational numbers between any two given rational numbers.
Method: Find a common denominator and scale up the fractions by multiplying numerator and denominator by a suitable integer to reveal numbers in between. Alternatively, taking the mean (average) of two rational numbers always gives a rational number between them.