NCERT Solutions
Class 7 Maths
Rational Numbers
1
Introduction:
If a and b are two integers, then each of a + b, a - b and a x b is also an
integer.
However, it is not necessary that a b or b a is also an integer.
For example:
Consider two integers 8 and 5.
Clearly:
(i) 8 + 5 = 13 is an integer. (ii) 8 - 5 = 3 is an integer 5 - 8 = -3 is also an
integer.
Here, and are fractions and are said to form the system of rational numbers.
RATIONAL NUMBERS
∙If p and q are two co-prime integers and q 0, then the number is called a
rational number.
∙p and q are co-primes, p and q do not have any common factor other than
1.
Example:
(i) is a rational number as 3 and 7 both are integers and 7 0.
(ii) is a rational number as -15 is an integer, 19 is an integer and 19 0.
Rational Numbers in standard form:
A rational number is in standard form, if n is positive and m and n have only 1
as their common factor.
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Class 7 Mathematics ICSE | Rational Number | Notes
(i) Rational number is in standard form as 4 is positive and both 3 and 4 have
only 1 as their common factor.
(ii)Rational number is not in standard form as 24 is positive but 15 and 24 have
1 and 3 as their common factors.
Remember:
1. Every integer is a rational number but the converse is not true
(i) 5= is a rational number as 5 and 1 as integers and 1 0.
(ii) -8 = is a rational number as -8 and 1 as integers and 1 0.
(iii) Every natural number is an integer and so a rational number.
(iv) Every whole number is also an integer and so a rational number.
2. Each of 0 = , , …. etc. is a rational number.
3. In a rational number in the form of a fraction , p is called the numerator and
q is called the denominator.
In numerator = 7 and denominator = 11.
4. Positive rational number: A rational number is said to be positive, if its
numerator and denominator both are either positive or negative.
Each of the following rational numbers is a positive rational number:
etc.
5. Negative rational number: A rational number is said to be negative, if its
numerator and denominator are with opposite signs, i.e., if the numerator is
positive and the denominator is negative or if the numerator is negative and
the denominator is positive.
etc.
6. (i) Every positive integer (i.e., every natural number) is a positive rational
number.
e.g., natural numbers 5 = is a positive rational number.
(ii) Every negative integer is a negative rational number.
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Class 7 Mathematics ICSE | Rational Number | Notes
e.g., -5 = is a negative rational number.
Zero (0) is a rational number but it is neither positive nor negative.
REPRESENTATION OF RATIONAL NUMBERS ON NUMBER LINE
1. Draw a straight line. Mark a point O on this line and name it 0 (zero).
2. On the right side of O, mark points A, B, C, D, E, .... on the line drawn such
that the distance between consecutive points is the same.
i.e., OA =AB= BC= CD= ...........
3. In the same way, on the same line mark points A', B', C', D', etc. on the left
side of O such that:
OA' = A'B' = B'C' = ...........
On the whole, we must have;
......... B'C' = A'B' = OA' = OA = AB = BC = CD = .........
The points A, B, C, D, E, ........., which are on the right side of zero (0), represent
positive integers 1, 2, 3, 4, 5, ......... and the points A', B', C', D', E', ........., which
are on the left side of zero, represent negative integers -1, -2, -3, -4, -5, ......... .
Example:
Represent rational numbers on the same number line.
Answer:
Consider C'B' = B' A' = A'O = OA = AB = BC = one unit length Clearly, A
represents 1, B represents 2 and C represents 3.
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Class 7 Mathematics ICSE | Rational Number | Notes
In the same way, A' represents -1, B' represents = -2 and C' represents -3. Since
denominator of each given rational number is 2, mark middle points of each of
OA, AB, BC and OA', A'B', B'C' to divide each of OA, AB, BC and OA', A'B', B'C'
into two equal parts. Now the given points are marked on this number line.
Comparing Rational Number
1. Every positive rational number is greater than 0 (zero) and is greater than
every negative number:
e.g., 8 > 0, 8 > - 5, 8 > -93, 8 > -1235, etc.
2. Every negative rational number is less than 0 (zero) and is less than every
positive number:
e.g., -8 < 0, -8 < 5, -8 < 93, -8 < 123, etc.
3. Zero (0) is greater than every negative number and is smaller than every
positive number:
e.g., 0 > -5, 0 > , 0 < 5, 0 < etc. Example:
Compare
Answer:
First method: 1. Find the L.C.M. of denominators 5 and 7. L.C.M. of 5 and 7 =
35.
2. Make the denominators of each rational number equal to L.C.M. obtained
above, i.e., equal to 35.
On multiplying both the terms of a rational number by the same non-zero
number, the value of the rational number does not change.
=
=
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Class 7 Mathematics ICSE | Rational Number | Notes
3. For the same denominator, the rational number with greater numerator is
greater.
is greater than .
2nd method:
Upon cross-multiplying and , we get a × d and b × c
1. If a x d is greater than b x c ⇒ is greater than , i.e., > .
2. If a x d is less than b x c ⇒ is greater than , i.e., < .
Similarly on cross-multiplying and , we get:
3 × 7 and 5 × 5
= 21 and 25
= 21 < 25
is smaller than .
For any two numbers a and b:
(i) if a > b => -a < -b
(ii) if a < b => -a > -b
Important:
For any two rational numbers a and b marked on a number line, if:
1. a is on the left of b, then a is smaller than b i.e. a< b.
2. a is on the right of b, then a is greater than b i.e. a> b.
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Class 7 Mathematics ICSE | Rational Number | Notes
RATIONAL NUMBER(S) BETWEEN TWO GIVEN NUMBERS
An infinite (unlimited) number of rational numbers can be obtained (inserted)
between two given rational numbers.
Example:
L.C.M. of denominators 11 and 5 = 55
= = and = = [Making denominators equal]
30 < 33 ⇒ -30 > -33
is greater than
> > >
PROBLEMS ON RATIONAL NUMBERS (All operations)
1. Addition of Rational Numbers:
Case 1: When denominators are equal:
• Keeping the denominator same, add the numerators.
• If required, express the rational number obtained in its lowest terms.
Case 1: When denominators are equal:
+ = =
Case 2: When denominators are unequal:
∙Make the denominators of all the given rational numbers equal and then
proceed as case 1, given above. Thus:
+ = = [L.C.M of 3 and 4 is 12]
2. Subtraction of Rational Numbers:
Case 1: When denominators are equal:
E.g., - =
Case 2: When denominators are unequal:
E.g., - = = [L.C.M of 4 and 5 is 20]
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Class 7 Mathematics ICSE | Rational Number | Notes
3. Multiplication of Rational Numbers:
Product (multiplication) of two or more rational numbers =
E.g., =
4. Multiplicative inverse (Reciprocal):
Multiplicative inverse of is .
5. Division of Rational Numbers:
If and are two rational numbers such that 0, then
= × (multiplicative inverse of )
= ×