Class 7 Maths Notes for Rational Numbers

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CLASS 7
MATHS

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RATIONAL NUMBERS
Important Facts and Concepts:
𝑝
A number that can be expressed in the form𝑞 , where p and q are integers and q ≠ 0, is called a
rational number.
All integers and fractions are rational numbers.
If the numerator and denominator of a rational number are multiplied or divided by a non-zero
integer, we get a rational number which is said to be equivalent to the given rational number.
Rational numbers are classified as positive, zero or negative rational numbers. When the
numerator and denominator both are positive integers or both are negative integers, it is a
positive rational number. When either the numerator or the denominator is a negative integer, it
is a negative rational number.
The number 0 is neither a positive nor a negative rational number.
There are unlimited number of rational numbers between two rational numbers.
A rational number is said to be in the standard form, if its denominator is a positive integer and
the numerator and denominator have no common factor other than 1.
Two rational numbers with the same denominator can be added by adding their numerators,
keeping with the same denominator.
Two rational numbers with different denominators are added by first taking the LCM of the two
denominators and then converting both the rational numbers to their equivalent forms having the
LCM as the denominator and adding them as above.
While subtracting two rational numbers, we add the additive inverse of the rational number to be
subtracted to the other rational number.
𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐨𝐟 𝐧𝐮𝐦𝐞𝐫𝐚𝐭𝐨𝐫𝐬
• 𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐨𝐟 𝐫𝐚𝐭𝐢𝐨𝐧𝐚𝐥 𝐧𝐮𝐦𝐛𝐞𝐫𝐬 =
𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐨𝐟 𝐝𝐞𝐧𝐨𝐦𝐢𝐧𝐚𝐭𝐨𝐫𝐬
𝒑 𝒒
• The reciprocal of a non-zero rational number 𝒒 𝒊𝒔 𝒑 .
• To divide one rational number by the other non-zero rational number, we multiply the first
rational number by the reciprocal of the other.
−45
EXAMPLE 1 Reduce 30 to the standard form.
−45 3×3×5 5
Solution: 30 = − 2×3×3 = − 2
4 16
EXAMPLE 3: Do −9 and − 36represent the same rational number?
4 4 −16 −16 4 4
Solution: Yes, because −9 × 4 = 36 , ÷4 = −9
36
EXAMPLE 3 List three rational numbers between – 2 and – 1.
10 20 10 19 18 17
Solution:−2 × 10 = − 10 , −1 × 10 , Any three numbers are, − 10 , − 10 , − 10
SHORT ANSWER QUESTIONS:
55
Q1. Reduce 66 into the standard form.
55 11 5
Solution:66 ÷ 11 = 6
3
Q2. Express 4 as a rational number with denominator as −80
Solution:−80 ÷ 4 = −20

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3 −20 − 60
So, 4 × −20 = −80
15 11
Q3. Find the product of ×
22 5
15 11 3×5 11 3
Solution:22 × 5 = × 5 =2
2×11
2 1
Q4. Simplify:5 − 2
Solution: LCM of 2 𝑎𝑛𝑑 5 is 10
2 2 4
× =
5 2 10
1 5 5
× =
2 5 10
4 5 4−5 1
So, 10 − 10 = 10 = − 10

LONG ANSWER QUESTIONS:
Q1.List five rational numbers between −4 𝑎𝑛𝑑 − 3.
6 24
Solution:−4 × 6 = − 6
6 −18
−3 × = −
6 6
five rational numbers between −4 𝑎𝑛𝑑 − 3 are,
19 20 21 22 23
− ,− ,− ,− ,−
6 6 6 6 6
29 30
Q2. Solve: 4 − 7
Solution: LCM of 4 𝑎𝑛𝑑 7 𝑖𝑠 28.
29 7 203
× =
4 7 28
23 7 161
× =
4 7 28
So,
203 161 203 − 161 42 2 × 3 × 7 3
− = = = =
28 28 28 28 2 × 2 × 7 2
PRACTICE QUESTIONS:
MCQ:
Q1. In the standard form of a rational number, the common factor of numerator and denominator
is always:
(a) 0 (b) 1 (c) – 2 (d) 2
Q2: Which of the following rational numbers is equal to its reciprocal?
1
(a) 1 (b) 2 (c) 2 (d) 0
Q3. How many rational numbers are there between two rational numbers?
(a) 1 (b) 0 (c) unlimited (d) 100
Q4. In the standard form of a rational number, the denominator is always a
(a) 0 (b) negative integer (c) positive integer (d) 1

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Q5. To reduce a rational number to its standard form, we divide its numerator and denominator
by their
(a) LCM (b) HCF (c) product (d) multiple
Q6. Which is greater number in the following?
1 1
(a) – 2 (b) 0 (c) 2 (d)– 2
Q7. Which of the following rational numbers is negative?
−3 −5 9 3
(a) – (b) − 8 (c) 8 (d) − 7
7
𝑝
Q8. A rational number is defined as a number that can be expressed in the form 𝑞 , where are
integers and
(a)𝑞 = 0 (b)𝑞 = 1 (c)𝑞 ≠ 1 (d) 𝑞 ≠ 0
Q9. Find the odd one out of the following and give reason.
4 3 −3 −2 1 1 3
(a) 3 × 4 (b) 2 × 3 (c) 2 × 2 (d) − 3 × 1
4 −1
Q10. The value of − 3 − 3
is
(a) – 2 (b)– 3 (c)2 (d)–1
SHORT ANSWER QUESTIONS:
60
Q1. Reduce the following rational number in its lowest form: − 72 .
8 32
Q2. Are the rational numbers − 28 , 𝑎𝑛𝑑 −112 equivalent? Give reason.
7 5 2 1 3
Q3. Arrange the rational numbers − 10 , −8 , −3 , − 4 , − 5 in ascending order.
5 𝑥
Q4. If – 7 = 28 , find the value of 𝑥 ?
7
Q5. Give two rational numbers equivalent to: 8
1
Q6. What should be added to – 2to obtain the nearest natural number?
−25 −5
Q7. What‘s the Error? Chhaya simplified a rational number in this manner −30 = , what error
6
did the student make?

LONG ANSWER QUESTIONS:
1 1 1
Q1. Find the reciprocal of the following: 2 × 4 + 2 × 6
𝑝
Q2. Write each of the following numbers in the form𝑞 , where p and q are integers:
(a)six-eighths (b)three and half (c)opposite of 1 (d)one-fourth
1
Q3. From a rope 68 𝑚 𝑙𝑜𝑛𝑔, pieces of equal size are cut. If length of one piece is 4 4 𝑚 , find
the number of such pieces.
Q4. 150 students are studying English, Maths or both. 62 𝑝𝑒𝑟 𝑐𝑒𝑛𝑡 of the students are studying
English and 68 𝑝𝑒𝑟 𝑐𝑒𝑛𝑡 are studying Maths. How many students are studying both?
2
Q5. A body floats9of its volume above the surface. What is the ratio of the body submerged
volume to its exposed volume? Re-write it as a rational number.
1 3
Q6. If 𝑥 = 10 and 𝑦 = – 8, then evaluate 𝑥 + 𝑦, 𝑥 – 𝑦, 𝑥 × 𝑦 𝑎𝑛𝑑 𝑥 ÷ 𝑦.

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ANSWERS:
MCQ:
Q1 (b) 1 Q2 (a) 1 Q3 (c) unlimited Q4 (c) positive integer
1 3
Q5 (b) HCF Q6 (c) 2 Q7 (d) − 7 Q8 (d) 𝑞 ≠ 0
1 3
Q9 (d) − 3 × 1 Q10 (d)–1
SHORT ANSWER QUESTIONS:
5
Q1. − 6 Q2. Yes, as both are equivalent rational numbers.
7 2 5 3 1
Q3. − 10 , −3 , −8 , − 5 , − 4 Q4. 𝑥 = −20
14 21
Q5. There are many possibilities. Some of them are: 16 , 24
3
Q6. Q7. Error is of negative sign.
2
LONG ANSWER QUESTIONS:
8 3 7 1 1
Q1. 25 Q2. (a) 4 (b) 2 (c) -1 (d) 4 Q3. 16 Pieces Q4. 45
7 11 19 3 4
Q5. 7: 2 𝑎𝑛𝑑 2 Q6. − 40 , 40 , − 80 , − 15

TEST-1 (20 Marks)
Q1. Fill in the blanks to make the statements true: There are _______ number of rational
numbers between two rational numbers. (1)
2
Q2. Write equivalent rational numbers for ? (2)
3
3 −7
Q3. Represent the following rational numbers on a number line: 8 , 3 (2)
5 7
Q4. List two rational numbers between7 𝑎𝑛𝑑 8. (2)
7 −4
Q5. Find the sum of3 + 3 . (2)
3 7
Q6. Which is greater in the following? 4 𝑎𝑛𝑑 8 (2)
−4 −5
Q7. Find the product of: 5 𝑎𝑛𝑑 12 . (2)
1 1
Q8. Find a rational number exactly halfway between: 6 𝑎𝑛𝑑 9. (2)
29 30
Q9. Solve : 4 − 7 .
13 −14 13 −7 −13 34
Q10. Simplify: 11 × 5 + 11 × 5 + 11 × 5 (3)

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TEST-2 (30 Marks)
Q1. Fill in the blanks to make the statements true. The rational number _________ is neither
positive nor negative. (1)
12
Q2.Write equivalent rational number for 17 (2)
1 2 3
Q3. Write four more rational numbers to complete the pattern: − 3 , 6 , 9 , … , … , … , … (4)
5 3
Q4. Find the sum of −4 6 𝑎𝑛𝑑 − 7 4 (3)
3 6
Q5. Find the product of −2 4 𝑎𝑛𝑑 5 7 (3)
Q6. If 12 𝑠ℎ𝑖𝑟𝑡𝑠 of equal size can be prepared from 27𝑚 𝑐𝑙𝑜𝑡ℎ, what is length of cloth required
for each shirt? (3)
1 1
Q7. Insert 3 equivalent rational numbers between − 2 𝑎𝑛𝑑 5 (3)
2 5 7
Q8. Represent the following rational numbers on a number line: − , , (3)
5 −6 −3
6 3 1 3
Q9. Simplify: 5 × 5 − 5 × 7 (3)
Q10. Match column I to column II in the following: (5)
COLUMN-I COLUMN-II
i. 3 3 (a) −1
÷
4 4
ii. 1 4 2
÷ (b) − 3
2 3
iii. 2 3
÷ −1 (c) 2
3
iv. 3 1 3
÷ (d)
8
4 2
v. 5 5 (e) 1
÷ −
7 7

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