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NUMBER SYSTEM & SIMPLIFICATIONS
Digits: 0 to 9 are the digits.
Number: By using digits we write numbers.
Classification of Numbers:
Even Numbers
Numbers which are divisible by 2
eg: 2, 4, 6, 8, ...
Odd Numbers
Numbers which are not divisible by 2
eg: 1, 3, 5, 7, ...
Natural Numbers
All counting numbers
N = {1, 2, 3, ...}
n(n+1)
Sum of first 'n' natural numbers is ⎯⎯⎯
2
Sum of squares of first 'n' natural numbers is
n(n + 1)(2n + 1)
⎯⎯⎯⎯⎯
6
Sum of cubes of first 'n' natural numbers is
2
[⎯⎯⎯ ]
n(n + 1)
2
common factors. eg: 4, 7 and 9 are co-prime to
each other.
Some important Identities:
1) (a + b)2 = a2 + b2 + 2ab
2) (a - b)2 = a2 + b2 - 2ab
3) (a + b)(a - b) = a2 - b2
4) (a + b)3 = a3 + b3 + 3a2b + 3ab2
5) (a - b)3 = a3 - b3 - 3a2b + 3ab2
6) a3 + b3 = (a + b) (a2 - ab + b2)
7) a3 - b3 = (a - b) (a2 + ab + b2)
8) (a+b+c)2 = a2 + b2 +c2 + 2ab +2bc + 2ac
9) a3 + b3 + c3 - 3abc
= (a + b + c) (a2 + b2 + c2 - ab - bc - ac)
1.23 × 1.23 - 0.77 × 0.77
eg: ⎯⎯⎯⎯⎯⎯⎯⎯⎯ = ?
1.23 − 0.77
A) 0.46 B) 1
C) 2
D) 2.03 E) None
Sol: a2 - b2 = (a + b)(a - b)
a2 - b2
⇒ ⎯⎯ = a + b
a-b
1.23 × 1.23 − 0.77 × 0.77
∴ ⎯⎯⎯⎯⎯⎯⎯⎯ = 1.23 + 0.77 = 2
1.23 − 0.77
1
2. 60% of 25% of
5
⎯
6
of ? = 630
A) 5200
B) 5060
C) 4500
D) 4880
E) None of these
5
Sol: 60% of 25% of ⎯
6 of ? = 630
60
25 5
⇒ ⎯ × ⎯ ×⎯ × ?
100 100 6
= 630
100 100
6
⇒? = 630 × ⎯
× ⎯ × ⎯ = 5040
60
25
5
Hence answer is (E)
3. One-seventh of a number is 51. What will be
64% of that number?
A) 219.38
B) 235.41
D) 218.36
E) None of these
1
⎯
7 ×x
Sol:
Hence answer is (C)
= 57
C) 255.36
⇒ x = 399
5 6
Irrational Numbers: Can't be written in p/q
form, where q ≠ 0.
⎯
⎯
e.g.: √⎯
2 , √5 , 3 + √7 , π
Composite Numbers: Numbers having more
than two factors. eg: 4, 6, 8, 9, 10 ... etc
Prime Numbers: Numbers having only two
factors or Numbers divisible by one and the
number itself. eg: 2, 3, 5, 7, 11, 13, ... etc
Some points about prime numbers
★ 2 is the only even prime number
★ '1' is neither a prime nor composite number. It is a unitary.
★ There are 25 prime numbers upto 100.
Twin prime numbers: The difference between
two prime numbers is 2.
eg: (3, 5), (5, 7), (11, 13) etc
Co-prime numbers: Numbers not having any
1. 384 +116 ÷ 4 × 12 = ?
B) 732
D) 1200
C) 10.4
E) None of these
2
6
3
2. ⎯ of ⎯ of ⎯ of 7920 = ?
5
11
4
A) 1186
B) 1236
D) 1296
E) None of these
C) 1396
3. 16.45 × 2.8 + 4.5 × 1.6 = ?
A) 53.26
B) 51.46
D) 4926
E) None of these
C) 63.36
4. 5128 + 3893 - 2194 = ? − 1282
B) 8156
C) 8109
E) None of these
5. 1575 ÷ 25 ÷ 9 = ?
A) 567
B) 14
C) 49
D) 7
E) None
6. Difference between the digits of a two digit
number is 5 and the digit in the unit's place
is six times the digit in the ten's place. What
is the number?
A) 61
B) 64
C) 16
4. When 3888 is divided by the
square of a number and the
The numerator of a fraction is written in the
following form
2
2x
Two-third of a number = ⎯ × x = ⎯
3
3
3
3x
Three-fourth of a number = ⎯ × x = ⎯
4
4
30
3x
30% of a number = ⎯ × x = ⎯
100
10
2
25% of ⎯ of one-fourth of a number
3
2
1
25
=x × ⎯ × ⎯ × ⎯
3
4
100
3
4
5
eg.-1: ⎯ of ⎯ of ⎯ of 490 = ?
5
8
7
A) 115 B) 108 C) 116 D) 105 E) None
3
4
5
⎯ of 490
Sol: ⎯
7 of ⎯
5 of 8
3
4
5
⇒⎯
⎯ ⎯
7 × 5 × 8 × 490 = 105
Hence answer is (D)
D) 14
E) None
7. 80% of a number is equal to three fifth of
another number. What is the ratio between
answer so obtained is multiplied by
A) 2 : 3
B) 4 : 3
D) 5 : 7
E) None of these
C) 1 : 2
25
A) ⎯
117
23
B) ⎯
116
23
D) ⎯
117
E) None of these
25
C) ⎯
119
9. If (92)2 is added to the square of a number,
the answer so obtained is 10768. What is
the number?
A) 46
B) 2304
D) 2116
E) None of these
C) 48
C) 72
D) 80
E) None
Answers: 1) B 2) D 3) A 4) C 5) D 6) C 7) E
8) A 9) E 10) D
EXPLANATIONS
6. Let the ten's digit is 'x'
A) 324
B) 36
C) 144
D) 18
E) None of these
Sol: Let the number 'x'
3888
∴ ⎯⎯
2
x
× 21 = 252
3888
⇒ x2 = ⎯⎯
⇒
× 21 = 324
252
x = √⎯⎯
324 = 18
Hence answer is (D)
5. If (16)3 is subtracted from the square of a
number, the answer so obtained is 3825.
What is the number?
A) 89
B) 69
D) 59
E) None of these
C) 49
Sol: Let the number be x
x2 - 163 = 3825 ⇒ x = 89
Hence answer is (A)
∴ x + 5 = 6x ⇒ x = 1
Units place digit is 1 + 5 = 6
eg: 56 × 52 = 56 + 2 = 58
2.
am
⎯n
a
=
eg:
⎯2 = 37
am - n
37
-2
= 35
3
n
3. (am) =
amn
2
eg:
(73)
=7
3×2
4. (ab)m = am × bm
3
3
3
eg: (3 × 4) = 3 × 4
1
5. a-m = ⎯m
1
⎯
⎯
6. m√ a = a m 1
⎯
⎯
eg: 7√ 9 = 9 7
7. a0 = 1
eg: 60 = 1
eg: 6411 ÷ 644 = (8)?
6411
Sol: ⎯⎯
= 8?
644
⇒ 6411−4 = 8 ?
⇒ (82)7 = 8 ?
⇒ 814 = 8 ?
∴? = 14
All simplification questions are to be solved
based on BODMAS rule only.
B - Bracket
O - Of
D - Division
M - Multiplication
A - Addition
S - Subtraction
eg: (74 × 786) ÷ ? = 1211.75
A) 42
B) 36
C) 56
D) 54
E) None of these
Sol: (74 × 786) ÷ ? = 1211.75
74 × 786
⇒ ⎯⎯⎯ = 1211.75
?
74 × 786
⇒ ? = ⎯⎯⎯ = 48
1211.75
Hence answer is (E)
2. (3675 ÷ 75) × (7480 ÷ 80) = ?
A) 4581.5
B) 4852.9
C) 4651.5
D) 4321.8
E) None of these
Sol: (3675 ÷ 75) × (7480 ÷ 80)
3675
7480
⇒ ⎯⎯ × ⎯⎯ = 4581.5
75
80
Hence answer is (A).
3
80
3
∴ 80% x = ⎯ y ⇒ ⎯ x = ⎯ y
5
100
5
x 3
100
3
⇒ ⎯=⎯ × ⎯ = ⎯
y 5
80
4
∴x:y=3:4
9. Let the number be x
2
x +
922
= 3825 ⇒ x = 48
10. Let the first even number be 'x'
x+x+2+x+4+x+6+x+8
10. Sum of five consecutive even numbers is
380. What is the biggest number?
B) 74
252. What is the number?
∴ Required number is 16
7. Let the first and second number be x and y.
28 × 5 - 15 × 6
8. ⎯⎯⎯⎯⎯⎯⎯
=?
⎯
72 + √ 256 + (13)2
A) 76
21, the final answer so obtained is
Unit's place digit = x + 5
the first and second numbers?
A) 1500
A) 8913
D) 8297
64
× 399 = 255.36
100
Hence answer is (C)
★ To solve the equations based on
G.S.GIRIDHAR
numbers, the number is written in
algebraic form. For example let the number
be x.
am × an = am+n
1
1
eg: 4-3 = ⎯
= ⎯
43
64
∴ 64% of 399 = ⎯
Quantitative Aptitude
1.
a
All counting numbers are 'Natural'!
Sum of first 'n' odd numbers = n2
Sum of first 'n' even numbers = n(n + 1)
Whole Numbers
0 and all natural numbers
W = {0, 1, 2, 3, ...}
Integers: Natural numbers, zero and negative
of natural numbers
I = {..., −3, -2, -1, 0, +1, +2, +3 ...}
Rational Numbers: These are of the form of
p/q, where q ≠ 0.
1 5
, −, 0, -4 etc
Q =−
Laws of Indices
∴ ⎯⎯⎯⎯⎯⎯⎯⎯ = 380
5
⇒ x = 72
∴ Numbers are 72, 74, 76, 78 and 80
Biggest number is 80
380
Shortcut: Middle number = ⎯⎯ = 76
5
Numbers are 72, 74, 76, 78 and 80
∴ Biggest number is 80.
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