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Important Formulae

Basic Formulas:
1. (a + b)(a - b) = (a2 - b2)

2. (a + b)2 = (a2 + b2 + 2ab)

3. (a - b)2 = (a2 + b2 - 2ab)

4. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

5. (a3 + b3) = (a + b)(a2 - ab + b2)

6. (a3 - b3) = (a - b)(a2 + ab + b2)

7. (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)

8. When a + b + c = 0, then a3 + b3 + c3 = 3abc.

Practice Questions

1.

 A number is as much greater than 36 as is less than 86. Find the number? A.60 B.61 C.62 D.63

Let, the number be X. Then
X – 36 = 86 – X
2X = 122
X = 61

2.

 Find a number such that when 15 is subtracted from 7 times the number, the result is 10 more than twice the number. A.5 B.10 C.15 D.20

Let, the number be X. Then
7X – 15 = 2X + 10
5X = 25
X = 5

3.

 The sum of rational number and its reciprocal is 13/6 . Find the number? A.2/3 or 3/2 B.4/5 or 5/4 C.1/2 or 2/5 D.5/2 or 4/3

Let the number be X.
Then, X + 1/X = 13/6 ⇔(X² + 1)/X = 13/6 ⇔6X² - 13X + 6 = 0
⇒ 6X² - 9X – 4X +6 = 0
⇒ (3X - 2)(2X - 3) = 0
⇒ X = 2/3 or 3/2
Hence, the required number is 2/3 or 3/2

4.

 The sum of two numbers is 184. If one third of the one exceeds one seventh of the other by 8, find the smaller number? A.70 B.71 C.72 D.73

Let the numbers be X and (184 - X). Then,
⇒ X/3 – (184 - X)/7 = 8
⇒ 7X – 3(184 - X) = 168
⇒ 10X = 720
⇒ X = 72

5.

 The sum of two numbers is 15 and the sum of their squares is 113. Find the numbers? A.7 and 8 B.8 and 9 C.9 and 10 D.10 and 11

Let the numbers be X and (15 - X)
Then, X² + (15 - X)² = 113
⇒ X² + 225 + X² - 30X = 113
⇒ 2X² - 30X + 112 = 0
⇒ X² -15X + 56 = 0
⇒ (X - 7)(X - 8) = 0
⇒ X = 7 or X = 8
So, the numbers are 7 and 8.

6.

 The average of four consecutive even numbers is 27. Find the largest of these numbers? A.25 B.30 C.35 D.40

Let, the consecutive even numbers be X, X+2, X+4 and X+6.
Then, sum of these numbers = (27 x 4) = 108
So, X + X + 2 + X + 4 + X + 6 = 108
4X = 96
X = 24
∴ Largest number = (X + 6) = 30

7.

 The ratio between a two digit number and the sum of the digits of that number is 4:1. If the digit in the unit's place is 3 more than the digit in the ten's palace, what is the number? A.33 B.34 C.35 D.36

Let, the ten's digit be X. then unit's digit = (X + 3).
Sum of digits = X + (X + 3) = 2X + 3
Number = 10X + (X + 3) = 11X + 3
∴ (11X + 3) ÷ (2X + 3) = 4/1
⇒ 11 X + 3 = 4(2 X +3)
⇒ 3X =9
⇒ X = 3
Hence, Required number = 11 X + 3 = 36

8.

 50 is divided in two parts such that the sum of their reciprocals is 1/12. Find the two parts? A.30 and 20 B.40 and 30 C.40 and 20 D.20 and 10

Let, the two parts be X and (50 - X).
Then, 1/X + 1/(50-X) = 1/12
⇒ (50 – X + X) ÷ X(50 – X) = 1/12
⇒ X² - 50X + 600 = 0
⇒ X² - 30X – 20X + 600 = 0
⇒ (X - 30) (X - 20) = 0
⇒ X = 30 or X = 20
So, the parts are 30 and 20.