**Addition **

Addition is the process of finding out a single number or fraction equal to two or more quantities taken together. .

Subtraction is the process of finding out the quantity left when a smaller quantity (number/fraction) is reduced from a larger one. .

Multiplication signifies repeated addition. If a number has to be repeatedly added then that number is the Multiplicand. The number of times the addition is to be done is the Multiplier. The sum of repetition is the Product. For example, in the multiplication 3 x 4 = 12, 3 is Multiplicand, 4 is Multiplier and 12 is Product. .

Division is the reverse, of multiplication. In this we find how often a given number called Divisor is contained in another given number called Dividend. The number expressing this is called the Quotient and the excess of the dividend over the product of the divisor and the quotient is called the Remainder. .

For example, in the division 32/5, 32 is the Dividend, 5 is the Divisor, 6 is the Quotient and 2 is the Remainder. .

The same operations are performed in algebra also. Algebra treats quantities as in aritlunctie but, with greater generality, for while the quantities used in arithmetical processes are denoted by figures which have single definite value, algebraic quantities are denoted by symbols which may have any value we choose to assign to them. .

(i) The sum of a number of like terms is a like term (like terms are the terms which differ only in their numerical components). .

(ii) If the terms are not all of the same sign, add together separately the coefficient of positive terms and the coefficient of all the negative terms. The difference of these two results preceded by the sign of the greater will give the coefficient of the sum required. .

2a^{2}b - 7a^{2}b
+ 4a^{2}b + 5a^{2}b - 3a^{2}b = a^{2}b(2 + 4 +
5) = a^{2}b (7 + 3) = 11 a^{2}b - 10a^{2}b - a^{2}b

(iii) When expression within the brackets is preceded by the sign “+”, the sign of every term within the bracket remains unaltered even if the bracket is removed. .

However, if the bracket is preceded by the sign “-” the bracket may be removed if the sign of every item within the bracket is changed. .

a + (b – c + d) = a + b – c + d;

a - (b – c + d) = a – b + c - d

The product of two terms with like signs is positive; the product of two terms with unlike signs is negative. .

-1 x -1 = +1; +1 x -1 = -1;

+1 x +1 = -1 x +1 =-1;

In Staff Selection Commission exams, we get question on simplification. .

In simplifying an expression, various operations must be performed as per the following order. .

V → Vinculum

B → Remove Brackets - in the order ( ), { }, [ ]

O → Of

D → Division

M → Multiplication

A → Addition

S → Subtraction

While performing mathematical operations on numbers, speed is very important, Given unlimited time all problems can be solved. To develop speed, it is necessary to know certain simple guidelines. The basis for such rules is the assumption that the student is comfortable and fast in .

(1) additions than in multiplication

(2) multiplying by smaller numbers than bigger numbers. .

(3) dividing by a small number than multiplying by big numbers. .

You have to know by-heart Multiplication Tables up to 20 and Squares of numbers up to 25. .

(i) Multiplication by a number close to 10, 100, 1000, etc. .

For e.g. 9 = 10 - 1 ; 101 = 100 + 1; .

To multiply with such numbers, convert the number into (10 ± k), k = 1, 2, 3 and perform the operation. .

1. 175 x 12 = 175 x (10 + 2) = 175 X 10 + 175 X 2 = 1750 + 350 = 2100 .

2. 46 x 98 = 46(100 - 2) - (46 x 100)-(46 x 2) = 4600 - 92 = 4508 .

3. 13456 x 9899 = (13456 x 10000) - (13456 x 101) = 13456 x 10000 - 13456 (100 + 1) .

= 133200944

(ii) Multiplication
by 5 or powers of 5 can be simplified into multiplication by 10 and its powers
and dividing by 2 and its powers. i.e., 5^{k} = (10/2)^{k}.

1. 68 x 5 = (68 x 10)/2 = 680/2 = 340 .

2. 2345 x 125 = 2345 (10/2)^{3}
= 2345000/8 or 2345 (100 + 100/4) .

The student may choose a method which is more suitable to him/her. .