I) Ratio : The ratio of two quantities in the same units is the fraction that one quantity is of the other . thus , the ratio a to b is the fraction a/b, written as a : b. where  a = antecedent                 b = consequent Example : In  the ratio 2 : 3, represents    2/3   with antecedent = 2 , consequent = 3                                                    Rule :   The value of a ratio remain unchanged , if each one of its terms is multiplied or divided by a same non-zero number . 1) If a > b , then a : b is called a ratio of greater inequality . 2) If a < b , then  a : b is called a ratio of less inequality. 3) If a > b   and same  positive number is added to each term of a : b , then the ratio is diminished . 4) If a < b and same positive number is added to each term of a : b , then the ratio is increased . 5) a 2 : b 2 is called duplicate ratio of a : b . 6) a 3  : b 3    is called triplet ratio of a : b . 7)  √ a : √b is called sub - duplicate ratio of a

I.) Factors When a divides b exactly , then we say that a is a factor of b and we write , a/b . Example : 1) Prime factorisation of  22475 ? Solution :      5  |  22475      5  |    4495    29  |      899    31  |        31          |           1 Thus, prime factorisation of  22475 = 5×5×29×31. II.) Multiplies When a divides b exactly , then we say that b is a ,multiple of a . III.) H.C.F. (Highest Common Factor ) or G.C.D. (Greatest Common Factor ) or  G.C.M. (Greatest common Multiple ) The H.C.F. of two or more than two numbers is greatest number that divides each one of them exactly. This is also Known as greatest cimmon divisor (G.C.D. ) or greatest common measure (G.C.M.) . To Find H.C.F. of Given Numbers : 1) H.C.F. By Factorisation : Express each number as the product of primes . Now, take the product of least powers of common factors to get the H.C.F. . Example : a) Find the H.C.F. of 96 , 528 and 792 .  Sol.  96   = 2X2X2X2X2X3 528 = 2X2X2X2X3X11 792 = 2X2X2X3X3X11 H.C.F. = 2X

Percentage    Percent mean that many hundredths. To express      x % as a fraction :                                x %  =  x/100 Thus                   20 %  = 20/100 = 1/5 To express    a/b as a percent :                              a/b = [(a X 100 )/b ] % Thus               1/5 = [( 1 X 100 )/ 5] %                                      = 20 % Example :- 1) Express as a fraction  64 %. Sol.  64 % =  64/100                        =   16/25 2) Express as a decimal 34 % Sol.  34 % =  34/100                       =    0.34 3) Express as a rate percent 3/7 Sol.  3/7    =  [(3 X 100 )/ 7 ]                                6                    =    42-------%                                  7       Rule I. If the price of a commodity increases by r%, then reduction in consumption , so as not to increase the expenditure , is : or If A s income is r % more than that of B , then : B s income is less than that of A by : We use same formula in above both condition      r X 100  ----------------- 

Square and Square Root 1) Square :- A number multiply by itself is called square of a number. let a number be y , y X y = y^2 Example :- 1) 10 X 10 = 100 2)  5 X 5 = 25 3) 11 X 11 =121 2) Square Root :- If x^2 = y, we say that square root of y is x and we write, √y = x Rule :- If a given number is a perfect square, we resolve it into the product of prime factors and take the product of prime factors choosing one out of every pair of the same primes. Example :- 1) Evaluate :  √225 Sol. Resolving 225 into prime factors , we get √ 225 =√( 3 X 3 X 5 X 5 )           =  3 X 5           =  15 2) Find the square root of 1734489. Sol.                           1317               I-------------------------------------            1   I     01  73  44  89       + 1   I      -1     -------- I-------------------------------------       23    I       0  73       +3    I        -  69     -------- I--------------------------------------     261    I             444      + 1    I          -  261     ---