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  www.talentsprint.com TalentSprint©2016 Number system 1.   When sum and difference of two numbers (X and Y) are given, then X = (sum + difference)/2 Y = (sum + difference)/2 2.   Difference between two digits of two digit number is = (Difference in srcinal and interchanged number)/9 3.   Sum of first n odd numbers is n 2  4.   Sum of first n even numbers n(n+1) 5.   Sum of squares of first n natural numbers is n(n+1)(2n+1)/6 6.   Sum of cubes of first n natural numbers is [n(n+1)/2] 2   Algebra 1.   (a+b) 2  = a 2 + 2ab + b 2 2.   (a-b) 2  = a 2 - 2ab + b 2 3.   (a+b) 2  = (a-b) 2 + 4ab 4.   (a-b) 2 = (a+b) 2 - 4ab 5.   (a+b) 3 = a 3 +b 3 +3ab(a+b) = a 3 +b 3 +3a 2 b+3ab 2 6.   (a-b) 3 = a 3 - b 3 -3ab(a-b) = a 3 - b 3 - 3a 2 b + 3ab 2  7.   a 3 +b 3  =(a+b) 3  - 3ab(a+b) 8.   a 3 - b 3  =(a-b) 3  + 3ab(a-b)  www.talentsprint.com TalentSprint©2016 9.   a 2 -b 2  = (a-b)(a+b) 10.a 3 +b 3  = (a+b)(a 2 -ab+b 2 ) 11.a 3 - b 3  = (a-b)(a 2 +ab+b 2 ) 12.a m  x a n  = a m+n  13.a m  / a n  = a m –  n  14.(a/b) (m / n)  =(b/a) -(m / n)  15.a m  / b   –  n  = a m x b n   Ratio and Proportion  1.   If four quantities are in proportion, then Product of Means = Product of Extremes. In the proportion a:b::c:d, we have bc = ad 2.   If a:b::c:x, x is called the fourth proportional of a, b, c. a/b = c/x or, x = bc/a. 3.   If two numbers are in a:b ratio and the sum of these numbers is x, then numbers will be ax/(a+b) and bx/(a+b) respectively 4.   If three numbers are in the ratio a:b:c and the sum of these numbers is x, then these numbers will be ax/(a+b+c), bx/(a+b+c) and cx/(a+b+c) respectively 5.   The ratio of two numbers is a : b. If n is added to each of these numbers, the ratio becomes c : d. The two numbers will be given as an(c-d)/(ad-bc) and bn(c-d)/(ad-bc) respectively  www.talentsprint.com TalentSprint©2016 6.   The ratio of two numbers is a : b. If n is subtracted from each of these numbers, the ratio becomes c : d. The two numbers are given as an(d-c)/(ad-bc) and bn(d-c)/(ad- bc) respectively 7.   If the ratio of two numbers is a: b, then the numbers that should be added to each of the numbers in order to make this ratio c:d is given by (ad-bc)/(c-d) 8.   If the ratio of two numbers is a:b, then the number that should be subtracted from each of the numbers in order to make this ratio c:d is given by (bc-ad)/(c-d) 9.   The CP of the item that is cheaper is CP cheaper  and the CP of the item that is costlier (dearer) is CP Dearer . The CP of unit quantity of the final mixture is called the Mean Price and is given by       =  ℎ −             −    ℎ   Percentage 1.   a % of b = a x b/100 2.   If A is x% more than B, then B is less than A by %100100    x x  3.   If A is x% less than B, then B is more than A by %100100    x x  4.   If A is x% of C and B is y% of C, then A = x/y   B  www.talentsprint.com TalentSprint©2016 5.   If two numbers are respectively x% and y% more than a third number, then first number is %100100100       y x of the second number and the second number is %100100100       x y  of the first number 6.   If two numbers are respectively x% and y% less than a third number, then the first number is %100100100       y x  of the second number and the second number is %100100100       x y  of the first number 7.   If the price of a commodity decreases by P %, then the increase in consumption so that the expenditure remains same is %100100       P  P   8.   If the price of a commodity increases by P%, then the reduction in consumption so that the expenditure remains same is %100100       P  P    9.   If a number is changed (increased/decreased) successively by x% and y%, then net% change is given by [x+y+(xy/100)]%, which represents increase or decrease in value according as the sign is positive or negative 10.If two parameters A and B are multiplied to get a product and if A is changed by x% and another parameter B is changed by y%, then the net% change in the product (A   B) is given [x+y+(xy/100)]% 11.In an examination, the minimum pass percentage is x%. If a student secures y marks and fails by z marks, then the maximum marks in the examination is 100(y+z)/x
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