Usefull Number system Short-cuts.

1. When Sum and Diff of two numbers (X and Y) are given, then X = (sum + diff)/2 Y = (sum - diff)/2 . 2. Diff between two digits of two digit number is =(Diff in original and interchanged number)/9 . 3. Sum of first n odd nos is n2 . 4. Sum of first n even nos n(n+1) . 5. Sum of squares of first n natural no’s is n(n+1)(2n+1)/6 . 6. Sum of cubes of first n natural numbers is [n(n+1)/2]2 . 7. If the sum of squares of two numbers is x and the square of their diff is y, then the product of the two numbers is [(x-y)/2] . *********Ratio and Proportion Short-cuts********* . 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. 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) . 6. 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)