Useful Number system Shortcuts.
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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)