Speed and accuracy are of utmost importance in the ultra competitive examination of CAT, especially in the Quantitative Section (Math). The only way of achieving this is by extensive practice and using various tricks, shortcuts and surefire ready-to-work formulas. Hence, it is of utmost importance for any CAT aspirant that he musters all of the below-mentioned formulas if he or she aspires of achieving the top rank at CAT or any other management test viz. XAT/ATMA/MAT etc.







Below is the first part of our new series of articles to be published on this website subsequently wherein we will bring out the most useful and comprehensive formula list plus various important tricks and shortcuts in the Quantitative Ability section with the help of easy-to follow examples.







1. To find the number of factors of a given number, express the number as a product of powers of prime numbers.



In this case, 48 can be written as 16 * 3 = (2^4 * 3)



Now, increment the power of each of the prime numbers by 1 and multiply the result.



In this case it will be (4 + 1)*(1 + 1) = 5 * 2 = 10 (the power of 2 is 4 and the power of 3 is 1)



Therefore, there will 10 factors including 1 and 48. Excluding, these two numbers, you will have 10 – 2 = 8 factors.







2. The sum of first n natural numbers = n (n+1)/2



The sum of squares of first n natural numbers is n (n+1)(2n+1)/6



The sum of first n even numbers= n (n+1).



The sum of first n odd numbers= n^2







3. To find the squares of numbers near numbers of which squares are known



To find 41^2, Add 40+41 to 1600 = 1681



To find 59^2, Subtract 60^2-(60+59) = 3481







4. If an equation (i: e f(x) =0) contains all positive co-efficient of any powers of x , it has no positive roots then.



e.g.: x^4+3x^2+2x+6=0 has no positive roots.







5. For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x) .



Hence the remaining is the minimum number of imaginary roots of the equation (Since we also know that the index of the maximum power of x is the number of roots of an equation.)







6. For a cubic equation ax^3+bx^2+cx+d=o



sum of the roots = - b/a



sum of the product of the roots taken two at a time = c/a



product of the roots = -d/a







7. For a bi-quadratic equation ax^4+bx^3+cx^2+dx+e = 0



sum of the roots = - b/a



sum of the product of the roots taken three at a time = c/a



sum of the product of the roots taken two at a time = -d/a



product of the roots = e/a







8. If for two numbers x+y=k (=constant), then their PRODUCT is MAXIMUM if



x=y (=k/2). The maximum product is then (k^2)/4







9. If for two numbers x*y=k (=constant), then their SUM is MINIMUM if



x=y (=root (k)). The minimum sum is then 2*root (k).







10. Product of any two numbers = Product of their HCF and LCM.



Hence product of two numbers = LCM of the numbers if they are prime to each other.







11. For any regular polygon , the sum of the exterior angles is equal to 360 degrees



Hence measure of any external angle is equal to 360/n. (where n is the number of sides)



For any regular polygon, the sum of interior angles =(n-2)180 degrees



So measure of one angle in



Square =90



Pentagon =108



Hexagon =120



Heptagon =128.5



Octagon =135



Nonagon =140



Decagon = 144







12. If any parallelogram can be inscribed in a circle, it must be a rectangle.







13. If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i:e oblique sides equal).







14. For an isosceles trapezium, sum of a pair of opposite sides is equal in length to the sum of the other pair of opposite sides. (i:e AB+CD = AD+BC , taken in order) .







15. Area of a regular hexagon : root(3)*3/2*(side)*(side)



16. For any 2 numbers a>b







a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic menasa respectively)



(GM)^2 = AM * HM







17. For three positive numbers a, b ,c



(a+b+c) * (1/a+1/b+1/c)>=9







18. For any positive integer n



2<= (1+1/n)^n <=3 19. a^2+b^2+c^2 >= ab+bc+ca



If a=b=c , then the equality holds in the above.



a^4+b^4+c^4+d^4 >=4abcd







20. (n!)^2 > n^n (! for factorial)











21. If a+b+c+d=constant, then the product a^p * b^q * c^r * d^s will be maximum



if a/p = b/q = c/r = d/s .







22. Consider the two equations







a1x+b1y=c1



a2x+b2y=c2







Then ,



If a1/a2 = b1/b2 = c1/c2 , then we have infinite solutions for these equations.



If a1/a2 = b1/b2 <> c1/c2 , then we have no solution for these equations.(<> means not equal to )



If a1/a2 <> b1/b2 , then we have a unique solutions for these equations..







23. For any quadrilateral whose diagonals intersect at right angles , the area of the quadrilateral is



0.5*d1*d2, where d1,d2 are the lengths of the diagonals.











24. Problems on clocks can be tackled as assuming two runners going round a circle , one 12 times as fast as the other . That is ,



the minute hand describes 6 degrees /minute



the hour hand describes 1/2 degrees /minute .



Thus the minute hand describes 5(1/2) degrees more than the hour hand per minute .







25. The hour and the minute hand meet each other after every 65(5/11) minutes after being together at midnight.



(This can be derived from the above).