38 More Math Formulae, Tricks and Shortcuts in QA for CAT and GMAT

This article is the second one in our series of "Maths Tricks and Shortcuts for CAT and GMAT". The first article on Maths Tricks is posted here. This article features 38 of the most important ready made formula or tricks which come in handy while answering objective questions on Quantitative Ability (maths) in management entrance tests such as CAT, GMAT etc. Remembering them would help the aspiring candidates a lot in solving more questions in the QA section accurately in a shorter span of time.

1. If n is even , n(n+1)(n+2) is divisible by 24
   If n is any integer , n^2 + 4 is not divisible by 4
   
2. Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram , the coordinates of the meeting point of the diagonals can be found out by solving for
   [(a+e)/2,(b+f)/2] =[ (c+g)/2 , (d+h)/2]
   
3. Area of a triangle
   1/2*base*altitude = 1/2*a*b*sinC = 1/2*b*c*sinA = 1/2*c*a*sinB = root(s*(s-a)*(s-b)*(s-c)) where s=a+b+c/2
   =a*b*c/(4*R) where R is the CIRCUMRADIUS of the triangle = r*s ,where r is the inradius of the triangle .
   In any triangle
   a=b*CosC + c*CosB
   b=c*CosA + a*CosC
   c=a*CosB + b*CosA
   
4. If a1/b1 = a2/b2 = a3/b3 = .............. , then each ratio is equal to
   (k1*a1+ k2*a2+k3*a3+..............) / (k1*b1+ k2*b2+k3*b3+..............) , which is also equal to
   (a1+a2+a3+............./b1+b2+b3+..........)
   
5. (7)In any triangle
   a/SinA = b/SinB =c/SinC=2R , where R is the circumradius
   
6. x^n -a^n = (x-a)(x^(n-1) + x^(n-2) + .......+ a^(n-1) ) ......Very useful for finding multiples .For example (17-14=3 will be a multiple of 17^3 - 14^3)
   
7. e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + ........to infinity
   2 < e < 3
   
8. log(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 .........to infinity [ Note the alternating sign . .Also note that the logarithm is with respect to base e ]
   
9. In a GP the product of any two terms equidistant from a term is always constant.
   
10. For a cyclic quadrilateral , area = root( (s-a) * (s-b) * (s-c) * (s-d) ) , where s=(a+b+c+d)/2
    
11. For a cyclic quadrilateral, the measure of an external angle is equal to the measure of the internal opposite angle.
    (m+n)! is divisible by m! * n! .
    
12. If a quadrilateral circumscribes a circle , the sum of a pair of opposite sides is equal to the sum of the other pair .
    
    
13. The sum of an infinite GP = a/(1-r) , where a and r are resp. the first term and common ratio of the GP .
    
    The equation whose roots are the reciprocal of the roots of the equation ax^2+bx+c is cx^2+bx+a
    
    
14. The coordinates of the centroid of a triangle with vertices (a,b) (c,d) (e,f)
    is((a+c+e)/3 , (b+d+f)/3) .
    
    
15. The ratio of the radii of the circumcircle and incircle of an equilateral triangle is 2:1 .
    
    
16. Area of a parallelogram = base * height
    
    
17. APPOLLONIUS THEOREM:
    
    In a triangle , if AD be the median to the side BC , then
    AB^2 + AC^2 = 2(AD^2 + BD^2) or 2(AD^2 + DC^2) .
    
    
18. For similar cones, ratio of radii = ratio of their bases.
    The HCF and LCM of two nos. are equal when they are equal.
    
    
19. Volume of a pyramid = 1/3 * base area * height
    
    
20. In an isosceles triangle , the perpendicular from the vertex to the base or the angular bisector from vertex to base bisects the base.
    
    
21. In any triangle the angular bisector of an angle bisects the base in the ratio of the
    other two sides.
    
    
22. The quadrilateral formed by joining the angular bisectors of another quadrilateral is
    always a rectangle.
    
    
23. Roots of x^2+x+1=0 are 1,w,w^2 where 1+w+w^2=0 and w^3=1
    
    
24. |a|+|b| = |a+b| if a*b>=0
    else |a|+|b| >= |a+b|
    
    
25. 2<= (1+1/n)^n <=3
    
    
26. WINE and WATER formula:
    
    If Q be the volume of a vessel
    q qty of a mixture of water and wine be removed each time from a mixture
    n be the number of times this operation be done
    and A be the final qty of wine in the mixture
    then ,
    A/Q = (1-q/Q)^n
    
    
27. Area of a hexagon = root(3) * 3 * (side)^2
    
28. (1+x)^n ~ (1+nx) if x<<<1
    
29. Some Pythagorean triplets:
    
    3,4,5 (3^2=4+5)
    5,12,13 (5^2=12+13)
    7,24,25 (7^2=24+25)
    8,15,17 (8^2 / 2 = 15+17 )
    9,40,41 (9^2=40+41)
    11,60,61 (11^2=60+61)
    12,35,37 (12^2 / 2 = 35+37)
    16,63,65 (16^2 /2 = 63+65)
    20,21,29(EXCEPTION)
    
    
30. Apollonius theorem could be applied to the 4 triangles formed in a parallelogram.
    
    
31. Area of a trapezium = 1/2 * (sum of parallel sides) * height = median * height
    where median is the line joining the midpoints of the oblique sides.
    
32. When a three digit number is reversed and the difference of these two numbers is taken , the middle number is always 9 and the sum of the other two numbers is always 9 .
    
33. Any function of the type y=f(x)=(ax-b)/(bx-a) is always of the form x=f(y) .
    
34. Let W be any point inside a rectangle ABCD .
    Then
    WD^2 + WB^2 = WC^2 + WA^2
    
    Let a be the side of an equilateral triangle. Then if three circles be drawn inside
    this triangle touching each other then each has a radius = a/(2*(root(3)+1)) 
    
    
35. Let 'x' be certain base in which the representation of a number is 'abcd' , then the decimal value of this number is a*x^3 + b*x^2 + c*x + d
    
    
36. When you multiply each side of the inequality by -1, you have to reverse the direction of the inequality.
    
37. To find the squares of numbers from 50 to 59
    
    For 5X^2 , use the formulae
    
    (5X)^2 = 5^2 +X / X^2
    
    E.g. (55^2) = 25+5 /25
    
    =3025
    
    (56)^2 = 25+6/36
    
    =3136
    
    (59)^2 = 25+9/81
    
    =3481
    
38. a+b+(ab/100)
    This is used for successive discounts types of sums.
    Like in 1999 population increases by 10% and then in 2000 by 5%
    So the population in 2000 now is 10+5+(50/100)=+15.5% more that was in 1999
    and if there is a decrease then it will be preceded by a negetive sign and likewise.