Quant Exceptions
By ASAX
Arithmetic/Numbers:
1.
2.
3.
4.
Numbers means: -1,-(1/2),0,(1/2),1 [Unless specified]
All Prime numbers are not odd. 2 is the ONLY even prime number
Watch out for 0/1/-1 exceptions in all numerical related problems
Number-Reverse always divisible by 9. Number+Reverse always
divisible by 11.
5. AP:
a. Sequence a1, a2,…an, so that a(n)=a(n-1)+d (constant)
b. nth term an = a1 + d ( n – 1 )
c. Sn=n*(a1+an)/2 or Sn=n*(2a1+d(n-1))/2
6. GP:
a. The general sum of a N term GP with common ratio R
=A1*(RN – 1)/(r-1), A1 first term.
b. If an infinite GP (|r|<1) = A1/(1-r)
7. Number Series:
a. 1-N Sum of all numbers = N(N+1)/2
b. 1-N sum of odd numbers =N2 nth term=2n-1
c. 0-N sum of even numbers =N2-N nth term=2n
d. 1-N sum of squares of all numbers = N(N+1)(2N+1)/6
e. 1-N Sum of square of first N Odd numbers = N(2N-1)(2N+1)/3
f. 1-N sum of cubes of all numbers = (N(N+1)/2)2
Special Number 0:
8.
9.
10.
11.
12.
0 is neither positive nor negative.
0 is even integer.
0!=1
01=0
00=Undefined
Factors, Factorials & Percent:
13. Number=GCD*LCM
14. Consider 1 as one of the factors while counting them. E.g.
Factors of X2 are X2,X and 1
15. Sum of all factors of perfect square is odd
16. Factors ≤ N ≤ Multiples
17. Sum of N consecutive numbers always divisible by N, if N is odd.
18. Product of N consecutive numbers always divisible by N!
Sept, 2012
Quant Exceptions
By ASAX
19. (N! + 1) is not divisible by any positive integer less than or equal to
N.
20. Maximum number of factors <N is the Power of 2<N
21. If you notice 210=2*3*5*7=the product of the first four primes. So,
210+1=211 must be a prime. For example: 2+1=3=prime, 2*3+1=7=prime,
2*3*5+1=31=prime.
22. Remainders
a. X/p,q,r gives same remainder (s) X=kLCM(p,q,r)+s
b. X/p,q,r gives remainders s,t,u where p-s=q-t=r-u=v(say),
X=kLCM(p,q,r)-v
c. 9A+2,11B+7 Give values For A & B =0,1,2.. common number will
be X
d. (6!)7!/13 =(6!)12K/13 because power is12K(ie 13-1) remainder is 1
e. 254/7=[(2) 3] 16/(23-1) = Therefore remainder is 1.
23. 0!=1!=1
24. Number of Zeros in Factorial=(127/5)+(127/25)+(125/125)=25+5+1=31
25. % change not equal to %
26. X% inc, X% Dec total X2/100 Dec
27. Parallel Ratios Method: x/y = 5/3 ; (x-10)/(y+10) = 7/5
28. Unknown Multiplier Method: (5x – 10)/(3x + 10) = 7/5
29. Higher/greater than is different from times.
30. Divisor of a => factor of A => a number when a is divided by
results in 0 remainder
31. S.P=(100±P)/CP Discount=(S.P-M.P)/M.P
32. Successive Discounts=M.P(100-D1)(100-D2)/100
33. SI=PNR/100
34. CI=P(1+(R/100t)tn—t – number of times per year. Half yearly =>2 times
year. Etc
35. R=[(Final/Initial)(1/n) -1]
Special Interest – Perfect Squares:
36. Perfect square has odd number of DISTINCT factors
37. Sum of DISTINCT factors is odd
38. Odd number of Odd-factors and Even number of Even factors
(1,3,9) & (2,8,18,36) for 36
39. Even powers for PRIME factors
Exponents:
Sept, 2012
Quant Exceptions
By ASAX
40. n0=00=1-1=11
41. X2=(±X)2 - Even power masks the SIGN.
42. When raised to powers Positive number increase in value, except
those between 0 and 1.
43. x = bY
is the same as
y = logbx
2
2
2
44. (a+b) = a +b +2ab
45. (a3+b3)=(a+b)(a2-ab+b2)
46. (a3-b3)=(a+b)(a2+ab+b2)
47. (a+b)3=a3+b3+3ab(a+b)
48. Any positive integer root from a number more than 1 will be
more than 1. For example:
.
Algebraic expressions:
49. X(X-1)=X-1 => NEVER cancel out the like terms in algebraic
expressions. Expand to get maximum roots.
50. Equation with two unknown – we can always find out the value
of one or two of the unknowns if extra condition (implicit or
explicit) such as – real/whole number is given.
51. Root= -b±√(b2-4ac)/2a
Inequalities:
52. To multiply/Divide variables you need to know the SIGN of the
Variables.
53. Two inequalities can be added if pointed it at the same directions
54. Two inequalities can be multiplied if pointed it at the same
directions and are positive.
55. You cannot DIVIDE two inequalities.
56. One side is – other side is + you cannot square.
57. a1/x<ax for all a>1
58. X 2 > 0 if X≠0
59. √x > y & x2 > Y => x > y
60. |X|<1 => for X >0, X<1 for X<0 –X<1 => X>-1  -1<X<1
Absolute values
61. |x|+|y|=9 can be used for simultaneous equations with unknown |x|
& |y|
Sept, 2012
Quant Exceptions
By ASAX
62. Substitute the values in to the expression, for an absolute value
equation; both the solutions need not be valid.
63. An absolute Value is ALWAYS >0, whenever you see absolute value
on one side of the equation keep that in Mind!
a. 3|x2 – 4| = y – 2, LHS always +ve(absolute value) => RHS can’t be –ve => y≥2
64. - M = | - M | => -M never Negative => -M <=0 =====ALWAYS CONSIDER ZERO=====
65. Note that
. Next, since
and
then
and
.
Statistics:
66. Numbers below median/mean are ≤ median/mean.
67. Consecutive integers – mean=median is non-integer => total # =
even else odd.
68. Mean = 0, either all are 0 or sum is zero. [-1, 1] or [-1,0,1]
69. Standard Deviation: 3 ways to express:
a. Sqrt(mean squared distances of the numbers from the
mean of the numbers)
b. Sqrt(X2 –(x) 2) Square of mean-mean of squares of numbers
c. Sqrt(variance of the set)
70. PERCENTAGE bound by ±1SD,±2SD,±3SD = 62.8%,95.4%,99.7%
71. Maximize Range – X,X,Mean,Mean,Y -----Then <mean keep it least ie
equal to first one. >mean keep it least => equal to mean and then
largest
Sets: 3 Set diagram
72. In a set – 42 people in group, 29 employed, 24 students – employed
students? – Can’t say as we don’t know how many people are NOT
employees or NOT students.
73. X (total/Given number)=I+II+III [I is only, II is two over lapping, III All]
74. I=IA+IB+IC
75. S=I+2II+3III
76. S-X=II+2III
77. X is max when S=X
78. X is second Max when II(min)+2III
79. X is third Max when only III exists – all are over lapping.
80. III Min
100 (total)
A
90
100-90=10
B
80
20
C
70
30
Sept, 2012
Quant Exceptions
III Min= 100-60 = 40
By ASAX
60
Geometry
81. Parallel lines in a circle given one angle all the arcs lengths
made by chords can be found.
82. Imposter Parallel lines it may seem, but might not
be.
83. Maximum # 90degree angle in triangle is 1.
84. For lengths of triangles A-B<C<A+B
85. A2+B2>C2 for acute angle triangles.
86. A2+B2<C2 for obtuse angle triangles
87. A2+B2=C2 for perpendicular
88. Median,NOT altitude or angular bisector divides the side
equally. [Exception: equilateral]
89. Right Triangle, altitude divides triangle in to two similar
triangles.
90. Apply Similarity of triangles after finding corresponding Similar
ANGLES to resolve ratios of lengths
91. Imposter triangle - side 3, 4, Third side need not be 5, because need
not be right angled.
92. Isosceles means two sides are equal. Unless said, any two sides can
be equal
93. Diagonals perpendicular & bisect each other => Rhombus or
Square.
94. Diagonals equal => Rectangle or Square
95. All squares are rectangles
96. Maximize a product by equaling the numbers. Square has the
maximum area for any given perimeter A+B=Constant, AB]max when
A=B
97. Similarly, Square has minimum perimeter for given area.
AB=Constant, A+B] min when A=B
98. For diagonals, use √2 for Square and √3 for Cuboid
99. For an isosceles triangle, the area will be maximum when it is a
right angled triangle. An isosceles triangle can be considered
as one half of a rhombus with side lengths 'b'. Now a rhombus of
greatest area is a square, half of which is a right angled
isosceles triangle
100. Radius Circle inscribed in a right angle triangle. = (a+b-c)/2
Sept, 2012
Quant Exceptions
By ASAX
Probability
101. In simultaneous picks, pretend it not simultaneous, Break it
down. Work it as without replacement problems.
102. Probability of A cannot be used to determine Probability of B.
(independent events)
RTD
103.
104.
105.
Average Speed = Total Distance/Total time NOT (speed1+speed2)/2
Unit conversion – minutes hours => kmph=5/18 mps
Speed Proportional 1/Time
a. 40% Inc Speed == 28%Decrease Time or
b. (2/5) Inc Speed == (2/5+2) Decrease Time
106. Circular – total distance = distance of circular path for
both the cases – kiss and depart
107. Overtaking (Same direction) – A & B same distance
108. Meeting (Opposite directions) – A & B Same time
109. A=======B One Starting at A at certain time, another Starting
at B at another time, First meet= [Common Time x Speed component
(either First Starting/reaching) ] + Time of Late Starter
110. A=======B First Meet = D, Second Meet = 2D+1D…
111. Circular – First Meet = D, Second Meet = 1D+1D
112. Circular – Meeting Time:
Time
Same Direction
Opp Direction
First Meet
Distance/(a-b)
Distance/(a+b)
First Meet @
LCM (D/a, D/b)
LCM (D/a, D/b)
Starting Pt
113. Average Speed: Total.D/Total.Time
a. Same Distance , Avg.Speed= Harmonic Mean of Speeds =
2ab/(a+b)
b. Same Time, Avg, Speed = Arithmetic Mean of Speeds = (a+b)/2
RTW
114. rate1 +rate2= total Rate [we usually do, 1/time1 +1/time2=1/total time,
which is also right]
Co-ordinate Geometry
115. Area of Triangle: Determinants: XY1 (1,0,1)(2,3,1)(5,7,1)
Sept, 2012
Quant Exceptions
By ASAX
116. Parallel Lines m1=m2, perpendicular lines m1*m2=-1
117. A certain square is to be drawn on a coordinate plane. One of
the vertices must be on the origin, and the square is to have an
area of 100. If all coordinates of the vertices must be integers,
how many different ways can this square be drawn? 12 not 8.
118. Perpendicular bisector passes through the mid-point
119. +/- Slope
120. Parabola = ax2+bx+c. +/- value of a = Up/Downward. |a| High/low
= Narrow/wide
121. b2-4ac > 0 => 2 roots/intercepts =0 => equal roots <0 =>no
solution/no intercepts
122. Reflection along Y axis. X=-x
123. Reflection along X axis. Y=-y
124. Reflection along x=y line. Y=x
125. Reflection along y=|x| => v
126. Graph http://gmatclub.com/forum/quick-way-to-graphinequalities-76255.html
127. Using Graph to solve http://gmatclub.com/forum/graphicapproach-to-problems-with-inequalities-68037.html
128. Positive slope: if +ve X intercept => -ve Y intercept.
Mixtures:
129. Initial Volume P. Q Volume taken out and replaced n times.
Final Volume in Solution= ((P-Q)/P)n
Permutation & Combination:
130. Combination - Permutation where x elements are identical:
nPk/x!
131. Circular Permutation = (n-1)!
132. Consider 0s and 1s.
a. How many ways to give 12 chocolates among 3 children
(A,B,C). ABC can be 0
i. A+B+C=12 => One scenario - 0|00|000000000 1,2,9.
There are 12 0’s and 2 |’s =>Number of
ways=(12+2)C2=14C2 ways
b. 3 dice thrown. Probability sum=12. Here, A,B,C can take
values from (1-6)
Sept, 2012
Quant Exceptions
By ASAX
c. Routes North & south roads: Rows(r) Columns (c).
Number of routes from point A to B via R*C = (r+c)Cr
Other Misconceptions:
133. If it is said, A is preceded by B, B is preceded by C. Then, we
cannot, take it as. ABC,ABC, but only as ABC. It is not a pattern.
Numbers:
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16 2/3% = 1/6
33 1/3% = 1/3
66 2/3% = 2/3
83 1/3% = 5/6
40% = 2/5
60% = 3/5
80% = 4/5
12 1/2% = 1/8
37 1/2% = 3/8
62 1/2% = 5/8
87 1/2% = 7/8
Prime Numbers 2 3 5 7, 11 13 17 19, 23 29, 31 37, 41 43 47, 53 59, 61 67, 71 73 79,
83 89, 97 101
Perfect Numbers: 6,28,496,8128
√2=1.414 & √3=1.732 pi=3.14
Unique Number – 37K=111,222,333,..
Last Digit Patterns:
o 2 = 2,4,8,6
o 3 = 3,9,7,1
o 4 = 4,6
o 5=5
o 6=6
o 7 = 7,9,3,1
o 8 = 8,4,2,6
o 9 = 1,9
Sept, 2012