Quantitative Aptitude formulae in a quick look

Finding number of Factors
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 = (24 * 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.

Sum of n natural numbers
-> 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

Finding Squares of numbers
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

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

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

Finding number of Imaginary Roots
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 are 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.)

Reciprocal Roots
The equation whose roots are the reciprocal of the roots of the equation ax^2+bx+c is cx^2+bx+a

Roots
Roots of x^2+x+1=0 are 1,w,w^2 where 1+w+w^2=0 and w^3=1

Finding Sum of the rootsFor 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

For a biquadratic 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

Maximum/Minimum
-> 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

-> 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) .


Inequalities
-> x + y >= x+y ( stands for absolute value or modulus ) (Useful in solving some inequations)

-> a+b=a+b if a*b>=0 else a+b >= a+b

-> 2<= (1+1/n)^n <=3 -> (1+x)^n ~ (1+nx) if x<<<1> When you multiply each side of the inequality by -1, you have to reverse the direction of the inequality.


Product Vs HCF-LCM 
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

AM GM HM
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

Sum of Exterior Angles
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

Problems on clocks
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 .
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) .


Co-ordinates
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]

Ratio
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+..........)

Finding multiples
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)

Exponents
e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + ........to infinity 2 <>GP
-> In a GP the product of any two terms equidistant from a term is always constant .

-> The sum of an infinite GP = a/(1-r) , where a and r are resp. the first term and common ratio of the GP .

Mixtures
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

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)

Appolonius theorem
Appolonius theorem could be applied to the 4 triangles formed in a parallelogram.

Function
Any function of the type y=f(x)=(ax-b)/(bx-a) is always of the form x=f(y) .

Finding Squares
To find the squares of numbers from 50 to 59

For 5X^2 , use the formulae

(5X)^2 = 5^2 +X / X^2

Eg ; (55^2) = 25+5 /25
=3025
(56)^2 = 25+6/36
=3136
(59)^2 = 25+9/81
=3481

Successive Discounts
Formula for successive discounts
a+b+(ab/100)
This is used for succesive discounts types of sums.like 1999 population increses 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 preceeded by a -ve sign and likewise.

Rules of Logarithms:
-> loga(M)=y if and only if M=ay

-> loga(MN)=loga(M)+loga(N)

-> loga(M/N)=loga(M)-loga(N)

-> loga(Mp)=p*loga(M)

-> loga(1)=0-> loga(ap)=p

-> log(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 .........to infinity [ Note the alternating sign . .Also note that the ogarithm is with respect to base e ]


ALGEBRA :

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

2.      Sum of the squares of first n natural numbers = n(n+1)(2n+1)/6

3.      Sum of the cubes of first n natural numbers = [n(n+1)/2]²

4.      Sum of first n natural odd numbers = n²

5.      Average = (Sum of items)/Number of items

 

Arithmetic Progression (A.P.):

An A.P. is of the form a, a+d, a+2d, a+3d, ...

where a is called the 'first term' and d is called the 'common difference'

1.      nth term of an A.P. tn = a + (n-1)d

2.      Sum of the first n terms of an A.P. Sn = n/2[2a+(n-1)d] or Sn = n/2(first term + last term)

 

Geometrical Progression (G.P.):

A G.P. is of the form a, ar, ar², ar³, ...

where a is called the 'first term' and r is called the 'common ratio'.

1.      nth term of a G.P. t_n = ar^n-1

2.      Sum of the first n terms in a G.P. Sn = a|1-rⁿ|/|1-r|

 

Permutations and Combinations :

1.      nPr = n!/(n-r)!

2.      nPn = n!

3.      nP1 = n

 

1.      nCr = n!/(r! (n-r)!)

2.      nC1 = n

3.      nC0 = 1 = nCn

4.      nCr = nCn-r

5.      nCr = nPr/r!

 

Number of diagonals in a geometric figure of n sides = nC2-n

 

Tests of Divisibility :

1.      A number is divisible by 2 if it is an even number.

2.      A number is divisible by 3 if the sum of the digits is divisible by 3.

3.      A number is divisible by 4 if the number formed by the last two digits is divisible by 4.

4.      A number is divisible by 5 if the units digit is either 5 or 0.

5.      A number is divisible by 6 if the number is divisible by both 2 and 3.

6.      A number is divisible by 8 if the number formed by the last three digits is divisible by 8.

7.      A number is divisible by 9 if the sum of the digits is divisible by 9.

8.      A number is divisible by 10 if the units digit is 0.

9.      A number is divisible by 11 if the difference of the sum of its digits at odd places and the sum of its digits at even places, is divisible by 11.

 

H.C.F and L.C.M :

H.C.F stands for Highest Common Factor. The other names for H.C.F are Greatest Common Divisor (G.C.D) and Greatest Common Measure (G.C.M).

The H.C.F. of two or more numbers is the greatest number that divides each one of them exactly.

The least number which is exactly divisible by each one of the given numbers is called their L.C.M.

Two numbers are said to be co-prime if their H.C.F. is 1.

H.C.F. of fractions = H.C.F. of numerators/L.C.M of denominators

L.C.M. of fractions = G.C.D. of numerators/H.C.F of denominators

 

Product of two numbers = Product of their H.C.F. and L.C.M.

 

PERCENTAGES :

1.      If A is R% more than B, then B is less than A by R / (100+R) * 100

2.      If A is R% less than B, then B is more than A by R / (100-R) * 100

3.      If the price of a commodity increases by R%, then reduction in consumption, not to increase the expenditure is : R/(100+R)*100

4.      If the price of a commodity decreases by R%, then the increase in consumption, not to decrease the expenditure is : R/(100-R)*100

 

PROFIT & LOSS :

1.      Gain = Selling Price(S.P.) - Cost Price(C.P)

2.      Loss = C.P. - S.P.

3.      Gain % = Gain * 100 / C.P.

4.      Loss % = Loss * 100 / C.P.

5.      S.P. = (100+Gain%)/100*C.P.

6.      S.P. = (100-Loss%)/100*C.P.

 

Short cut Methods:

1.      By selling an article for Rs. X, a man loses l%. At what price should he sell it to gain y%?       (or)

A man lost l% by selling an article for Rs. X. What percent shall he gain or lose by selling it for Rs. Y?

 

(100 – loss%) : 1^st S.P. = (100 + gain%) : 2^nd S.P.

 

2.      A man sold two articles for Rs. X each. On one he gains y% while on the other he loses y%. How much does he gain or lose in the whole transaction?

In such a question, there is always a lose. The selling price is immaterial.

Formula for loss %   =

RATIO & PROPORTIONS:

1.      The ratio a : b represents a fraction a/b. a is called antecedent and b is called consequent.

2.      The equality of two different ratios is called proportion.

3.      If a : b = c : d then a, b, c, d are in proportion. This is represented by a : b :: c : d.

4.      In a : b = c : d, then we have  a* d = b * c.

5.      If a/b = c/d then ( a + b ) / ( a – b  ) = ( d + c ) / ( d – c ).

 

TIME & WORK :

1.      If A can do a piece of work in n days, then A's 1 day's work = 1/n

2.      If A and B work together for n days, then (A+B)'s 1 days's work = 1/n

3.      If A is twice as good workman as B, then ratio of work done by A and B = 2:1

 

PIPES & CISTERNS :

1.      If a pipe can fill a tank in x hours, then part of tank filled in one hour = 1/x

2.      If a pipe can empty a full tank in y hours, then part emptied in one hour = 1/y

3.      If a pipe can fill a tank in x hours, and another pipe can empty the full tank in y hours, then on opening both the pipes,

 

the net part filled in 1 hour = (1/x-1/y)  if y>x

the net part emptied in 1 hour = (1/y-1/x) if x>y

 

TIME & DISTANCE :

1.      Distance = Speed * Time

2.      1 km/hr = 5/18 m/sec

3.      1 m/sec = 18/5 km/hr

4.      Suppose a man covers a certain distance at x kmph and an equal distance at y kmph. Then, the average speed during the whole journey is 2xy/(x+y) kmph.

 

PROBLEMS ON TRAINS :

1.      Time taken by a train x metres long in passing a signal post or a pole or a standing man is equal to the time taken by the train to cover x metres.

2.      Time taken by a train x metres long in passing a stationary object of length y metres is equal to the time taken by the train to cover x+y metres.

3.      Suppose two trains are moving in the same direction at u kmph and v kmph such that u>v, then their relative speed = u-v kmph.

4.      If two trains of length x km and y km are moving in the same direction at u kmph and v kmph, where u>v, then time taken by the faster train to cross the slower train = (x+y)/(u-v) hours.

5.      Suppose two trains are moving in opposite directions at u kmph and v kmph. Then, their relative speed = (u+v) kmph.

6.      If two trains of length x km and y km are moving in the opposite directions at u kmph and v kmph, then time taken by the trains to cross each other = (x+y)/(u+v)hours.

7.      If two trains start at the same time from two points A and B towards each other and after crossing they take a and b hours in reaching B and A respectively, then A's speed : B's speed = (√b : √

 

SIMPLE & COMPOUND INTERESTS :

Let P be the principal, R be the interest rate percent per annum, and N be the time period.

1.      Simple Interest = (P*N*R)/100

2.      Compound Interest = P(1 + R/100)^N – P

3.      Amount = Principal + Interest

 

LOGORITHMS :

If a^m = x , then m = log_ax.

Properties :

1.      log _xx = 1

2.      log _x1 = 0

3.      log _a(xy) = log _ax + log _ay

4.      log _a(x/y) = log _ax - log _ay

5.      log _ax = 1/log _xa

6.      log _a(x^p) = p(log _ax)

7.      log _ax = log _bx/log _ba

Note : Logarithms for base 1 does not exist.

 

AREA & PERIMETER :

Shape                          Area                            Perimeter

Circle                           ∏ (Radius)²                 2∏(Radius)

Square                         (side)²                          4(side)

Rectangle                    length*breadth                        2(length+breadth)

 

1.      Area of a triangle = 1/2*Base*Height or

2.      Area of a triangle = √ (s(s-(s-b)(s-c)) where a,b,c are the lengths of the sides and s = (a+b+c)/2

3.      Area of a parallelogram = Base * Height

4.      Area of a rhombus = 1/2(Product of diagonals)

5.      Area of a trapezium = 1/2(Sum of parallel sides)(distance between the parallel sides)

6.      Area of a quadrilateral = 1/2(diagonal)(Sum of sides)

7.      Area of a regular hexagon = 6(√3/4)(side)²  

8.      Area of a ring = ∏(R²-r²) where R and r are the outer and inner radii of the ring.

 

VOLUME & SURFACE AREA :

Cube :

Let a be the length of each edge. Then,

1.      Volume of the cube = a³ cubic units

2.      Surface Area = 6a² square units

3.      Diagonal = √ 3 a units

Cuboid :

Let l be the length, b be the breadth and h be the height of a cuboid. Then

1.      Volume = lbh cu units

2.      Surface Area = 2(lb+bh+lh) sq units

3.      Diagonal = √ (l²+b²+h²)

Cylinder :

Let radius of the base be r and height of the cylinder be h. Then,

1.      Volume = ∏r²h cu units

2.      Curved Surface Area = 2∏rh sq units

3.      Total Surface Area = 2∏rh + 2∏r² sq units

Cone :

Let r be the radius of base,  h be the height, and l be the slant height of the cone. Then,

1.      l² = h² + r²

2.      Volume = 1/3(∏r²h) cu units

3.      Curved Surface Area = ∏rl sq units

4.      Total Surface Area = ∏rl + ∏r² sq units

Sphere :

Let r be the radius of the sphere. Then,

1.      Volume = (4/3)∏r³ cu units

2.      Surface Area = 4∏r² sq units

Hemi-sphere :

Let r be the radius of the hemi-sphere. Then,

1.      Volume = (2/3)∏r³ cu units

2.      Curved Surface Area = 2∏r² sq units

3.      Total Surface Area = 3∏r² sq units

Prism : 

      1.   Volume = (Area of base)(Height)