GRE USEFUL FORMULAE........

GEOMETRY - TRIANGLES
1. Area of a triangle with base b and height h = (1/2)*b*h
2. The area of an equilateral triangle with side a is [sqrt(3)/4]*a^2
3. The area of any triangle given the length of its 3 sides a, b and c:is sqrt[s(s-a)(s-b)(s-c)] where s= (a+b+c)/2


SOME USEFUL FACTS

Number Theory

The product of any three consecutive integers is divisible by 6.

Similarly, the product of any four consecutive integers is divisible by 24.

Permutation and Combination

When n dice (n > 1) are rolled simultaneously, the number of outcomes in which all n dice show the same number is 6, irrespective of the value of n.

Similarly, when n fair coins (n > 1) are tossed simultaneously, the number of outcomes in which all n coins turn up as heads or as tails is 1, irrespective of the value of n.

Speed Time

When an object travels the first x hours at p km/hr and the next x hours at q km/hr, the average speed of travel is the arithmetic mean of p and q.

However, when the object travels the first x kms at p km/hr and the next x kms at q km/hr, the average speed is the harmonic mean of p and q.
Number Theory

Any perfect square has an odd number of factors including 1 and the number itself and a composite number has an even number of factors including 1 and itself.

Any perfect square can be expressed in the form 4n or 4n+1.

Profit Loss

If the selling price of 2 articles are equal and 1 of them is sold at a profit of p% and the other at a loss of p%, then the 2 trades will result in a cumulative loss of ((p^2)/100)%.

If the cost of price of 2 articles are equal and 1 of them is sold at a profit of p% and the other at a loss of p%, then the 2 trades will result in no profit or loss.

Progressions

Arithmetic mean of 'n' numbers will always be greater than the geometric mean of those 'n' numbers which will be greater than the Harmonic mean of those 'n' numbers.

Arithmetic mean of 2 numbers = geometric mean of '2' numbers = harmonic mean of '2' numbers if both the numbers are equal.
TIME AND WORK
If A can do a piece of work in x days, then A’s one day’s work=1/x
• If the ratio of time taken by A and B in doing a work is x:y, then, ratio of work done is 1/x :1/y=y:x. And the ratio in which the wages is to be distributed is y:x
• If A can do a work in x days and B can do the same work in y days, then A and B can together do the work in (xy)/(x+y) days
• If “a” men or “b” women can do a piece of work in x days, then “m” men and “n” women can together finish the work in (abx)/(an+bm) days
• If A is x times efficient than B, and working together, they finish the work in y days, then Time taken by A=y(x+1)/(x), Time taken by B=y(x+1)
• If A and B can finish a work in “x” and “ax” days respectively, that is if A is “a” times efficient than B, then working together, they can finish the work in (ax)/(a+1) days
• If A and B working together can complete a work in x days, whereas B working alone can do the same work in y days, ten, A alone will complete the work in (xy)/(y-x) days.
• Pipe A can fill a tank in x hrs and B can empty a tank in y hrs.If both pipes are opened together, the tank will be filled in (xy)/(y-x) hrs
• A pipe can fill a cistern in x hrs but due to leakage in the bottom, it is filled in y hrs, then the time taken by the leak to empty the cistern is (xy)/(y-x) hrs
GEOMETRY STUFF

Prisms



Volume = Base area X Height
Surface = 2b + Ph (b is the area of the base P is the perimeter of the base)

Cylinder



Volume = r*2 h
Surface = 2rh

Pyramid



V = 1/3 bh

b is the area of the base
Surface Area: Add the area of the base to the sum of the areas of all of the triangular faces. The areas of the triangular faces will have different formulas for different shaped bases.



Cones



Volume = 1/3 r*2 x height = 1/3 r2h
Surface = r2 + rs = r2 + r





Sphere



Volume = 4/3 r^3
Surface area = 4r^2

Distance of a Point from a Line

The perpendicular distance d of a point P (x 1, y 1) from the line ax +by +c = 0 is given by:
d =| ax1 +by1 +c|/[ (a² +b²)]




Simple And Compound Interest

1. Simple Interest = PNR/100

where, P --> Principal amount
N --> time in years
R --> rate of interest for one year

2. Compound interest (abbreviated C.I.) = A -P =

where A is the final amount, P is the principal, r is the rate of interest compounded yearly and n is the number of years

3. When the interest rates for the successive fixed periods are r1 %, r2 %, r3 %, ..., then the final amount A is given by A =


4. S.I. (simple interest) and C.I. are equal for the first year (or the first term of the interest period) on the same sum and at the same rate.

5. C.I. of 2nd year (or the second term of the interest period) is more than the C.I. of Ist year (or the first term of the interest period), and C.I. of 2nd year -C.I. of Ist year = S.I. on the interest of the first year.

STATISTICS
1. Mean.
(i) Mean (for ungrouped data) = where x1, x2, x3, ..., xn are the observations and n is the total no. of observations.
(ii) Mean (for grouped data) = , where x1, x2, x3, ..., xn are different variates with frequencies f1, f2, f3, ..., fn respectively.
(iii) Mean for continuous distribution.
Let there be n continuous classes, yi be the class mark and fi be the frequency of the ith class, then
mean = (Direct method)
Let A be the assumed mean, then
mean = A + , where di = yi -A (Short cut method)
If the classes are of equal size, say c, then
mean = A +c x , where ui = (Step deviation method)
ANALYTICAL GEOMETRY

LINES - BASICS:

1. The equation of X axis: y =0
2. The equation of Y axis: x = 0
3. Equation of straight line parallel to X axis: y =a, where a is any constant
4. Equation of straight line parallel to Y axis: x =a, where a is any constant
5. Equation of a straight line through a given point (x1, y1) and having a given slope m: y -y1 = m (x - x1)
6. Equation of a straight line through a given point (0, 0) and having a given slope m: y = m x
7. Equation of a straight line with a slope m and y-intercept c is: y = mx + c
8. Equation of a straight line passing through two points (x1, y1) and (x2, y2) is:
(y -y1)/(y2 - y1) = (x -x1)/(x2 -x1)
9. Equation of a straight line whose x and y intercepts are a and b is:
x/a + y/b = 1
10. The length of the perpendicular drawn from origin (0,0) to the line Ax + By + C =0 is :
C/ sqrt(A^2 + B^2)
11. Length of the perpendicular from (x1, y1) to the line Ax + By + C =0 is:
Ax1 + By1 +C / sqrt(A^2 + B^2)
12. The point of intersection of two lines a1x + b1y +c1 = 0 and a2x + b2y +c2 = 0 is :
([b1*c2 - b2*c1]/[a1*b2 - a2*b1], [c1*a2 - c2*a1]/[a1*b2 - a2*b1])
13. The condition for concurrency of three lines a1x + b1y +c1 = 0, a2x + b2y +c2 = 0 and a3x + b3y +c3 = 0 is (in determinant form)
| a1 b1 c1 |
| a2 b2 c2 | = 0
| a3 b3 c3 |
14. The angle between two lines y = m1x + c1 and y = m2x + c2 is tan inverse of the modulus of :
[(m1 - m2)/(1 + m1*m2)]
15. Condition for parallelism of two lines with slopes m1 and m2 is m1 = m2
16. Condition for perpendicularity of two lines with slopes m1 and m2 is m1*m2=-1

CIRCLES:

17. General equation of a circle with centre (x1, y1) and radius r is:
(x - x1)^2 + (y - y1)^2 = r^2
18. The equation of a circle whose diameter is the line joining the points (x1, y1) and (x2, y2) is :
(x - x1)(x - x2) + (y - y1)(y - y2) = 0
19. The equation of the tangent to the circle x^2 + y^2 = a^2 (where a is the radius of the circle) at the point (x1, y1) on it is :
x*x1 + y*y1 + a^2
20. The condition for y = mx + c to be a tangent to the circle x^2 + y^2 = a^2 is :
c^2 = a^2 (1 + m^2)

Multiplication of 2digit by 2 digit number

ab
x cd
------
pqrs

1. first multiply bd - write down the unit fig at s carry over the tens fig.
2. Multiply axd & bxc add them together and also add the carry over from step 1 write down the units fig at r and carry over the tens fig.
3. Multiply axc and add the carry over from step 2. write down at pq.

TRIGONOMETRY

For angle A = 0, 30° (π/6), 45° (π/4), 60° (π/3), 90° (π/2):

sin A = (sqrt0)/2, (sqrt1)/2, (sqrt2)/2, (sqrt3)/2, (sqrt4)/2
cos A = (sqrt4)/2, (sqrt3)/2, (sqrt2)/2, (sqrt1)/2, (sqrt0)/2
tan A = 0, (sqrt3)/3, 1, sqrt3, undefined

In any triangle:
sine = (opposite side) / hypotenuse
cosine = (adjacent side) / hypotenuse
tan = (opposite side)/(adjacent side) = (sine/cosine)

Probability - 'The Rules'

1. If two events are mutually exclusive (i.e. they cannot occur at the same time), then the probability of them both occurring at the same time is 0. then: P(A and B) = 0 and P(A or B) = P(A) + P(B)

2. if two events are not-mutually exclusive (i.e. there is some overlap) then: P(A or B) = P(A) + P(B) - P(A and B)

3. If events are independent (i.e. the occurrence of one does not change the probability of the other occurring), then the probability of them both occurring is the product of the probabilities of each occurring. Then: P(A and B) = P(A) * P(B)

4. If A, B and C are not mutually exclusive events, then P(A or B or C) = P(A) + P(B) + P(C) - P(A and B) - P(B and C) - P(C and A) + P(A and B and C)

and = intersection
or = union
sorry for making the formulae clusmsy using this 'and' and 'or'.


STATISTICS – HARMONIC MEAN

The harmonic mean of x1,...,xn is
n / (1/x1 + ... + 1/xn)
As the name implies, it's a mean (between the smallest and largest values). An example of the use of the harmonic mean: Suppose we're driving a car from Amherst (A) to Boston (B) at a constant speed of 60 miles per hour. On the way back from B to A, we drive a constant speed of 30 miles per hour (damn Turnpike). What is the average speed for the round trip?

We would be inclined to use the arithmetic mean; (60+30)/2 = 45 miles per hour. However, this is incorrect, since we have driven for a longer time on the return leg. Let's assume the distance between A and B is n miles. The first leg will take us n/60 hours, and the return leg will take us n/30 hours. Thus, the total round trip will take us (n/60) + (n/30) hours to cover a distance of 2n miles. The average speed (distance per time) is thus:
2n / {(n/60) + (n/30)} = 2 / (1/20) = 40 miles per hour.
The reason that the harmonic mean is the correct average here is that the numerators of the original ratios to be averaged were equal (i.e. n miles at 60 miles/hour versus n miles at 30 miles/hour). In cases where the denominators of two ratios are averaged, we can use the arithmetic mean.

SOME USEFUL SHORTCUTS

* Product of 2 numbers is the produst of their LCM & HCF.

* LCM of a fraction = LCM of numerator/HCF 0f denominator.

*HCF of a fraction = HCF of numer./LCM of denom.
Ratio & Proportion:

* if a/b = c/d = e/f = .....

then, a/b = c/d = e/f =(a+c+e+...)/(b+d+f+...)

* If a/b = c/d,

Then,

i) b/a = d/c

ii) a/c = b/d

iii) (a+b)/ b = (c+d)/d

iv) (a-b)/b = (c-d)/d

v) (a+b)/(a-b) = (c+d)/(c-d)


FORMULAE ON INTEREST

Money in Compound Interest gets doubled in 70/r years (approximately)

ie. P(1+r/100)^N = 2P when N=70/r

DIVISIBILITY RULES

Divisibility by:

2 If the last digit is even, the number is divisible by 2.
3 If the sum of the digits is divisible by 3, the number is also.
4 If the last two digits form a number divisible by 4, the number is also.
5 If the last digit is a 5 or a 0, the number is divisible by 5.
6 If the number is divisible by both 3 and 2, it is also divisible by 6.
7 Take the last digit, double it, and subtract it from the rest of the number; if the answer is divisible by 7 (including 0), then the number is also.
8 If the last three digits form a number divisible by 8, then so is the whole number.
9 If the sum of the digits is divisible by 9, the number is also.
10 If the number ends in 0, it is divisible by 10.
11 Alternately add and subtract the digits from left to right. If the result (including 0) is divisible by 11, the number is also.
Example: to see whether 365167484 is divisible by 11, start by subtracting:
3-6+5-1+6-7+4-8+4 = 0; therefore 365167484 is divisible by 11.
12 If the number is divisible by both 3 and 4, it is also divisible by 12.
13 Delete the last digit from the number, then subtract 9 times the deleted digit from the remaining number. If what is left is divisible by 13,then so is the original number.