All Necessary Formulas for GRE/GMAT Math combined

All Necessary Formulas for GRE/GMAT Math combined

Sequence

\(1+2+3+………..+n = \frac{(n (n+1))}{2}\)

\( 1^2+2^2+3^2+……..+n^2 =\frac{ n (n+1)(2n+1)}{6}\)

\( 1^3+2^3+3^3+……..+n^3 =\frac{ \{n (n+1)\}^2}{4}\)

Sum of an arithmetic Series, \(S = {n \over2}\{ {2a+(n-1)d} \}\)

a = first term , d = common difference, n = number of terms ;

e.g.: 11+18+25+… … … ; where,  a=11, d=18-11=7.  If n=29, the arithmetic sum of first 29 terms is 3161. [\(S = {29 \over2}\{ {2 \times 11+28\times 7} \}=3161\) ]

Sum of an geometric Series, \(S = \frac{a (q^n-1)}{(q-1)} , q > 1\ or\ S = \frac {a (1-q^n)}{(1-q)} , q < 1 \)

a = first term, q = common ratio, n = number of terms

e.g.: 3+1+1/3+1/9+ 1/27+ … … … ; where , a = 3 , q =1/3 < 1 . If, n = 6, the geometric sum of first 6 terms of the series is

\(S = \frac {3 (1-(1/3)^6)}{(1-1/3)}=364/81\)

Work rates

\(\frac{1}{time}=\frac{1}{time1}+\frac{1}{time2}\)

Here, time is the time to completion of a task when two workers are combining effort.

 

Distance, Rate, and Time

• \(Distance=  Rate \times Time; d=rt\)

• \( Average\ Speed = \frac{Total\ distance\ traveled}{Total\ time}\)

Interest

• Simple Interest = \(P (1+{rt \over 100});\) P is the principal, r is the rate, and t is time
• Compound Interest = \(P [1 + (r / 100n)]^{nt}\)  P is the principal, r is the rate, t is the time, n is the

Slope

• A slope is the steepness of a line in a coordinate system.
• Slope intercept formula: \(y = mx + b\) ,Here, m = slope & b = y-intercept
• Two point [(x₁,x_{2) ;} (y₁,y₂₎] slope formula = rise/run = (Δy)/(Δx) = (y₂-y₁)/(x₂-x₁)
• Parallel Slopes are equal slopes.
• Perpendicular slopes are the opposite reciprocal, for example, perpendicular slope to 5 is -1/5

Circle

• Area of a Circle = π r²
• Circumference of a Circle  = 2 π r
• Arc Length = (x/360 ) × 2 π r
• Area of Sector = (x/360) × π r²

Cube  

If each side of a cube is s, then the volume= s³  and the area = 6s²

Polygon

• The sum of the angles in a polygon:  (n-2) × 180° where n = number of sides
• Average degrees per side = \(180°\times {(n-2)\over n}\)
• Area of Triangle =  \({1\over 2}\times bh\) , b = base , h =height
• An isosceles triangle is the one which has two equal sides.
• An equilateral triangle is the one which has all (3) equal sides and the 3 angles are equal and 60°.
• Area of Trapezoid =  \({1\over2} \times {Height} \times {(Parallel\ Side1 + Parallel\ Side2)}\)
• Area of a Square = s², s = side (length) of the square
• Perimeter of a Square = 4s
• Area of a Rectangle = l×w, l =length , w= width
• Perimeter of a Rectangle = 2l+2w

Three-dimensional shape

• Volume of a Cylinder = πr²h  , r= radius, h= height
• Surface Area of Cylinder = 2πr² + 2πrh
• Volume of rectangular solid = lwh , w=width
• Surface area of rectangular solid = 2(l×w)+2(l×h)+2(w×h)
• Surface of circular cylinder = 2πr² + 2πrh

Distance

• Distance Formula : \(d = \sqrt {(y₂ - y₁)² + ( x₂ - x₁)²}\)
• Finding the Midpoint: when the endpoints are (x₁, y₁) and (x₂, y₂), the midpoint is: ((x₁+x₂)/2), ((y₁+y₂)/2)

Pythagorean Theorem

• This theorem only applies to right triangles i.e. triangles with a 90-degree angle.
• c² = a² + b² where a and b are the sides and c is the hypotenuse.
• a and b are the two shorter sides, or “legs,” and c is the hypotenuse (the longest side of a right triangle).
• Certain triangle-side combinations (a:b:c), called Pythagorean triples, are easy & useful to memorize: [3:4:5] , [5:12:13] , [8:15:17]

Combinations

• ⁿC_r= n! / [r! (n - r)!] ; n>r,  The formula is  used when order does not matter, e.g.:  picking any 5 fruits from a collection of 15. n is the total number, r is the number one is choosing.
• ⁿC_r = ⁿC_n-r

Permutations 

• nPr = n! / (n - r)! ; used when order does matter.
• n!  = n ( n-1) (n-2) ……………1

Angles

• If one angle is 90° then it is called right angle.
• The angle which is less than  90° is defined as the acute angle.
• The angle which is greater than  90° is defined as the obtuse angle.

Percent

• Percent Increase or Decrease = (difference between initial & final / initial) × 100
• B% of A = A×

Average

Average = (Sum of numbers) / (Total of numbers being averaged)

= \(\frac{\sum_{i=1}^{i=n}x_i }{n}\)

Quadratic Equations

• x²-y² = (x+y)(x-y)
• (x+y)² = x² + 2xy + y²
• (x-y)² = x² - 2xy - y²
• The format of ideal quadratic equations is ax²+bx+c

Inequality

If, X<Y then -X>-Y

Order of Operations

Order of Operations: PEMDAS

1. Parentheses

2. Exponents

3. Multiplication/Division

4. Addition/Subtraction

Prime Number

• Only even prime number 2
• It is useful to memorize all primes below 60

 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59

Properties of Even & Odd Number

Even + Even = Even; (4+6= 10);     Even + Odd = Odd; (4+5= 9);

Odd + Even = Odd; (1+4= 5);           Odd + Odd= Even;  (5+7= 12);

Even x Even = Even; (4x6= 24);     Even x Odd = Even; (4x5= 20);

Odd x Even = Even; (1x4= 4);           Odd x Odd= Odd;  (5x7= 35).

Rules of Divisibility

• 3: sum of digits divisible by 3
• 4: the last two digits of the number are divisible by 4
• 5: the last digit is either a 5 or zero
• 6: even number and sum of digits is divisible by 3
• 8: if the last three digits are divisible by 8
• 9: the sum of digits is divisible by 9 

 

Useful table to Utilize in Concise Calculation

Fraction	Decimal	Percent
1/100	0.01	1%
1/50	0.02	2%
1/25	0.04	4%
1/20	0.05	5%
1/10	0.1	10%
1/9	0.1111111…	11.11%
1/8	0.125	12.5%
9/8 (1+1/8)	1.125	1.125x100%
1/6	0.1666666….	16.67%
1/5	0.2	20%
1/4	0.25	25%
1/3	0.3333333…. = 3.3̅3̅3	33.33%
2/3	0.6666666….	66.67%
2/5	0.4	40%
1/2	0.5	50%
3/5	0.6	60%
3/4	0.75	75%
4/5	0.8	80%

 

Exoponents  

• (X^a )(X^b) = X ^a+b
• (X^a ) / (X^b) = X ^a-b
• (X^a )(X^b) = X ^ab

 

Statistics &  Probability

• Normal Distribution Curve: The Bell Curve

 

Figure 1: Bell Curve (Normal Distribution Curve); Source: www.ck12.org

 

• MODE = 3 median – 2 mean
• Quartile deviation, QD = (Q3-Q1)/2  = 5/6  Mean deviation
• Interquartile range = Q3-Q1

 

Figure 2: Normal Distribution Curve; Source: www.skillstairway.com

 

• Mean: A ‘mean’ is the average of a set of numbers.
• Median: The value that falls in the middle of a set of values
• Mode: The value that appears most often in a set of values
• To easily find the median/mode, one has to put the numbers in order from least to greatest.
• Standard Deviation: The bigger the standard deviation is, the more dispersed the values are; the smaller the standard deviation is, the more closely the values in a set are to the mean.

 

Example set of playing cards

Figure 3: Example set of playing cards; Source: www.wikipedia.com  [Important to know the chart and cards with stipulated numbers of each set and subset]

 

Probability Calculation 

• The probability of an even A to happen:

P(A) = Favorable outcomes where A occurs / Total possible number of outcomes

• P(A) definitely won't occur 0
• P(A) definitely will occur = 1

 

Complement of an Event

• P(Event happening) + P(event not happening) = 1
• P(A and B) [if the events are independent] = P(A) x P(B)
• P(A OR B) [if the events are independent] = P(A) + P(B) - P(A and B)

 

Set & Venn diagram

• Total = n(No Set) + n(Exactly one set) + n(Exactly two sets) + n(Exactly three sets)
• Total = n(No Set) + n(At least one set)
• Total = n(A) + n(B) + n(C) – n(A and B) – n(B and C) – n(C and A) + n(A and B and C) + n(No Set)
• n(At least one set) = n(A) + n(B) + n(C) – n(A and B) – n(B and C) – n(C and A) + n(A and B and C)

 

Perpetual and potential practices will aid anyone to merge the mechanisms and memory to utilize the formulas at the right time in the right places so that the purpose will sail him/her towards a smart score. It is highly recommended to be adept at applying the formulas by means of regular exercises bolstered by the power of memory.