Calendar - Important Formulas | Aptitude

1. Odd Days:
   We are supposed to find the day of the week on a given date.
   For this, we use the concept of 'odd days'.
   In a given period, the number of days more than the complete weeks are called odd days.
2. Leap Year:
   (i). Every year divisible by 4 is a leap year, if it is not a century.
   (ii). Every 4^th century is a leap year and no other century is a leap year.
   Note: A leap year has 366 days.
   Examples:
   1. Each of the years 1948, 2004, 1676 etc. is a leap year.
   2. Each of the years 400, 800, 1200, 1600, 2000 etc. is a leap year.
   3. None of the years 2001, 2002, 2003, 2005, 1800, 2100 is a leap year.
3. Ordinary Year:
   The year which is not a leap year is called an ordinary years. An ordinary year has 365 days.
4. Counting of Odd Days:
   1. 1 ordinary year = 365 days = (52 weeks + 1 day.)
       1 ordinary year has 1 odd day.
   2. 1 leap year = 366 days = (52 weeks + 2 days)
       1 leap year has 2 odd days.
   3. 100 years = 76 ordinary years + 24 leap years
        = (76 x 1 + 24 x 2) odd days = 124 odd days.
        = (17 weeks + days)  5 odd days.
       Number of odd days in 100 years = 5.
      Number of odd days in 200 years = (5 x 2)  3 odd days.
      Number of odd days in 300 years = (5 x 3)  1 odd day.
      Number of odd days in 400 years = (5 x 4 + 1)  0 odd day.
      Similarly, each one of 800 years, 1200 years, 1600 years, 2000 years etc. has 0 odd days.
5. Day of the Week Related to Odd Days:

   No. of days:	0	1	2	3	4	5	6
   Day:	Sun.	Mon.	Tues.	Wed.	Thurs.	Fri.	Sat.

Volume and Surface Area - Important Formulas |Aptitude

1. CUBOID
   Let length = l, breadth = b and height = h units. Then
   1. Volume = (l x b x h) cubic units.
   2. Surface area = 2(lb + bh + lh) sq. units.
   3. Diagonal = l² + b² + h² units.
2. CUBE
   Let each edge of a cube be of length a. Then,
   1. Volume = a³ cubic units.
   2. Surface area = 6a² sq. units.
   3. Diagonal = 3a units.
3. CYLINDER
   Let radius of base = r and Height (or length) = h. Then,
   1. Volume = (r²h) cubic units.
   2. Curved surface area = (2rh) sq. units.
   3. Total surface area = 2r(h + r) sq. units.
4. CONE
   Let radius of base = r and Height = h. Then,
   1. Slant height, l = h² + r² units.
   2. Volume = r²h cubic units.
   3. Curved surface area = (rl) sq. units.
   4. Total surface area = (rl + r²) sq. units.
5. SPHERE
   Let the radius of the sphere be r. Then,
   1. Volume = r³ cubic units.
   2. Surface area = (4r²) sq. units.
6. HEMISPHERE
   Let the radius of a hemisphere be r. Then,
   1. Volume = r³ cubic units.
   2. Curved surface area = (2r²) sq. units.
   3. Total surface area = (3r²) sq. units.
      Note: 1 litre = 1000 cm³.
   4. 
      

Average - Important Formulas |Aptitude

1. Average:

   Average =		Sum of observations
   		Number of observations

2. Average Speed:
   Suppose a man covers a certain distance at x kmph and an equal distance at y kmph.

   Then, the average speed druing the whole journey is		2xy		kmph.
   		x + y

Numbers - Important Formulas | Aptitude

Some Basic Formulae:

1. (a + b)(a - b) = (a² - b²)
2. (a + b)² = (a² + b² + 2ab)
3. (a - b)² = (a² + b² - 2ab)
4. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
5. (a³ + b³) = (a + b)(a² - ab + b²)
6. (a³ - b³) = (a - b)(a² + ab + b²)
7. (a³ + b³ + c³ - 3abc) = (a + b + c)(a² + b² + c² - ab - bc - ac)
8. When a + b + c = 0, then a³ + b³ + c³ = 3abc.

Problems on H.C.F and L.C.M - Important Formulas

1. Factors and Multiples:
   If number a divided another number b exactly, we say that a is a factor of b.
   In this case, b is called a multiple of a.
2. Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.):
   The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly.
   There are two methods of finding the H.C.F. of a given set of numbers:
   1. Factorization Method: Express the each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.
   2. Division Method: Suppose we have to find the H.C.F. of two given numbers, divide the larger by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is required H.C.F.
      Finding the H.C.F. of more than two numbers: Suppose we have to find the H.C.F. of three numbers, then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given number.
      Similarly, the H.C.F. of more than three numbers may be obtained.
3. Least Common Multiple (L.C.M.):
   The least number which is exactly divisible by each one of the given numbers is called their L.C.M.
   There are two methods of finding the L.C.M. of a given set of numbers:
   1. Factorization Method: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors.
   2. Division Method (short-cut): Arrange the given numbers in a rwo in any order. Divide by a number which divided exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.
4. Product of two numbers = Product of their H.C.F. and L.C.M.
5. Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1.
6. H.C.F. and L.C.M. of Fractions:

   1. H.C.F. =	H.C.F. of Numerators
   	L.C.M. of Denominators

   2. L.C.M. =	L.C.M. of Numerators
   	H.C.F. of Denominators

7. H.C.F. and L.C.M. of Decimal Fractions:
   In a given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.
8. Comparison of Fractions:
   Find the L.C.M. of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with L.C.M as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.

Compound Interest Formulas

1. Let Principal = P, Rate = R% per annum, Time = n years.
2. When interest is compound Annually:

   Amount = P		1 +	R		n
   			100

3. When interest is compounded Half-yearly:

   Amount = P		1 +	(R/2)		2n
   			100

4. When interest is compounded Quarterly:

   Amount = P		1 +	(R/4)		4n
   			100

5. When interest is compounded Annually but time is in fraction, say 3 years.

   Amount = P		1 +	R		3	x		1 +	R
   			100						100

6. When Rates are different for different years, say R₁%, R₂%, R₃% for 1^st, 2^nd and 3^rd year respectively.

   Then, Amount = P

   1 +

   R1

   1 +

   R2

   1 +

   R3

   .

   100

   100

   100

7. Present worth of Rs. x due n years hence is given by:

   Present Worth =	x	.
   	1 + R 100