1) A number is divisible by 2, if its unit’s place digit is 0, 2, 4, or 8.
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, starting from the RHS,(Sum of its digits at the odd place) – (Sum of its digits at even place) is equal to 0 or 11x.
1) Natural Numbers : Counting numbers 1, 2, 3, 4, 5, .. are called natural numbers.
2) Whole Numbers : All counting numbers together with zero form the set of whole numbers. Thus, a) 0 is the only whole number which is not a natural number. b) Every natural number is a whole number.
3) Integers : Counting numbers …..-2, -1, 0, 1, 2 ….. are called integers.
4) Rational Numbers : Any number which can beexpressed as a ratio of two integers for example a p/qformat where ‘p’ and ‘q’ are integers. Proper fractionwill have (p
5) Factors : A positive integer ‘f’ is said to be a factor ofa given positive integer 'n' if f divides n withoutleaving a remainder. e.g. 1, 2, 3, 4, 6 and 12 are thefactors of 12.
6) Prime Numbers : A prime number is a positivenumber which has no factors besides itself and unity.Composite Numbers: A composite number is anumber which has other factors besides itself andunity.
7) Factorial : For a natural number 'n', its factorial isdefined as: n! = 1 x 2 x 3 x 4 x .... x n (Note: 0! = 1)Absolute value: Absolute value of x (written as |x|) isthe distance of 'x' from 0 on the number line. |x| isalways positive. |x| = x for x > 0 OR -x for x < 0
1) To express A% as a fraction, we have A% = A / 100
2) To express A/B as a percent, we have A/B = (A/B × 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
5) If the population of a town is ‘P’ in a year, then its population after ‘N’ years is : P(1 + R/100)N
6) If the population of a town is ‘P’ in a year, then its population ‘N’ years ago is : P / [(1 + R/100)N]
Cost Price : The price at which an article is purchased, is called its cost price, abbreviated as C.P.
Selling Price : The price at which an article is purchased, is called its cost price, abbreviated as C.P.
Profit or Gain : The price at which an article is purchased, is called its cost price, abbreviated as C.P.
Loss : If Selling Price is less than Cost Price, the seller is said to have incurred a loss.
1) Gain = (S.P.) - (C.P.)
2) Loss or gain is always reckoned on C.P.
3) gain% = [Gain*100/C.P.]
4) Loss = (C.P.) - (S.P.)
5) Loss% = [Loss*100/C.P.]
6) Selling Price = [(100+Gain%) * C.P.]/100
7) Selling Price = [(100-Loss%)* C.P.]/100
8) Cost Price = 100/(100+Gain%) * S.P.
9) Cost Price = 100/(100-Loss%) * S.P.
10) If the value of a machine is ‘P’ in a year, then its value after ‘N’ years at a depreciation of ‘R’ p.c.p.a is : P(1 - R/100)N
11) If the value of a machine is ‘P’ in a year, then its value ‘N’ years ago at a depreciation of ‘R’ p.c.p.a is : P / [(1 - R/100)N]
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 proportion, the first and fourth terms are known as extremes, while the second and third are known as means.
5) Product of extremes = Product of means
6) Compounded Ratio of two ratios a/b and c/d is ac/bd
7) Duplicate ratio of a : b is a2 : b2
8) Triplicate ratio of a : b is a3 : b3
9) Sub-duplicate ratio of a : b is √a : √b
10) Sub-triplicate ratio of a : b is ∛a : ∛b
11) Reciprocal ratio of a : b is b : a
12) If a / b = c / d, then, (a + b) / b = (c + d) / d, which is called the componendo
13) If a / b = c / d, then, (a - b) / b = (c - d) / d, which is called the dividendo
14) If a / b = c / d, then, (a + b) / (a - b) = (c + d) / (c - d), which is called the componendo & dividendo
15) Variation: We say that x is directly proportional to y if x = ky for some constant k and we write, x α y
16) Also, we say that x is inversely proportional to y if x = k / y for some constant k and we write x α 1 / y
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.
Product of two numbers = Product of their H.C.F. and L.C.M.
Let F = Face Value of the Bill, TD = True Discount, BD = Banker's Discount, BG = Banker’s Gain, R = Rate of Interest, PW = True Present Worth and T = Time in Years
1) Banker’s Gain (B.G.) = (B.D.) – (T.D.) for the unexpired time
Note : When the date of the bill is not given, grace days are not to be added
2) BD = Simple Interest on the face value of the bill for unexpired time =
3) TD = Simple Interest on the present value for unexpired time =
4) TD =
5) PW = F - TD
6) F =
7) BG = BD – TD = Simple Interest on TD =
8) TD = √PW × BG
9) TD =
Let rate = R% per annum and Time = T years. Then,
1) P.W.=[100 × Amount /100 + (R × T) =100 × T.D./ R × T
2) T.D.=[(P.W.) × R × T /100] = [ Amount × R × T/100 + (R × T)]
3)Sum=[(S.I.)×(T.D.)] /[(S.I.) - (T.D.)]
4) (S.I.) - (T.D.) = S.I. on T.D.
5) When the sum is put at compound interest, then P.W. = Amount/[1 +R/100]T
1) Stock Capital : The total amount of money needed to run the company is called the stock capital.
2) Shares or Stock : The whole capital is divided into small units, called shares or stock.
For each investment, the company issues a 'share-certificate', showing the value of each share and the number of shares held by a person.
The person who subscribes in shares or stock is called a share holder or stock holder.
3) Dividend : The annual profit distributed among share holders is called dividend.
Dividend is paid annually as per share or as a percentage.
4) Face Value : The value of a share or stock printed on the share-certificate is called its Face Value or Nominal Value or Par Value.
5) Brokerage : The broker’s charge is called brokerage.
6) When stock is purchased, brokerage is added to the cost price.
7) When the stock is sold, brokerage is subtracted from the selling price.
8) The selling price of a Rs. 100 stock is said to be :
a) at par, if S.P. is Rs. 100 exactly;
b) above par(or at premium), if S.P. is more than Rs. 100;
c) below par(or at discount), if S.P. is less than Rs. 100.
5) By ‘a Rs. 800, 9% stock at 95 ’, we mean a stock whose face value is Rs. 800, annualinterest is 9% of the face value and the market price of a Rs. 100 stock is Rs. 95.
1) Rectangle :
a) Area of a rectangle = (length × breadth)
b) Perimeter of a rectangle = 2 ×(length + breadth)
2) Square :
a) Area of square =(side)2
b) Area of a square = ½ (diagonal)2
3) Area of 4 walls of a room = 2 ×(length + breadth) × heigh
4) Triangle :
a) Area of Triangle = ½ × base × height
b) Area of a triangle = √s(s-a)(s-b)(s-c) , where s = ½ (a +b +c ), anda ,b ,c are the sides of the triangle
c) Area of an equilateral triangle = √3/4 × (side)2
d) Radius of incircle of an equilateral triangle of side a = a / 2√3
e) Radius of circumcircle of an equilateral triangle of side a = a /√3
5) Parallelogram/Rhombus/Trapezium :
a) Area of a parallelogram = Base × Height
b) Area of a rhombus = ½ × (Product of diagonals)
c) The halves of diagonals and a side of a rhombus form a right angled triangle withside as the hypotenuse.
d) Area of trapezium = ½ × (sum of parallel sides) × (distance between them)
6) Circle/Arc/Sector, where R is the radius of the circle :
a) Area of a circle =πR2
b) Circumference of a circle = 2πR
c) Length of an arc =Ө/360 × 2πR
d) Area of a sector = ½ ×(arc × R)=Ө/360 ×πR2
1) Cuboid :
Let length =l , breadth =b & height =h units Then,
a)Volume = (l ×b ×h ) cu units
b) Surface Area = 2 (lb +bh +hl ) sq. units
c) Diagonal =√l2+b2+h2 units
2) Cube :
Let each edge of a cube be of length a. Then,
a) Volume = a3 cu units
b) Surface Area = 6a2 sq. units
c) Diagonal = (√3 × a) units
3) Cylinder :
Let radius of base =r & height (or length) =h. Then,
a) Volume = (πr2h) cu. units
b) Curved Surface Area = (2πrh ) sq. units
c) Total Surface Area = 2πr (r + h) sq. units
4) Cone :
Let radius of base = r & height =h. Then,
a) Slant height,l = √h2+r2units
b) Volume = (1/3πr2h) cu. units
c) Curved Surface Area = (πrl ) sq. units
d) Total Surface Area =πr(r +l) sq. units
5) Sphere :
Let the radius of the sphere be r. Then,
a) Volume = (4/3πr3) cu units
b) Surface Area = (4πr2) sq. units
6) Hemi-sphere :
Let the radius of the sphere be r. Then,
a) Volume = (2/3πr3) cu units
b) Curved Surface Area = (2πr2) sq. units
b) Total Surface Area = (3πr2) sq. units
1) Inlet : A pipe is linked with a tank or a cistern or a reservoir, and it fills with water, is known as an inlet.
Outlet : A pipe is linked with a tank or cistern or reservoir, emptying it, is known as an outlet.
2) If a pipe can fill a tank in x hours, then:part filled in 1 hour =
3) If a pipe can empty a tank in y hours, then:part emptied in 1 hour =
4) If a pipe can fill a tank in x hours and another pipe can empty the full tank in y hours (where y > x), then on opening both the pipes, thenthe net part filled in 1 hour =(1/x-1/y)
5) If a pipe can fill a tank in x hours and another pipe can empty the full tank in y hours (where x > y), then on opening both the pipes, thenthe net part emptied in 1 hour =(1/y-1/x)
1) Work from Days :
If X can do a piece of work in n days, then X's 1 day's work =
2) Days from Work :
If X's 1 day's work=, then X can finish the work in n days
3) Ratio :
If X is thrice as good a workman as Y, then:
Ratio of work done by X and Y = 3 : 1.
Ratio of times taken by X and Y to finish a work = 1 : 3
The money borrowed or lent out for a certain period is called the principal or the sum.
Extra money paid for using other's money is called interest.
3) Simple Interest (S.I.):
If the interest on a sum borrowed for certain period is reckoned uniformly, then it is called simple interest.Let Principal = P, Rate = R% per annum (p.a.) and Time = T years. Then
(i) Simple Intereest =
(ii) P =; R=and T=
Let Principle =P , Rate = R % per annum and Time = T years. Then,
1) When interest is compounded Annually :
Amount = P[1 + R/100]N
2) When interest is compounded Half-yearly :
Amount = P[1 + (R/2)/100]2N
3) When interest is compounded Quarterly :
Amount = P[1 + (R/4)/100]4N
4)When interest is compounded Annually but time is in fraction, say 3 years.
Amount = P[1+R/100]3 x [1+((2/5)R)/100]
5) When Rates are different for different years, say R1%, R2%, R3% for 1st, 2nd and 3rd year respectively.
Then, Amount = P[1 +(R1/100)][1 +(R2/100)][1 +(R3/100)]
6) Present worth of Rs. X due n years hence is given by:
Present Worth = X/[1+R/100]