MyMaths MyLife

An equation which has the unknown quantity raised only to powers which are whole numbers and the highest power being the square of the unknown quantity, is called a quadratic equation.
The most general form of a quadratic equation is ax^2 + bx + c = 0.
There are two values of x that satisfy such a quadratic equation. These values are called the roots of the quadratic equation.

The roots of the above quadratic equation are given by (-b±√(b^2-4ac))/2a

For ax^2 + bx + c = 0, sum of the roots = -b/a; Product of the roots = c/a

cube and cuboid:

Consider the fallowing objects: a brick, a box of matches, a die, a text book, a room in the house. They have a common shape, though their sizes are different. The geometrical name that we give to each of these objects is the cuboid.
It has six rectangular faces. There are in all 12 edges of the cuboid. A cuboid has 8 corners called vertices.
The total area of all the six faces of a cuboid is called the total surface area of the cuboid.
Let l,b, and h, be the length, the breadth and the height of a cuboid,
then the lateral surface area= 2h(l+b)

The total surface area
=(the lateral surface area)+(area of ABCD)+(area of EFGH)
=2h(l+b)+lb+lb
=2lh+2bh+2lb
=2(lb+bh+hl)

The Relation Between G.C.D and L.C.M:

For GCD concept click here:http://business-maths.blogspot.com/2009/02/greatest-common-divisor.html
For LCM concept click here:http://business-maths.blogspot.com/2009/02/least-common-multiple-lcm.html
Find the G.C.D and L.C.M of 30 and 48 and it shows that the product of GCD and LCM is equal to the product of the two given numbers.
GCD of 30,48 is 6.
And LCM of 30,40 is 240.
LCM*GCD=240*6=1440
Product of 30 and 48= 30*48=1440.
Hence the product of the two numbers is equal to the product of their G.C.D and L.C.M.
If a and b are any two natural numbers and L and G are respectively their L.C.M and G.C.D., then a*b=L*G

->(a+b)²=a²+b²+2ab
->(a-b)²=a²+b²-2ab
->(a+b)²-(a-b)²=4ab
->(a+b)²+(a-b)²=2(a²+b²)
->a²-b²=(a+b)(a-b)
->(a-+b+c)²=a²+b²+c²+2(ab+b c+ca)
->a³+b³=(a+b)(a²+b²-ab)
->a³-b³=(a-b)(a²+b²+ab)
->a³+b³+c³-3a b c=(a+b+c)(a²+b²+c²-ab-b c-ca)
->If a+b+c=0 then a³+b³+c³=3a b c

Addition and Multiplication are said to be commutative, because
A+B = B+A
A*B = B*A

Addition and Multiplication are said to be Associative, because
(A+B)+C = A+(B+C)
(A*B)*C = A*(B*C)

Multiplication is Distributive over Addition, because
A*(B+C) = (A*B)+(A*C)

here A, B, C represent any Real Number

Example problems:

1.Find the average of all these numbers.142,147,153,165,157.

Solution:
142 147 153 165 157
Here consider the least number i.e, 142
comparing with others,
142 147 153 165 157
+5 +11 +23 +15
Now add 5+11+23+15 = 52/5 = 10.8
Now add 10.8 to 142 we get 152.8
(Average of all these numbers).
Answer is 152.8


2.Find the average of all these numbers.4,10,16,22,28
Solution:
4,10,16,22,28
As the difference of number is 6
Then the average of these numbers is central one i.e, 16.
Answer is 16.

3.Find the average of all these numbers.4,10,16,22,28,34.

Solution:
Here also difference is 6.
Then middle numbers 16,22 take average of these
two numbers 16+22/2=19
Therefore the average of these numbers is 19.
Answer is 19.

4.The average marks of a marks of a student in 4 Examination
is 40.If he got 80 marks in 5th Exam then what is
his new average.

Solution:
4*40+80=240
Then average means 240/5=48.
Answer is 48.

5.In a group the average income of 6 men is 500 and that
of 5 women is 280, then what is average income of the group.

Solution:
6*500+5*280=4400
then average is 4400/11=400.
Another Method: here consider for 6 men
6 men – each 500.
so 5th women is 280.
then 500-280=220.
then 220*6/11=120.
therefore 120+280=400.
Answer is 400.

6.The average weight of a class of 30 students is 40 kgs if the
teacher weight is included then average increases by 2 kgs then
find the weight of the teacher?

Solution:
30 students average weight is 40 kgs.
So,when teacher weight is added it increases by 2 kgs
so total 31 persons ,therefore 31*2=62.
Now add the average weight of all student to it
we get teachers weight i.e, 62+40=102 kgs.
Answer is 102 kgs.

7.The average age of Mr and Mrs Sharma 4 years ago is 28 years .
If the present average age of Mr and Mrs Sharma and their son
is 22 years. What is the age of their son.

Solution:
4 years ago their average age is 28 years.
So their present average age is 32 years.
32 years for Mr and Mrs Sharma then 32*2=64 years.
Then present age including their son is 22 years.
So 22*3 =66 years.
Therefore son age will be 66-64 = 2 years.
Answer is 2 years.

8.The average price of 10 books is increased by 17 Rupees when
one of them whose value is Rs.400 is replaced by a new book.
What is the price of new book?

Solution:
10 books Average increases by 17 Rupees
so 10*17= 170.
so the new book cost is more and by adding its cost average
increase,therefore the cost of new book is 400+170=570Rs.
Answer is 570 Rs.

9.The average marks of girls in a class is 62.5. The average marks
of 4 girls among them is 60.The average marks of remaining girls
is 63,then what is the number of girls in the class?

Solution:
Total number of girls be x+4.
Average marks of 4 girls is 60.
therefore 62.5-60=2.5
then 4*2.5 =10.
the average of remaining girls is 63
here 0.5 difference therefore 0.5*x=10(since we got from 4 girls)
(this is taken becoz both should be equal)
x=10/0.5
x=20.
This clear says that remaining are 20 girls
therefore total is x+4=20+4=24 girls
Answer is 24 girls.

10.Find the average of first 50 natural numbers.

Solution:
Sum of the Natural Numbers is n(n+1)/2
therefore for 50 Natural numbers 50*51/2=775.
the average is 775/50=15.5
Answer is 15.5 .

11.The average of the first nine prime number is?

Solution:
Prime numbers are 2,3,5,7,11,13,17,19,23
therefore 2+3+5+7+11+13+17+19+23=100
then the average 100/9= 11 1/9.
Answer is 11 1/9.

12.The average of 2,7,6 and x is 5 and the average of and the
average of 18,1,6,x and y is 10 .what is the value of y?

Solution:
2+7+6+x/4=5
=>15+x=20
=>x=5.
18+1+6+x+y/5=10
=>25+5+y=50
=>y=20.

13.The average of a non-zero number and its square is 5 times the
number.The number is

Solution:
The number be x
then x+x2/2=5x
=>x2-9x=0
=>x(x-9)=0
therefore x=0 or x=9.
The number is 9.

14.Nine persons went to a hotel for taking their meals . Eight of
them spent Rs.12 each on their meals and the ninth spent Rs.8 then
the average expenditure of all the nine. What was the total money
spent by them?

Solution:
The average expenditure be x.
then 8*12+(x+8)=9x
=>96+x+8=9x.
=>8x=104
=>x=13
Total money spent =9x=>9*13=117
Answer is Rs.117


15.The average weight of A.B.C is 45 Kgs.If the average weight of
A and B be 40 Kgs and that of Band C be 43 Kgs. Find the weight of B?

Solution:
The weight of A,B,Care 45*3=135 Kgs.
The weight of A,B are 40*2=80 Kgs.
The weight of B,C are 43*2=86 Kgs.
To get the Weight of B.
(A+B)+(B+C)-(A+B+C)=80+86-135
B=31 kgs.
Answer is 31 Kgs.

16.The sum of three consecutive odd number is 48 more than the average
of these number .What is the first of these numbers?

Solution:
let the three consecutive odd numbers are x, x+2, x+4.
By adding them we get x+x+2+x+4=3x+6.
Then 3x+6-(3x+6)/3=38(given)
=>2(3x+6)=38*3.
=>6x+12=114
=>6x=102
=>x=17.
Answer is 17.

17.A family consists of grandparents,parents and three grandchildren.
The average age of the grandparents is 67 years,that of parents is 35
years and that of the grand children is 6 years . What is the average
age of the family?

Solution:
grandparents age is 67*2=134
parents age is 35*2=70
grandchildren age is 6*3=18
therefore age of family is 134+70+18=222
average is 222/7=31 5/7 years.
Answer is 31 5/7 years.

18.A library has an average of 510 visitors on Sundays and 240 on
other days .The average number of visitors per day in a month 30
days beginning with a Sunday is?

Solution:
Here specified that month starts with Sunday
so, in a month there are 5 Sundays.
Therefore remaining days will be 25 days.
510*5+240*25=2550+6000
=8550 visitors.
The average visitors are 8550/30=285.
Answer is 285.

19.The average age of a class of 39 students is 15 years .
If the age of the teacher be included ,then average
increases by 3 months. Find the age of the teacher.

Solution: Total age for 39 persons is 39*15=585 years.
Now 40 persons is 40* 61/4=610 years
(since 15 years 3 months=15 3/12=61/4)
Age of the teacher =610-585 years
=>25 years.
Answer is 25 years.

20.The average weight of a 10 oarsmen in a boat is increases
by 1.8 Kgs .When one of the crew ,who weighs 53 Kgs is
replaced by new man. Find the weight of the new man.

Solution: Weight of 10 oars men is increases by 1.8 Kgs
so, 10*1.8=18 Kgs
therefore 53+18=71 Kgs will be the weight of the man.
Answer is 71 Kgs.

21.A bats man makes a score of 87 runs in the 17th inning
and thus increases his average by 3. Find the average
after 17th inning.

Solution: Average after 17 th inning =x
then for 16th inning is x-3.
Therefore 16(x-3)+87 =17x
=>x=87-48
=>x=39.
Answer is 39.

22.The average age of a class is 15.8 years .The average age
of boys in the class is 16.4 years while that of the girls
is 15.4 years .What is the ratio of boys to girls in the class.

Solution: Ratio be k:1 then
k*16.4 + 1*15.4 = (k+1)*15.8
=>(16.4-15.8)k=15.8-15.4
=>k=0.4/0.6
=>k=2/3
therefore 2/3:1=>2:3
Answer is 2:3

23.In a cricket eleven ,the average of eleven players is
28 years .Out of these ,the average ages of three groups
of players each are 25 years,28 years, and 30 years
respectively. If in these groups ,the captain and the
youngest player are not included and the captain is
eleven years older than the youngest players ,
what is the age of the captain?

Solution: let the age of youngest player be x
then ,age of the captain =(x+11)
therefore 3*25 + 3*28 + 3*30 + x + x+11=11*28
=>75+84+90+2x+11=308
=>2x=48
=>x=24.
Therefore age of the captain =(x+11)= 24+11= 35 years.
Answer is 35 years.

24.The average age of the boys in the class is twice
the number of girls in the class .If the ratio of
boys and girls in the class of 36 be 5:1, what is
the total of the age (in years) of the boys in the class?

Solution: Number of boys=36*5/6=30
Number of girls =6
Average age of boys =2*6=12 years
Total age of the boys=30*12=360 years
Answer is 360 years.

25.Five years ago, the average age of P and Q was
15 years ,average age of P,Q, and R today is
20 years,how old will R be after 10 years?

Solution: Age of P and Q are 15*2=30 years
Present age of P and Q is 30+5*2=40 years.
Age of P Q and R is 20*3= 60 years.
R ,present age is 60-40=20 years
After 10 years =20+10=30 years.
Answer is 30 years.

26.The average weight of 3 men A,B and C is 84 Kgs.
Another man D joins the group and the average now
becomes 80 Kgs.If another man E whose weight is
3 Kgs more than that of D ,replaces A then the
average weight B,C,D and E becomes 79 Kgs.
The weight of A is.

Solution:Total weight of A, B and C is 84 * 3 =252 Kgs.
Total weight of A,B,C and Dis 80*4=320 Kgs
Therefore D=320-252=68 Kgs.
E weight (68+3)=71 kgs
Total weight of B,C,D and E = 79*4=316 Kgs
(A+B+C+D)-(B+C+D+E)=320-316 =4Kgs
A-E=4Kgs
A-71=4 kgs
A=75 Kgs
Answer is 75 kgs

27.A team of 8 persons joins in a shooting competition.
The best marksman scored 85 points.If he had scored
92 points ,the average score for the team would
have been 84.The team scored was.

Solution: Here consider the total score be x.
therefore x+92-85/8=84
=>x+7=672
=>x=665.
Answer is 665

28.A man whose bowling average is 12.4,takes 5 wickets
for 26 runs and there by decrease his average by 0.4.
The number of wickets,taken by him before his last match is:

Solution: Number of wickets taken before last match be x.
therefore 12.4x26/x+5=12(since average decrease by 0.4
therefore 12.4-0.4=12)
=>12.4x+2612x+60
=>0.4x=34
=>x=340/4
=>x=85.
Answer is 85.

29.The mean temperature of Monday to Wednesday was 37 degrees
and of Tuesday to Thursday was 34 degrees .If the
temperature on Thursday was 4/5th that of Monday.
The temperature on Thursday was:


Solution:
The total temperature recorded on Monday,Wednesday was 37*3=111.
The total temperature recorded on Tuesday,
Wednesday,Thursday was 34*3=102.
and also given that Th=4/5M
=>M=5/4Th
(M+T+W)-(T+W+Th)=111-102=9
M-Th=9
5/4Th-Th=9
Th(1/4)=9
=>Th=36 degrees.

30. 16 children are to be divided into two groups A and B
of 10 and 6 children. The average percent marks obtained
by the children of group A is 75 and the average percent
marks of all the 16 children is 76. What is the average
percent marks of children of groups B?

Solution: Here given average of group A and whole groups .
So,(76*16)-(75*10)/6
=>1216-750/6
=>466/6=233/3=77 2/3
Answer is 77 2/3.

31.Of the three numbers the first is twice the second and
the second is twice the third .The average of the reciprocal
of the numbers is 7/72,the number are.

Solution:Let the third number be x
Let the second number be 2x.
Let the first number be 4x.
Therefore average of the reciprocal means
1/x+1/2x+1/4x=(7/72*3)
7/4x=7/24
=>4x=24
x=6.
Therefore
First number is 4*6=24.
Second number is 2*6=12
Third number is 1*6=6
Answer is 24,12,6.

32.The average of 5 numbers is 7.When 3 new numbers
are added the average of the eight numbers is 8.5.
The average of the three new number is:

Solution: Sum of three new numbers=(8*8.5-5*7)=33
Their average =33/3=11.
Answer is 11.

33.The average temperature of the town in the first
four days of a month was 58 degrees. The average
for the second ,third,fourth and fifth days was
60 degree .If the temperature of the first and
fifth days were in the ratio 7:8 then what is
the temperature on the fifth day?

Solution :
Sum of temperature on 1st 2nd 3rd
and 4th days =58*4=232 degrees.
Sum of temperature on 2nd 3rd 4th
and 5th days =60*4=240 degrees
Therefore 5th day temperature is 240-232=8 degrees.
The ratio given for 1st and 5th days be 7x and 8x degrees
then 8x-7x=8
=>x=8.
therefore temperature on the 5th day =8x=8*8=64 degrees.

Least Common Multiple (L.C.M):


Rules:

The smallest of the common multiples of two natural numbers (a and b) is called the least common multiple(LCM) of the numbers a and b.

The smallest of the common multiples of two or more natural numbers is called the least common multiple(LCM).

If two numbers are c0-prime, then their LCM is equal to their product.

Given two numbers, if the first number is a multiple of the second number, then their LCM is equal to the first number.

Relationship Between GCD and LCM:
If a and b are any two natural numbers and L and G are respectively their LCM and GCD, then a*b=L*G

Example:

LCM of 30,48

2 |30,48
_______
3 |15,24
_____
5,8

LCM of 30,48 = 2*3*5*8= 240

Factorial is defined for any positive integer. It is denoted by !. Thus “Factorial n” is written as n!. n! is defined as the product of all the integers from 1 to n.

Thus n! = 1.2.3.. ... (n-1),n.

Example 5! = 1*2*3*4*5 = 120

0! is defined to be equal to 1.
Therefore 0! = 1 and 1! = 1

n!=n*(n-1)!
eg: 10!=10*(10-1)!=10*(9)!

Area of the four walls of a room:

If we look around and observe the walls of a room, we find that generally the walls are in the shape of a rectangle the floor and the ceiling of the room are also of rectangular shape.
Let l, b, h be the lengths of AB,AD and AE as shown in the figure. Here l and b are the length and breadth of the floor and h the height of the room.
For the rectangle ABFE, the lengths of two adjacent sides are l and h. its area = lh
For the rectangle BCGF, the lengths of two adjacent sides are b and h. its area = bh
For the rectangle CDHG, the lengths of two adjacent sides are l and h. its area = lh
For the rectangle DAEH, the lengths of two adjacent sides are b and h. its area = bh
Observe the opposite walls being of the same size and shape have the same area too.

Hence the total area of the four walls = lh+bh+lh+bh
= 2lh+2bh
= 2h(l+b)

A decimal in which a digit is repeated continuously is called a Recurring decimal. Recurring decimals are written in a shortened form, the digits which are repeated being marked by dots placed over the first and the last of them, thus




The digit, or set of digits, which is repeated is called the period of the decimal. In the decimal equivalent to 8/3, the period is 6. In 21/22 it is 54.
behaves

mensuration :Areas

* The area of simple closed figure is the measure of the region closed by the boundary of the figure.

* The area is measured in square units. A square meter is the area of square whose side is one meter.

* A square centimeter is the area of square whose side is one centimeter.

* If l and b denote the length and breadth of a rectangle and A its area, then A=l*b=lb

* If s denotes the side of a square and A its area, then A=s^2.

Quantities are said to be in arithmetic progression(A.P) when they increase or decrease by a common difference to get the next or the previous term respectively.

An arithmetic progression be represented by a, a + d, a+ 2d, ...., a + (n-1)d, where a is the first term; n is the number of terms in the progression and d is the common difference.
In an Arithmetic progression, n'th term = a + (n-1)d

Sum of n terms = (n/2) * [2a + (n-1)d]
If three numbers are in arithmetic progression, the middle number is called the Arithmetic mean.
Arithmetic Mean = (a+b+c)/3 where a,b and c are in Arithmetic Progression

Arithmetic Mean of 'n' terms in Arithmetic progression =
(first term + last term)/2
(or)
1/2{2a + (n-1)d}

Work to be done is usually considered as one unit. It may be constructing a wall or laying a road, filling up or emptying a tank or eating certain amount of food. If there is more than one person carrying out the work, it is assumed that each person does the same amount of work each day and all the persons do exactly the same amount of work.

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

2.If A's 1 day's work=1/n, then A can finish the work in n days.

Ex: If A can do a piece of work in 4 days,then A's 1 day's work=1/4.
If A's 1 day’s work=1/5, then A can finish the work in 5 days

3.If A is thrice as good workman as B,then: Ratio of work done by A and B =3:1. Ratio of time taken by A and B to finish a work=1:3.


4.Definition of Variation: The change in two different variables follow some definite rule. It said that the two variables vary directly or inversely.Its notation is X/Y=k, where k is called constant. This variation is called direct variation. XY=k. This variation is called inverse variation.

5.Some Pairs of Variables:
i)Number of workers and their wages. If the number of workers increases, their total wages increase. If the number of days reduced, there will be less work. If the number of days is increased, there will be more work. Therefore, here we have
direct proportion or direct variation.

ii)Number workers and days required to do a certain work is an example of inverse variation. If more men are employed, they will require fewer days and if there are less number of workers, more days are required.

iii)There is an inverse proportion between the daily hours of a work and the days required. If the number of hours is increased, less number of days are required and if the number of hours is reduced, more days are required.

Distance
Distance covered per unit time is called speed
Speed = Distance/Time
Time = Distance/speed
Distance = speed*time

Distance is normally measured in Km, meters or miles; Time in hours or seconds and speed in km/hr, miles/hr, meter/second.
1km/hr = 5/18 m/s
1 m/s = 18/5 Km/hr

If the ratio of the speed of A and B is a:b,then the ratio of the time taken by them to cover the same distance is 1/a : 1/b or b:a

Relative Speed
suppose a man covers a distance at x kmph and an equal distance at y kmph.then the average speed during the whole journey is (2xy/x+y)kmph

The most difficult mathematics is that which you do not know.

A surprising amount of mathematics is actually easy once you've learned it. Of course, once you learn the easy stuff, then you have to start tacking the deep stuff, and that gets harder.

One teacher I had, was introducing a new concept, and we did an example in class. (and this was a class for good mathematicians -- not your average students) There was a lot of blank stares, and not everybody seemed to follow all the way through.

The very next thing he asked was for us to differentiate the function x² with respect to x. Of course, everybody could do that very easily.

His response? "The reason you can do differentiation, but not the other thing, is that you've differentiated things hundreds of times, but you haven't done this other thing very much yet."

We will have equations of one or two unknowns invariably in every problem. Some times we get three equations in three unknowns. In general, we need as many equations as the variables we will have to solve for. So, for solving for the values of two unknowns, we need two equations (or two conditions given in the problem) and for solving for the values of three unknowns, we need three equations.

One equation in one unknown:
An equation like 2x + 6 = 36 is an equation in one variable. We have only one variable x whose value we have to find out.

Two equations in two unknowns:
A set of equations like
2x + 3y = 10 ………… (1)
3x + 5y = 12 ………… (2)
Is called simultaneous equations in two unknowns. Here, we have two variables ( or unknowns) x and y whose values we have to find out.

And also we use three equations in three unknowns.

The basic number system we use in maths are:

Natural Numbers:
The numbers we use for counting are called natural numbers. The set of all natural numbers is denoted by “N”.
Hence N= {1,2,3,4……..}.

Whole Numbers:
When all the natural numbers and zero are put together, we get a new set of whole numbers. The set of whole numbers is denoted by “W”.
Hence W={0,1,2,3,4………}.

Integers:
The set containing positive numbers(1,2,3….), negative numbers(-1,-2,-3,…..) together with zero is called as set of integers. We denote the set of integers with “Z”.
Hence Z={……-3,-2,-1,0,1,2,3,……..}.

Rational Numbers:
A rational number is a number of the form a/b. Where a and b are integers, b≠0.
Example: {2/3, 4/7,7/9……..}

Prime Numbers:
Numbers which do not have any other factors except 1 and itself are called prime numbers.
Examples: {1,2,3,5,7,11,13,17,19,23,29,31,37,41……….}

Composite Numbers:
All numbers greater than 1 and except prime numbers are called composite numbers.
Examples: {4,6,8,9,10,12…….}.

Even Numbers:
Numbers which are exactly divisible by 2 are called even numbers.
Examples: {2,4,6,8,10…….}

Odd Numbers:
Numbers which are not exactly divisible by 2 are called even numbers.
(or)
Numbers which are not even numbers are called odd numbers.
Examples: {1,3,5,7,9……..}.

Important Facts and Formulae:

Results On Triangle

1.Sum of the angles of a triangle is 180 degrees.

2.The sum of any two sides of a triangle is greater
than third side.

3.Pythagoras Theorem:

In a right angled triangle (Hypotenuse)2 = (Base)2 +(Height)2

4.The line joining the mid point of a side of a triangle
to the opposite vertex is called the MEDIAN.

5.The point where the three medians of a triangle meet,
is called CENTROID. The centroid divides each of the
medians in the ratio 2:1

6.In an isosceles triangle, the altitude from the
vertex bisects the base

7.The median of a triangle divides it into two triangles
of the same area.

8.The area of the triangle formed by joining the mid points
of the sides of a given triangle is one-fourth of the area
of the given triangle.

Results On Quadrilaterals

1.The diagonals of a Parallelogram bisect each other.

2.Each diagonal of a Parallelogram divides it into two
triangles of the same area.

3.The diagonals of a Rectangle are equal and bisect
each other

4.The diagonals of a Square are equal and bisect each
other at right angles.

5.The diagonals of a Rhombus are unequal and bisect
each other at right angles.

6.A Parallelogram and a Rectangle on the same base
and between the same parallels are equal in area.

7.Of all he parallelogram of given sides the parallelogram
which is a rectangle has the greatest area.




Formulae:

1.Area of a RECTANGLE = length * breadth

Length = (Area/Breadth) and Breadth = (Area/Length)

2.Perimeter of a RECTANGLE = 2(Length + Breadth)

3.Area of a SQUARE = (side)2 = ½ ( Diagonal)2

4.Area of four walls of a room = 2(length + breadth) * height

5.Area of a TRIANGLE = ½ * base * height

6.Area of a TRIANGLE = √[s * (s-a) * (s-b) * (s-c)],
where a,b,c are the sides of the triangle and s = 1/2(a+b+c)

7.Area of EQUILATERAL TRIANGLE = √(3/4)* (side)2

8.Radius of in circle of an EQUILATERAL TRIANGLE of
side a = r / 2√3

9.Radius of circumcircle of an EQUILATERAL TRIANGLE
of side a = r / √3

10.Radius of incircle of a triangle of area ∆ and
semi perimeter S = ∆ / s

11.Area of a PARALLELOGRAM = (base * height)

12.Area of RHOMBUS = 1/2 (product of diagonals)

13.Area of TRAPEZIUM =
=1/2 * (sum of parallel sides)* (distance between them)

14.Area of a CIRCLE =  r2 where r is the radius

15.Circumference of a CIRCLE = 2r

16.Length of an arc = 2 rø / 360, where ø is central angle

17.Area of a SECTOR = ½ (arc * r) = r2ø / 360

18.Area of a SEMICIRCLE = r2 / 2

19.Circumference of a SEMICIRCLE = r

In any business environment the most important concern is about profit or loss of the transaction conducted.

The money paid by a shopkeeper to buy goods is called the cost price(C.P.) of the
shopkeeper. The price at which he sells the goods is called the selling price(S.P.) of the shopkeeper.

If the selling price(S.P.) is greater than the cost price(C.P.) then the shopkeeper gets profit. Profit = Selling price - Cost price.

If the selling price is less than the cost price then the shopkeeper gets loss.
Loss = Cost price - Selling price.

4. For comparison of profit or loss obtained in different transactions, they are they are expressed percentages.
Suppose
S.P = Selling price, C.P = Cost price,
P = Profit, L = Loss

Ratio of profit to cost price = P : C.P.
Percentage of profit = (P/c.p)*100.
Percentage of loss = (L/c.p)*100.

When two or more than two persons run a business jointly, they are called partners and the deal is known as partnership.


Ratio of Division of Gains:
1.When the investments of all the partners are do the same time, the gain or loss is distributed among the partners in the ratio of their investments.

Suppose A and B invest Rs x and Rs y respectively for a year in a business, then at the end of the year:(A's share of profit):(B's share of profit)=x:y

2.When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking (capital*number of units of time). Now gain or loss is divided in the ratio of these capitals.

Suppose A invests Rs x for p months and B invests Rs y for q months, then (A's share of profit):(B's share of profit)=xp:yq

3.Working and sleeping partners:A partner who manages the business is known as working partner and the one who simply invests the money is a sleeping partner.

Formula:
1.When investments of A and B are Rs x and Rs y for a year in a
business ,then at the end of the year
(A's share of profit):(B's share of profit)=x:y

2.When A invests Rs x for p months and B invests Rs y for q months,
then A's share profit:B's share of profit=xp:yq

Some Important definitions we use in sets.

Null Set:
A set is said to be a null set if it has no elements. It is also called an empty set or a void set and is denoted by ø.
Example:
{x | x is a perfect square and 4< x <9}

Finite and infinite sets:
A set ‘A’ is said to be finite if it is either an empty set or contains finite number of elements. Otherwise it is called an infinite set.
Example:
Set of natural numbers less than 100 is finite.

Cardinality of a finite set:
The number of distinct elements in a set is called the cardinality of the set. If a finite set A has n distinct elements, the cardinality of the set is n and is denoted by O(A) or n(A). The cardinality of an empty set is zero.
Example:
Cardinality of A = {a,e,I,o,u} is 5.

A set is a well defined collection objects. The objects of the set are called its elements. Sets are usually denoted by capital letters and the elements of the set are denoted by lower case. If an element x belongs to set A, it is denoted by x Є A. If x is not an element of A, it is denoted by striped(/)Є A.

A set, in general is represented in two forms:
1) In this form, a set is described by actually listing out all the elements. For example, the set of all odd natural numbers less than 10 is represented by {1,3,5,7,9}.

2) In this form, a set is described by a characterizing property. For example, the set of all odd natural numbers less than 10 is represented by {x Є N | x < 10 and x is odd}. The symbol | is read as “such that”.

My opinion is that mathematics is indeed a descriptive language. It is somewhat unique in this role though. The reason for this uniqueness is the fact that when people created this abstract system they wanted to use the most basic elements inherent in our reasoning, rather than just describe the observed. The result of this was a system that had very few and very basic axioms that seemed to match precisely the most basic patterns we observe in the Universe. It also meant that this system could evolve and create many patterns that appear to be greatly similar to those that we find in nature and can describe those.

What do I mean by basic elements ?
Well, one basic element of all reasoning systems we had so far is the existence of separate entities. Others are space, time, laws/relations that control
the entities and possibly more.

Percent implies "for every hundred". This concept is developed to facilitate easier comparison of fractions by equalizing denominator to 100.

Percentages can also be represented as decimal fractions. In such a case it is effectively equivalent to proportion of the original quantity. Any percentage can be expressed as a decimal fraction by dividing the percentage figure by hundred.

Percentages increase or decrease of a quantity is the ratio expressed in percentage of the actual increase or decrease of the quantity and the original amount of the quantity.

1.To convert a common fraction into per cent, we multiply the fraction with 100. And put % symbol to the result.

2.To convert a percent into fraction we must remove % symbol., divide it with 100 and simplify the product.

Factorial is defined for any positive integer. It is denoted by !. Thus “Factorial n” is written as n!. n! is defined as the product of all the integers from 1 to n.

Thus n! = 1.2.3.. ... (n-1),n.

Example 5! = 1*2*3*4*5 = 120

0! is defined to be equal to 1.
Therefore 0! = 1 and 1! = 1

An equation which has the unknown quantity raised only to powers which are whole numbers and the highest power being the square of the unknown quantity, is called a quadratic equation.
The most general form of a quadratic equation is ax^2 + bx + c = 0.
There are two values of x that satisfy such a quadratic equation. These values are called the roots of the quadratic equation.

The roots of the above quadratic equation are given by (-b±√(b^2-4ac))/2a

For ax^2 + bx + c = 0, sum of the roots = -b/a; Product of the roots = c/a

In the previous post we have discussed little bit about algebra. Here take a look about variables and constants.

In arithmetic, the numbers used have definite values. These values do not change. But the letters used in algebra have no particular value and may have any value assigned to them.

Consider,
The perimeter 'p' of a rectangle of length 'l' and breadth 'b' is p = 2(l+b). Here '2' is a fixed number. The letters p, l and b have no fixed value. They can take any positive value depending upon the size of the rectangle.

So, a letter symbol which can take any value of a certain set is called a variable. Above p, l and b are variables. Quantities which have only one fixed value are called constants. Above 2 is a constant.

Algebra is my favorite part in maths because it is easy. You will not find any difficulty when comparing with other parts.

Algebra is the part of the maths. In algebra, letters are used to represent numbers. The letters which are used to represent numbers are called literal numbers or literals. Using of literals in place of numerals helps us to think in more general terms and obtain a rule.

Consider 5*6 = 6*5 , -2*3 = 3*-2 , (1/2)*(3/4) = (3/4)*(1/2)
We can generalize the fact by the statement that the product of two rational numbers remains same in whichever order they are multiplied.
This can further be simplified by using symbols as, a*b = b*a , where a and b are any two rational numbers.

Permutations and combinations is one of the important areas in many exams because of two reasons. The first is that solving questions in this area is a measure of students reasoning ability. Secondly, solving problems in areas like probability requires through knowledge of permutations and combinations.

If one operation can be performed in 'm' ways and, a second operation then can be performed in 'n' ways, the number of ways of performing the two operations will be m*n.

Permutations:
The different arrangements of a given number of things by taking some or all at a time,are called permutations.
eg:- All permutations( or arrangements)made with the letters
a,b,c by taking two at a time are (ab,ba,ac,ca,bc,cb).

ombinations:
Each of the group or selections which can be made by taking some or all of a number of items is called a combination.
eg:- All permutations( or arrangements)made with the letters
a,b,c by taking two at a time are (ab,bc,ca).

Probability is nothing but chances of occurrences. I go to my college by bus. I find my college bus for every 5 buses. So the Probability of getting my bus is 1 out of 5.

Probability is an important topic for the entrance exams. This is a topic that you will be requiring in your management courses also. Hence the basic concepts that we are going to learn should be understood thoroughly. Because their usefulness goes beyond the entrance exams.

The word Probability is used, in a broad sense, to indicate a vague possibility that something might happen.

Terms we use in Probability are
Experiment:
An operation which can produce some well-defined outcome is called an experiment.

Random Experiment:
An experiment in which all possible out comes are known and the exact output cannot be predicted in advance is called a random experiment.

Basic Mathematical Symbols:

+ Plus

- Minus

× Multiply

÷ Divide

= Equal

% percent

: Ratio

> Greater than

< Less than

.∙. Therefore

∙.∙ Because

± Plus- minus sign

≠ Not equal to

≤ Less than or equal to

≥ Greater than or equal to

∞ Infinity

≈ Almost equal to

I am a big fan of cricket, i rate cricketers on the basis of their averages. Average is nothing but consistency.

Average is very simple but effective way of representing an entire group by a single value.

Average of any number of quantities will always be higher than the lowest value and lower than the highest value of items whose average is being taken.

1.Average = Sum of quantities/Number of quantities.

2.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.

Sometimes it so happens that the borrower and the lender agree to fix up a certain unit of time ,say yearly or half-yearly or quarterly to settle the previous account.
In such cases ,the amount after the first unit of time becomes the principal for the 2nd unit ,the amount after second unit becomes the principal for the 3rd unit and so
on. After a specified period ,the difference between the amount and the money borrowed is called Compound Interest for that period.

Formulae:
As we discussed in the simple interest
Let principal=p,Rate=R% per annum Time=nyears

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 in fractions
say 3 2/5 yrs Amount=P[1+(R/100)]3[1+((2R/5)/100)]
5.When rates are different for different years R1%,R2%,R3%
for 1st ,2nd ,3rd yrs respectively 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)]n

When you borrow money from some one he will give you money for interest. Usually money lender calculates simple interest and gives you money.

The interest is calculated only on the principal through out the loan period, it is called simple interest.

Principal or Sum:- The money borrowed or lent out for a certain period is called Principal or the Sum.

Interest:- Extra money paid for using others money is called Interest.

Simple Interest:- If the interest on a sum borrowed for a certain period is reckoned uniformly,then it is called Simple Interest.

Formulae:
Principal = P
Rate = R% per year
Time = T years. Then,

(i)Simple Interest(S.I)= (P*T*R)/100

(ii) Principal(P) = (100*S.I)/(R*T)
Rate(R) = (100*S.I)/(P*T)
Time(T) = (100*S.I)/(P*R)

You feel difficult when you are learning this type of maths specially at the age of 11 or 12 years. When i was studying my 6th class, i was first introduced this maths.

Business maths is one of the branches of mathematics just as algebra, geometry, statics. You can also call business maths as arithmetic.

Topics like simple interest, compound interest, percentages, distances will definitely create problems. Even though it is difficult it is useful in real life. When you are buying or selling some goods, you can not calculate your profit or loss if you don't know business mathematics. When it comes to money borrowing from others(generally persons who you don't know), you should know simple interest. otherwise they may cheat you. I too have seen many such cases.

In my experience, main reason is "The Fear Factor".

The first is, because the feelings of inferiority and outright fear that many, probably most, students feel when they confront mathematics, severely inhibit students’ natural intelligence and creativity. It is as though every mathematical subject, and every concept within a subject, is surrounded by a kind of “force field” that radiates, “Not for you!”, “You aren’t smart enough!”. The origin of this force field may be early experiences in a family in which, say, a father had always been good at mathematics, and had made it clear he expected his children to likewise be good at the subject. In the case of women, the origin might be subtle messages sent by teachers through out the primary and secondary school years — perhaps without conscious intention — that technical subjects are too hard for girls. Or, it might be the atmosphere that surrounds mathematics and indeed all technical subjects in the nation’s most schools.

I have been studying mathematics since my childhood, but i was never asked about "what is maths?". Even i too never tried.
Today my neighbor who is studying 9th standard has asked me the question. At that time i had no answer to tell.

Really, what is maths?

There doesn't seem to be a clear cut, definitive answer. I'm happy about that. At this stage I'm writing down some things I have discovered so far. Not definitive, rather preliminary, but a start.

Maths is NOT about formulas and cranking out computations - or rather that's a very small part of maths.

Maths is about perceiving and acting in the world in an enhanced way, about perceiving the world in a different way and being able to act more powerfully within it.

Basic Numbers-Number System:

Natural Numbers:
The numbers we use for counting are called natural numbers. The set of all natural numbers is denoted by “N”.
Hence N= {1,2,3,4……..}.

Whole Numbers:
When all the natural numbers and zero are put together, we get a new set of whole numbers. The set of whole numbers is denoted by “W”.
Hence W={0,1,2,3,4………}.

Integers:
The set containing positive numbers(1,2,3….), negative numbers(-1,-2,-3,…..) together with zero is called as set of integers. We denote the set of integers with “Z”.
Hence Z={……-3,-2,-1,0,1,2,3,……..}.

Rational Numbers:
A rational number is a number of the form a/b. Where a and b are integers, b≠0.
Example: {2/3, 4/7,7/9……..}

Prime Numbers:
Numbers which do not have any other factors except 1 and itself are called prime numbers.
Examples: {1,2,3,5,7,11,13,17,19,23,29,31,37,41……….}

Composite Numbers:
All numbers greater than 1 and except prime numbers are called composite numbers.
Examples: {4,6,8,9,10,12…….}.
Even Numbers:
Numbers which are exactly divisible by 2 are called even numbers.
Examples: {2,4,6,8,10…….}

Odd Numbers:
Numbers which are not exactly divisible by 2 are called even numbers.
(or)
Numbers which are not even numbers are called odd numbers.
Examples: {1,3,5,7,9……..}.