MyMaths MyLife

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……..}.