EDUCATIONAL GURU MATH

Permutation and Combination

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Permutation : Permutation means arrangement of things. The word arrangement is used, if the order of things is considered.

Combination: Combination means selection of things. The word selection is used, when the order of things has no importance.

Example: Suppose we have to form a number of consisting of three digits using the digits 1,2,3,4, To form this number the digits have to be arranged. Different numbers will get formed depending upon the order in which we arrange the digits. This is an example of Permutation.

Now suppose that we have to make a team of 11 players out of 20 players, This is an example of combination, because the order of players in the team will not result in a change in the team. No matter in which order we list out the players the team will remain the same! For a different team to be formed at least one player will have to be changed.

Now let us look at two fundamental principles of counting:

Addition rule : If an experiment can be performed in ‘n’ ways, & another experiment can be performed in ‘m’ ways then either of the two experiments can be performed in (m+n) ways. This rule can be extended to any finite number of experiments.

Example: Suppose there are 3 doors in a room, 2 on one side and 1 on other side. A man want to go out from the room. Obviously he has ‘3’ options for it. He can come out by door ‘A’ or door ‘B’ or door ’C’.

Multiplication Rule : If a work can be done in m ways, another work can be done in ‘n’ ways, then both of the operations can be performed in m x n ways. It can be extended to any finite number of operations.

Example.: Suppose a man wants to cross-out a room, which has 2 doors on one side and 1 door on other site. He has 2 x 1 = 2 ways for it.

Factorial n : The product of first ‘n’ natural numbers is denoted by n!.

n! = n(n-1) (n-2) ………………..3.2.1.

Ex. 5! = 5 x 4 x 3 x 2 x 1 =120

Note 0! = 1

Proof n! =n, (n-1)!

Or (n-1)! = [n x (n-1)!]/n = n! /n

Putting n = 1, we have

O! = 1!/1

or 0 = 1

Permutation

Number of permutations of ‘n’ different things taken ‘r’ at a time is given by:-

ⁿP_r = n!/(n-r)!

Proof: Say we have ‘n’ different things a₁, a₂……, a_n.

Clearly the first place can be filled up in ‘n’ ways. Number of things left after filling-up the first place = n-1

So the second-place can be filled-up in (n-1) ways. Now number of things left after filling-up the first and second places = n - 2

Now the third place can be filled-up in (n-2) ways.

Thus number of ways of filling-up first-place = n

Number of ways of filling-up second-place = n-1

Number of ways of filling-up third-place = n-2

Number of ways of filling-up r-th place = n – (r-1) = n-r+1

By multiplication – rule of counting, total no. of ways of filling up, first, second -- rth-place together :-

n (n-1) (n-2) ------------ (n-r+1)

Hence:
ⁿP_r = n (n-1)(n-2) --------------(n-r+1)

= [n(n-1)(n-2)----------(n-r+1)] [(n-r)(n-r-1)-----3.2.1.] / [(n-r)(n-r-1)] ----3.2.1

ⁿP_r = n!/(n-r)!

Number of permutations of ‘n’ different things taken all at a time is given by:-

ⁿP_n = n!

Proof :
Now we have ‘n’ objects, and n-places.

Number of ways of filling-up first-place = n

Number of ways of filling-up second-place = n-1

Number of ways of filling-up third-place = n-2

Number of ways of filling-up r-th place, i.e. last place =1

Number of ways of filling-up first, second, --- n th place
= n (n-1) (n-2) ------ 2.1.

ⁿP_n = n!

Concept.

We have ⁿP_r = n!/n-r

Putting r = n, we have :-

ⁿP_r = n! / (n-r)

But ⁿP_n= n!

Clearly it is possible, only when n! = 1

Hence it is proof that 0! = 1

Note : Factorial of negative-number is not defined. The expression –3! has no meaning.

Examples

Q. How many different signals can be made by 5 flags from 8-flags of different colours?

Ans. Number of ways taking 5 flags out of 8-flage = ⁸P₅

= 8!/(8-5)!

= 8 x 7 x 6 x 5 x 4 = 6720

Q. How many words can be made by using the letters of the word “SIMPLETON” taken all at a time?

Ans. There are ‘9’ different letters of the word “SIMPLETON”

Number of Permutations taking all the letters at a time = ⁹P₉

= 9! = 362880.

Number of permutations of n-thing, taken all at a time, in which ‘P’ are of one type, ‘g’ of them are of second-type, ‘r’ of them are of third-type, and rest are all different is given by :-

n!/p! x q! x r!

Example: In how many ways can the letters of the word “Pre-University” be arranged?

13!/2! X 2! X 2!

Number of permutations of n-things, taken ‘r’ at a time when each thing can be repeated r-times is given by = n^r.

Proof.

Number of ways of filling-up first –place = n

Since repetition is allowed, so

Number of ways of filling-up second-place = n

Number of ways of filling-up third-place

Number of ways of filling-up r-th place = n

Hence total number of ways in which first, second ----r th, places can be filled-up

= n x n x n ------------- r factors.

= n^r

Example: A child has 3 pocket and 4 coins. In how many ways can he put the coins in his pocket.

Ans. First coin can be put in 3 ways, similarly second, third and forth coins also can be put in 3 ways.

So total number of ways = 3 x 3 x 3 x 3 = 3⁴ = 81

Metric Conversion Charts

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Approximate Conversions To Metric Measures

Symbol	When you Know	Multiply by	To Find	Symbol
LENGTH
in ft yd mi	inches feet yards miles	2.5 30.0 0.9 1.6	centimeters centimeters meters kilometers	cm cm m km
AREA
in² ft² yd² mi²	square inches square feet square yards square miles acres	6.5 0.09 0.8 2.6 0.4	square centimeters square centimeters square meters square kilometers hectares	cm² cm² m² km² ha
MASS
oz lb	ounces pounds short tons (2000 lb)	28 0.45 0.9	grams kilograms metric ton	g kg t
VOLUME
tsp Tbsp in³ fl oz c pt qt gal ft³ yd³	teaspoons tablespoons cubic inches fluid ounces cups pints quarts gallons cubic feet cubic yards	5 15 16 30 0.24 0.47 0.95 3.8 0.003 0.76	milliliters milliliters milliliters milliliters liters liters liters liters cubic meters cubic meters	mL mL mL mL L L L L m³ m³
TEMPERATURE (exact)
F	Farenheit Degrees	subtract 32, multiply by 5/9	Celsius Degrees	C

Approximate Conversions from Metric Measures

Symbol	When you Know	Multiply by	To Find	Symbol
LENGTH
mm cm m m km	millimeters centimeters meters meters kilometers	0.004 0.4 3.3 1.1 0.6	inches inches feet yards miles	in in ft yd mi
AREA
cm² m² km² ha	square centimeters square meters square kilometers hectares (10,000 m²)	0.16 1.2 0.4 2.5	square inches square yards square miles acres	in² yd² mi²
MASS
g kg t	grams kilograms metric ton (1,000 kg)	0.035 2.2 1.1	ounces pounds short tons	oz lb
VOLUME
mL mL L L L m³ m³	milliliters milliliters liters liters liters cubic meters cubic meters	0.03 0.06 2.1 1.06 0.26 35.0 1.3	fluid ounces cubic inches pints quarts qallons cubic feet cubic yards	fl oz in³ pt qt gal ft³ yd³
TEMPERATURE (exact)
C	Celsius Degrees	multiply by 5/9, add 32	Farenheit Degrees	F

Approximate Liquid and Dry Measure Equivalencies

Customary	Metric
1/4 teaspoon	1.25 milliliters
1/2 teaspoon	2.5 milliliters
1 teaspoon	5 milliliters
1 tablespoon	15 milliliters
1 fluid ounce	30 milliliters
1/4 cup	60 milliliters
1/3 cup	80 milliliters
1/2 cup	120 milliliters
1 cup	240 milliliters
1 pint (2 cups)	480 milliliters
1 quart (4 cups, 32 ounces)	960 milliliters (0.96 liters)
1 gallon (4 quarts)	3.84 liters
1 ounce (by weight)	28 grams
1/4 pound (4 ounces)	114 grams
1 pound (16 ounces)	454 grams
2.2 pounds	1 Kilogram (1,000 grams)

Oven Temperature Equivalencies

Description	Farenheit	Celsius
Cool	200	90
Very Slow	250	120
Slow	300-325	150-160
Moderately Slow	325-350	160-180
Moderate	350-375	180-190
Moderately Hot	375-400	190-200
Hot	400-450	200-230
Very Hot	450-500	230-260

Cooking Measurement Equivalents

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The information below shows measuring equivalents for teaspoons, tablespoons, cups, pints, fluid ounces, and more. This page also includes the conversions for metric and U.S. systems of measurement.

1 tablespoon (tbsp) =	3 teaspoons (tsp)
¹/16 cup =	1 tablespoon
¹/8 cup =	2 tablespoons
¹/6 cup =	2 tablespoons + 2 teaspoons
¹/4 cup =	4 tablespoons
¹/3 cup =	5 tablespoons + 1 teaspoon
³/8 cup =	6 tablespoons
¹/2 cup =	8 tablespoons
²/3 cup =	10 tablespoons + 2 teaspoons
³/4 cup =	12 tablespoons

1 cup =	48 teaspoons
1 cup =	16 tablespoons
8 fluid ounces (fl oz) =	1 cup
1 pint (pt) =	2 cups
1 quart (qt) =	2 pints
4 cups =	1 quart
1 gallon (gal) =	4 quarts
16 ounces (oz) =	1 pound (lb)
1 milliliter (ml) =	1 cubic centimeter (cc)
1 inch (in) =	2.54 centimeters (cm)

Source: United States Dept. of Agriculture (USDA).

U.S.–Metric Cooking Conversions

U.S. to Metric

Capacity		Weight
¹/5 teaspoon	1 milliliter	1 oz	28 grams
1 teaspoon	5 ml	1 pound	454 grams
1 tablespoon	15 ml
1 fluid oz	30 ml
¹/5 cup	47 ml
1 cup	237 ml
2 cups (1 pint)	473 ml
4 cups (1 quart)	.95 liter
4 quarts (1 gal.)	3.8 liters

Metric to U.S.

Capacity		Weight
1 milliliter	¹/5 teaspoon	1 gram	.035 ounce
5 ml	1 teaspoon	100 grams	3.5 ounces
15 ml	1 tablespoon	500 grams	1.10 pounds
100 ml	3.4 fluid oz	1 kilogram	2.205 pounds = 35 ounces
240 ml	1 cup
1 liter	34 fluid oz = 4.2 cups = 2.1 pints = 1.06 quarts = 0.26 gallon

Scale Factors - I

2009-02-25T02:53:00.001+05:30

or 'Why There Are No Giant Spiders!

Giant creatures the stuff of many science fiction movies ... from giant ants and spiders to 10-storey tall babies. But is it really possible for creatures to be so large? If so, why aren't there any 6 metre tall spiders?

On this page we'll try to explain why ants and spiders could never be as big as an elephant ... and still look the same.

The reason for this has to do with scale factors ...

Here's a simple box. Beside it is one twice as big.
You can tell it's twice as big because we've put a scale beside it.

Actually, to be more precise, the second box is twice as long as the first one.
The correct way to say this is that it has been scaled up by a factor of two.

By making the box twice as long, we've also doubled the width and height.

Here's the same diagram, only this time we've shaded one face.
The second box is still twice as long as the first one.

Notice what happens to the area of one face of the box when it's twice as long ...

The AREA has been increased by a factor of FOUR.

Here's the same diagram again, only this time we've shaded all the boxes.
The second box is still twice as long as the first one.

Notice what happens to the volume of the box when it's twice as long ...

The VOLUME has been increased by a factor of EIGHT.

Increasing the length of an object by a factor of 2 increases the area by a factor of 4 and the volume by a factor of 8.

Let's start over with another set of boxes.

Here's a simple box. Beside it is one three times as big.
You can tell it's three times as big because we've put a scale beside it.

To be more precise, the second box is three times as long as the first one.
The correct way to say this is that it has been scaled up by a factor of three.

By making the box three times as long, we've also tripled the width and height.

Here's the same diagram, only this time we've shaded one face.
The second box is still three times as long as the first one.

Notice what happens to the area of one face of the box when it's three times as long ...

The AREA has been increased by a factor of NINE.

Here's the same diagram again, only this time we've shaded all the boxes.
The second box is still three times as long as the first one.

Notice what happens to the volume of the box when it's three times as long ...

The VOLUME has been increased by a factor of TWENTY-SEVEN.

Increasing the length of an object by a factor of 3 increases the area by a factor of 9 and the volume by a factor of 27.

Do you see the pattern? Let's summarize what happened (and add a few more examples) in a table:

	LENGTH	AREA	VOLUME
Scale Factor 2	2 times	4 times	8 times
Scale Factor 3	3 times	9 times	27 times
Scale Factor 4	4 times	16 times	64 times
Scale Factor 5	5 times	25 times	125 times
Scale Factor X	X times	X² times	X³ times

Increasing the length of an object by factor X increases the area by factor X² and the volume by factor X³

Scale Factors -II

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Page Two
The reason that scale factors are important is that some physical properties of bodies depend on area, and some depend on volume. This has important consequences for the sizes of those bodies, and how they function.

Mass

The mass of an object depends on its volume. For example, if you increase an object's volume by eight times, its mass will also be eight times greater.

Here's a spider that has grown in size by a factor of three. The big one is three times as long as the small one.

From the previous page, you know that this will give the bigger spider a volume that is 27 times the volume of the small spider. This means the bigger spider's mass is 27 times the mass of the smaller one. The big one is 27 times heavier.

This principle holds true whenever you enlarge someting. If a dog were to grow to 5 times its normal size, it would weigh 125 times as much. The weight of an object increases as the cube of the scale factor. Another example? Suppose an ant were to grow 200 times bigger. It would weigh 200³, or 8,000,000 times as much.

Strength

The strength of an organism depends on its cross-sectional area. For example, you might measure your strength by how much weight you could lift with your biceps muscle. If you wanted to get stronger, you would exercise that muscle to make it bigger. But what will increase is the cross-sectional area of the muscle. It will get wider in the middle, but won't get longer.

For example, a muscle that is twice as wide might be four times stronger.
The strength of an organism increases as the square of the scale factor.
Another example? A leg that is twice as big will support four times as much weight. An ant that is 200 times bigger will be 200², or 40,000 times stronger.

Do you see the problem now? As an organism gets bigger, its weight increases as the cube of the scale factor, but its strength increases only as the square. Its strength isn't increasing as fast as its weight.

Let's use an actual example to make it clear what's happening. We'll use the example of an ant that has grown 200 times in length.

Let's make the normal ant's strength 1 unit, and its weight 1 unit. We can then make a ratio of strength to weight by dividing these numbers. We get 1/1 = 1. We'll call this ratio the relative strength ... for the normal ant it's 1.

Now let's work it out for the big ant.
The big ant is 200 times as long, so it will be 200², or 40,000 times stronger.
The big ant's weight is 200³, or 8,000,000 times heavier.
The big ant's strength to weight ratio is 40,000/8,000,000 = 0.005, so it's relative strength is 0.005, or 1/200^th.

Although the ant's size has increased, its strength hasn't kept pace with its weight. It has only 1/200^th of the strength needed to support itself. The result would be an giant ant that couldn't stand up! In fact, since its body shell strength won't have increased enough either, it would end up as a squashed puddle on the floor, totally unable to support itself or its internal organs.

Oxygen Requirements

Cells in living organisms require oxygen to function. If all the cells in an organism were to grow by a factor of X, their surface area would now be X² times as large, but their volume and mass would be X³ times bigger.

The amount of oxygen required depends on the mass of the cells, so the bigger cells would need X³ as much oxygen. However, oxygen gets into the cells through their surface membranes, which have only increased by a factor of X².

The bigger cells' oxygen-aquiring ability won't have increased as fast as their need for oxygen ... so all the cells will die. An organism that increases in size will suffocate!

You may be wondering how any large animals can exist at all, given what we've shown you. For example, when a young animal grows to adulthood, how do its bigger cells breathe?
The answer is simple ... the cells don't get bigger ... they divide. As the organism grows, the number of cells grows, but the cell size remains about the same.

Consider animals that are really large, like the elephant. What do you notice about its body? An elephant isn't shaped like a dog or mouse that's been expanded many times. It's legs are much thicker, in proportion to its body size, than smaller animals.

In order for an organism to be this big, its supporting bones and muscles in the legs must be bigger in proportion to the rest of its body than organisms that are small. Otherwise it wouldn't be strong enough to stand up.

If a spider were as big as an elephant, it would have to look something like this. Notice the very thick legs ... far thicker in proportion to its body than a regular-sized spider.

But of course this spider could never exist. Not only would it suffocate because its cells couldn't get enough oxygen, but it would also be unable to breathe because, although its shell would have to be proportionately thicker to support its inner organs, its area would have only increased as the square of the scale factor, while its mass will have increased as the cube of that value. And a spider breathes through its skin.

So despite what you see in really bad movies, if anyone ever manages to come up with a device to instantly make insects a hundred times bigger, or to enlarge a baby to skyscraper height, none of these enlarged organisms could ever survive. They'd be unable to stand up, and they'd suffocate!

How to find the surface area and volume of prisms.

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Theory:

A 'prism' is a solid figure with a uniform cross section. Here are some examples of prisms:

Rectangular based prism:

Circular based prism: (Cylinder)

Triangular based prism:

The surface area of any prism equals the sum of the areas of its faces, which include the floor, roof and walls. Because the floor and the roof of a prism have the same shape, the surface area can always be found as follows:

From now on we will call the 'floor' the 'base'. The height of the prism = 'H'.

The surface area of a prism = 2 × area of base + perimeter of base × H

The actual formula used to find the surface area will depend on the shape of the base of the prism.

For example:

Rectangular based prism Base shape: Rectangle, length 'L' and width 'W' Area of base: L × W Perimeter of base: 2(L+W) Surface area = 2LW + 2(L+W)H

Circular based prism Base shape: Circle, radius 'R' Area of base: pR² Perimeter of base: 2pR Surface area = 2pR² + 2pRH

Triangular based prism Base shape: Triangle: base 'b', height 'h', and sides S₁, S₂ and S₃ Area of base: ½b×h Perimeter of base: S₁+ S₂ + S₃ Surface area = bh + (S₁+ S₂ + S₃)H

Surface Area Formulas

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in general, the surface area is the sum of all the areas of all the shapes that cover the surface of the object.

Surface Area of a Cube = 6 a²

(a is the length of the side of each edge of the cube)

In words, the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a² . Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.

Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac

(a, b, and c are the lengths of the 3 sides)

In words, the surface area of a rectangular prism is the are of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.

The area of the top and bottom (side lengths a and c) = a*c. Since there are two of them, you get 2ac. The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc. The left and right side have side lengths of a and b, so the surface area of one of them is a*b. Again, there are two of them, so their combined surface area is 2ab.

Surface Area of Any Prism

(b is the shape of the ends)

Surface Area = Lateral area + Area of two ends

(Lateral area) = (perimeter of shape b) * L

Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)

Surface Area of a Sphere = 4 pi r²

(r is radius of circle)

Surface Area of a Cylinder = 2 pi r² + 2 pi r h

(h is the height of the cylinder, r is the radius of the top)

Surface Area = Areas of top and bottom +Area of the side

Surface Area = 2(Area of top) + (perimeter of top)* height

Surface Area = 2(pi r²) + (2 pi r)* h

In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle.

You can find the area of the top (or the bottom). That's the formula for area of a circle (pi r²). Since there is both a top and a bottom, that gets multiplied by two.

The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So the area of the rectangle is (2 pi r)* h.

Add those two parts together and you have the formula for the surface area of a cylinder.

Surface Area = 2(pi r²) + (2 pi r)* h

Tip! Don't forget the units.

These equations will give you correct answers if you keep the units straight. For example - to find the surface area of a cube with sides of 5 inches, the equation is:

Surface Area = 6*(5 inches)²

= 6*(25 square inches)

= 150 sq. inches

Special Right Triangles 45º-45º-90º

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Certain triangles possess "special" properties that allow us to use "short cut formulas" in arriving at information about their measures. These formulas let us arrive at the answer very quickly.

One such triangle is the 45º-45º-90º triangle.

There are two "special" formulas that apply ONLY to the 45º-45º-90º triangle.

45º-45º-90º (Isosceles Right Triangle)
"Special" Formulas

You must remember that these formulas can be used ONLY in a 45º-45º-90º triangle.

What if I forget the formulas?
What should I do?

The nice thing about mathematics is that there is always another way to do the problem. If you forget these formulas, you could always use the Pythagorean Theorem or a Trigonometry formula.

Let's look at 3 solutions to this problem where you are asked to find x:

Special Formula solution

Pythagorean Theorem solution

Trigonometric solution

We are looking for the hypotenuse so we will use the formula that will give the answer for the hypotenuse:

Substituting the leg = 7, we arrive at the answer:

A nice feature of these special formulas is that the answer is already in reduced form.

Since a 45º-45º-90º, also called an isosceles right triangle, has two legs equal, we know that the other leg also has a length of 7 units.
c²= a²+ b²
x² = 7² +7²x² = 49 + 49
x² = 98

Use either 45º angle as the reference angle (where your stick figure will stand). One possible solution is shown below:

rounded
rounded

Special Right Triangles 30º-60º-90º

2009-02-14T05:00:00.001+05:30

Certain triangles possess "special" properties that allow us to use "short cut formulas" in arriving at information about their measures. These formulas let us arrive at the answer very quickly.

One such triangle is the 30º-60º-90º triangle.

There are two "special" formulas that apply ONLY to the 30º-60º-90º triangle.

30º-60º-90º "Special" Formulas
H = hypotenuse
LL = long leg (across from 60º)
SL = short leg (across from 30º)

You must remember that these formulas can be used ONLY in a 30º-60º-90º triangle.

Using the formulas:

EASY:

Find x and y.

x is the short leg

Answer

y is the long leg

Answer

HARDER:

Find x and y.

6 is the short leg and
x is the hypotenuse
(start with what you have given)

Answer

y is the long leg

Answer

HARDEST:
(requires more algebraic manipulation)

Find x and y.

8 is the long leg and
x is the hypotenuse
(start with what you have given)

Answer

y is the short leg

Answer

What if I forget the formulas?
What should I do?

There is always more than one way to tackle a problem. If you forget these formulas, you could always use the Trigonometry formulas to find one of the missing sides and then use the Pythagorean Theorem to find the last side OR you could use the Trigonometric formulas to find both missing sides.

Unfortunately, the Pythagorean Theorem by itself,
will not help you find both of the missing sides. Remember that you need to know TWO sides of a triangle in order to engage the Pythagorean Theorem.

Finding the Equation of a Line

2009-02-12T04:18:00.001+05:30

To find the equation of a line you need to know either two points or one point and the slope. If you are given two points, the first step in any of the methods is to find the slope from the two given points. So first memorize the slope formula.

Given: A line passes through the two points (-4, 1) and (2, 5). Find an equation of this line.

Finding the Slope

All three methods require the slope of the line, so first find the slope:

Note: Either point may be considered ( x₁, y₁ )

Point-Slope Method

Plug in the slope for m, and either of the given points for ( x₁, y₁). So here we are finding the equation of a line with slope 2/3 passing through the point (2, 5).

Then convert to standard form:

Eliminate denominator

Rearrange terms

Change signs, done! The standard form equation is

Slope-Intercept Method

Plug in the slope for m, and either of the given points for ( x, y ). This time we'll use the other point, so here is the method for finding the equation of a line with slope 2/3 passing through the point (-4, 1).

Now solve for b, and rewrite with x, y.

The slope-intercept equation is

Converting between Forms

The previous methods are all that is ever needed. It is helpful to have a shortcut, or quick method, but not necessary. Before you ever apply the third method you should be able to easily convert between the standard and slope-intercept forms of linear equations.

Converting to Slope-Intercept Form

Converting to slope-intercept from standard form:

Add the opposite of the x-term to both sides, and if necessary divide by the coefficient of the y-term, and simplify.

a)	b)	c)

Converting to Standard Form

Converting to standard from slope-intercept form: Multiply through by the LCD, if necessary, then move the x-term to the left side, and adjust the signs.

a)	b)	c)

Study the examples above, and compare both forms. Look at the slope in slope-intercept form, and compare that with the left-hand side of the standard form.

Practice, and mastery of these conversions leads to the understanding that

a line with equation Ax + By = C has a slope of

Standard Form Shortcut

The shortcut is very simple: Create the left-hand side of a standard form equation with the given slope. Then find the value of the constant C by plugging in any given point. So, to find the equation of a line with slope 2/3 passing through the point (2, 5):
    Slope 2/3
    Plug in (2, 5)
        Therefore, the standard form equation is

Conversion Tables

2009-02-07T06:31:00.001+05:30

To Convert

Multiply By

To Obtain

Acres

Acres

Acres

Acres

Acre-feet

Acre-feet

Acre-feet

Bar

Bar

Bar

Bushels (dry)

Centimeters (cm)

Centimeters

Centimeters

Centimeters

Centimeters

cm/sec

cm/sec

cm (cubed)

Cubic feet

Cubic feet

Cubic feet

Cubic feet

Cup

Feet (ft)

Feet

Feet per minute

Feet head of water

Foot candle

Gallons (gal)

Gal

Gal

Gal/acre

Gal/acre

Gal/1000 ftsquared

Gal/minute

Grams (g)

Gram

G/ha

Grams per liter

Grams per liter

Grams/sq.meter

G/cm (cubed)

G/cm (cubed)

Hectares (ha)

Inches

Inches

Inches

Insquared

In (cubed) 43,560

0.00405

4047

4840

325,851

43560

1233.5

14.5

1019.7

29.53

0.03524

0.03281

0.3937

0.1094

0.01

10

1.9685

0.0223694

0.0610237

0.0283

7.4805

1728

0.037

8

30.48

0.3048

0.01136

0.433

10.764

3.785

3785

128

9.354

2.938

4.0746

2.228 x 10 (-3)

0.002205

0.035274

0.000893

1000

10

0.00020481

0.036127

62.428

2.471

2.540

0.0254

25.40

6.4516

16.3871 Sq. feet

Sq. kilometer

Sq. meter

Sq. yards

Sq. feet

Cu. feet

m (cubed)

Lb/sq.in.

g/cm (cubed)

inches Hg at 0 degrees C

msquared

Feet

Inches

Yards

Meters

Millimeters (ml)

ft/min

MPH

inch (cubed)

Cu. meter

Gallons

Cubic inches

Cubic yards

fl oz

Centimeters

Meters

MPH

PSI

Lux

Liters

Millimeters

Ounces (liquid)

Liters/hectare

Oz/1000 ftsquared (liquid)

L/100 msquared

Cubic feet/second

Pounds

oz

lbs/a

PPM

Percent

lb/sq.feet

lb/in (cubed)

lb/ft (cubed)

Acres

Centimeters

Meters

Millimeters

cmsquared

cm (cubed)

Kilograms (kg)

Kg/hectare

Kg/ha

Kg/L

Kilometers (Km)

Kilometers

Kilometers

Kilometers

Kilometers

Km/h

Km/h

Kilopascals (kPa)

Liters (l)

Liters

Liters

Liters

L/100 msquared

Liters/hectare

Meters (m)

Meters

Meters

Meters

Meters

Meters

Meters/sec

Msquared

M (cubed)

M (cubed)

Miles (statute)

Miles

Miles

Miles

Miles/hour (mph)

Miles/hour

Miles/hour

Miles/hour

Milliliters (ml)

Milliliters

Millimeters (mm)

1 mm Hg @ 0 C

Ounces (fluid)

Ounces (fluid)

Ounces (weight)

Parts per million (ppm)

PPM

PPM

PPM

PPM

PPM

PPM

Percent (%)

Pint

pt/A

pt/A 2.2046

0.892

0.02048

8.3454

100,000

3281

1000

0.6214

1094

0.62137

54.6807

0.145

0.2642

33.814

2.113

1.057

0.2454

0.107

3.281

39.37

1.094

100

0.001

1000

2.2369

10.764

35.3147

1.30795

160,900

5280

1.609

1760

1.467

88

1.61

0.447

0.0338

0.0002642

0.03937

0.13332

0.02957

29.573

28.35

2.719

0.001

8.34

1

0.013

0.3295

8.345

10

0.473

1.1692

0.3673 Pounds

Pounds/acre

lb/1000 ftsquared

lb/gal

Centimeters

Feet

Meters

Miles

Yards

MPH

ft/min

Pounds/sq.in. (psi)

Gallons

Ounces

Pints

Quarts

gal/1000 ftsquared

Gallons/acre

Feet

Inches

yards

Centimeters

Kilometers

Millimeters

MPH

ftsquared

ft (cubed)

yd (cubed)

Centimeters

Feet

Kilometers

Yards

Feet/second

Feet/minute

Kilometers/hour

meter/second

Ounces (fluid)

Gallons

Inches

kPa

Liters

Milliliters

Grams

lb ai/acre foot of water

Grams/l

Lb/million gal

mg/kg

Ounces/100 gal of water

Gal/acre-foot of water

lbs/million gal of water

g/kg

liter

L/ha

oz/1000 ftsquared

Pounds

Pounds

Pounds/acre

Pounds/A

Pounds/sq.ft.

Pounds/1000 ftsquared

Pounds/yd (cubed)

Pounds/gallon

PSI (lbs/sq.in.)

PSI

PSI

PSI

PSI

Quarts

Qt/A

Qt/A

Sq. centimeters

Sq. centimeters

Sq. feet

Sq. feet

Sq. feet

Sq. inch

Ton (2000 lbs)

Yards

Yards

Yards

yd (cubed)

yd (cubed) 0.4536

453.6

1.12

0.02296

4883

43.5597

0.0005937

0.12

6.9

0.06895

0.068046

2.31

6.89

0.9463

2.3385

0.7346

0.001076

0.1550

929

0.0929

9.294 x 10 (-6)

6.452

907

91.44

0.9144

914.4

27

0.7645 Kilograms

Grams

Kg/hectare

lb/1000 ftsquared

Grams/sq.meter

lb/A

G/cm (cubed)

Kg/liter

Kilopascals

Bar

atm

feet head of water

kPa

Liters

L/ha

oz/1000 ftsquared

Sq. feet

Sq. inches

Sq. centimeters

Sq. meters

Hectares

Sq. centimeters

kg

Centimeters

Meters

Millimeters

ft (cubed)

m (cubed)

Area Equivalents

1 acre = 43,560 ft squared = 4840 yd 2 = 0.4047 hectares = 160 rods
squared = 4047 m 2 = 0.0016 sq. mile

1 acre-inch = 102.8 m 3 = 27,154 gal = 3630 ft 3

1 hectare (ha) = 10,000 m 2 = 100 are = 2.471 acres = 107,639 ft squared

1 cubic foot (ft 3 ) = 1728 in 3 = 0.037 yd 3 = 0.02832 m 3 = 28,320 cm
3

1 square foot (ft 2 ) = 144 in 2 = 929.03 cm 2 = 0.09290 m 2

1 square yard (yd 2 ) = 9 ft 2 = 0.836 m 2

1 cubic yard (yd 3 ) = 27 ft 3 = 0.765 m 3

Units Sq. In. Sq. Ft. Sq Yd. Sq. cm Sq. m

Sq. In. 1 0.006944 0.0007716 6.452 0.000645

Sq. Ft. 144 1 0.1111 929 0.0929

Sq Yd. 1296 9 1 8361 0.8361

Sq. cm 0.155 0.001076 0.0001196 1 0.0001

Sq. m     1550 10.76 1.196 10,000 1

Liquid Equivalents

1 ft (cubed) of water = 7.5 gal = 62.4 lbs. = 28.3 liters

1 acre-inch of water = 27,154 gal = 3630 ft 3

1 liter (l) = 2.113 pts. = 1000 ml = 1.057 qts. = 33.8 fl.oz. = 0.26 gal

1 US gallon=4 qt.=8 pt. = 16 cups = 128 fl.oz. = 8.337 lbs of water =
3.785 L = 3785 ml = 231 in 3 = 256 tbsp. = 0.1337 ft3

1 quart = 0.9463 liters = 2 pt. = 32 fl. oz. = 4 cups = 64 tablespoons
(tbsp.)=57.75 in 3 = 0.25 gal = 946.4 ml

1 pint = 16 fl. oz. = 2 cups = 473.2 ml = 32 level tablespoons = 0.125
gal = 0.5 qt

1 cup = 8 fl. oz. = = ½ pt. = 16 tablespoons = 236.6 ml

1 tablespoon = 14.8 ml = 3 teaspoons (tsp.) = 0.5 fl.oz.

1 milliliter (ml) = 1 cm 3 = 0.34 fl.oz. = 0.002 pts

1 teaspoon = 4.93 ml = 0.1667 fl. oz. = 80 drops

1 US fluid ounce = 29.57 ml = 2 tablespoons = 6 tsp. = 0.03125 qt

Temperature Equivalents

degrees Centigrade = (°F-32)x5/9

degrees Fahrenheit = (°Cx9/5)+32

Pressure Equivalents

1 lb per square inch (PSI) = 6.9 kilopascal (kPa)

1 PSI = 2.31 feet head of water

Units Dyne Lb. Newton

Dyne 1 2.248x10-6 10-5

Lb. 444,823 1 4.448

Newton 100,000 0.2248 1

Length Equivalents

centimeter (cm) = 0.3937 inch = 0.01 m = 0.03281 ft.

meter (m) = 3.28 feet = 39.4 inches = 100 cm = 1.094 yds = 1000 mm

kilometer = 0.621 statute mile = 1000 meters = 100,000 cm = 3281 ft =
39,370 in.

inch = 2.54 cm = 25.4 mm = 0.0254 m = 0.08333 ft.

foot = 0.3048 meters = 30.48 cm = 12 inches

yard = 0.9144 meters = 3 feet = 36 inches = 91.44 cm

statute mile = 1760 yards = 5280 feet = 1.61 kilometers = 1609 meters

Mixture Ratios

1 mg/g = 1000 ppm

1 fl.oz./gal = 7490 ppm

1 fl.oz./100 gal = 75 ppm

1 pt/100 gal = 1 teaspoons/1gal

1 qt/100 gal = 2 tablespoons/1 gal

Flow

1 gpm = 0.134 ft 3 /minute

1 ft (cubed) /min. (cfm) = 449 gal/hr. (gph) = 7.481 gal/min.

Weight Equivalents

1 ton (US) = 2000 lb = 0.907 metric tons = 907.2 kg

1 metric ton = 10 6 g = 1000 kg = 2205 lb

1 lb = 16 oz = 453.6 grams (g) = 0.4536 kg

1 oz (weight) = 28.35 g = 0.0625 lb

1 gram = 1000 mg = 0.0353 oz = 0.001 kg = 0.002205 lb

milligrams (mg) = 0.001 grams

1 kilogram (kg) = 1000 grams = 35.3 oz = 2.205 lbs

microgram (mg) = 10 -6 grams = 0.001 mg

nanogram (ng) = 10 -9 grams = 0.001 micrograms (mg)

picogram = 10 -12 grams

1 ppm= 0.0001%= 0.013 fl oz in 100 gal =1 mg/kg=1 mg/L=1 mg/g= 0.379 g
in 100 gal water= 8.34 x 10 -6 lb/gal=1ml/l

10 ppm = 0.001% = 10 mg/L 100 ppm = 0.01% = 100 mg/L1000 ppm = 1mg/g =
0.1% = 1000 mg/L

1 ppb = 1 ug/kg or 1 ug/L or 1 ng/g

1 ppt = 1 picogram/g

1 % = 10,000 ppm = 10g/L = 1g/100ml = 10g/kg = 1.33 oz by weight/gal
water = 8.34 lbs/100 gal water

Leather Substance

The measurement of the thickness of a finished leather, in

millimeters (1 mm. = 0.03937 inch),

irons (1 iron = 1/48 inch),

ounces (1 ounce = 1/64 inch):

Inch

Ounces

Irons

Millimeters

1/64

1/32

3/64

1/16

5/64

3/32

7/64

1/8

9/64

5/32

11/64

3/16

13/64

7/32

15/64

1/4

17/64

9/32 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18 0.75

1.5

2.25

3.0

3.75

4.5

5.25

6.0

6.75

7.5

8.25

9.0

9.75

10.5

11.25

12.0

12.75

13.5 0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

4.0

4.4

4.8

5.2

5.6

6.0

6.4

6.8

7.2

Multiplication Table

2009-01-29T03:35:00.001+05:30

Multiplication Table

x 0 1 2 3 4 5 6 7 8 9 10 11 12

0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11 12

2 0 2 4 6 8 10 12 14 16 18 20 22 24

3 0 3 6 9 12 15 18 21 24 27 30 33 36

4 0 4 8 12 16 20 24 28 32 36 40 44 48

5 0 5 10 15 20 25 30 35 40 45 50 55 60

6 0 6 12 18 24 30 36 42 48 54 60 66 72

7 0 7 14 21 28 35 42 49 56 63 70 77 84

8 0 8 16 24 32 40 48 56 64 72 80 88 96

9 0 9 18 27 36 45 54 63 72 81 90 99 108

10 0 10 20 30 40 50 60 70 80 90 100 110 120

11 0 11 22 33 44 55 66 77 88 99 110 121 132

12 0 12 24 36 48 60 72 84 96 108 120 132 144

Monomials, Binomials, and Polynomials

2009-01-25T05:39:00.001+05:30

A monomial
is a constant, a variable, or the product of a constant and one or more variables.

A polynomial
is the sum of one or more monomials.

"... this means that a
monomial
is a single term which has NO plus sign (+) or minus sign (-) between
entries. It could be things like

13, 3x, -57, x², 4y², -2xy, or
520x²y²

but NOT

(2x+7)
or
(4x²-2x)."

"A polynomial
can be one monomial or a bunch of
monomials hooked together with plus signs and/or minus signs."

Name
Number
of Terms
Example

Monomial

1 (mono implies one)

4x²

Binomial
2(bi implies two)

4x² + 3x

Trinomial
3(tri implies three)

4x² + 3x + 5



The expression
4x² is a monomial because it has one term.

The expression
4x² + 3x is a
binomial
because it has two terms.

The expression 4x² +3x + 5 is
a trinomial
because it has three terms.

All three
expressions can be
called polynomials.

Solids

2009-01-25T04:58:00.001+05:30

Volume
is measured in cubic units.

Surface area is measured in square units.

Volume (V) and
Surface Area (SA) Formulas

Rectangular
Solid

V=lwh

SA=2lh + 2hw + 2lw

Cylinder

Sphere

Cone

If the
cross-sectional shape of a solid is the same at every level,

the volume of the solid is the product of

the area of its base and
its height:
V = Bh



Prism

V = Bh

Pyramid

volume & surface areas of solids

2009-01-25T04:11:00.001+05:30

Space Figure

A space figure or three-dimensional figure is a figure that has depth in addition
to width and height. Everyday objects such as a tennis ball, a box, a bicycle, and
a redwood tree are all examples of space figures. Some common simple space figures
include cubes, spheres, cylinders, prisms, cones, and pyramids. A space figure having
all flat faces is called a polyhedron. A cube and a pyramid are both polyhedrons;
a sphere, cylinder, and cone are not.

Cross-Section

A cross-section of a space figure is the shape of a particular two-dimensional
"slice" of a space figure.
Example:
The circle on the right is a cross-section of the cylinder on the left.

The triangle on the right is a cross-section of the cube on the left.

Volume

Volume is a measure of how much space a space figure takes up. Volume is used
to measure a space figure just as area is used to measure a plane figure. The volume
of a cube is the cube of the length of one of its sides. The volume of a box is
the product of its length, width, and height.
Example:
What is the volume of a cube with side-length 6 cm?

The volume of a cube is the cube of its side-length, which is 6³ = 216
cubic cm.
Example:
What is the volume of a box whose length is 4cm, width is 5 cm, and height is
6 cm?

The volume of a box is the product of its length, width, and height, which is 4 × 5 × 6 = 120
cubic cm.

Surface Area

The surface area of a space figure is the total area of all the faces of the figure.
Example:

What is the surface area of a box whose length is 8, width is 3, and height is
4? This box has 6 faces: two rectangular faces are 8 by 4, two rectangular faces
are 4 by 3, and two rectangular faces are 8 by 3. Adding the areas of all these
faces, we get the surface area of the box:
8 × 4 + 8 × 4 + 4 × 3 + 4 × 3 + 8 × 3 + 8 × 3 =

32 + 32 + 12 + 12 +24 + 24=

136.

Cube

A cube is a three-dimensional figure having six matching square sides. If L
is the length of one of its sides, the volume of the cube is L³ = L × L × L.
A cube has six square-shaped sides. The surface area of a cube is six times the
area of one of these sides.
Example:
The space figure pictured below is a cube. The grayed lines are edges hidden from
view.

Example:
What is the volume and surface are of a cube having a side-length of 2.1 cm?

Its volume would be 2.1 × 2.1 × 2.1 = 9.261
cubic centimeters.

Its surface area would be 6 × 2.1 × 2.1 = 26.46
square centimeters.

Cylinder

A cylinder is a space figure having two congruent circular bases that are parallel.
If L is the length of a cylinder, and r is the radius of one of the
bases of a cylinder, then the volume of the cylinder is L × pi × r²,
and the surface area is 2 × r × pi × L + 2 × pi × r².
Example:
The figure pictured below is a cylinder. The grayed lines are edges hidden from
view.

Sphere

A sphere is a space figure having all of its points the same distance from its
center. The distance from the center to the surface of the sphere is called its
radius. Any cross-section of a sphere is a circle.

If r is the radius of a sphere, the volume V of the sphere is given
by the formula V = 4/3 × pi ×r³.

The surface area S of the sphere is given by the formula S = 4 × pi ×r².
Example:
The space figure pictured below is a sphere.

Example:
To the nearest tenth, what is the volume and surface area of a sphere having a
radius of 4cm?

Using an estimate of 3.14 for pi,

the volume would be 4/3 × 3.14 × 4³ = 4/3 × 3.14 × 4 × 4 × 4 = 268
cubic centimeters.

Using an estimate of 3.14 for pi, the surface area would be 4 × 3.14 × 4² = 4 × 3.14 × 4 × 4 = 201
square centimeters.

Cone

A cone is a space figure having a circular base and a single vertex.

If r is the radius of the circular base, and h is the height of the
cone, then the volume of the cone is 1/3 × pi × r² × h.
Example:
What is the volume in cubic cm of a cone whose base has a radius of 3 cm, and
whose height is 6 cm, to the nearest tenth?

We will use an estimate of 3.14 for pi.

The volume is 1/3 × pi × 3² × 6 = pi ×18 = 56.52,
which equals 56.5 cubic cm when rounded to the nearest tenth.
Example:
The pictures below are two different views of a cone.

Pyramid

A pyramid is a space figure with a square base and 4 triangle-shaped sides.
Example:
The picture below is a pyramid. The grayed lines are edges hidden from view.

Tetrahedron

A tetrahedron is a 4-sided space figure. Each face of a tetrahedron is a triangle.
Example:
The picture below is a tetrahedron. The grayed lines are edges hidden from view.

Prism

A prism is a space figure with two congruent, parallel bases that are polygons.
Examples:
The figure below is a pentagonal prism (the bases are pentagons). The grayed lines
are edges hidden from view.

The figure below is a triangular prism (the bases are triangles). The grayed lines
are edges hidden from view.

The figure below is a hexagonal prism (the bases are hexagons). The grayed lines
are edges hidden from view..

Tricky Math Questions

2009-01-24T17:04:00.001+05:30

Have a go at solving these 15 math questions.

Questions:

1) There are 8 Apples on the table, you take 3. How many do you have?

2) 10 Birds in a field. 2 were shot, how many were left?

3) Take away the first letter, take away the last letter, then take away all the other letters. What do you have left?

4) If you have 4 melons in one hand, and 7 apples in the other - What do you have?

5) A box has nine ears of corn in it. A Squirrel carries out three ears
a day, and yet it takes him nine days to carry out all the corn.
Explain?

6) Why do white sheep eat more than black sheep?

7) A man wanted to plant 4 trees, but all 4 had to be equal distances from each other. How did he do it?

8) I have 2 coins in my hand that add up to 60c. One of the coins isn't a 50c piece. What are the coins?

9) A fisherman was asked how long was the fish he had caught. He said "it is 30cms plus half its length" How long was the fish?

10) A Hammer and a Nail cost $31. If the Hammer cost $30 more than the Nail, what is the cost of each?

11) It takes 7 men 2 hours to build a wall. How long does it take 3 men to build the same wall?

12) "I will bet you $1" said Fred, "that if you give me $2, I will give you $3 in return."
"Done," replied Tom. Was he?

13) "How much will one cost?"

"25 cents"

"How much will fifteen cost?"

"50 cents"

"OK then, I'll take one hundred and sixteen"

"Thank you, that will be 75 cents please"

Explain.

14) What comes next in the following sequence ?
1, 4, 5, 6, 7, 9, 11,...

15) In a scientific context, what could the following phrase mean?
How I want a drink, alcoholic of course, after the heavy chapters involving quantum mechanics

Dont Look for the Answers.

Answers:

1) If you take 3 then you have 3.

2) 2 - the others flew away.

3) The Mailman.

4) Big hands.

5) He has 2 of his own ears, so he carries out only 1 ear of corn per day.

6) There are more white sheep than black sheep.

7) He planted 3 trees at the corners of an equilateral triangle. He
built a mound in the middle and planted the 4th on the top of the mound
so that it was the same distance from the other 3 trees. (on the points
of a tetrahedron.)

8) 50c and a 10c (The coin that isn't a 50c piece is a 10c. The other coin is the 50c).

9) 60 cms.

10) Hammer $30.50, Nail $0.50

11) No need to bother, the 7 men have already built it.

12) Tom accepts the bet, gives Fred $2. Fred does not give Tom $3 so loses the bet and has to pay Tom $1. Result Fred gains $1.

13) He is buying house numbers, each separate digit costs 25c, so 116 is three digits so 3 x 25c = 75c.

14) 100 (The next number that doesn't contain a "T" in the spelling).

15) The number of letters in each word refers to pi to 14 decimal places, i.e. 3.14159265358979

Marvellous Answer

2009-01-24T16:52:00.001+05:30

A mechanic was removing the
cylinder heads from the motor of a car when he spotted the famous heart
surgeon in his shop, who was standing off to the side, waiting for the
service manager to come to take a look at his car. The mechanic shouted
across the garage,"Hello Doctor!! Please come over here for a minute."
The famous surgeon, a bit surprised, walked over to the
mechanic.

The mechanic straightened up, wiped his hands on a
rag and asked argumentatively, "So doctor, look at this. I also open
hearts, take valves out, grind 'em, put in new parts, and when I finish
this will work as a new one. So how come you get the big money, when you
and me is doing basically the same work? "
The doctor leaned over
and whispered to the mechanic...................

*
*
*
*
*
*

He said:
"Try to do it when the engine is running ".

How observant r u??

2009-01-24T16:50:00.001+05:30

Readout loud the text inside the triangle below.

More than likely yousaid, "A bird in the bush," and........

if thisIS what YOUsaid, then youfailed to see

that the word THE is repeated twice!

Sorry, look again.

Next, let's play with some words.

What do yousee?

In
black youcanreadthe word GOOD, in white the word EVIL (inside each
black letter is a white letter). It's all very physiological too,
because it visualize the concept that good can't exist without evil (or
the absence of good is evil ).

Now, what do yousee?

Youmay
not see it at first, but the white spaces readthe word optical, the
blue landscape reads the word illusion. Look again! Canyousee why
thispainting is called an optical illusion?

What do yousee here?

Thisone is quite tricky!

The word TEACH reflects as LEARN.

Last one.

What do yousee?

Youprobably readthe word ME in brown, but.......

when youlook through ME

youwill see

YOU!

Do youneed to look again?

Test Your Brain
Thisis really cool. The second one is amazing so please readall the way though.

ALZHEIMERS' EYE TEST

Count every "F" in the following text:

FINISHED FILES ARE THE RE
SULT OF YEARS OF SCIENTI
FIC STUDY COMBINED WITH
THE EXPERIENCE OF YEARS...

(SEE BELOW)

HOW MANY ?

WRONG, THERE ARE 6 -- no joke.
READIT AGAIN !
Really, go Back and Try to find the 6 F's before youscroll down.

The reasoning behind is further down.

The brain cannot process "OF".

Incredible or what? Go back and look again!!

Anyone who counts all 6 "F's" on the first go is a
genius.

Three is normal, four is quite rare.

Send thisto your friends.
It will drive them crazy.!
And keep them occupied
For several minutes..!

More Brain Stuff . . From Cambridge University.

O lny srmat poelpe canraed tihs.

cdnuolt blveiee taht I cluod aulaclty uesdnatnrd waht I was rdanieg. The
phaonmneal pweor of the hmuan mnid, aoccdrnig to a rscheearch at Cmabrigde Uinervtisy,

it
deosn't mttaer in waht oredr the ltteers in a wrod are, the olny
iprmoatnt tihng is taht the frist and lsat ltteer be in the rghit
pclae. The rset canbe a taotl mses and youcansitll raed it wouthit a
porbelm.

Tihs
is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but
the wrod as a wlohe. Amzanig huh? yaeh and I awlyas tghuhot slpeling
was ipmorantt! if
youcanraed tihs psas it on !!

Roman Numerals

2009-01-12T06:01:00.001+05:30

The Basics

A smaller number in front of a larger number means subtraction, all else means addition. For example, IV means 4, VI means 6.

You would not put more than one smaller number in front of a larger number to subtract. For example, IIV would not mean 3.

You must separate ones, tens, hundreds, and thousands as separate items. That means that 99 is XCIX, 90 + 9, but never should be written as IC. Similarly, 999 cannot be IM and 1999 cannot be MIM.

I The numeral one. II is two, III is three. You seldom see IIII as 4, since IV can also mean 4, plus its shorter to write.

V The numeral 5. IV is 4, VI is 6, VII is 7, VIII is 8.

X The numeral 10. IX is 9, XI is 11, etc.

L The numeral 50. XL would be 40.

C The numeral 100. Think of Century having a hundred years. C is short for the Latin word Centum, but that's not very easy to remember.

D The numeral 500.

M The numeral 1000.

Sometimes you will see a numeral with a line over it. That means to multiply it by 1000. A numeral V with a line over it means 5000.

Scatter Plots

2009-01-12T04:55:00.001+05:30

Scatter plots are similar to line graphs in that they
use horizontal and vertical axes to plot data points.
However, they have a very specific purpose. Scatter
plots show how much one variable is affected by another.
The relationship between two variables is called their
correlation .

Scatter plots usually consist of a large body of data.
The closer the data points come when plotted to making
a straight line, the higher the correlation between the
two variables, or the stronger the relationship.

If the data points make a straight line going from the
origin out to high x- and y-values, then the variables
are said to have a positive correlation . If
the line goes from a high-value on the y-axis down to
a high-value on the x-axis, the variables have a
negative correlation .

A perfect positive correlation is given the value of 1.
A perfect negative correlation is given the value of -1.
If there is absolutely no correlation present the value
given is 0. The closer the number is to 1 or -1, the
stronger the correlation, or the stronger the relationship
between the variables. The closer the number is to
0, the weaker the correlation. So something that seems
to kind of correlate in a positive direction might have
a value of 0.67, whereas something with an extremely weak
negative correlation might have the value -.21.

An example of a situation where you might find a perfect positive
correlation, as we have in the graph on the left above, would be
when you compare the total amount of money spent on tickets at
the movie theater with the number of people who go. This means
that every time that "x" number of people go, "y" amount of money
is spent on tickets without variation.

An example of a situation where you might find a perfect negative
correlation, as in the graph on the right above, would be if you
were comparing the speed at which a car is going to the amount of
time it takes to reach a destination. As the speed increases, the
amount of time decreases.

On the other hand, a situation where you might find a strong but not
perfect positive correlation would be if you examined the number of
hours students spent studying for an exam versus the grade received.
This won't be a perfect correlation because two people could spend the
same amount of time studying and get different grades. But in general
the rule will hold true that as the amount of time studying increases
so does the grade received.

Let's take a look at some examples. The graphs that were shown
above each had a perfect correlation, so their values were 1 and
-1. The graphs below obviously do not have perfect correlations.
Which graph would have a correlation of 0? What about 0.7? -0.7? 0.3?
-0.3?

Bar Graphs

2009-01-12T04:52:00.003+05:30

Bar graphs are used to display data in a similar way to line graphs.
However, rather than using a point on a plane to define a value, a bar
graph uses a horizontal or vertical rectangular bar that levels off at
the appropriate level.

There are many characteristics of bar graphs that make them useful.
Some of these are that:

They make comparisons between different variables very easy to see.

They clearly show trends in data, meaning that they show how one variable
is affected as the other rises or falls.

Given one variable, the value of the other can be easily determined.

Line Graphs

2009-01-12T04:52:00.001+05:30

Line graphs compare two variables. Each variable is plotted along an
axis . A line graph has a vertical axis and a horizontal axis.
So, for example, if you wanted to graph the height of a ball after you
have thrown it, you could put time along the horizontal, or x-axis, and
height along the vertical, or y-axis.

As I mentioned before, each type of graph has characteristics that
make it useful in certain situations. Some of the strengths of line
graphs are that:

They are good at showing specific values of data, meaning that given
one variable the other can easily be determined.

They show trends in data clearly, meaning that they visibly show how
one variable is affected by the other as it increases or decreases.

They enable the viewer to make predictions about the results of data
not yet recorded.

Unfortunately, it is possible to alter the way a line graph appears to
make data look a certain way. This is done by either not using consistent
scales on the axes, meaning that the value in between each point along
the axis may not be the same, or when comparing two graphs using different
scales for each. It is important that we all be aware of how graphs
can be made to look a certain way, when that might not be the way the
data really is.

Let's take a look at an example.

In a few years, you might be interested in getting some kind of parttime
job. You find the following line graph, which plots the minimum wage
versus time from Ocober, 1938, to September, 1997. What kinds of things
might you be able to tell from it?

Answer the following questions by looking at the graph, and then click
on Answers to compare your responses.

1. What was the minimum wage in January, 1978?

2. When did the minimum wage reach $3.35?

3. Between what time periods was the largest increase in minimum
wage?

4. Based on your observations of the graph, make a prediction about
what the wage might be in the year 2000.

5. What about the scales used on the graph might make the data appear
differently than how it really is?

CIRCLE/PIE GRAPHS

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Circle or pie graphs are particularly
good illustrations when considering how many parts of a whole are inception.
In the table below both the number of hours in a whole day devoted to certain
activities is listed as well as the percent of time for each of these activities.
The pie chart is then divided very much as a baker's pie would be into slices
that represent the proportional amounts of time spent on each activity.

To the right of the pie chart is
a legend that tells which color stands for which category. In addition, the
percents are also near the pie slice that stands for that particular amount
of time spent.

Percent of Hours of a Day Spent
on Activities

ACTIVITY HOURS PERCENT OF DAY

Sleep 6 25

School 6 25

Job 4 17

Entertainment 4 17

Meals 2 8

Homework 2 8

BAR GRAPHS

2009-01-12T04:30:00.001+05:30

Bar graphs are an excellent way to
show results that are one time, that aren't continuous - especially samplings
such as surveys, inventories, etc. Below is a typical survey asking students
about their favorite after school activity. Notice that in this graph each column
is labeled - it is also possible to label the category to the left of the bar.

In this case, the numbers for each
category are across the bottom of the chart.

A bar chart is marked off with a
series of lines called grid lines. These lines typically mark off a numerical
point in the series of numbers on the axis or line. In this case, each grid
line going up and down marks a multiple of 20 as the graph is divided such.More
gridlines can make it easier to be exact with the amounts being shown on the
bar graph, but too many can make it confusing.

Notice that for data that does not
fall evenly on a multiple of 20, the bar is in between two grid lines.

Bar graphs are useful to get an overall
idea of trends in responses - which categories get many versus few responses.

Favorite Student After
School Activity

Activity Number

Visit
W/Friends 175

Talk on
Phone 168

Play Sports 120

Earn Money 120

Use Computers 65

LINE GRAPHS 1

2009-01-12T04:29:00.001+05:30

A line graph is most useful in displaying
data or information that changes continuously over time. The example below shows
the changes in the temperature over a week in January. Notice that the title
of the graph is "Average Daily Temperature for January 1-7 in degrees Fahrenheit".

To the left is a table that shows
the date in one column and the corresponding temperature in the second column.
The line graph on the right shows the degrees of temperature going up the vertical
axis (up and down numbers on the left of the graph) and the days of the week
on the horizontal axis (going sideways from left to right). The points for the
temperature for each day are connected by a line - thus the graph is a line
graph.

Average Daily Temperature
for January 1-7 in Degrees Fahrenheit

Date

Temperature

1 10

2 25

3 30

4 42

5 23

6 25

7 40

x	1	2	3	4	5	6	7	8	9	10	11	12
0	0	0	0	0	0	0	0	0	0	0	0	0
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144

	A monomial is a constant, a variable, or the product of a constant and one or more variables.
	A polynomial is the sum of one or more monomials.

Name	Number of Terms	Example
Monomial	1 (mono implies one)	4x²
Binomial	2(bi implies two)	4x² + 3x
Trinomial	3(tri implies three)	4x² + 3x + 5

	Rectangular Solid *V=lwh* *SA=2lh + 2hw + 2lw*
	Cylinder
	Sphere
	Cone

A smaller number in front of a larger number means subtraction, all else means addition. For example, IV means 4, VI means 6. You would not put more than one smaller number in front of a larger number to subtract. For example, IIV would not mean 3. You must separate ones, tens, hundreds, and thousands as separate items. That means that 99 is XCIX, 90 + 9, but never should be written as IC. Similarly, 999 cannot be IM and 1999 cannot be MIM.
I	The numeral one. II is two, III is three. You seldom see IIII as 4, since IV can also mean 4, plus its shorter to write.
V	The numeral 5. IV is 4, VI is 6, VII is 7, VIII is 8.
X	The numeral 10. IX is 9, XI is 11, etc.
L	The numeral 50. XL would be 40.
C	The numeral 100. Think of Century having a hundred years. C is short for the Latin word Centum, but that's not very easy to remember.
D	The numeral 500.
M	The numeral 1000.
Sometimes you will see a numeral with a line over it. That means to multiply it by 1000. A numeral V with a line over it means 5000.

ACTIVITY	HOURS	PERCENT OF DAY
Sleep	6	25
School	6	25
Job	4	17
Entertainment	4	17
Meals	2	8
Homework	2	8

x	1	2	3	4	5	6	7	8	9	10	11	12
0	0	0	0	0	0	0	0	0	0	0	0	0
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144

x	1	2	3	4	5	6	7	8	9	10	11	12
0	0	0	0	0	0	0	0	0	0	0	0	0
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144