Math Homework Help

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2008-02-11T05:45:00.000-12:00

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A Sales Chart

2008-01-19T18:40:00.001-12:00

Sarah went to a grocery shop. There is a sales chart.

Cabbage

$1.8 for 2

Carrot

$0.6 for 1 k.g

Onion

$1.75 for 1 carton

Potato

$2.05 for 1 carton

Tomato

$1 for 1 k.g

Sarah bought 4 carton potatoes and 1 cabbage. If she gave $10, how much money she got back?

Hint: 10-{(4x2.05)+.9}

The Fundamental Principle of Counting

2008-01-13T05:06:00.000-12:00

If an event can happen in ecactly m ways, and if following it, a second event can happen in exactly n ways, then the two events in succession can happen in exactly mn ways.

Illustration.

Suppose there are five routs from A to B and three routs from B to C. In how many ways a person can go from A to C?

Since there are five different routs from A to b, the person can go the first part of his journey in 5 different ways. Having completed in any one of the 5 different ways , he has 3 different ways to complete the second part of the journey fro B to C. Thus each way of going from A to B give rise to 3 different ways of going from B to C.

There fore the total number of ways of completing the whole journey = number of ways for the first part x number of ways for the second part.
= 5 x 3=15.

Generalisation

If an event can occur in m different ways, a second event in n different ways, a third event in exactly p different ways and so on, then the total number of ways in which all events can occur in succession is mnp....

Some defnitions of average

2008-01-09T20:36:00.000-12:00

One of the most important objectives of statistical analysis is to get one single value that describes the characteristic of the entire mass of unwieldy data. Such a value is called central value or an average or the expected value of the variable.

The term " average " has been defined by various authors. Some important definitions are given below.

"Average is an attempt to find one single figure to describe whole of figures" -Clark

" An average is a single value selected from a group of values to represent them in some way-a value which is supposed to stand for whole group, of which it is a part, as typical of all the values in he group" -A.E Waugh

" An average is a typical value in the sense that it is sometimes employed to represent all the individual values in a series or of a variable" -Ya-Lun-Chou

"An average is a single value within the range of the data that is used to represent all the values in the series.Since an average is somewhere within the range of the data. It is also called a measure of central value" - Croxton &Cowden

Belated New Year Wishes

2008-01-05T20:37:00.000-12:00

I had been busy with some personal matters for the last five six weeks. In between gone for a vacation as well. But the new year day was a bit dull as my kids and I was not well after coming back from the vacation.

I wish you all a very prosperous and happy new year.

A Problem of Combination

2007-11-20T04:07:00.001-12:00

A box contains 12 marbles of three different colors. Green, yellow and blue- 4 each.

If you were to close your eyes and pick them at random, how many marbles must you take out to be sure that there are at least two of one color among the marbles picked out?

Answer:

In the first three pickings you may get 1 of each color. in the 4 th pick there will be at least two of one color.
So, the answer is 4.

Right Foot Forward

2007-11-20T03:48:00.001-12:00

A short man takes three steps to a tall man's two steps. They both start out on the left foot. How many steps do they have to take before they are both are stepping out on the right foot together?

Ans:

Very simple.

A Puzzle In Paradox

2007-11-01T06:24:00.001-12:00

Thirteen teachers are in Paradox, a math conference. When they arrive at the Enigma Hotel to check in, they are told that only 12 rooms are available. Since their school had made reservations for 13 rooms, the teachers are a bit upset that they will have to find another place to stay. As they are preparing to leave and find another hotel, the manager comes out and asks if there is a problem. When she hears of their situation she assures them that the Enigma Hotel has enough space to accommodate each teacher in his or her own room. She takes two of the teachers to room #1 and promises to come back in a few minutes and take one of them to another room. She takes the third teacher to room #2, the fourth teacher to room #3, the fifth teacher to room #4 and so on, taking the twelfth teacher to room #11. She then returns to room #1 and escorts the extra teacher waiting there to room #12. All of the teachers are now happily settled in their own rooms.

Is this possible? Why or why not?

O.K Here is the solution.

Actually, it is not possible. When the manager is putting the 12 th teacher in #11, the 13 th teacher is still waiting there and he or she must go to the #12 room. So, all the rooms are now occupied and in the first room there are two teachers and the teacher waiting there can not go to the #13 room.

Just Relax

2007-10-31T05:17:00.000-12:00

We have been discussing about serious math topics and problems for a long time. We also need some rest from that. So today we can have some math jockes.
Here we go.

Q: Why do you rarely find mathematicians spending time at the beach?
A: Because they have sine and cosine to get a tan and don't need the sun!

Q: How does one insult a mathematician?
A: You say: "Your brain is smaller than any eps1ilin >0!"

Q: How does a mathematician induce good behavior in her children?
A: `I've told you n times, I've told you n+1 times...'

Q: Why do mathematicians often confuse Christmas and Halloween?
A: Because Oct 31 = Dec 25.

Q: What does a mathematician present to his fiancee when he wants to propose?
A: A polynomial ring!

Two math students, a boy and his girlfriend, are going to a fair. They are in line to ride the ferries wheel when it shuts down.
The boy says: "It's a sin for those people to keep us waiting like this!"
The girl replies: "No - it's a cosin, silly!!!"

HAPPY HALLOWEEN!

Find this two digit number

2007-10-04T05:52:00.001-12:00

In a two digit number, the ten’s digit is three times the units digit.

If 36 is subtracted from the number, the digits interchange their places.
Find this two digit number.

Ans:

We can do this problem by algebraic method.
Let x be the unit's place digit.
Then the ten's place digit is 3x

Also, the number is
When interchanging their digits, the new number will be 10x+3x
So, when 36 is subtracted from the number,

That is units place digit is 2. So then's place digit is 3 times 2 =6

Therefore the number is 62.
Hope you understood this

Limit of a function at a pont.

2007-09-25T03:38:00.000-12:00

Let f(x) be a function of x. Let a and l be constants such that as , as we have
.
In such case we say that the limit of the function f(x) as x approaches a is l. we write this as

In case , no such number l exist, then we say that does not exist finitely.

Illustration

Let a regular polygon of n sides be inscribed in a circle. The area of the polygon cannot be greater than the area of the circle., however large the number of sides of the polygon increases indefinitely the area of the polygon continually approaches the area of the circle. Thus the difference between the area of the circle and the polygon can be made as small as we please by sufficiently increasing the number of sides of the polygon.

We have ,

(Area of the polygon of n sides)=Area of the circle

Mathematics Of Ice To Aid Global Warming Forecasts

2007-09-20T16:31:00.000-12:00

University of Utah mathematicians have arrived at a new understanding of how salt-saturated ocean water flows through sea ice -- a discovery that promises to improve forecasts of how global warming will affect polar icepacks.

In the current issue of the journal Geophysical Research Letters, math Professor Ken Golden and colleagues show that brine moving up or down through floating sea ice follows "universal transport properties."
"It means that almost the exact same formulas describing how water flows through sedimentary rocks in the Earth's crust apply to brine flow in sea ice, even though the microstructural details of the rocks are quite different from sea ice," says Golden, who currently is on an Australian research ship in Antarctica.

(University of Utah mathematician Ken Golden stands in front of sea ice melt ponds in the Arctic near Barrow, Alaska. His research on sea ice's permeability to salt water promises to help improve forecasts of the effects of global warming. )

The study suggests similar porous materials -- including ice on other worlds, such as Jupiter's icy ocean-covered moon Europa -- should follow the same rules, he adds.
Golden has made several trips to Antarctica and the Arctic for his studies.

The American Geophysical Union, which publishes the journal carrying Golden's study, says sea ice is important because it is both "an indicator and regulator of climate change; its thinning and retreat show the effects of climate warming, and its presence greatly reduces solar heating of the polar oceans."

"Sea ice also is a primary habitat for microbial communities, sustaining marine food webs," the group adds. "The permeability of sea ice and its ability to transport brine are important to many problems in geophysics and biology, yet remain poorly understood."

The AGU says Golden's study presents "a unified picture of sea ice permeability," and how that permeability to brine flow varies with the temperature and salinity of the ice.

Icy math and climate change

"One of the most important aspects of the polar sea ice packs is the role they play in Earth's albedo -- whether Earth absorbs or reflects incoming solar radiation," says Golden. "White sea ice reflects; the ocean absorbs. In the late spring, melt ponds [atop the ice] critically affect the albedo of the polar ice packs. The drainage of these melt ponds is again largely controlled by the permeability of the ice."

The Intergovernmental Panel on Climate Change's predictions "that the summer Arctic ice pack may disappear sometime during 2050-2100 depend in part on these types of considerations," he adds. "Now that we have a much firmer understanding of how permeability depends on the variables of sea ice, namely temperature and salinity, our results can help to provide more realistic representations of sea ice in global climate models, helping to hone the predictions for world climate and the effects of warming."

The results "can also help in understanding how polar ecosystems respond to climate change," Golden says. "Biological processes in the polar regions depend on brine flow through sea ice. For example, the rich food webs in the polar oceans are based on algae and bacteria living in the ice, and their nutrient intake is controlled by brine flow."

"In the Antarctic, ice formed from flooding of ice surfaces is an important component of the ice pack, and this formation is dependent on brine flow," he adds. "Brine drainage out of sea ice and the subsequent formation of Antarctic bottom water is an important part of the world's oceans."
Golden says the formulas that describe brine flow through sea ice and groundwater flow through sediments arose from abstract solid-state physics models used to describe atomic-scale phenomena in metals.

"These formulas exhibit universality, meaning that the end result doesn't depend on the details of the model or system, only on the dimension of the system," he says. "While large classes of abstract models obey this principle, real materials often do not. So it is surprising that a complex, real material like sea ice actually obeys these formulas."

To conduct the study, Golden and colleagues analyzed sea ice and "modeled" or simulated its behavior mathematically, and also made field and laboratory measurements of sea ice, including using X-rays to make CT-scan images of how the microscopic pore structure of ice varies with temperature.

Golden conducted the study with University of Utah colleagues Amy Heaton, a chemistry graduate student, and Jingyi Zhu, an associate professor of mathematics. Other co-authors are from the University of Alaska Fairbanks.

Courtesy: Science Daily.

Mathematical Signs

2007-09-14T17:49:00.000-12:00

If you look at any mathematics book written before 1500's, it will be very hard to understand.The Hindu -Arabic numerals familiar to us may have been used. Every thing else was different. The signs and symbols that make up the rest of the language of mathematics as we study it today had not yet been invented.

The sign and sign for subtraction first appeared in 1489 in a German arithmetic handbook. They may have been borrowed from signs used by merchants to mark certain packages . A +was marked on packages with too much of whatever the package contained, while a - meant too little.

The sign for multiplication was invented by an Englishman William Oughtredin 1631.

The sign for division was invented earlier by a German mathematician Johann Heinrich Rahn.

The = for "equals" was invented by the English mathematician Robert Recorde in 1557.

A cricket problem

2007-09-09T04:57:00.000-12:00

A group consists of 50 students. Out of these 20 are girls .There are 10 Australian students. Out of 50 students, 25 students like cricket.
What is the probability of selecting an Australian girl who likes cricket?

Hint:

Probability of selecting a girl= 20/50
Probability of selecting an Australian student = 10/50
Probability of a student like cricket=25/50

Therefore the the probability of selecting an Australian girl who likes cricket= (20/50 )(10/50)(25/50) = 1/25 =0.25

Net Problem

2007-08-29T05:19:00.000-12:00

The following is the net representation of a cube.

How will you place the letters L, A, F on the figure so that it should spell LEAF around the sides of the cube?

A football problem

2007-08-21T06:07:00.000-12:00

At a football championship 600 tickets were sold .
Child ticket cost $2 each and adult ticket cost $5 each. The total money collected for the game was $1650.

Find the number of tickets sold in each category.

Hint:

Let x be number of children and y be number of adults.

x+y = 600

2x+5y= 1650

==>x= 450, y=150

What is statistics?

2007-08-09T06:47:00.000-12:00

The word statistics conveys many meanings to people. to some it is another form of mathematics and for some other it suggests charts, graphs and tables which they find in newspapers and many other such things.

The study of statistics involves methods of refining numerical and non-numerical information into useful forms. In addition to meaning data, statitics also refers to a subject just as mathematics or physics. In this sense statistics is a body ofmethods of obtainining and analysing in order to make decisions on them.

Math Problem Solving Techniques

2007-06-17T01:31:00.000-12:00

Life is an arena of problems. L.A Averill has said,"The only worthwhile life is a life which contains its problems; to live without any longings and ambitions is to live only half way". A human child has to meet and solve problems as he grows-problems which present in his physical surroundings, his intellectual associations and in his social contacts. These problems grow in number and complexity as he or she grows older and older. His success in life is in large measuredetermined by the individual's capasity and competence to solve them.

Mathematics is a subject of problems.Stydying mathematics is different from studying other subjects. Math is learned by doing problems. Efficiency and ability in solving problems is a guarantee for success in learning this subject.

The first and most important step in solving a problem is to understand the problem.Read the problem clearly and grasp its meaning. Superficial or careless reading does not pay in mathematics. Be sure that you understand clearly what is given and what you are expected to find or prove. Keep these things in mind throughout your work.
Take sufficient time to think.
Plan thoroughly before you start.I dentify which skills and techniques you have learned can be applied to solve the problem at hand.

READ MORE.......

Ana-The bearer

2007-05-30T02:47:00.000-12:00

Today we have a questions on percentage.

Ana is working in a restaurant as a bearer.

As a penalty Ana’s wages were decreased by 50%.
After one month the reduced wages were increased by 50%.

Find her loss.

Hint: The salary increased is the 50% of the decreased salary.

Answer:

New salary is 50+25=75
Therefore loss=25%

Triangle Problem

2007-05-05T03:41:00.000-12:00

Today we have a problem on triangle

Hint:

Since AB=BC, angle C= angle A (Angles opposite to equal sides are equal)

Also, Sum of the angles is 180 º

Answer

55 º

Everything is Number

2007-04-22T17:17:00.000-12:00

Pythagoras had the habit of thinking about the numbers always. He had a passion for numbers.
He mentioned each number has its own qualities like this:-

One: It represents reason. It is the source of all numbers
Two: Represents Man
Three: Represents Woman
Four: It represents justice because it is the product of two equal numbers
Five: Represents wedding. Because it is the sum of 2(Man) and 3(Woman)

Mathematician suggests extra dimensions are time-like

2007-04-18T04:21:00.000-12:00

In a recent study, mathematician George Sparling of the University of Pittsburgh examines a fundamental question pondered since the time of Pythagoras, and still vexing scientists today: what is the nature of space and time? After analyzing different perspectives, Sparling offers an alternative idea: space-time may have six dimensions, with the extra two being time-like.

Read More

Sourse: www.physorg.com

Area Problem

2007-04-18T03:40:00.000-12:00

John has a rectangular lawn of dimensions 50 meter by 40 meter.

It has two roads each 2 meter wide running in the middle of it one parallel to the length and one parallel to the breadth. He wants to tar the road.

Find area of the roads to be tarred.

Hint:

We have two roads to be tarred. One is horizontal and the other is vertical.

So, area to be tarred is the area of the two roads which are rectangular in shape.

Area of a rectangle is length x breadth.

Area of the horizontal road = 2 x 50=100 sq.m

Area of the vertical road = 2 x 40=80 sq.m

So,the total are of the two roads=100+80=180 sq.m

But, the roads are crossing at the middle. It is a square portion of sides 2m by 2m in dimensions and we can see that two times we added that area. So, we should subtract that area from the total area of the two rectangular roads.

Area of the square portion is 2 x 2=4 sq.m

Therefore, the area to be tarred = Total area of the two rectangles - The common area

Ans: 176 sq.m

Temparature Problem

2007-04-17T01:59:00.000-12:00

The average temperature of for Monday, Tuesday and Wednesday was 30ºC.
The average temperature for Tuesday, Wednesday and Thursday was 29º C. The Temperature on Thursday was 32º C.

What was the temperature on Monday?

Hint:

Total temperature on Mon,Tue and Wed = 30ºC*3 = 90ºC
Total temperature on Tue,Wed and Thu = 29ºC*3 = 87ºC

Total temperature on Tue and Wed = Total temperature on Tue, Wed and Thu -Temperature on Thu = 87ºC-32ºC= 55ºC

Now, Temperature on Monday can be calculated as follows

Temperature on Monday = Total temperature on Mon,Tue and Wed-Total temperature on Tue and Wed

Ans: 35ºC

Fundamental Theorem of Arithmetic

2007-04-12T20:09:00.000-12:00

Fundamental Theorem of Arithmetic :

Every composite number can be expressed ( factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

The Fundamental Theorem of Arithmetic says that every composite number can be factorised as a product of primes. Actually it says more. It says that given any composite number it can be factorised as a product of prime numbers in a‘unique’ way, except for the order in which the primes occur. That is, given any composite number there is one and only one way to write it as a product of primes,as long as we are not particular about the order in which the primes occur.

So, for example, we regard 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other possible order in which these primes are written. This fact is also stated in the following form:

The prime factorisation of a natural number is unique, except for the order of its factors.

In general, given a composite number x, we factorise it as x = p1p2 ... pn, where p1, p2,..., pn are primes and written in ascending order

If we combine the same primes, we will get powers of primes.

Once we have decided that the order will be ascending, then the way the number is factorised, is unique.

The Fundamental Theorem of Arithmetic has many applications, both within mathematics and in other fields.

The Gong

2007-04-10T20:04:00.000-12:00

If a clock takes 6 seconds to strike 6. How long the same clock take to strike 12.

Can you find the answer?

Euclid’s Division Lemma

2007-04-09T17:52:00.000-12:00

Euclid's Division Lemma
Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0≤r
This result was perhaps known for a long time, but was first recorded in Book VII of Euclid’s Elements. Euclid’s division algorithm is based on this lemma.
Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b.
Let us see how the algorithm works, through an example first. Suppose we need to find the HCF of the integers 455 and 42.

We start with the larger integer, that is, 455.
Then we use Euclid’s lemma to get 455 = 42 × 10 + 35.
Now consider the divisor 42 and the remainder 35, and apply the division lemma to get 42 = 35 × 1 + 7.
Now consider the divisor 35 and the remainder 7, and apply the division lemma to get 35 = 7 × 5 + 0.

Notice that the remainder has become zero, and we cannot proceed any further. We claim that the HCF of 455 and 42 is the divisor at this stage, i.e., 7. You can easily verify this by listing all the factors of 455 and 42.Why does this method work? It works because of the following result. So, let us state Euclid’s division algorithm clearly.To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ≤rStep 2 : If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

This algorithm works because HCF (c, d) = HCF (d, r) where the symbol HCF (c, d) denotes the HCF of c and d, etc.Euclid’s division algorithm is not only useful for calculating the HCF of very large numbers, but also because it is one of the earliest examples of an algorithm that a computer had been programmed to carry out.

Remarks :
1. Euclid’s division lemma and algorithm are so closely interlinked that people often call former as the division algorithm also.
2. Although Euclid’s Division Algorithm is stated for only positive integers, it can be extended for all integers except zero, i.e., b ≠ 0.

Algorithm & Lemma

2007-04-03T05:33:00.000-12:00

An algorithm is a series of well defined steps
which gives a procedure for solving a type of
problem।

The word algorithm comes from the name
of the 9th century Persian mathematician
al-Khwarizmi. In fact, even the word ‘algebra’
is derived from a book, he wrote, called Hisab
al-jabr w’al-muqabala.

A lemma is a proven statement used for
proving another statement.

Theoretical cloaking device is created

2007-04-03T04:07:00.000-12:00

U.S. scientists, taking a tip from Star Trek, have used nanotechnology to create a theoretical optical "cloaking" device that can make objects invisible.
The Purdue University engineers, following mathematical guidelines devised by British physicists, created the theoretical device that can render objects invisible by guiding light around anything placed inside the "cloak."
The design uses an array of tiny needles radiating outward from a central spoke. The device would bend light around the object being cloaked. Background objects would be visible, but not the object surrounded by the cylindrical array of nano-needles, said Vladimir Shalaev, a professor of electrical and computer engineering.
The design does, however, have a major limitation: It works only for any single wavelength, and not for the entire frequency range of the visible spectrum, Shalaev said.
"But this is a first design step toward creating an optical cloaking device that might work for all wavelengths of visible light," he said.
The research is detailed in a paper appearing this month in the journal Nature Photonics.

Copyright 2007 by United Press International. All Rights Reserved.

Ref: science daily

Biggest number

2007-03-21T05:53:00.000-12:00

Can you find the biggest number you can make with three 3s?

Meaningful learning

2007-03-20T05:16:00.000-12:00

Before we try to understand the concept of meaningful learning, it will be better here to know what learning stands for. according to R.S Wood worth "an activity may be called learning in so far it develops the individual in any way good or bad and makes his environment and experiences different from what it would otherwise have been".

Learning can produce both good and bad developments in the learner. But the learner, hie parents, his teachers and the society in general want the process of learning to lead to good results and healthy outcomes.
Meaningful learning is therefore that learning which is oriented towards good experiences and outcomes. In it there is no place for meaningless and harmful experiences. It must ensure positive results. It is constructive, productive, purposeful and progressive in nature.

Meaningful learning in mathematics can consist of the mathematical experiences of the following nature;-

Which are helpful in mental, emotional and social development.
which have utilitarian, practical and behavioral values.
Which are useful in learning higher and advanced aspects of the subject.
Which are helpful in the proper learning of other subjects and activities of the curriculum.
Which stimulate and maintain interest in the subject.

Applied Mathematics

2007-02-26T05:04:00.000-12:00

Applied mathematics is the application of pure mathematics in the service of a given purpose. It has some direct or practical application to objects and happenings in the material world. It plays a great role in the development of various subjects. Every discovery in science owes much to applied mathematics. Principles of applied mathematics have been useful in the investigation of such phenomenon as heat , sound, light, optics,navigation and astronomy. applied mathematics is a part of mathematics definitely related to or suggested by some tangible situations, though not always intended for practical use. It is the connecting link between pure mathematics on one side, physical, biological, social sciences and technology on the other.

Pure Mathematics

2007-02-20T04:56:00.000-12:00

Pure mathematics involves systematic and deductive reasoning. It treats only theories and principles without regard to their application to concrete things. It is developed on an abstract, self contained basis without any regard to any possible kind of practical applications that may follow. It consists of all those assertions as that if such and such proposition is true of anything,such and such another proportion is true of that thing.

The Chess Board

2007-02-03T03:37:00.000-12:00

We all know that a chess board has 64 squares. This can be completely covered by 32 cardboard rectangles, each cardboard covering just 2 squares.

Supposing we remove 2 squares of the chess board at diagonally opposite corners, can we cover the modified board with 31 rectangles? If it can be done, how can we do it? And if it cannot be done, prove it impossible.

Answer

No.It can not be done.

Each rectangle covers one white square and one black square, because on a chess board the white and black squares are always adjacent.

I think now you gor the reason.

Limits

2007-01-09T07:34:00.000-12:00

The notion of limit is one of the most basic and powerful concepts in all of mathematics. Differentiation and Integration, which comprise the core of study in calculus, are both products of the limit. The concept of limit is the foundation stone of calculus and as such is the basis of all that follows it. It is extremely important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus at the entry level.

You can find some basic results on limits here

A Puzzling Birthday

2007-01-03T03:57:00.000-12:00

I met a girl some days before when I was walking through a park with my children. My children and she became friends and started playing together. I decided to talk to her and during the conversation I asked her age. The smart girl replied, Two days back I was ten years old, and next year I shall be 13. If you know what is taday, you will be able to figure out my age!!!

It took some time for me to calculate her age. What about you?

Solution
The date on which I met the girl was 1st January 2007. The girl's birthday was on 31st December 1995. So The girl was 11 years old on the date I met her.

New Website

2007-01-02T04:06:00.000-12:00

Today I have launched a new math help website.

Its name is www.mathsupporter.com

I will add some basic math topics later.

Please visit my site MATH SUPPORTER.COM

Merry Christmas

2006-12-24T03:25:00.000-12:00

Guess Your Birthday

2006-12-24T03:03:00.000-12:00

Now It is Christmas time. We are celebrating the birthday of Jesus. On this occasion we can have a birthday puzzle.

Take the month number of your birthday; Jan=1, Feb=2......
Multiply that by 5
Then add 6
Then multiply that by 4
Then add9
Then multiply that total by 5
Finally add the day you were born on
From that number , subtract 165

Then you will get the month and the day you were born.

You can ask your friends to do it tell their birth dates and become a hero.

A number Puzzle

2006-12-18T04:57:00.000-12:00

There are two numbers with the difference of 3 between them and the difference of their squares is 51.

Can you find the numbers?

Gold Puzzle

2006-12-18T04:45:00.000-12:00

Which is worth more, a bucket full of half a sovereign gold pieces or an identical bucket full of 1 sovereign gold pieces?

If you have the answer please post in the comments section.

Reccurring Number Magic

2006-12-15T04:51:00.000-12:00

You write down the following 8 digit number on a piece of paper
1 2 3 4 5 6 7 9
Then ask your friend to circle one of the digits. Say 5

You then ask your friend to multiply the 8 digit number by 45, and magically the result ends up being:

1 2 3 4 5 6 7 9
x 4 5
---------------
5 5 5 5 5 5 5 5

with the answer as a raw of the chosen number 5

The Secret!!!

When your friend circles a number, you need to multiply the chosen number by 9 in your head. Then you need to ask your friend to multiply the 8 digit number by the number you have worked out and you magically get the answer as a raw of the chosen number.

Angles

2006-12-12T03:50:00.000-12:00

An angle is the union of two non- collinear rays with a common initial point.

Two rays forming an angle are called the 'arms' of the angle and the common initial point is called the 'vertex' of the angle. sometimes, it will be convenient to refer to angle BAC, simply angle A. However this can not be done if there are more than one angle, with the same vertex A.

Interior of an angle: The interior of an angle BAC, is the set of all points P in its plane, which lie on the same side of line AB as C, and also on the same side of line AC as B.

Exterior of an angle: The exterior of an angle BAC is the set of all points Q in its plane, which do not lie on the angle or in its interior.

Types of Angles
An angle whose measure is 90 degrees is called a right angle.
An angle whose measure is less than 90 degree is called an acute angle.
An angle whose measure is more than 90 degrees is called an obtuse angle.

Division by '0'-The problem is solved

2006-12-09T17:09:00.000-12:00

Schoolchildren from Caversham have become the first to learn a brand new theory that dividing by zero is possible using a new number - 'nullity'. But the suggestion has left many mathematicians cold.
Read More

Some Properties Of Ratios

2006-12-09T05:05:00.000-12:00

1) Let a:b>c:d and c:d be two ratios. Then,

i) a:b > c:d, if ad>bc,

ii) a:b <>

iii) a:b = c:d, if ad=bc

2) A ratio a:b is called a ratio of

i) greater inequality ifa>b,

ii) less inequality if a < b

iii) equality if a=b

3) If the same positive quantity is added to both the terms of a ratio of greater inequality, then the ratio is decreased.

4) If the same positive quantity is added to both the terms of a ratio of less inequality, then the ratio is decreased.

5) If the same positive quantity is subtracted to both the terms of a ratio of greater inequality, then the ratio is increased.

6) If the same positive quantity is added to both the terms of a ratio of greater inequality, then the ratio is increased.

John's Birthday

2006-12-09T04:48:00.000-12:00

John was born in 19ab. The two digit number ab when divided by 2 gives his age on his birthday in 1999. Can you tell me how old was he?

Solution
33 years

Ratio

2006-12-05T03:24:00.000-12:00

The ratio of two quantities of the same kind and in the same units is a comparison by division of the measure of two quantities.
In other words ,the ratio of two quantities of the same kind is the relation between their measures and determines how many times one quantity is greater than or less than the other quantity.
The ratio of a to b is the fraction a/b, and is generally written as a:b.

Example 1: The ratio of $25 to $50 is 25:50 or25/50 or 1:2
Example 2: The ratio of 2m to 80 cm is 200:80 or 200/80 or 5:2
Example 3: There is no ratio between $10 and 5 meter.

Since the ratio of two quantities of the same kind determines how many times one quantity contains other, is an abstract quantity. In other words, ratio has no unit or it is independent of the units used in the quantities compared.

For the ratio a:b, a and b are called terms of the ratio. The former a is called the first term or antecedent and the later b is known as the second term or consequent.

Geometry in Court

2006-12-03T21:16:00.000-12:00

In real life, math can be relevant in the courtroom or come up in legal disputes.

Last year, the Pythagorean theorem was a deciding factor in a case before the New York State Court of Appeals. A man named James Robbins was convicted of selling drugs within 1,000 feet of a school. In the appeal, his lawyers argued that Robbins wasn't actually within the required distance when caught and so should not get the stiffer penalty that school proximity calls for.

The arrest occurred on the corner of Eighth Avenue and 40th Street in Manhattan. The nearest school, Holy Cross, is on 43rd Street between Eighth and Ninth Avenues.

Law enforcement officials applied the Pythagorean theorem to calculate the straight-line distance between the two points. They measured the distance up Eighth Avenue (764 feet) and the distance to the church along 43rd Street (490 feet), using the data to find the length of the hypotenuse, 907.63 feet.

Read More

UW wants to boost math enrollment

2006-12-03T20:58:00.000-12:00

On the surface, math may seem less glamorous than biology or engineering, but the field, which deals with abstract concepts and equations, is integral to all other scientific fields.

As the United States faces a critical shortage of American scientists, the University of Wisconsin-Madison will lead an effort to attract more students to mathematics, ultimately providing the sciences with a stronger, smarter workforce.

How Can We Draw The Graphs Of Trigonometric Functions?

2006-12-02T23:11:00.000-12:00

We have four ways to alter the graph of a function: vertical translation, vertical scaling, horizontal translation, and horizontal scaling. Note that some or all four of these can be applied to a function at once in the sense that you may have to deal with a function of the form y = AsinB(x - C) + D. You should simply work these graphs out one step at a time, concerning yourself first with the affect of B and C, then the affect of the presence of A, and finally the affect of D.

Example:
Draw the graph of y =2 sin(x/2+π/6)+3

Solution
Let’s change this is of the form y = AsinB(x - C) + D
So y =2 sin(x/2+π/6)+3
can be written as y =2 sin ½ [x-(-π/3)]+3
Step 1: Draw sinx

B= ½ .so period is 4pi. That means it completes one cycle after 4pi.so stretch the graph horizontally so that its period is 4pi.

Next, we see that C=-pi/3. This will horizontally shift our graph pi/3units to the left. Thus, we now have

Now A=2, which will vertically scale our graph by a factor of 2. Thus, at this point, our range should now be [-2,2]

Finally, we must shift our graph vertically 3 units due to the presence of D=3. Hence, our range will now move to [1,5]

We can adopt the same method for drawing the graphs of other trigonometric functions. But remember first you should convert the given functions of the form

y = A sinB(x - C) + D

y = A cosB(x - C) + D

y = A tanB(x - C) + D

or accordingly, depending upon which of the trigonometric function is given.

On The Line

2006-11-28T22:38:00.000-12:00

Today I have a very simple puzzle for you.

It is a small town railway station and there are 20 stations on that line. At each part of the 2 stations the passengers can get tickets for any of the other 19 stations.

How many different kinds of tickets do you think the booking clerk has to keep?

Solution
The answer is simple. 19x20=380

Phase Shift Of Trigonometric Functions

2006-11-27T04:48:00.000-12:00

The phase shift is the horizontal shift away from the standard graph of the trigonometric function.

In y = AsinB(x - C) + D,C is the phase shift.
For cosine function y = AcosB(x - C) + D and tangent function y = AtanB(x - C) + D also C is the phase shift.

If the phase shift is positive, there has been a horizontal shift to the right and if it is negative, there has been a horizontal shift to the left.

In reading off the phase shift, make sure you have the function in the form above.

For example, the phase shift of y = sin(2x - π /2) is not π/2. Rewrite the expression for the function in the required form to get
y = sin2(x - π/4).
Now we see the correct phase shift, is π/4.

In the graph we can see graph of sin(2x - π /2) is shifted to π/4 units to the right.

Period Of Trigonometric Functions

2006-11-26T03:53:00.000-12:00

We know trigonometric functions are periodic functions. Sinx and cos x are periodic functions with period 2π or 360° . But tan and cot remain unchanged when x is increased by π or 180° .
So, they are periodic functions with period π.

The general form of the sine function is y = A sinB(x - C) + D
Here, the period is 2π/IBI.

The general form of the cosine function is y = A cosB(x - C) + D
We know cosine functions are identical to the sine functions. So, the period of cosine function is also 2π/IBI.
But for a tangent function. It is π instead of 2π because the period of tan x is π.
If the general form of a tangent function is y = A tanB(x - C)+ D,
its period is π/IBI

There fore, the value of B is the key factor in determining the period of tangent functions. Change in its value changes horizontal stretching. When drawing the graph we have to “stretch” or ;“shrink” the graph horizontally by a factor of B.
Also, the period is unchanged by vertical scaling or shifting or by horizontal shifting.

Hello Again

2006-11-25T18:52:00.000-12:00

I couldn't post any articles last week. My daughter was not well. She is in her kinder garden classes. This year she already missed lots of classes due to various reasons. She will be five in next January. She is under aged when compared to other children. The average age of children in her class is above 5 ½ years. She is just above average in her studies. In math she is O.K. But she has to improve in language studies. I do not force her to study at all. If she likes, she study, otherwise not. Now I am thinking about repeating her in the same class next year also. But her teacher says there is no need to repeat her. So, I am in confusion.

The Mystery Of The Missing Dollar

2006-11-18T05:31:00.000-12:00

Kathy and Jane were selling sweets in the market place. Kathy at 3 for a dollar and Jane at 2 for a dollar. One day both of them were obliged to return home when each had thirty sweets unsold. They put together the two lots of sweets and and handing them over to a friend, asked her to sell them at 5 for two dollars. According to their calculation, after all, 3 for 1 dollar and 2 for 1 dollar was exactly same as 5 for 2 dollars.

Now they were expecting to get 25 dollars for the sweets as they would have got, if sold separately. But much do their surprise they got only 24 dollars for the entire lot.

Now where did the 1 dollar go? Their friend was a cheat?

Solution
There isn't really any mystery. While the two ways of selling are only identical, when the number of sweets sold at 3 for 1 dollar and 2 for 1 dollar is in proportion of 3:2. So , if Kathy had sold 36 sweets and Jane 24, they would have fetched 24 dollars (12 dollars each) , immaterial of, whatever sold separately or at 5 for 2 dollars. But if they had the same number of sweets which led to loss of 1 dollar when sold together, in every 60 sweets. So if they had 60 each (120 altogether) , there would be a loss of 2 dollars and so on.

In the case of 60, the missing 1 dollar arises from the fact that Kathy gains 2 dollars and Jane losses 3 dollars(If they share $12 each).

Kathy receives $9.5 and Jane $14.5, so that each loses $.50 in the transaction.

Periodic Functions

2006-11-16T06:06:00.000-12:00

Periodic functions are functions that repeat its values over and over, after some definite period or cycle on a specific period. This can be expressed mathematically that A function f is said to be periodic if there exists a real T>0 such that f (x+T) = f(x) for all x.

The fundamental period of a function is the length of a smallest continuous portion of the domain over which the function completes a cycle. That is, it's the smallest length of domain that if you took the function over that length and made an infinite number of copies of it, and laid them end to end, you would have the original function.

If a function is periodic, then the smallest t>0 ,if it exists such that f (x+t) = f(x) for all x, is called the fundamental period of the function.

The trigonometric functions sine and cosine are common periodic functions, with period 2π.
ie. sin (x+2π)= sin x , cos(x+2π)=cos x

But tan and cot remain unchanged when x is increased by pi.
ie. tan(x+π)=tan x, cot(x+π)= cot x
So, they are periodic functions with period π .

An aperiodic function (non-periodic function) is one that has no such period

Amplitude Of Trigonometric Functions

2006-11-15T06:33:00.000-12:00

We know basic trigonometric functions are sinx, cosx, tanx.
These functions are periodic functions.( The period is the shortest interval over which the function runs through one complete cycle of its graph.)
Sinx and cos x are periodic functions with period 2π.
But tan and cot remain unchanged when x is increased by pi.So, they are periodic functions with period π.

Amplitude

See the graph of sinx . We know its range is [-1, 1].

It is clear from the graph that its amplitude is 1

When we draw the graph of 2 sin x, we can see that its range is [-2, 2]
The multiplication factor 2 has “stretched'' the graph of sinx vertically by a factor of 2, while retaining the same x-intercepts.

This vertical scaling factor is known as the amplitude of the function.

The amplitude of the sine and cosine functions is half the vertical distance between its minimum value and its maximum value.

For a function A sin x, its y values range from –A to +A
So amplitude is 1/2 of [A-(-A) ]=A

The vertical shifts do alter the greatest and least values that the function attains but do not alter the amplitude.

We can verify this by taking the examples 2sin x and 2sinx+3 For 2sinx,the minimum and maximum values are -2 and 2 .

Amplitude is ½ . 2-(-2)=2
For 2sinx +3, minimum and maximum values are 1 and 5 .

Amplitude is ½ (5-1)=2

In general , the amplitude of

y = A sinB(x - C) + D and
y = A cosB(x - C) + D ,where B is a non-zero real number, is IAI

The tangent function has no amplitude, because the tangent function has no minimum or maximum value.its range is (-infinity, infinity)

Finding The Way To Freedom

2006-11-14T02:45:00.000-12:00

Today let's relax a bit.

A peasant is convicted. He gets the death penalty. The judge allows him to say a last sentence in order to determine the way the penalty will be carried out. If the peasant lies, he will be hanged, if he speaks the truth he will be beheaded. The peasant speaks a last sentence and to everybody surprise some minutes later he is set free because the judge cannot determine his penalty.

What did the peasant said?

The peasant said: "I shall be hanged!"

If the peasant was lying, he would be hanged. But that's what the peasant was saying. So he speaks the truth. But if he speaks the truth, he would be beheaded, so then he was not speaking the truth. So it is impossible for the judge to determine whether the peasant speaks the truth or not. So therefore the judge cannot determine the penalty and sets the peasant free.

Don't go away....... I need your help.......

My 2 year old son put a coin in an empty bottle and insert a cork into the neck of the bottle.
The bottle has a very nice shape. So I don't want to break it.
How could I remove the coin without taking the cork out ?

Answer

Don't worry for a solution. He himself found a way and took it out. He pulled the cork into the bottle and the coin came out.

Rational Root(Zero) Theorem

2006-11-08T19:41:00.000-12:00

Roots of a Polynomial

A root or zero of a function is a number that, when plugged in for the variable, makes the function equal to zero. Thus, the roots of a polynomial P(x) are values of x such that P(x) = 0.

The Rational Zeros (Roots) Theorem

We can use the Rational Zeros Theorem to find all the rational zeros of a polynomial.

Arrange the polynomial in descending order

Write down all the factors of the constant term. These are all the possible values of p.

Write down all the factors of the leading coefficient (The coefficient of the first term of a polynomialwhen writing in descending order.)
These are all the possible values of q.

Write down all the possible values of p/q . Remember that since factors can be negative p/q, and - (p/q)must both be included. Simplify each value and cross out any duplicates.

Use synthetic division or remainder theorem to determine the values of p/q for which P(p/q) = 0. These are all the rational roots of P(x).

Example:

Find all the possible rational roots of

Mysterious Number 1089

2006-11-08T04:17:00.000-12:00

Think of a 3 digit number where all the digits are different.
Reverse the digits
Subtract the smaller number from the larger number
Reverse the digits again
Add the numbers in steps 3 and 4

Whatever 3 digit number you choose the number will always be 1089!

Here is an example

543
345
543 - 345 = 198
891
891 +198 =1089

Online Mathematics Textbooks

2006-11-06T04:27:00.000-12:00

I have found a site that offers many free math ebooks. Most of them are in PDF format. If you don't have a PDF reader download it here.
Here is the link to free Online Mathematics Textbooks.

Some useful formulea from Analytical Geometry

2006-11-06T03:26:00.000-12:00

Approach To School Affects How Girls Compare With Boys In Math

2006-11-02T02:13:00.000-12:00

More women are pursuing higher education and doctoral degrees than ever before, but women still are rare in the math-oriented professions. Yet, researchers say, girls perform just as well as boys on achievement tests and tend to earn better grades in math than do boys during the earlier school years.

A new study in the journal Developmental Psychology indicates that how girls and boys approach their schooling underlies the differences in math grades. It also suggests that although the girls' approach to school may give them an edge in the grades they earn in math, it may not buy them much when it comes to math scores on achievement tests because girls are not more confident than the boys about their skills in math.

The study examined 518 boys and girls as they went through fifth and seventh grades in three primarily white, middle- to upper-class school districts in Illinois. Using children's reports, researchers looked at how the children approached their schoolwork, including their goals and in-class behavior. The children also reported on how confident they were about their ability to do well in math. Researchers also reviewed the young students' grades and achievement test scores in math.

In the classroom, girls outperformed boys at both time points of the study, with the girls' grades rising over time, while the boys' grades remained the same, said Eva Pomerantz, a professor of psychology at the University of Illinois at Urbana-Champaign. The study was part of the doctoral work done by Gwen A. Kenney-Benson, who now is at Allegheny College in Pennsylvania.

Concerns with outperforming others and engaging in disruptive behaviors while in the classroom, both of which characterized boys more than girls, were tied to lower grades in math by the researchers.

"This was due in large part to the fact that such competitive and disruptive leanings were associated with decrements in learning strategies such as preparing for tests, seeking help, and persisting even when things were challenging that led to higher grades," Pomerantz said.

Girls consistently used these learning strategies more than the boys did, the researchers found. It appears that, in contrast to boys, girls are more concerned with learning than with outperforming their classmates. They also engaged in less disruptive classroom behavior. As a consequence, girls used more focused learning strategies, giving them an edge over boys in terms of grades, Pomerantz said.

The researchers noted that the differences in grades between girls and boys disappeared once children's concerns with learning versus outperforming others, engagement in disruptive behavior and learning strategies were taken into account.

At achievement test time, however, girls' lost their advantage in math; their scores were the same as those of boys. After examining various factors, what stood out, Pomerantz said, was children's confidence in their ability to do well in math.

In the classroom, she said, children may be less likely to feel that they will be judged based on their gender, believing instead that their own behavior, knowledge and effort will determine their grades. Thus, she added, the girls' approach to schoolwork will pay off in the classroom, while the boys' approach will not. It also could be, the researchers theorized, that higher grades given to the girls reflect rewards from their teachers for better behavior.

During achievement tests, the researchers suggest, the environment changes. Removed for girls is the familiarity of the classroom, which is replaced with uncertainty and increased stress. In such a situation, confidence mattered more than in the classroom.

Because confidence was found to be a predictor of scores on math achievement tests, Pomerantz said, girls may not have kept the edge they had while in the classroom because confidence levels did not differ along gender lines.

It may be that while many girls are going on into higher education, they continue to steer away from "stereotypically masculine fields, such as science and engineering" because the "more competitive environment of these fields is not a good fit with how girls approach school," the researchers wrote.

"Consequently, even if the topic is of interest," Pomerantz said, "the girls' more learning-oriented approach may not match the work environment, where the atmosphere in these fields may provide a better fit to boys' more competitive approach."

Courtesy: Science News

Distance Formula

2006-11-01T04:59:00.000-12:00

Distance formula is useful to find the distance between any two given points in a cartetian plane.

Remainder Theorem and Factor Theorem

2006-10-30T05:05:00.000-12:00

Remainder Theorem
Let p(x) be any polynomial of degree n>0 ,and a any real number. If p(x) is divided by ( x-a), then the remainder is p(a).

The Remainder Theorem can be proved as follows.

Proof
Let us suppose when p(x) is divided by ( x-a), the quotient is q(x) and remainder is r(x).

So we have,
p(x)=(x-a) q(x)+r(x), where r(x)=0 or degree of r(x)< degree of x-a.
Since degree of ( x-a) is1, either r(x)=0 or degree of r(x)=0
So r(x) must be a constant,say r.

Thus for all values of x,
p(x)=(x-a) q(x)+r ........(1), where r is a constant.

In particular, when x=a,
p(a)=0.q(x)+r
=0+r
=r
Hence the theorem.

Factor Theorem
Let p(x) be a polynomial of degree n>0. If p(a)=0 for a real number a, then (x-a) is a factor of
p(x). Conversely, if (x-a) is factor of p(x), then p(a)=0.

Proof

First part:
Let p(a)=0
Then by remander theorem, r=0
So equation (1) becomes
p(x)=(x-a) q(x)
==>(x-a) is a factor of p(x).

Second Part:
By remainder theorem,

p(x)=(x-a) q(x)+r
ie. p(x)=(x-a) q(x)+p(a)
Since (x-a) is a factor, p(a) must be zero.
This proves the theorem.

Example
Find the remainder when p(x)= x^2 +3x+1 is divided by x+1.
Determine whether( x-2) is a factor of p(x) or not.

Solution
x+1= x-(-1)
[Always check whether the divisor is in the form of (x-a)or not. Otherwise rewrite that in the form of(x-a)]

So, here a=-1
There fore by remainder theorem, the required remainder is
p(a)= p(-1)
=(-1)^2+3(-1)+1
=1-3+1
=-1

We know, by factor theorem,

if (x-2) is a factor of p(x), then p(2) must be zero.

Here p(2)=2^2+3(2)+1
=4+6+1
=11 which is not equal to zero.
So (x-2) is not a factor of p(x).

An Interesting E-Mail

2006-10-29T01:22:00.000-12:00

Today I received an interesting E-Mail. Here is the Mail and the reply I have sent.

Plot the point with coordinates (3, –5).

Graph 2x – y = 4.

Graph using the intercept method: 2x + y = 4.

Write the equation of the line with slope 4 and y-intercept (0, –5). Then
graph the line.

Math For You to P

Hi P
Thanks for writing to me
I think you have four questions. So my fee willbe $6. If that is O.K for you, I will send the detailed solutions for you.

P to me
You don't have to send me a solution for my problems because the site said
that it was free.

Math For You to P
No. Where did you see that? I have clearly mentioned about my service at the home page.Here also you can see more details. http://math-homeworkhelp.blogspot.com/2006/07/i-dont-do-math-homework-for-them.html.Why did you send the questions then?

P to me
Like I mentioned earlier the site said something about getting free help
witrh your math. I got the site from another math student so maybe I
clicked on the wrong site. That is why I posted the questions. Once again
you can disregard the questions and I will found help elsewhere. Sorry if
I wasted your time.

Math For You to P
No P,
You came from MSN for the keyword 'Math Genius'.(http://search.msn.com/results.aspx?q=math ) Not even for math help.So you are not a targeted visitor for me.So it do not worry me.(otherwise also no problem.I have been getting enough clients.) You are from Bastrop,Louisiana, U.S.You go to anywhere for help.(are you really a student?)That is not my topic. But I have a humble request : Don't lie.I have the detailed track record of your visit.
Don't worry.Take a deep breath and Relax.

P to me
I think I misread the article it said free lessons. Like I say disregard
the questions because I already have the solution.

Math For You to P
That is O.K. No problem.

Train Problem

2006-10-27T06:26:00.000-12:00

Two trains start from two different stations A and B. One is going from station A to station B and another from station B to station A. If first train takes 8 hours to complete the journey and the second one takes 12 hours, when do the two trains cross each other?

Answer
The first and second train travel 1/8 and 1/12 respectively of the total distance of AB in one hour . So they cover 1/8 +1/12=5/24 of the distance AB in one hour. That means they will cross each other after 24/5 (4 hours 48 minutes) hours.

Points and Lines: Incidence Properties

2006-10-24T02:56:00.000-12:00

Given a point l and a point P. If P is an element of l,
then we say that P lies on l, or
P is incident on l, or
l passes through P.

Now we can try to find answers to the following questions on the basis of our experience.

Given a point P, is there a line that passes through P? how many such lines are there?
Given two distinct points A and B, is there a line that passes through both a and B? How many such lines are there?
Given a line l, is there a point that lies on it? How many such points are there?
Given two distinct lines l and m, is there a point that lies on both l and m? how many such points are there?

Based on the results of the above questions we can arrive at the following conclusions. These conclusions have to be taken as axioms.

Incidence Axiom 1: A line contains infinitely many points.

Incidence Axiom 2: Through a given point, there pass infinitely many lines.

Incidence Axiom 3: Given two distinct points A and B, there is one and only one line that contains both the points.

According to the third axiom, any two distinct points of the plane determine a line uniquely and completely.

Definition: Three or more than three points are said to be collinear, if there is a line which contains them all.

Definition: Three or more than three lines are said to be concurrent if there is a point which lies them all.

Age Of Demochares

2006-10-20T08:44:00.000-12:00

Yesterday I couldn't write. Today I hoped to write about basic geometric concepts. But today also I am very busy and I haven't enough time to write a long topic.

So we can have a small puzzle today. This is an ancient problem dating back to about 310 A.D.
Demochares had one-fourth of his life as a boy, one-fifth as a youth , one-third as a man, and has spent 13 years in his dotage. How old was Demochares?

Another problem of this kind:
A stone weights one Kilogram and half of its weight. What is the actual weight of the stone?

Solutions:
First Question
Let x be Demochares's age.
Then, x/4+x/5+x/3+13=x
ie 47x/ 60+13=60
==>47x/60=47
==>x=60
So, Demochares was 60 years.

Second Question
2 kilogram

Basic Concepts of Geometry

2006-10-18T03:33:00.000-12:00

It is very important for a math student to learn the basic concepts in mathematics. It is a common fact that most children find math is very hard and in particular Geometry. The main reason behind this is that they don't have the basics in maths.
and in Geometry the concepts are more abstract. If the students get some good basic help in math, I am sure most of them do better in maths. So let us learn some basics of Geometry today.
There are three basic concepts of geometry. These are "point", "line" and "plane". I am not attempting to define them as it is not possible to define them precisely. We can however, have a good idea of these three by considering examples. A fine dot made by a sharp pencil on a sheet of paper, resembles a geometrical point very closely. The sharper the pencil, the closer is the dot to the concept of a point.

The surface of a sheet of paper or the surface of a smooth table are examples of plane. But these surfaces limited in extent. The geometrical plane extends endlessly in all directions.
A straight line, drawn on a sheet of paper with a sharp pencil, is a close example of a geometrical straight line. A geometrical line is a set of points and extends endlessly in both the directions. To emphasize this we use two arrowheads.
It is impossible to find exact example for point,line and plane in nature. The geometrical point, line and plane are ideal concepts. but for practical purpose it is enough to deal with close examples.
So, a plane is a set of points, line is a subset of plane. Moreover all the other figures in geometry are sets of points. But they are not just set of points. They are special set of points possessing some properties.

Notation
We use capital letters such as A, B, C, P, Q, R, X, Y, Z etc. to denote points.
We use small letters (lower case) such as l,m,n,p,q,r etc. to denote lines.

What is the peculiarity of 1729?

2006-10-17T07:48:00.000-12:00

What is the peculiarity of 1729?

This number is known in the name of the famous mathematician Ramanujan.

Answer

This is the one and only one number which can be written as the sum of the cubes of two numbers in two different ways.

10^3 + 9^3 =1729
12^3 + 1^3 = 1729

Math Topics Offered

2006-10-16T03:43:00.000-12:00

I can help you in the following topics of Mathematics
You can send in me your math problems at getmath@gmail.com

All math solutions will be clearly demonstrated.

Algebra

Middle school and high school algebra
Functions
Linear Equations, Inequalities.
Quadratic Equations
Simultaneous System of Equations
Arithmetic and Geometric Progressions.
Sequences and Series
Binomial Theorem
Probability, Permutations, Combinations.
Logarithms
Matrices and Determinants

Trigonometry

Trigonometric Identities.
Trigonometric Functions
Solutions of Triangles
Heights and Distance Problems

Geometry

Middle school and High school Geometry
Prisms
Similar triangles,
Congruent Triangles

Analytic Geometry

The Straight Line.
The Circle.
Conic Sections (Parabola,Ellipse,Hyperbola)

Statistics

Probability
Mean Deviation
Standard Deviation

Calculus

Functions, Limits and Continuity.
Differentiation
Integration
Definite Integral
Definite Integral Applications (Areas and Volumes)

Around The Equator.

2006-10-16T02:37:00.000-12:00

Today again I have a puzzle. This one is not a typical math puzzle. You should use your common sense to answer this simple puzzle. Here we go......

Two identical trains, at the equator start travelling round the world in opposite directions. They start together, run at the same speed and are on different tracks.
Which train will wear out its wheel treads first?

You can post your answers in the comments section.

Update
The train travelling against the spin of the earth will wear its wheel more quickly, as the centrifugal force is less in this train.

Online Math Tutoring via E-Mail

2006-10-14T06:30:00.000-12:00

I offer math tutoring via email.This is a great option for students looking for an online tutor, or students who don't necessarily need to sit down with a tutor for a long time. This can also be a good option for homeschooling students and those working in independent study situations.

For a long session, I will likely ask you to email me a sample or two of the types of questions you have, so that I can be sure that it looks like something I can help you with effectively and I can calculate my fee.Once you agreed to puchase my service,you can pay me through my Paypal account getmath@gmail.com. You do not need a Paypal account in order to use PayPal, as long as you have a credit card. (If you are under 13 years, you can ask your parents to open an account.)

IMPORTANT NOTE
It is highly recommended that, when you e-mail your questions to me, you include the area of study , the topic of study and your grade level.
Please feel free to describe all the difficulties you've been having and what you hope to achieve with tutoring.

A Matter of Time

2006-10-13T03:21:00.000-12:00

Fifty minutes ago if it was four times as many minutes past 3 O' clock, how many minutes is it to 6 O' clock?

Got the answer?
Please feel free to put it in the comments section.

Wait for the correct answer till tomorrow.

Update
The answer was posted in the comments section.

History of Analytical Geometry

2006-10-13T02:54:00.000-12:00

We know Geometry is one of the most ancient branches of mathematics. The big step forward in geometry after the Greeks, was the development of a new method called Co-ordinate geometry or Analytical geometry. Modern analytical geometry is also called "Cartesian" after the name of Rene Decartes (1596-1665). But the fundamental principles and methods were already discovered by Pierre de Fermat(1601-1665). Unfortunately, Fermats treatise on the subject entitled "Ad locus planos et so lidos Isagoge"(Introduction to plane and solid loci) was published only published posthumously in 1679. So Decartes came to be regarded as the unique inventor of the analytical geometry.
In co-ordinate geometry, we enlist the services of Algebra in aid of Geometry.

The Peculiar Number

2006-10-12T09:35:00.000-12:00

There is a number which is very peculiar. This number is three times the sum of its digits. Can you find the number?
Answer
It is 27
2+7=9
3.9=27

Area of a Triangle

2006-10-11T01:39:00.000-12:00

We know the formula for finding the area of a triangle is 1/2 bh, where b is the base and h is the height of the altitude from the opposite vertex to the base.

Another important formula is given below. We can use this formula when two sides and the included angle are given. We can derive Hero's formula from this.

The area of triangle is given by

Similarly other results can also prove.

The Missing Dollar

2006-10-10T05:05:00.000-12:00

Today I have a very interesting puzzle for you. Some of you might have heard about this before. Do you know the answer? Here is the puzzle once again.

Three friends went to a restaurant for lunch. When the bill arrived, it came exactly $30. So each person put in $10 towards the bill. The waiter took the money but the manager pointed out that the customers had been overcharged by $5. The dishonest waiter took five $1 coins from the till and kept 2 for himself and gave back the other 3 back to the customers.

Each of the friends had paid $10 minus the $1 they got back for the meal. That is each paid $9 which in total is $27 for all the 3 friends. The waiter had kept $2 making a total of $27 + $2 = $29.

So what happened to the other $1?

Answer

There is no missing dollar here. The friends spent $ 27 together. From that $25 they paid for the bill and the remaining $2 had taken by the waiter. But I have one question for you -Who told you to add $2 to $27?

Projection Formulea

2006-10-09T05:18:00.000-12:00

In a triangle ABC,

a = b cosC + c cosB
b = c coaA + a cosC
c = a cosB + b cosA

These are known as projection formulea.

( To know the the parameters refer here. )

Proof
Let ABC be any of the triangles in the above figures

In fig (i) we have
a= BC = BD+DC.................(a)
But BD/DA= cosB andDC/CA=cosC
==>BD=c cosB
and
DC=b cosC
Putting these values in (a)
a=c cosB+ bcosC

In fig(ii) BC=BD-CD ........................... (b)
Here CD/CA=cos(180-C)=-cosC
==>CD=-bcosC
Also,BD=ccosB
Putting these values in (b)
a= c cosB-(-b cosC)
=c cosB+ b cosC

Other results can also be proved.

Similarly the results can also be proved for Fig(iii) also.

Example:

In any triangle ABC, prove that
(b +c )cos A+(c +a) cos B+ (a + b )cos C = a+ b+ c

Answer

Formula required: Projection Formula

In any triangle ABC, a = b cos C+c cos B

We have L.H.S=(b +c )cos A+(c +a) cos B+ (a + b )cos C

=b cos A+c cos A+c cos B+a cos B+ a cos C+ b cos C

=(b cos C+c cos B) + (a cos C+ c cos A) +(a cos B+b cos A) [ using projection formula]

=a+b+c

=R.H.S

History of Trigonometry

2006-10-07T07:07:00.000-12:00

The study of Trigonometry was first started in India. The ancient Indian mathematicians Aryabhatta (A.D 476),Bhaskara I(A.D 600), Bhaskara(A.D 1114)and Brahmagupta(A.D 598) got important results. All this knowledge first went from India to Middle East and from there to Europe. The Greeks had also started the study of trigonometry but their approach was so clumsy that when the Indian approach became known, it was immediately adopted throughout the world.
In India, the predecessor of the modern trigonometric function, known as the sine of an angle, and the introduction of the sine function represents the main contribution of the sidhantas to the history of mathematics.
Baskara I gave formula to find the values of sine function for angles more than 90 degree.
The name of Thales (A.D 600) is associated with height and distances problems. He is credited with the determination of the height of pyramid in Egypt by measuring shadows of the pyramid using similarity property.

Value of a Function

2006-10-06T17:21:00.000-12:00

Let f:A-->B be a function. Let y0 be an element in B. then y0 is called a value of f provided there is some element, x0 in A, such that y0 = f(x0); that is, y0 is a value of
the function f if it corresponds, with respect to the rule of f, to some x0 in the set A = Dom(f).

Example
Find the value of the following function when x=-2

Find the Domain and Range of this function.

Range and Domain of Functions

2006-10-04T00:38:00.000-12:00

Yesterday we have learned what is a function.
Today let's discuss about range and domain of a function.

Answers

Try to do more problems from your text.

Functions

2006-10-03T02:46:00.000-12:00

Today we can discuss a topic from functions.

Definitions:

Function
Let A anb B be two non empty sets. A function "f" from a set A to a set B is a rule so that to each element x in A there corresponds exactly one element y in B, under f ,then we say that f is a functin from A to B and write
f:A -> B

y is called the image of x under f and is denoted by f(x). x and y are respectively called the independent variable and the dependent variable. We also say that y is a function of x and write
y=f(x)

Examples:

In the family of circles, the area A of the circle is a function of radius r of the circle.
Here radius r is the independent variable and area A is the dependent variable
The speed of a chemical reaction increases 2 times with the addition of every 5 milligrams of a catalyst. Here the amount of catalyst is the independent variable and speed of the chemical reaction is the dependent variable.

Parents can help their children in learning Math effectively.

2006-10-02T04:05:00.000-12:00

Mathematics is a difficult subject for many students. Children struggle with math because they don't enjoy it or they don't see how it is relevant to their future.

As a parent, you should encourage your student to spend time learning mathematics and pointing out how you use math in your everyday life - at home or at work. Even though math is difficult and requires a good deal of study time, parents should maintain a high level of expectations for their children to ensure success. Studies have shown that children perform better in subjects that interest them.

Generally I have seen many people who proudly say "I hate math". When hearing this young children also think it is some thing great that if they do bad in maths.So unknowingly they began to hate math.So, don't tell your child "math is hard" because they will believe you.Inspire confidence in them.

Be practical. Find something your child likes and relate it to math.

When doing problems,take it step-by-step. Do not get overwhelmed by the entire problem at once; break it down into small manageable pieces.

Encourage your child to be creative when problem solving. There is always more than one way to skin a cat and math isn't that different. Let your child explore alternative viewpoints - the teacher's way is not the only way.

How to send me your math problems?

2006-10-01T17:13:00.000-12:00

My readers are asking me, how they can type math symbols when sending questions. You can use any of the following methods.

For those who are having a scanner. You can write your questions in a paper and scan it and save it as a JPGE file. Attach this file with your E-mail and send it to me at: getmath@gmail.com
If you don't have a scanner, you can send it as a word file. For typing Math symbols you can use Equation Editor. If you are using MS word it is already there. In the tool bar, click insert. Now you can see object option. Click object. In the 'create new' option, select Microsoft equation. With this you can type the Math symbols you want. Save your question as a .doc file and attach it with your E-mail and send it to me at: getmath@gmail.com.

I have mentioned about my service in an earlier post . Please read that before sending the questions. Read that here

If you have Math Type you can use that also.

How Many Faces

2006-09-27T05:16:00.000-12:00

There are 11 faces in the picture. Can you find them all?
For a clearer view you can click the picture and download it. Or press F11.

Normal people find 5 or 6.If you find 8 you have a good sense of observation.
(Of course observation is very essential for learning Mathematics)

If you find 9, your sence of observation is above average.

If you find 10,you are a very good observer.

If you find 11, you are an extreamly good observer.

Some Math Jokes

2006-09-26T04:51:00.000-12:00

You may be amazed to hear that people have bothered to dream of jokes with a mathematical theme. Here is a few to keep you going.

• ‘Old mathematicians never die’-they just lose their functions.
• ‘Mathematicians don’t sin-they sine, and always have a nice tan but are forever going off on a tangent.
• Math teacher mum talking to her son-…..’ if I have told you n times I have told you n+1 times’
• What do you call a tea pot of boiling water on top of Mount Everest- H ypotenuse( high pot in use) >• WHY IS 6 AFRAID OF 7? -Because 789 (7 ate 9)
• What is the sine of 40?- Over the hill.
• What did one math book say to another book?- I have lots of problems.
• SON: Dad, will you do maths homework for me tonight ?
FATHER:No son, it wouldn’t be right.
SON: Well, you can try.

An Accident

2006-09-25T07:25:00.000-12:00

Today we can have a number problem.O.K?

On his way to school, Jim is knocked down by a car.

He noticed the number of the car.

• The digit in thousand’s place is 4 times the digit in unit place
• The digit in ten’s place is half of the digit in thousand’s place
• The digit in hundred’s place is 3 less than the digit in ten’s place
• Two is in the unit place

Find the number.

I think it is very easy to find out.

Ans: 8142

The Law of Cosines

2006-09-23T23:01:00.000-12:00

The Law of Cosines

In any triangle ABC ,

If we know the three sides of any triangle, we can find the angles of the triangles using this formula.

Example:

In a triangle ABC, given a=25, b=52, c=63. Find A

Ans.

We have

Solutions of Triangle

2006-09-20T02:58:00.000-12:00

I think yesterday you had a big laugh after reading math genius. So today we are fresh and can have some lessons from Trigonometry.

We know the three sides and three angles of a triangle are called its parts. The process of finding the unknown parts of a triangle from its known parts is known as solution of a triangle. This has many applications in surveying, navigation, astronomy and other sciences.
In a Triangle ABC, The angles are denoted by A,B,C and the lengths of the corresponding opposite sides BC, CA and AB by a,b, and c respectively.

The Law of Sines

Using this formula we can find solutions of a triangle.If any three parts are given, we can find the other three parts.

Homework Help from India

2006-09-19T04:19:00.000-12:00

Many and many people from USA and UK are looking India for getting tutoring.You will get a tutor for very low fee. And quality tutoring is guaranteed.Read more

Math Genius

2006-09-19T03:34:00.000-12:00

Yesterday I visited a blog. I found this post. I couldn't stop laughing.
Click here to view that post.

The Magic Triangle?

2006-09-18T04:59:00.000-12:00

See the picture below

Can you tell me the answer?

It's an optical illusion.The hypotenuse of both triangles seem to be straight lines but they're not.
I think the answer lies in the fact that neither shape is actually a triangle, they are quadrilaterals. They have four sides.

The side that looks like it is the hypotenuse is actually two sides, the corner being where the green and red triangle.
I have searched the web and find an answer here.

If you have any good explanation about this please let me know.

10 Trees Problem

2006-09-11T17:21:00.000-12:00

There are 10 trees spaced out equally along my street with one at each end. If I run at a steady speed from one end of the street, I can reach the 5 th tree in just 5 seconds.

How long would it take me to run down the whole street at the same pase?
( No, it is not 10 seconds)

Did you get the answer?

When I reach the 5 th tree I cover 4 trees.
So,time taken to cover 4 trees is 5 seconds.
So, time taken to cover 1 tree is 5/4 seconds.
When I reach the 10 th tree I cover 9 trees.
So, time taken to cover 9 trees( and this is same as the time taken to run down the whole street) is 9 multiplies 5/4
ie 45/4
or
11.25 seconds

Mobius Strip

2006-08-20T03:59:00.000-12:00

A sphere has two surfaces. A sheet of paper has two surfaces. The first one sided surface was discovered by A.F Mobius and bears his name Mobius strip.

A model can be obtained by taking a paper strip ,giving it a half twist(180 degree), and then attaching the two ends. If you start drawing a line down its center along a line parallel to its edge of the strip, you'll end up right back where you started. You will never cross an edge. To compare this you can make a cylinder by attaching the ends of the paper without any twist and draw a line like you did before. See the difference.In Euclidean space there are two types of mobius strips depending upon the direction(clockwise or anticlockwise) of the twist.

This illustration shows interlocked gears along the length of a mobius strip.

The mobius strip has several properties. Try cutting the strip along the line you drew. Instead of getting two separate strips you get one strip with two half twists. Then draw a center line around the resulting band and cut along it. You will get two strips wound around each other.

Paradox

2006-08-19T02:49:00.000-12:00

In my last post I mentioned about paradox. Let's see what a paradox is. A paradox is a situation in which something seems both true and false. in other words a paradox is a sentence or a sentiment that is seemingly contradict or opposed to common sense and is yet perhaps true in fact.

Examples:

When we increase our knowledge, we know how little we know and there are lots of things we really don't know about. So when we really know a lot we say "I know that I know nothing"
A Mobius strip (wait for my next post to know what a Mobius strip is) is a topological paradox.Actually a mobius strip is a surface with only one side and one boundary; but it seems that it has two surfaces and two boundaries.

Paradoxes are as old as humankind. The ancient Greeks studied them intensely and resulted in the discovery of irrational numbers.

Mathematical Logic

2006-08-14T05:24:00.000-12:00

Mathematical logic is branch of mathematics. The idea of logic was first given by an English man George Boole in 1854 in his book ‘An investigation of the Laws of Thoughts. Logic is the study of reasoning.

In practical life, we express our ideas by means of sentences. A sentence may be either true of false or neither.In Mathematics we are concerned with only those sentences which are either true or false. These sentences are called logical statements.
A sentence which is both true and false simultaneously is not a sentence, rather, it is a paradox.(I will give details about paradox in my next post)

Example:

The angles of a rectangle are right angle-Since this sentence is true it is a statement.

3+6=10-Since this sentence is false, it is a statement.

x is a prime number-this is true for x=2,3,5 etc but false for x=1,4,6 etc.So this sentence is not a logical statement.

Learning linked to sleep

2006-07-30T04:36:00.000-12:00

Students who study hard all week and party on the weekends may lose a lot what they learned. Sleep deprivation on critical nights after learning may cause a 30% loss. Carlyle Smith, a professor of psychology at Trent University in Peter borough, Ontario researched the effect of sleep deprivation on learning by controlling the sleep of students the night they were taught a complex logic game and a list of paired words.

Smith found that when students were tested on paired words, a week later there was no learning deficit among the students deprived of sleep, but when tested on complex logic game, the students deprived of sleep showed a 30%
learning deficit when compared to the group of students not deprived of sleep.
Sleep deprivation on the 3rd night after learning had the same results showing a deficit of about 30% for the complex logic game. Sleep deprivation the 2nd night after learning seemed to have no effect.

This means that if you party all night on Friday after a rough week in the school you will lose 30% of the learning you acquired on Wednesday and Friday. If you lose sleep on Saturday night then Thursday's learning is also affected.
Ref: Mathematics today

Sides of a Triangle

2006-07-28T08:51:00.000-12:00

Today let’s discuss one problem from geometry.

I‘ll ask you one question. What is the area of a right triangle with sides 4cm 5cm and 9cm? Before trying to answer, please read the question once more.

Answer
It is not possible to make a triangle with the given measurements.
In a triangle, the sum of the two sides will be greater than the third side. Here in this case sum of two sides 4cm+5cm=9cm, which is not greater than the third side of measurement 9cm.