Online CAT Preparation

Thursday, March 24, 2011

Is 0.999…= 1?

Posted: 24 Mar 2011 07:49 AM PDT

Many a times we have made 0.999….= 1. But we always thought it's an approximation, they are not equal though.

It might be surprising for many of us to know that 0.999….. is actually EQUAL to the integer 1. It can be proved like this,

If x = 0.999..., then 10*x = 9.999... so by subtracting the first equation from the second, we get

9*x = 9.000...

Therefore, x=1.

Here's another proof - The number 0.1111... = 1/9, so if we multiply both sides by 9, we obtain 0.9999...=1.

So by similar arguments, every rational number with a terminating decimal expansion has another expansion that ends in a never-ending string of 9's. So, for instance, the rational 9/20 can be represented as 0.45 (the same as 0.35000...) or 0.44999...

General mathematical proof

Any rational number can be expressed in such a way that the digit in each place of a decimal expansion is associated with a positive or negative power of 10. The k-th place to the left of the decimal corresponds to the power 10^k. The k-th place to the right of the decimal corresponds to the power 10^(-k) or 1/10^k.

If the digits in each place are multiplied by their corresponding power of 10 and then added together, one obtains the real number that is represented by this decimal expansion.

So the decimal expansion 0.9999... actually represents the infinite sum

9/10 + 9/100 + 9/1000 + 9/10000 + ...

Using the formula for finding the sum of infinite G.P. series i.e. {a/(1-r)}, we get,

(9/10)/(1 - 1/10) = 1

Hence 0.999... equals 1.

There can be many other proofs. Math enthusiasts are welcome to suggest more proofs. Alternatively if you can prove that they not equal, please post.

Friends let me know your suggestions/feedback on the type of article I post on Quickermaths. You can also suggest me a topic to write. You can also write it yourself and send it across to me to be posted on QuickerMaths.

Tuesday, March 22, 2011

Email Subscription Confirmation

Posted: 22 Mar 2011 02:15 AM PDT

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Monday, March 21, 2011

Brain Puzzle Question

Posted: 21 Mar 2011 02:39 AM PDT

Solve this interesting brain teaser or puzzle and exercise your mind.

Thief in the Jhootas' Club - Brain Teaser

An expensive painting was stolen from the Jhootas' Club, but the CID is having a hard time identifying the culprit because every statement made by a member of the Jhootas' Club is false. Only four members visited the club on the day that the painting was stolen. This is what they told to the CID inspector Fredrick:

Anand: None of us took the painting. The painting was here when I left.
Bobby: I arrived second. The painting was already gone.
Chirantan: I was the third to arrive. The painting was here when I arrived.
Dinesh: Whoever stole the painting arrived before me. The painting was already gone.

Obviously neither Fredrick nor Daya could figure out who's the thief out of the four members of Jhootas' club. They requested you (an expert on such subjects) to solve this puzzle for them. According to you who out of these four liars stole the painting? Please explain to all of us.

Saturday, March 12, 2011

How to convert from decimal to other number systems

Posted: 12 Mar 2011 07:26 AM PST

This post will be of special interest for people who are regularly in touch with mathematics. Students preparing for competitive examinations usually have Base System (Number Systems) in the list of their topics under quantitative aptitude. You can suggest any addition to the post below by posting a comment or mailing me at vineetpatawari[at]gmail[dot]com. If you have any queries post it as comment.

Conversion from decimal to binary and other number bases

In order to convert a decimal number into its representation in a different number base, we have to be able to express the number in terms of powers of the other base. For example, if we wish to convert the decimal number 100 to base 4, we must figure out how to express 100 as the sum of powers of 4.

100 = (1 x 64) + (2 x 16) + (1 x 4) + (0 x 1)

= (1 x 4^3) + (2 x 4^2) + (1 x 4^1) + (0 x 4^0)

Then we use the coefficients of the powers of 4 to form the number as represented in base 4:

100 = 1 2 1 0 base 4

Take another example; convert 117 into binary system –

Now since we have to convert 117 into binary we have to express 117 as the sum of the powers of 2. Obviously all the powers need to be less than 128 (=2^7)

117 = (1 x 64) + (1 x 32) + (1 x 16) + (0 x 8 ) + (1 x 4) + (0 x 2) + (1 x 1)

117 in decimal = 1110101 in binary

This method is less of calculation and more of application of mind and needs a lot of practice to master.

The other way to do this, which is more frequently used, is to repeatedly divide the decimal number by the base in which it is to be converted, until the quotient becomes zero. As the number is divided, the remainders - in reverse order - form the digits of the number in the other base.

Example: Convert the decimal number 82 to base 6:

Solution: 82/6 = 13 remainder 4

13/6 = 2 remainder 1

2/6 = 0 remainder 2

The answer is formed by taking the remainders in reverse order: 214 in base 6

In my next post, I will write about converting other number bases to decimal number system.

Author – Vineet Patawari

Friday, March 4, 2011

How to express fractions as decimals

Posted: 04 Mar 2011 12:54 AM PST

In the past few days I didn't enough time to think and write about a any topic on QuickerMaths.com. Today the suggestion of this topic came from one of you, so half the work was done. Friends I will request you to keep suggesting new topics on which I can write for everyone's benefit on QuickerMaths.com

How to express fractions as decimals or percentage

This post will help you to learn:

Express a given percent as a decimal or fraction.
Solve a given problem that involves finding a percent.
Determine the answer to a given percent problem where the answer requires rounding, and explain why an approximate answer is needed (e.g., total cost including taxes).
Work problems involving pie charts and percents.
Work problems involving tables and percents.

Decimal Equivalents of Fractions

You should know these:

1/2 = .5 = 50%

1/3 = .333... = 33.33%

1/4 = .25 = 25%

Starting with the thirds, of which you already know one:

1/3 = .333... = 33.33%

2/3 = .666... = 66.66%

You also know 2 of the 4ths, as well, so there's only one new one to learn:

1/4 = .25

2/4 = 1/2 = .5

3/4 = .75

Fifths are very easy. Take the numerator (the number on top), double it, and stick a

decimal in front of it.

1/5 = .2

2/5 = .4

3/5 = .6

4/5 = .8

There are only two new decimal equivalents to learn with the 6ths:

1/6 = .1666...

2/6 = 1/3 = .333...

3/6 = 1/2 = .5

4/6 = 2/3 = .666...

5/6 = .8333...

One-seventh is an interesting number. Read the comments on Cyclic Numbers

1/7 = .142857142857142857...

For now, just think of one-seventh as: 0.142857

See if you notice any pattern in the 7ths:

1/7 = .142857...

2/7 = .285714...

3/7 = .428571...

4/7 = .571428...

5/7 = .714285...

6/7 = .857142...

Notice that the 6 digits in the 7ths ALWAYS stay in the same order and the starting digit is the only thing that changes.

If you know your multiples of 14 up to 6, it isn't difficult to work out where to begin the decimal number. Look at this:

For 1/7, think "1 * 14", giving us .14 as the starting point.

For 2/7, think "2 * 14", giving us .28 as the starting point.

For 3/7, think "3 * 14", giving us .42 as the starting point.

For 4/14, 5/14 and 6/14, you'll have to adjust upward by 1:

For 4/7, think "(4 * 14) + 1", giving us .57 as the starting point.

For 5/7, think "(5 * 14) + 1", giving us .71 as the starting point.

For 6/7, think "(6 * 14) + 1", giving us .85 as the starting point.

8ths aren't that hard to learn, as they're just smaller steps than 4ths. If you have trouble

with any of the 8ths, find the nearest 4th, and add .125 if needed:

1/8 = .125

2/8 = 1/4 = .25

3/8 = .375

4/8 = 1/2 = .5

5/8 = .625

6/8 = 3/4 = .75

7/8 = .875

9ths are almost too easy:

1/9 = .111...

2/9 = .222...

....

8/9 = .888...

10ths are very easy, as well. Just put a decimal in front of the numerator:

1/10 = .1

2/10 = .2

...

9/10 = .9

Remember how easy 9ths were? 11th are easy in a similar way, assuming you know your multiples of 9:

1/11 = .090909... = 9.09%

2/11 = .181818... = 18.18%

3/11 = .272727... = 27.27%

.....

10/11 = .909090...

As long as you can remember the pattern for each fraction, it is quite simple to work out

the decimal place as far as you want or need to go!

Lesson Summary

In this lesson, the learner has learnt about:

The decimal equivalents of everything from 1/2 to 10/11
Author: Vineet Patawari

Wednesday, February 16, 2011

How to write letters and email effectively?

Posted: 15 Feb 2011 09:58 AM PST

The purpose of this article is to help professionals and future managers in email and letter writing process.

How to give your written communication an extra punch?

Some pointers to help you write hard-hitting business or personal documents:

Time spent on planning your communications will pay dividends. Make a rough draft of what you want to write or say, so that you can experiment with various versions. Remember that language is important because the words you choose convey your attitudes as well as information.
Get to the point from the beginning. Cut the small talk and make a good impression by being crisp and business-like. Documents that do not do this waste the readers’ time and may end up in the waste bin or 'Recycle Bin'.
Use straightforward language rather than jargon. People prefer to be treated as human beings, not computers! Technical language has its place, but it is impersonal and should be used only when necessary.
Use sentences that are short and to the point, not sentences that ramble on and cannot quite decide what they want to say or how to say it – like this one!
Steer clear of the passive voice, since it is an indirect way of speaking and creates distance between you and your audience or reader. For example, if you say, "We will attend to your order promptly," that promotes more confidence than if you say, "Your order will be attended to soonest." This lacks the personal touch and may give the impression that you do not want to accept responsibility for your work.
It is very important that you think about the audience you are writing or speaking to and make a real effort to communicate with them.
A basic issue that business people sometimes ignore is spelling. Incorrect spelling makes a poor impression. If you are unsure about the spelling of any words you have used it is worth the trouble of running a spell check on your computer. However, computer dictionaries are often limited and therefore many technical terms may still need to be checked manually.
Correct grammar is as important as spelling. Some word processors now have grammar checkers that operate in the same way as spell checkers.
Finally, always read carefully through a talk or letter to check for typographical and other errors. Are the facts and dates accurate? Reading aloud is a good idea, because you can hear how the communication sounds: the ear provides a cross-check for what the eye may have missed.

If you want a book on writing effective emails (my favorite) for improving your written communication you can consider ordering the book titled – The Business Letter Handbook (How to Write Effective Letters & Memos For Every Business Situation) – Purchase Online from flipkart.com

Author: Vineet Patawari

Understanding Platonic Solids with Modular Origami

Posted: 24 Nov 2010 09:36 PM PST

A guest post by Maria Rainier

Understanding Platonic Solids with Modular Origami

Solid geometry is perhaps one of the best mathematical applications of origami, but of course, there are many other ways to use it in improving students' understanding of math's processes, concepts, and underpinnings. For anyone who has difficulty with the abstract components of math, origami can help provide both visual aids and the opportunity to arrive at mathematical conclusions through trial and error. It's an especially effective way to help visual and kinesthetic learners to understand basic geometric concepts.

You can teach two of the platonic solids with a simple demonstration or a more elaborate project, depending on how much time you'd like to spend. With a demonstration, you'll be doing most of the origami module construction, allowing students to experiment with it. If you assign a project, you can have different groups working to construct their own modular components and the more difficult module itself. Either way, it will help to become familiar with the model before you use it to teach solid geometry, but constructing the components isn't difficult and you'll be able to envision the model easily. Take a look at the following instructions and images to determine how you would adapt this idea to your teaching style.

5 Intersecting Tetrahedra = 1 Dodecahedron

Constructing the five tetrahedra is a relatively easy task, but weaving them together to form a dodecahedron is both challenging and fascinating. Your students will almost certainly need your help if you decide to have them complete this part, but accomplishing something so difficult is great for self confidence and a stronger grasp of solid geometry.

Basic Unit

You'll need ten squares of paper to complete this model – two for each tetrahedron. Divide each square into equal thirds, then cut them into strips so that you have 30 small 1X3 rectangular pieces. To create one modular unit, fold one of the pieces in half along the longer side, unfold, and bring the edges into the center crease. To form a 60° pointed end, fold the top right edge into the center and give the resulting new edge a light pinch (this is just to form a guidance crease). Now, fold the top left corner to meet the crease you've just made on the right side, taking care to form a corner at the top of your midline crease. Fold the top right corner down over it to get a triangular point. Now, unfold both of the flaps you've just made and reverse fold the left flap so that it's inside of your unit, creating a small pocket. Fold the top edge of the right flap down to meet the 60° crease and unfold. Turn the unit 180° and repeat at the other end to finish your first unit, then give it a good crease along the midline. Make five more, and you'll be ready to make your first tetrahedron – see this helpful Merrimack College page for diagrams.

Tetrahedron

To construct a tetrahedron, simply insert the right-hand projection of one unit into the left-hand pocket of another. Now, add a third unit to join the first two, forming one of the tetrahedral frame's four points and three of its six edges. Use the remaining three units to complete the tetrahedron.

Dodecahedron

Now, the tricky part is weaving your five tetrahedra together to form a dodecahedron. The rule of thumb is that the peak of each tetrahedron should come through the base of another – it's also helpful to keep in mind that the 20 points of the combined tetrahedra form the pentagonal points of the dodecahedron. The diagrams described above are especially helpful in assembling the final platonic solid, but the peak-base rule can also be used to successfully weave the dodecahedron.

Wrap-Up Questions

Can you make any other platonic solids using the modular units that form the tetrahedra?
Why is the 60° angle important? Could you complete this model with units formed by any other angles?
Could thinner units be made with the 60° angles intact?
Why does the method used to form the 60° angle in the construction of the basic units work? (Hint: Check out Huzita's fifth axiom.)

About Author

Maria Rainier is a freelance writer and blog junkie. She is currently a resident blogger at First in Education, where recently she's been researching online mechanical engineering degrees and blogging about student life. In her spare time, she enjoys square-foot gardening, swimming, and avoiding her laptop.

Tuesday, February 8, 2011

List of Best MAT Books

Posted: 07 Feb 2011 09:57 PM PST

1. MAT (Topicwise Analysis & Solutions) by G. K. Publishers - Purchase Online

2. Barron's MAT: Miller Analogies Test by Robert Sternberg, Sternberg Karin Ph. D. Purchase Online

3. MAT Guide by Dr R P Datason - Purchase Online

4. MAT (Management Aptitude Test) Entrance Exams. With Practice CD by Mittal - Purchase Online

5. 501 Word Analogy Questions by Learning Express Llc - Purchase Online

6. MAT - Prev. Papers Solved by Rph Editorial Board - Purchase Online

7. Mat Management Aptitude Test: Fast Track Preparat by Ravi Chopra, Kant Chopra A, Ghosh D - Purchase Online

Sunday, February 6, 2011

How to calculate EMI?

Posted: 06 Feb 2011 09:00 AM PST

Calculation of EMI

In our daily life we face enormous application of mathematics. Calculation of equated monthly installments (EMI) for car or home loan is one such common application of mathematics.

EMI or equated monthly installments is the most popular form of loan payment. It is a fixed amount of repayment made every month towards the loan, which includes payment towards both principal and interest. Most of us always believe the bank executives blindly on the figure which they quote as EMI.

This post is to explain the mathematics behind EMI and how to calculate it in excel using inbuilt excel function.

Calculation of EMI

EMI= P x r x (1 + r)^n / ((1+r)^n -1)

Here p = principal amount (loan taken)

r = interest rate per month (ex: if interest rate per annum is 10% then 10/(12*100))

n= tenure in months

For example,

EMI = 100000*0.01*(1+0.01)^24 /((1+0.01)^24 -1) = 4707

Where,

p = loan taken = 1,00,000

r = interest rate per month = 1% = 0.01

n= tenure in months = 2 Years = 24 months

This formula assumes, EMI payment is made at the end of each period (month). This is also called EMI in arrears. If EMI is paid at the beginning of each period it is called EMI in advance.

Further additions will be done on EMI for any other processing fee or possible charges which may be applicable as per the rules of financing institutions (bank).

Calculation of EMI in excel

In excel it is very simple to calculate EMI. There is an inbuilt formula for EMI calculation called PMT

PMT(rate,nper,pv)

Where,

Rate – Interest rate for the loan.

nper – Total number of payments for the loan.

PV – Present value/principal or loan taken.

FV – Future value (you can omit it)

Type – we have to put the value either 0 or 1. If payments are made at the beginning (EMI in advance) of each period, 1 is used. If EMI payments are made at the end of the period (EMI in arrears) put 0. If omitted 0 is taken a default value.

Lot more can be discussed about EMI. Please share your knowledge, doubts or experiences with EMI calculations by posting comments below.

Author - Vineet Patawari

Thursday, March 24, 2011

Is 0.999…= 1?

Is 0.999…= 1?

Tuesday, March 22, 2011

Email Subscription Confirmation

Email Subscription Confirmation

Monday, March 21, 2011

Brain Puzzle Question

Brain Puzzle Question

Saturday, March 12, 2011

How to convert from decimal to other number systems

How to convert from decimal to other number systems

Friday, March 4, 2011

How to express fractions as decimals

How to express fractions as decimals

Wednesday, February 16, 2011

How to write letters and email effectively?

How to write letters and email effectively?

Thursday, February 10, 2011

Understanding Platonic Solids with Modular Origami

Understanding Platonic Solids with Modular Origami

Tuesday, February 8, 2011

List of Best MAT Books

List of Best MAT Books

Sunday, February 6, 2011

How to calculate EMI?

How to calculate EMI?

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