Manorama Yearbook 2012 (Free CD) – This best seller has a long history of success. It is India's best General knowledge update covering almost everything that a student needs in competitive examinations – Purchase Online

General Awareness and Current Affairs Digest 2012 by Krishna Reddy, M Laxmikanth is a Tata Mcgraw Hill publication. This is a comprehensive book full with information and latest data useful for various competitive examinations in segments like civil services, banking, insurance, railways, SSC, Indian Forest Service, NDA/NA, CDs etc. – Purchase Online

Competition Success Review: Year Book 2012 (Wall Maps India & World Inside) by P. N. Kapil can also be very useful for people preparing for competitive examinations like Bank PO, Bank Clerical and Railways and so on.  – Purchase Online

The Pearson General Knowledge Manual 2012 by Edgar Thorpe, Showick Thorpe– Another awesome compilation of the most useful current affairs and general awareness topics and Q&A for competitive examinations. – Purchase Online

The Pearson Concise General Knowledge Manual 2012 by Edgar Thorpe, Showick Thorpe – This is the concise   form of the above book. Overall, this is a good book for Government competitive examinations Purchase Online

The links above are affiliate links to Flipkart. I would request you to read the reviews and take opinion of friends and teachers before ordering any book. I shall be glad to add books suggested by you, to the above list. Please feel free to add your suggestions by posting a comment below.

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Life Is A Gamble: Mathematics Of Expected Value And Insurance Explained

It’s always interesting when someone makes the claim that they don’t gamble. Even this early in the year most people will have made several decisions where the outcome was uncertain and those outcomes were mostly financial - just like a bet. Whether it was deciding to start the year with a new job or take out some kit car insurance on the newly built love of their lives they can’t be sure whether their financial position will be stronger or weaker as a result of their decision.

We’ll first explore the simple mathematics behind these gambles using a gambling example and explain why some decisions you should take each and every time they are offered to you. Interestingly products like insurance rely on our fear of ruin and are not always a fair gamble; we’ll explain why that is and how that leads to profit for insurance companies.

Assessing The Expected Value Of A Decision

When faced with a decision which has primarily financial outcomes it is natural that you would want to choose the option which provides the greatest expected financial return. Assessing this is simple and is shown below:

Expected Value = Probability 1 x Value of Outcome 1 + Probability 2 x Value of Outcome 2...

A simple example is a game of dice. The drunken gambler at the local bar offers you the chance to play a game. If the dice rolls a 4, 5 or 6 he will give you 15. If the dice rolls a 1, 2 or 3 you will give him 10.

EV = (0.5 x 15) + (0.5 x -10)
= 2.5

A normal casino game is always designed so the casino will win in the long run. In this game the player is being offered by the drunk gambler a game where the expected win is 2.5 per game by paying more when a 4, 5 or 6 comes despite the chance of them rolling is exactly the same as a 1, 2 or 3. Since this is a positive expected value the player should sit and play all night. We can calculate the expected value of a thousand games simply by multiplying the expected value of one game by one thousand.

EV(1000 Games) = 1000 x 2.5
= 2,500

The Profits Of Insurance

Insurance has sometimes been called a ‘sucker bet’ because it is perceived to have a negative expected value to the buyer of the policy and the likelihood of claiming is relatively low. The combination of these two factors usually makes it extremely profitable for insurance companies. The annual results of the big life insurance firms make this point very nicely!

“I got an online quote from an insurance company in the UK and was given the figure of £10 a month for a 40-year package of £100,000 cover. So (breaks out calculator) I pay the insurance company £4800 over the course of the policy and when (because it's not an if) I die they pay my beneficiary £100,000.”
http://boards.straightdope.com/sdmb/showthread.php?t=473242

This question shows a big part of why insurance looks to be better value to most people than it actually is. In this case by giving the insurance company £10 per month the customer is forgoing the ability to invest that money elsewhere. If investment rates were 0% so all we lost by buying the policy was £4,800 we can calculate the break even (EV=0) chance of death at that policy price as below where Pd is the probability of death.

EV = Pd x £100,000 + probability of surviving x -4800

=> 0 = (Pd x £100,000) - £4800(1 - Pd)

=> 0 = £100,000Pd - 4800 + £4800Pd

=> 4800 = 104800Pd

=> Pd = 4.58%

In reality though we lose the total value we could have received for that money. If the £10 per month had been invested and returned £10,000 at the end of the period then the probability of death to break even is 9.09%. This is important because insurance companies will be investing the money and will know what your probability of death is fairly well based on their actuarial calculations. If the applicants actual probability of death is 7% during that term let’s compare the perceived EV(1) seen from the perspective of the person asking the question with the actual EV(2) based on the value of the money if it had been invested.

EV(1) = 0.07 x 100,000 - 0.93 x 4800
= £7,000

EV(2) = 0.07 x 100,000 - 0.93 x 10000
= -£2300

As you can see our applicant perceives a positive EV situation when in fact the insurer is set to make a nice profit over their thousands of customers by calculating their investment returns and probabilities of death correctly.

Fear Of Ruin

The final factor to consider when understanding why consumers will take these negative EV bets by taking insurance is the insurers attitude to risk and their corresponding fear of ruin. If our applicant owns a nice house with a £100,000 mortgage in the above example and wants to protect their family from losing their home should they die the £10 per month is a very small commitment in the event they survive whereas the £100,000 mortgage is a very large commitment for the surviving family members.

Consumers are prepared to pay (through taking a negative EV bet) for this peace of mind and this allows the insurers to make a larger profit than they could simply through better calculating investment returns and risk of death than consumers are able to - although naturally as we saw in the question earlier this better ability to perceive these factors is a significant part of their profitability.

This is a guest post by Izzy Woods.

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The Rule of 72 is a good quick math shortcut to find out the following –

• Time required for an amount to double itself, at a given rate of interest
• Rate at which an amount should grow to double itself in given time

This formula can be applied for “Doubling Problems” related to money, population, etc. which grows at an annual compounded rate.

Formulae

1. To calculate the time; T = 72/R
2. To calculate the rate of interest; R= 72/T

T = Time required to double a sum of money at the rate of R% per annum.

R = Rate of interest at which a sum of money gets doubled in T years.

Explanation of the formula  

To find out the number of years required to double an investment in a fixed deposit which gives you 9% rate of interest compounding annually, divide 72 by 9.

For example, if you invest Rs. 10000 with compounding interest at a rate of 9% per annum, the rule of 72 gives 72/9 = 8 years required for the investment to become Rs. 20000; an exact calculation gives 8.0432 years. So there is small margin of approximation.

The above formula is more accurate at lower interest rates (say up till 10%). The approximation error starts increasing after that.

In case of continuous compounding, 69 instead of 72, gives more accurate results. However, in our day to day life the concept of continuous compounding is rarely used.

Guide to Online Schools has information on math classes.

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Algebraic Equations for Fun!

There was a lot of positive feedback about the alphametic cryptatrithms, I posted a few months ago: so here is a little game that is based on the same idea, but with a different execution. This version is fun for those with a propensity towards algebra, and geared more toward middle school or high-school ages; however, if you are an adult, please enjoy them as well. There are a few less steps here than in the cryptarithms, but I’m sure y’all will get a kick out of them all the same!

1. (DD)^^E  = DEED

First, identify what you know. D=D and E=E. The sum is four digits. If we begin our trial and error and start with the numeral 1, we can infer that (11)^E  = 1EE1. Now, to find E: We know that it can’t be 1, since D=1. 11² = 121. But, 11³ = 1331. There is the answer!

You can utilize a few math tricks here to eliminate some of the trial and error. Call upon old favorites like the square root trick, and don’t be fooled by simple mistakes (anything multiplied by 0 is 0!).

Here is another one:

ABCD x D = DCBA

I’ll let you work your way through this one. Here are some more for you to work through and if you solve all of them, you can find more at http://www.mathsisfun.com/puzzles/algebra-puzzles-index.html.

Solve them all? Try writing one yourself and share it.

1.         CAT = (C + A + T) × C × A × T

 2.         ABC × DEF = 123456, if A = 1

 3.         C^L=LOGIC

 4.         COW × COW = DEDCOW

Answers to the problems can be found on the link above.

Author Bio: As a law student, Danielle often uses conference call technology to attend virtual networking events, as well as to interview professionals in her field about case studies for homework. Visit her blog, professionalintern.com, to read about student-friendly technology and innovation.

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Values of Trigonometric Angles

Let’s start with most commonly used angles of Sin. The angles are 0°, 30° (π/6), 45° (π/4), 60° (π/3), 90° (π/2). For these angles we’ve to make fractions for which we’ve to write 0, 1, 2, 3 and 4 in the numerators and write 4 in the denominator of each fraction. After that take the square root of each of these fractions and there you are. Refer to the table below for better understanding.

For Cosine we simply have to write the results of Sin in reverse order. Refer to the table below. The values in the Cos row is in reverse order to that of Sin row.

Tan is very simple. You just have to remember that Tan=Sin/Cos. Hence, to get the value of Tan we’ve to divide values of angles of Sin from SinA row by values of angles of Cos from CosA row.

I am glad that of late, many QuickerMaths.com readers have started participating actively in discussions (comments) under various posts. A quick calculation trick/technique/method from your side can also be posted on QuickerMaths.com. You just have to write it properly and send it to me at vineetpatawari@gmail.com.  Don't forget to write a brief description (2-3 lines) of yours, to be added at the end of the post.

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To use this memory trick, you need to memorize this simple mnemonic -

Some people have curly brown hair turned permanently black

That’s all you need to memorize to register the trigonometrical ratios in your mind forever. So here you go,

Some People Have

S = P/H

Sin = Perpendicular / Hypotenuse

Curly Brown Hair

C = B/H

Cos = Base/Hypotenuse

Turned Permanently Black

T = P/B

Tan = Perpendicular/Base

There are 3 more ratios: Cosec, Sec and Cot. For these, just remember that -

Cosec is the reciprocal of Sin; or Cosec = 1/Sin = H/P

Sec is the reciprocal of Cos; or Sec = 1/Cos = H/B

Cot is the reciprocal of Tan; or Cot= 1/Tan = B/P

In my next post I’ll write about a simple memory trick for memorizing the value of all major angles of different trigonometry ratios like sin30, cos45, tan60, etc.

Update on 4th September, 2011 - as promised above here's the new post - http://www.quickermaths.com/memory-tricks-for-trigonometry/

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For many math students, word problems are a cause for anxiety and stress. After all, how can you solve a math problem that is written in words, with few numbers to rely on? In my college trigonometry class, I was guilty of this, feeling like an imbecile as I struggled over the most basic of problems. However, what many don’t realize is that all of the numbers you need are contained in the puzzle; you simply have to know how to recognize and implement those numbers in order to find your solution.

The best way to do this is to draw a sketch of the scenario laid out in the puzzle—in essence to illustrate the story being told—in order to solve the problem. Below are three story problems and a step-by-step guide for finding their solutions:

First, an easy one: Sir Patrick and his page Jake could travel 12 miles in 1 day. They were travelling from Gaul heading back to Rome, which is 600 miles away. A big flood flooded their path so they had to wait 4 days for conditions to improve before they could continue. A band of outlaws attacked them and stopped their travels for another 2 days. How many days did it take them in all to get from Gaul to Rome? (I have taken the liberty of emphasizing all of the numbers needed to solve this problem.)

Once you have a basic diagram, you can see that this is far easier than originally anticipated!

First, how many days would the trip have taken, were it not for the delays?

600 ÷ 12 = 50 days

How many days tacked on due to unbelievably bad luck and lack of modern amenities? 6

50 + 6 = 56 days total for trip.

As you can see, once you are able to select the pertinent information from the puzzle and plug them into the right spots, even the most confusingly worded problem becomes easy.

How about one that’s a little more challenging?

Two bowls and three plates cost $1421. The cost of the plate is half the cost of the bowl. What is the cost of the bowl?

OO – 2 bowls   +   ||| – 3 plates   =   $1421

| = ½ O                 cost of O = x

½ x + ½x + ½x + x + x = 1421

3.5x=1421

1421 ÷ 3.5 = $406 (cost of bowl)

406 ÷ 2 = $203 (cost of plate)

Easy-peasy.

Now, for a doozie: A fireman rests his ladder against a building, making a 57⁰ angle with the ground.  The bottom of the ladder is 28 feet from the base of the building.  How long is the ladder? (For this, I confess that I have already forgotten my trig, so I will be quoting from the website where this problem was originally posted.)

|\

|  \

|    \ ßx

|      \

|        \ ß57

28
According to Math with Larry, “The first thing to determine is what trig ratio we should use to solve this problem.  We have an angle, we have an adjacent side to that angle, and we need to know the hypotenuse. Referring to SOH-CAH-TOA, we can see that the cosine will help us. We know that cos 57 = 28/x. Using a calculator, we can determine that cos 57 = 0.545 (rounded). Using basic algebra, it is easy to determine that x = 51.4 (rounded). That means that the ladder is 51.4 ft long.”

As you can see, with the proper illustrations, you can simplify any problem, thus relieving your math anxieties and lessening your chances for panic attacks or breaking down into tears during trig quizzes (as I have been known to do). Good luck!

BIO: Danielle realized early in life that she was not a mathematician, and as such is studying business litigation. However, she has learned the importance of a basic understanding of math and its application to real-world problems. Whether she’s looking for sales and using coupons to save money, or figuring out the height of a building for whatever obscure reason she would need to do so, understanding how to interpret numbers plays a vital role in her life.

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Question

I have a total of rs 1000. Item A costs rs 110,item B costs rs 90, item C costs rs 70 ,item D costs rs 40 and item E costs rs 45. For every item D that i purchase, I must also buy only 2 items of B. For every item A, I must buy one item of C. For every item E, I must also buy 2 of item D and one of item B. For every item purchased  I earn  1000 points and for every rupee not spent I earn a penalty of 1500 points. My objective is to maximise the points I earn. What is the number of items that I purchase to maximise my points?

options....
a) 13   b)14   c)15   d)16

"Give me the procedure to solve this with in 2-3 mins . It takes to much time 2 solve this for me." -  Satish Kumar

Pay attention to Satish's problem and suggest time reduction techniques along with the solution to the above problem.

I would latter include the best answer(s), posted as comment, in the post itself.

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Through this article I would like to share certain ideas, which I gathered over the time, to reduce stress and increase efficiency during examinations. The underlying ideas can also be applied to manage stress in other aspects of our life.  I hope it will be beneficial for all the readers of QM, either directly or indirectly. The intention of writing this article is to help readers willing to increase their efficiency and perform better, by making the exam process less tiring and little more enjoyable.

How to Manage Stress before Examination?

• Take a piece of paper and list down the main reasons of your stress and anxiety. Next to each problem, write possible solutions.  Take a break and discuss with your parents and (if possible) your teachers your fears and doubts
• Create a schedule – organize your time and prepare for the exams.
• Do your exam revisions with your friends. You will get to know few things which you might have missed.
• Test your knowledge with friends. Meet them and try asking questions to each other.

How to beat Examination Fear?

• Learn to face the exams. Shed away any fear of failure. Positive thinking is highly beneficial as it removes most of the stress out of your head.

Attempt a mental representation of the actual conditions that cause you stress in detail. When you do this exercise, you will notice that the more you imagine and visualize the exams, the more familiar they will seem to you.

• When you can deal with your fears in your imagination, you can deal with them in reality as well.
• If you are in school right now, you are yet to face many and most difficult examinations of your life. I will suggest you to read either of the two books -
  • Exam Stress, No Worries - Su Dorland
  • Beat Stress!: The Exam Handbook - by Anita Naik and Lynn Peppas

*for more information, the books are linked to their respective pages on flipkart.

Can the study environment affect stress?

A conducive study environment will definitely help to remove a lot of stress from you.  So organize your study environment appropriately.

• Create a pleasant environment without strong lighting and noise.
• Make sure that the room has enough fresh air.
• Turn off the TV and radio.
• Keep your mobile phone in silent mode a little away from you and do not check it every minute  for missed calls and messages received. Better still; switch it off for some time when you need to concentrate on some important topic.
• Find comfortable places to study.

Please share your personal experiences about, how you used to manage stress during exam days. Everyone had a different way of handling the stress. I am sure all the readers must have faced this stress at some stage of their life and we all can learn something from each other.

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A: Because they have sine and cosine to get a tan and don't need the sun

Maths is Fun

Mathematics can be real fun but that's only probable if you like the logic, the numbers and the beauty of the subject.  However, with this post I have tried to make maths fun even for people who don't love maths. This is done by making fun of maths and mathematicians. Of course, these are not my creation and have been collected from different sources. I hope you would like these jokes on mathematicians.

Maths Jokes

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

An infinite crowd of mathematicians enters a bar.

The first one orders a pint, the second one a half pint, the third one a quarter pint...

"I understand", says the bartender - and pours two pints.

Q: What does the zero say to the the eight?

A: Nice belt!

A man flying a hot air balloon got lost. So he descended and asked a woman walking in a field “Where am I?” She thought for some time and then replied. “In a hot-air balloon”. Immediately the balloonist realized she was a mathematician, for three reasons

(1) She thought before replying

(2) What she said was absolutely correct and

(3) Her reply was totally useless.

A mathematician, statistician and accountant were finalist for a position as VP in a large corporation. The hiring committee asked them all the same last question

The mathematician was first."How much is 500 plus 500 ?" , they asked"1000" he replied without hesitation."Thank you", they dismissed him.

Next the statistician."How much is 500 plus 500?""On the average, 1000 with 95 % confidence" replied the statistician"Thank you", they dismissed him.

Next the accountant."How much is 500 plus 500?""What would you like it to be?" responded the accountant.They hired the accountant.

Question: "How many seconds are there in a year?"Answer: "Twelve, January second, February second, March second, ..."    :D

Q: What is the definition of a polar bear?

A: A rectangular bear after a coordinate transformation :D

Q: What will a logician choose: a half of an egg or eternal bliss in the afterlife?

A: A half of an egg! Because nothing is better than eternal bliss in the afterlife, and a half of an egg is better than nothing. :D

If you have some joke(s) on maths or mathematicians, you can share with all by posting a comment below.

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