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Tuesday, November 2, 2010

Mathematical constructs, Brute Force and the Logical Approach

Today I posted a reply to a question someone had asked on a 'Quant Forum'. When I read the question, I said to myself that since this number has to be less than that so this number must be in this range and so on... That is exactly what I wrote down on the forum too, a brute force method of mind, which I would like to rephrase as 'using conceptual understanding' While I was typing, the moderator merged the topic with an old one where someone had asked the same question and people had already discussed it in detail. After posting my response, I read the former discussion - and, I was a little taken aback. Most people had solved it using multiple inequalities. After wards, while cooking Tandoori chicken from a recipe I found on the net (so basically I was blindly following it), I got reminiscing...

I remembered the time 6-7 years back when I used to believe that I was good at solving Math questions (I still like to believe that) I was an Engineer for crying out loud; I had a firm Mathematical foundation. My mom had spent hours everyday with me when I was little to ensure that my Math was up to the par. (I must thank my mom for it) She had insisted that Math is done by using a pen and paper, by writing down stuff. So I could pretty well manage my way out of tricky questions. And, whenever the Mathematical constructs failed me, I would use the said brute force and get the correct answer somehow. Then, six years back, I met this guy, my mentor, my friend, philosopher and guide. He had been working in the test prep sector for many years and I had just joined it. He used to get all excited and twinkly - for want of a better word - whenever I solved a problem using brute fo - conceptual understanding but didn't bat an eyelid after looking at my elaborate Mathematical solutions. He encouraged me to think in terms of the concept itself, not Mathematical constructs. He himself presented theory in a manner that made solving the toughest of questions, very simple.

Let me break my monologue here to present you with an example first: (taken from the same forum - the values are changed though)

A certain principal amounts to $130 in 3 yrs and $150 in 5 years on a certain fixed simple interest. In how many yrs would the amount be $121 if the same principal had been put at the same rate of interest, but compounded annually?

Now the typical Mathematical way to deal with this problem would be the following:
Let P - Principal, i - Rate of interest, n - Number of time periods

In case of Simple Interest:
Amount = P + P x i x n
130 = P + P x i x 3 ... (I)
150 = P + P x i x 5 ...(II)
Subtracting (I) from (II), we get 20 = P x i x 2 or P x i = 10... (III)
Substituting (III) in (I), we get 130 = P + 10 x 3 which gives P = 100
Substituting P = 100 in (III), we get 100 x i = 10, or i = 10%

In case of Compound Interest:
Amount = P(1 + i)^n
121 = 100(1 + 10%)^n
121/100 = (1 + 0.1)^n
1.21 = (1.1)^n
Since we know that square of 1.1 is 1.21, hence n in this case will be 2 years.

If this is how we teach our kids to solve such simple problems, no doubt they hate Math!

Look at the far more elegant conceptual approach now:

The amount at the end of 3 yrs is $130 and at the end of 5 yrs is $150. It increases by $20 in 2 yrs. Since simple interest remains the same every year, the interest earned each year is $10.
Then the amount at the end of two years must have been $130 - $10 = $120 and at the end of one year, must have been $110 and so on...

Amount in case of compound interest for same principal and same rate of interest at the end of one year would be the same as the amount in case of simple interest at the end of one year. At the end of two years, amount in case of compound interest would include interest on last year's interest too. So the sum at the end of 2 years will be $110 + $10 + 10% of $10 = $121
Then, answer must be 2 yrs.

If you were a little lost in the conceptual explanation but nevertheless, it piqued your curiosity, then don't worry. In the coming days, in this blog, I intend to present theory on a lot of concepts in such a way that at the end of it all, you will be naturally inclined to think, rather than to solve.

I must warn you though, many people don't appreciate these methods. After all they don't involve elaborate equations in multiple variables. You just have to talk to yourself for a few seconds and out comes the answer! But guess what, you arrive at the correct answer in less than half a minute! So if you are preparing for a test and have time on your hands (6 months - give or take), we are going to have a lot of fun together!

Sorry mom, but I am not writing down anything to solve my Math questions.

2 comments:

  1. "Amount in case of compound interest for same principal and same rate of interest at the end of one year would be the same as the amount in case of simple interest at the end of one year. At the end of two years, amount in case of compound interest would include interest on last year's interest too. So the sum at the end of 2 years will be $110 + $10 + 10% of $10 = $121
    Then, answer must be 2 yrs. "

    What you mentioned above is the single most important thing we need to remember about Compound Interest problems.

    SI remains the same each year and all we need to do is calc. interest on interest

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  2. Yes Vikram. It's these concepts that can help us solve tricky questions in 30 secs.

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