Pages

Monday, November 8, 2010

The Four Prongs

ADDITION, SUBTRACTION, MULTIPLICATION, DIVISION

By no means am I implying that you do not know how to carry out these operations. But, I am always surprised when I have to go back to these concepts with people applying for a graduate degree. Elementary school teaches us what these operations do, but over the years we forget the main idea, the core of the operation and just remember the execution. So, to ensure that these fundamentals are revisited, I will first discuss a few interesting things about these basic mathematical operators.

Addition:

Adding two numbers is probably the simplest mathematical operation, still you will be surprised at the amount of time people take to add 126 to 94. If you are unfortunate enough, you may even get to see the following:

1 2 6

+ 9 4

____________

2 2 0

Mind you, I am not talking about an eight year old but at least 18 year old. For the sake of your sanity while taking a quant based test, you should do this simple calculation orally. Simply add the 6 from 126 to 94 to get 100. Then add the rest of the 120 to 100 to get 220.

Let’s look at a few examples, which I promise, I will type at random:

95 + 48 = 100 + 43 = 143

64 +73 = 70 + 60 ( = 130) + 7 = 137

739 + 182 = 740 + 181 = 840 + 81 = 920 + 1 = 921

45 + 39 + 18 + 26 = 40 + 30 + 10 + 20 (= 100) + 5 + 9 + 8 + 6 = 100 + 14 + 14 = 128

In each of the cases above, you can calculate in multiple ways. Feel free to experiment with different sets of numbers. With practice, it keeps getting easier and easier.

Subtraction:
Most people find subtraction a little harder than addition. Again, subtract in parts. If you want to subtract 27 out of 132, subtract 20 first, to get 112 and then 7 to get 105. Or, subtract 30 out of 132 to get 102 and then add 3 back because you only had to subtract 27. You get 105. Though, in subtraction, many a times, looking at the numbers and imagining their place in the typical subtraction diagram may turn out to be quite effective. For example if I want to subtract 36 from 83, I look at the unit’s digits and say that 3 – 6 will be 7 and 1 borrowed and then after subtracting the 1 from the tens digit, 7 – 3 will be 4 so the answer will be 47. Essentially, I am doing what eight year olds do, that is

8 3

- 3 6

_________________

4 7

But I am doing it in my mind, which makes it quicker.

Now if my question is, how many numbers are there from 11 to 25, both inclusive, and your answer is 25 – 11 = 14, then well done, you just flunked fifth grade. When we say 25 – 11, we are saying that we have 25 numbers and we are throwing away 11 of them. But I want to keep the 11thnumber; I want all the numbers starting from 11 and ending at 25. We need to add 1 to the result i.e. 14 so ensure that the 11thnumber that we threw out, is retained. Consequently, the number of numbers from 11 to 25, both inclusive is 1 + 14 = 15 numbers.

Multiplication:

It is a basic mathematical operator but, essentially, multiplication is still addition. When I ask, “What is 3 × 5?” I am actually asking, “What is 3 + 3+ 3 + 3 + 3?” i.e. addition of 3s written 5 times or addition of 5s written 3 times (because 5 × 3 is also the same as 3 × 5). This fact is very useful when performing multiplication orally. Now, if you want me to show you how to perform 139 ×463 orally, I am not that talented. But, in your everyday life and in exams that test your core concepts rather than calculation skills, you will rarely need to do such a multiplication. More often than not, the tougher multiplication required will be 16 × 12 or 23 × 18 etc. This is quite manageable.

When I ask for 16 × 12, I am asking for addition of twelve 16s i.e. 16 + 16 + 16 + … 12 times. Can I easily add ten 16s? Sure. It is 160. What about two 16s? That’s 32. So twelve 16s would add up to give 160 + 32 = 192.

Let’s take another example: 19 × 26. Easiest way to do this would be to find 20 times 26 which is 520 and then subtract 26 out of it because we wanted 26, 19 times, not 20 times. How do we subtract 26 from 520? By subtracting 20 first, to get 500, and then subtracting 6, to get 494.

It seems to scare people when they see something like 34 × 42. Is it really a cause of worry? 34 is just a notation for 3 × 3 × 3 × 3 i.e. Four 3s multiplied together. Then, 42 must be just 4 × 4. This means that 34 × 42 = 3 ×3 × 3 × 3 × 4 × 4 which is again just a multiplication, a pretty long one, but nevertheless, just a multiplication. More on indices, later.

Division:

Intuition says that if multiplication is actually addition, division should be subtraction. It is! When I divide 6 by 2, what I am doing is from 6, making groups of 2 and subtracting them.

6 – 2 = 4 (One group of 2 subtracted.)

4 – 2 = 2 (Another group of 2 subtracted.)

2 – 2 = 0 (Third group of 2 subtracted.)

I subtracted three groups so I get 3 as my quotient. Nothing is leftover so remainder is 0.

If I want to divide 11 by 3, I would be able to subtract 3 groups of 3 giving me a quotient of 3, and 2 will be leftover. Then 2 will be the remainder.

So if I ask you, is 82 divisible by 4, my question is, can I subtract groups of 4 from 82 such that nothing is leftover? When I subtract 20 groups of 4 (=80) from 82, I am left with 2. Hence 82 is not divisible by 4.

What if I ask, "Is 6 divisible by 3?" Of course, you say? True. 6 is 3 × 2 so I can make 2 groups of 3. "Is 7 divisible by 3?" Of course not! I get it. But is 6 × 7 divisible by 3? Now please don’t think 42 and then try and divide to find out. No. No. No. Can I say 6 × 7 is nothing but 3 × 2 × 7 or in other words 3 × 14? Then, does it mean that I can make 14 groups of 3 and hence it is divisible by 3? By the same logic, is 3 × 1279456 divisible by 3? Absolutely yes!

That's all for now. I will refer to these concepts again and again in other discussions so ensure that you are comfortable with them.


No comments:

Post a Comment