Quick Calculations

OK.. So I am not sure that the fear of numbers is called 'Numberophobia'. But then, this is not a post on English and Etymology, is it?

Many students fear mathematics. Their fear and/or disdain for the subject is uniform for all aspects of this subject: arithmetic - algebra - geometry - calculus - You name it, we hate it!

Now I can't claim to have a liking for stuff like calculus and statistics. But then, who requires calculus and statistics to clear the kind of entrance exams that we are preparing for, eh?

So here are a few tips and tricks on how to hone your quick calculation skills.

Now, before we understand the tricks to do quick calculations, we must have a few thing ready before hand. Then include, a working knowledge of multiplication tables.... (it's ok if you remember the tables for only single digit numbers, it's ok if you remember the tables for only 1, 2, 5 and 10!)

You should be able to add and subtract reasonably well.

If you are OK with this, then proceed!

So, you are given 13 x 25
There are many ways in which we can orally calculate the answer:

1. 25 x 13 = 25 x (10+3) = (25x10)+(25x3) = 250 + 75 = 325

2. 13 x 25 = (13 x 100/4) = 1300/4 = 650/2 = 325

3. 13 x 25 = 13 x (10 + 10 + 5) = 130 + 130 + 65 = 325

Easy eh?

Remember these tricks:

To multiply a number with 2, just double it
Multiplication by 4 = Double of Double
Multiplication by 5 = Half of 10 times the number
Multiplication by 6 = (Multliplication by 5) + 1 times the number
Multiplication by 9 = 10 times the number minus 1 time the number

25 = 100/4
45 = 50 - 5
27 = 25 + 2 = (100/4) + 2
 and so on...

Regular practice and you will soon be competent to multiply any two 2-digit numbers orally!

Here's a little game that can help you practice....

On the road, you get to see many vehicles with number plates. Make it a regular habit of adding up all the digits on the plate and also finding their product.

Regular practice will soon make you a calculation champ!

Averages

One of the simple yet very useful topic in the section of Mathematics is Averages.

Average means the value which is representative of the various values in a set. Now, this is a very crude definition of the word. Mathematically, there are better definitions. However, this should serve ok for a beginner.

Let us say that there are 3 boys A, B and C. A scores 20 marks, B scores 15 and C scores 25. The total of all the three boys is 60. Now if B and C had also scored 20 each, then the total remains at 60.

The value '20' is a representative value of the set in general. Why does the total not change? Because the increase in B's score is offset by the same decrease in C's score. That is we took 5 from C and gave it to B.

To find the average of this set, we simply take the total of all marks and divide it by the number of students.

i.e. 60 / 3 = 20.

This is the basic concept in averages.

Average = (Total of all observations)/(number of observations)

To take another example, if 4 students score 17, 19, 21 and 23 respectively, then the total is 80 and the average is (80/4) = 20.

Isn't this the same as taking 3 from the 4th student and giving it to the first and taking 1 from the third and giving it to the second? So all the four have 20 each. i.e. Each has the same value without affecting the total.

Typically, we get problems where the weight of some boys is given. We need to find the average if one of the boys is replaced with another.

How do we solve such problems?

Let's say there are 10 boys whose average weight is 50 kg. One of the boys is replaced by another who weighs 100 kg. So, what is the new average?

If the average weight of each boy is 50, that means each boy contributes 50 kgs to the total weight. (Individual weights do not matter.) So the total weight here is 10 x 50 = 500 kg.

When we remove 1 boy, we are effectively removing 50 kg from the total. So now we have 9 boys with a total weight of (500-50) = 450 kg.

The new boy brings in 100 kgs to the total. So now, total weight = 550 and number of boys = 10. So the average weight is now 550/10 = 55 kg.

Another way of doing this is, the 100 kg boy replaces a 50 kg boy. So from 100, 50 kg go towards maintaining the original total. The excess 50 kgs is divided equally among the 10 boys. So each boy gets an excess 5 kg. So the 50 kg boys are now to be taken as 55 kg boys and hence the average is 55 kgs.

Simple!

Percentages

One of the math topics which is actually the easiest, but a pet peeve of many students, is percentages.

Let's see if we can simplify it.

The word percentage literally means out of one hundred. (per century) So, all we do in this is try and see how many parts of 100 are involved in the game.

Assume we have an apple. We cut it into two equal halves. One part would be half of the whole. This is equal to 50 portions out of 100. (as 50 is half of 100). So one part out of two works out to be 50 per century or 50 %

We may do our calculations like this

out of 2 - 1
out of 100 - ?

? = (100 x 1)/2 = 50

Similarly, if we had 40 students in the class and 10 of them were boys, then 10 out of 40 is 1/4th. We know that 25 is 1/4th of 100. So 10 out of 40 is the equivalent of 25 %

Remembering reciprocals is a good way to do percentages. Reciprocals are nothing but 1/x where x is any natural number. For instance the reciprocal of 2, i.e. 1/2 = 50%

Reciprocal of 4 (double of 2) = 1/4 = 25% (half the reciprocal of 2)
Reciprocal of 8 (double of 4) = 1/8 = 12.5% (half the reciprocal of 4)

Calculating percentages of numbers is a very simple process, as long as we are good at doubling or halving numbers.

For instance, to calculate 37% of 500, we can split it as 10 + 10 + 10 + 5 + 2
10 % of any number is easy. 10% of 500 is 50
5% is half of 10% = 25
and 2% = twice 1% Now, 1% = 5 so 2% = 10

So we get 37% = 50+50+50+25+10=185

Of course, calculating it by formula 500x37/100 is also ok. Use whichever method is faster and more convenient.