Aptitude –I

Prologue

Easy Calculation methods (You will surely enjoy it)

Ratio and Proportion (Basics only)

Probability (Basics only)

Problem (Solved & Explained)

Note – I am a learner like you so do forgive me for the mistakes

Introduction

Easy Calculations (I hope it is easy). As there’s a proverb “time and tide waits for none” and this is truer in case of Paper-2(CSAT) and for paper-1 “Time is an illusion” lots of time nothing to do (lolz)

Fast Multiplication

1) Quick square of 2 digit number ending with 5 (e.g. 15,25,35,45,55 etc)Step1: Multiply the first digit by (itself + 1), and put 25 on the end. Step2: Ex: 15 i.e. (1*(1+1)) &25 = 225

Another e.g. 35Step1: Multiply the first digit (In this case its 3) by itself + 1, and put 25 on the end. Step2: Ex: 35 i.e. (3*(3+1)) &25 = 1225

2) Forward MethodWe know method to find square of a number ending with 5,say Square of 25=625, then just have a look to find square of 26.25’s square=625(known)26’square=25’square+ (25+26) =625+51=676.So square of 26 is 676.Another e.g. 16From above example we know 15’s square = 225Now we have to find 16’s square= 15’s square + (15+16) = 225+31 =256

3) Reverse MethodReverse approach through which you will able to find out squares of a number which is one less than given number.Consider the following example:

Suppose we know square of a number, say, ; how to find square of 14?(15)’square=225(known)(14)’square=225-(14+15) =225-29 = 196  Please try it for 35 also

4) Now some fast calculation method (Within 5 sec you have to do this calculation)11’square=11+1/1’square=12/1=12112’square=12+2/2’square=14/4=14413’square=13+3/3’square=16/9=169.14’square=14+4/4’square=18/16 (Apply step no 6 here) 18/16=18+1/6=196.15’square=15+5/5’square=20/25=20+2/5=225.16’square=16+6/6’square=22/36=22+3/6=256

The formula is self explanatory. However, let me explain it in detail for more clarification.• The slash is used just as a operator.• Our operating zone is 10 X 1 or simply 10.• 11 is more than 10.• We add 1 to 11to make 12.• The number of digits after the slash can be only one.• If the number of digits after the slash exceeds one, then we place only the rightmost digit on the extreme right after the slash, and the remaining gets added to the number on the left hand side of the slash. This calculation can be done up to 19

5) But for 20 formulas is slightly different.The slight Change in formula as follows:21’square=2 X (21+1)/1’square= 2 X (22)/1=44/1=441.This change is because now we are operating in the 10 X 2 Zone.Another e.g. 25 = 2 X (25+5)/5’square = 2X30/25 = 60/25 = 6(0+2)5 = 625

6) Similarly we can calculate Square of 31 but with slight change as follows:31’Square=3 X (31+1)/!’Square= 3 X (32)/1=96/1=961.One more example = 35’square = 3 X (35+5)/5’square = 3X40/25 = 120/25 = 1(20+2)5= 1225

7) Fast Multiplication methods of square of numbers

52 X 52 = (52 + 2) X (52 - 2) + (2 X 2) =

54 * 50 + 4 = 2704

To square any two-digit number x =52(in our case), decide what number p=50(we have taken so it will lower to n= 2 to reach 50), the nearest multiple of 10. Fast Multiplication of numbers

8) Easy method to calculate 3 digit numbers

Step1: If you have a large number to multiply and one of the numbers is even, you can easily subdivide to get to the answer Step2: Ex: 32 x 125, is the same as: 16 x 250 is the same as: 8 x 500 is the same as: 4 x 1000 = 4,000

9) 3 digit multiplication (Uploading an image (Its self explanatory) from internet. It’s the best of the lot.)

341 X 562 = 191642 Note – First timers it might be a bit tough for all of you. But by practice you will surely master it

10) Talking of 99. A Simple multiplication technique of 99 with any numberE.g. 23 X 99 = 23(100-1)= 2300 – 23 = 2277

It’s just and extension of 9e.g. 5 X 9 = 5(10-1) = 50-5 = 45

Division tricks (Mind you these are not just for CSAT exam but any competitive exams)

1) To calculate the remainder when number is divided by 27 & 37e.g 34568276Step 1: make triplets of number starting from units place 034 …..568……276Step 2: now add all the triplets 034+568+276 = 878

Divide it by 27 we get remainder as 14Divide it by 37 we get remainder as 27

2) To calculate the remainder when number is divided by 27 & 37e.g 2387850765Step 1: make triplets of number starting from units place 002…..387 …..850……765Step 2: now add all the triplets 002+387 +850+765= 2004

Divide it by 27 we get remainder as 6Divide it by 37 we get remainder as 6

3) Dividing number by 3 Step 1: Just add up all the digits until you have a single number. Step 2: If THAT number is divisible by three, so is the original number. Case in point: 8787. If you add 8+7+8+7 you get 30. Then add 3+0, which equals three. Step 3: Three is definitely divisible by three, so you know that 8787 is too.

Continuation of this: Division by 6

If a number is both divisible by three (see the three rule) AND an even number (ending in 0, 2,4, 6 or 8) then it is divisible by six too. 312 is an even number and if you add up all the digits they equal six, which is divisible by three. Therefore 312 is divisible by six.

4) Divisible by 4 Step 1: Are the last two digits in your number divisible by 4? Step 2: If so, the number is too! Step 3: For example: 358912 ends in 12 which is divisible by 4, thus so is 358912

5) Divisible by 11Step :1 Find odd = sum of odd numbered digit and even = sum of even numbered digits.Step : 2 If (sum-odd) is 0 or multiple of 11, then the number is divisible by 11.

879197: odd = 8 + 9 + 9 = 26 even= 7 + 1 + 7 =15 odd – even = 26 – 15 = 11 Hence, 879197 is divisible by 11.

These are some simple techniques for fast calculation.

Ratio and Proportion

Q1) what is Ratio and Proportion?

A) Ratio - is a comparison of two numbers a: b For example, if 25 students are reading my article right now -- 10 boys and 15 girls --, then the ratio of boys to girls is 10 to 15. A ratio can be expressed in three ways. The ratio of boys to girls might be expressed as

1)10 to 15

2)10:15 

10 3) --- 15

Proportion - A proportion is simply a statement that two ratios are equal. It can be written in two ways: as two equal fractions a/b = c/d; or using a colon, a:b = c:d

The following proportion is read as "twenty is to twenty-five as four is to five."

In problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion.

Step – I = To find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

Most of the time ratio of 3 numbers problem asked in CSAT.

Ratio and Proportion formulas:

1) A ratio of “a” and “b” is denoted by a:b and is read as: “a is to b”.

in a ratio the first part ( “a”=3 in our example ) is called Antecedent and second part ( “b”=4 in our example ) is called Consequent.

2) A duplicate ratio is the ratio of second degree of the original ratio. For example the duplicate

ratio of  is

3) The ratio obtained by multiplying two or more ratios term wise is called compounded ratio.

for example compounded ratio of ratios , , is

4) The ratio obtained by adding two or more ratios term wise is called “Addendo”.

for example Addendo ratio of ratios , , is

5) A proportion of ratios “a:b” and “c:d” is denoted by: a:b :: c:dwhere “a” and “d” are called Extremes and “b” and “c” are called means.

6) A proportion a:b :: c:d or can also be re-written as:

i) and is called “Invertendo”.

ii) and is called “Alternando” .

iii) and is called “Componendo”.

iv) And is called “Dividendo”.

v) And is called “Componendo and Dividendo”.

Solved and Explained Problems

1) In a committee (e.g. Lokpal Bill committee) of 48 members, the ratio of Men & Women are 3:1. How many women should join the committee to make the ratio 9:5 ?

A) Solution :

Traditional MethodStep 1 = Total members in Lokpal Bill Committee (Just an hypothetical example) = 48The ratio of men & women = 3+1 =4 (Note ratio of men = (3/4)) & women (1/4) )Step 2 = Total number of men (48 X(3/4) = 36Total number of women (48 X 1/4) = 12Step 2 = to find the number of women required to make the ratio 9:5Let the total required women be w, so total women will be (12+w)

Step 3 = 36

12+w=9

5

Using cross multiplication 9(12+w) = 36*5108+9w=180

w = 8 (Total number of women added in Lokpal Committee)So total women = 12+8=20

Shortcut method

Total number of required women = total number

((difference f ratio1)+(difference of ratio 2))

For above example = 48

((9−5 )+(3−1)) =486 = 8

2) In a Mid Meal Scheme there are 2100 students. They have reserve food for 50 days. Some students among them are on leave. Now they can utilize the food reserves for 75 days.How many students are on leave.

A) SolutionNumber of students left = xTotal students = 2100 && Food reserves = 50Presently students = 2100 – k && Food reserves =75Note if the number of soldier reduces the number of day increases (Inversely proportional logic)

5075 = 2100−k

2100

By solving we get k=700 (Number of students left)

ShortCut Method :

Formulae = A (B−C )B

A=2100B=Number of days after students leave = 75C= Number of original days

=2100(75−50)75

= 700

Probability (शक्यता)

Probability is the chance that something will happen - how likely it is that some event will happen.For e.g. 1st January 2014 will happen 100 % (Certain)

Scaling of Probability (So its all about 0-1)

Probability

How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

 

Tossing a Coin 

When a coin is tossed, there are two possible outcomes:

heads (H) or tails (T)

We say that the probability of the coin landing H is ½.

And the probability of the coin landing T is ½ 

Throwing Dice 

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.

The probability of any one of them is 1/6.

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability =  1

6

Probability =

Q1) What is sample space & sample point ?

A1) The Sample Space is all possible outcomes.

A Sample Point is just one possible outcome. And an Event can be one or more of the possible outcomes.

 :  Q2) A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a red marble? a green marble? a blue marble? a yellow marble?

Outcomes:   The possible outcomes of this experiment are red, green, blue and yellow.

Probabilities:   P(red)  =  # of ways to choose red  =    6    =    3   total # of marbles 22 11  

P(green)  =  # of ways to choose green  =    5   total # of marbles 22  

P(blue)  =  # of ways to choose blue  =    8    =    4   total # of marbles 22 11  

P(yellow)  =  # of ways to choose yellow  =    3   total # of marbles 22

The outcomes in this experiment are not equally likely to occur. You are more likely to choose a blue marble than any other color. You are least likely to choose a yellow marble.

(Note – Hypothetical Problem – Dr Man Mohan Singh(Mohan) fans do forgive me) Q3) Mohan getting late for cabinet meeting(total 78 council of ministers (Not relevant for problem)).He dresses up and searches for his socks in a drawer consisting of (12 white , 8 red socks , and 10 yellow) What is the probability that Mohan will be able to pull a pair of white socks from his drawer in just two pulls? Reason for white socks = Solution on the issue of IFS officer treated badly

A3) Step 1 =total number of socks (30) =Sample Space

The event that Mohan wants to happen is that he pulls two white socks in a row, so pulling a white sock on the first pull and another white sock on the second pull would be called favorable outcomes.

Any other outcomes, such as pulling yellow socks or red socks on either pull aren't going to help Mohan, so we aren't interested in those outcomes.

Step 2 = As you can see, there are 12 chances out of thirty that Sam will pull out a white sock on the first pull. We make the ratio 12/30 to represent the probability that Sam will get a white sock. We can reduce 12/30 to 2/5 if we want.

Step 3 = Now there are only 11 white socks in Sam's drawer, but still 8 red and 10 yellow ones for a total of 29 socks in the drawer.

Sam has 11 chances out of 29 of pulling out a white sock on the second time he rummages in the drawer.

We call Sam's second pull a dependent event because what happened with the first pull influences what happens with the second pull.

Step 4 = To calculate the probability that Sam will come up with two white socks on the first two pulls, we multiply the ratio from the first pull with the ratio of the second pull and come up with a third ratio

P (white,white) =1230 X

1129

= 25 X

1129

= 22145 (Mohan happy and so is the IFS officer)

Note – Most of the probability concepts explained beautifully and comprehensively by Mrunal Sir. So I am not elaborating more on Probability

Name – Arun Chettiar

References –

1. www.Mrunal.org2. www.google.com 3. R S Agarwal Book4. www.IndiaBix.com