Showing posts with label quantitative aptitude. Show all posts
Showing posts with label quantitative aptitude. Show all posts

14 January, 2017

STATISTICS(basic concept) NOTES for RRB NTPC EXAM

The words ‘Statistics’ appears to have been derived from the latin word ‘status’ meaning a (political) state. In its origin, Statistic was simply the collect of data on different aspects of the life of the people.

Statistics deals with data collected for specific purposes. We can make decisions about the data by analysing and interpreting it.

Central line tendency

Mean: The mean or average of a number of observation is the sum of the value of all the observation divided by the total number of the observations.

It is denoted by the symbol,  read as x bar

Here n is a number of observation.

Example- people were asked about the time in a week they spend in doing social work in their community. They said 10, 7, 13, 20 and 15 hours respectively.Find the mean (or average) time in a week devoted by them in social work.

Sol. The mean =(Sum of all the observations)/(Total number of observations)

=(10 + 7 + 13 + 20 + 15)/5=65/5=13 

So, the time spent by these 5 people in doing social work is 13 hours in a week.

The median- the median is that value of the given number of observations, which divide it into exactly two parts.  when the data is arranged in ascending or decreasing order.

The median of ungrouped data is calculated as follows:

(i) When the number of observation (n) is odd, the median is the value of the ((n+1)/2)ith observation.    

Example: If n = 15, the value of the ((15+1)/2)ith i.e. the 8th observation will be median

(ii) When the number of observation (n) is even, the median is the mean of the (n/2)ith and (n/2+1)ith observations.

Example: If n = 16 the mean of the value of the (16/2)ith and (16/2+1)ith observations. 

Example: The height (in cm) of 9 students of a class follows.

155, 152, 160, 144, 145, 148, 150, 147,149

Find the median of the data

Sol. Arrange the data in ascending data

144, 145, 147, 148, 149, 150, 152, 155, 160

Since the number of students is 9, an odd number.

Median is the height of the   = 5th student which is 149 cm

So, median = 149 cm

 Modes: The mode is that value of the observation which occurs most frequently i.e. an observation with the maximum frequency is called the mode.

Example: Find the mode of the following marks (out of 10) obtained by 20 students.

4, 6, 5, 9, 3, 2, 7, 7, 6, 5, 4, 9, 10, 10, 3, 4, 7, 6, 9, 9

Sol. We arrange this data in the following form:

2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 9, 9, 9, 9,10, 10

Here 9 occurs most frequently i.e. four times. So, the mode is 9.

24 October, 2016

Quant Study Notes: Probability

Dear Readers,

Today we’ll discuss about Probability. This topic is can fetch you marks easily but you need to know the right concepts and types of questions to practice.

All about Probability

Probability

Probability theory deals in random events. The central objects of probability theory are random variables, and events.

Event is generally defined as outcome of an experiment. Events can be classified as deterministic or probabilistic (random).

Deterministic Event

When an experiment is repeated under homogeneous conditions and it produces the same results, then the experiment is known as deterministic experiment. 

For example if a car runs at a speed of 50 km/hr under uninterrupted condition it will reach a place 100 km distant in 2 hr.

Random or Probabilistic Event

An experiment when repeated under identical condition does not produce the same result every time but out come is one of the several possible out comes, then such an experiment is known as a probabilistic experiment or a random experiment and these events are known as random events.  

For example if it is raining today. It may not rain tomorrow. If a coin is tossed, outcome may be a head or tail.

Sample Space

The sample space or universal sample space, often denoted S or U (for “universe”), of an experiment or random trial is the set of all possible outcomes.

Independent Events

If one event does not have any effect on other, then these events are known as independent events. If A and B are independent events then,

Example: A man throws two identical unbiased die. If he gets 3 on the first dice or an even number on the second, he wins. Find the probability that the man wins.

Solution: A = Event of getting 3 on the first dice and B = Event of getting an even number on the second dice

Conditional Probability

A conditional probability is the probability of an event given that another event has occurred. Suppose a card is randomly drawn from the deck of 52 cards and it is found a red card, the probability of finding that this card is a KING, is an example of conditional probability.

If event E2 has occurred already then probability of occurrence of E1 is denoted by

Example: A number is selected randomly from the set of numbers ranging from 1 to 100 number is found to be a multiple of 3. Find the probability that it’s a multiple of 7 also.

Solution: Method 1: Sample space S = {1, 2, 3………..100}

Given that the selected number is a multiple of 3, hence conditional sample space S_1 = {3, 6,………99}.

Now the number should be a multiple of 3 as well as 7, hence it must be a multiple of 21. There are 4 such numbers: 21, 42, 63 and 84.

Required sample space S_2 = {21, 42, 63, 84}

Probability = S2/S1 =4/33

Method 2: Suppose E2 is the event that number is a multiple of 3 and E1 is the event that number is a multiple of 7, then

22 September, 2016

Quant Study Notes: Profit and Loss

PROFIT & LOSS

Profit and loss are determined by the value of cost price and selling price. Cost price is the price at which an article is purchased and selling price is the price at which article is sold

Profit = selling price - Cost price 

Loss = Cost price - Selling price 

Percentage profit and loss are always calculated on cost price. 

If a cost price of m articles is equal to the selling Price of n articles, then Profit percentage 

MARKED PRICE

Marked price is also known as the list price. It is the price which is marked on the article.

Where CP = cost price and MP = marked price

DISCOUNT

Shopkeepers devise several ways to attract customers (consumers). Sometimes they sell an article at a price lower than its list price (LP)/marked price (MP). Recall that reduction offered by retailer on the list price is called discount. We may recall that

Discount = MP - SP

Example 1: Marked price of a dining table is Rs 1350. It is sold at Rs. 1188 after allowing certain discount. Find the rate of discount.

Solution:

MP of the dining table = Rs. 1350

SP of the dining table = Rs. 1188

Discount allowed = Rs. (1350 - 1188) = Rs. 162

Discount percent =162/1350×100=12

This the rate of discount is 12%

SUCCESSIVE DISCOUNTS

Sometimes more than one discount are offered by the shopkeeper on a single item or article. When two or more discounts are applicable successively to the list price of an article, they form the discount series.

Suppose a shopkeeper is offering 3 successive discounts of 10%, 20% and 30% then to calculate effective discount we assume that marked price is 100, then final value becomes 0.90 × 0.80 × 0.70 × 100 = 0.54 × 100 = 50.4

Total discount = 49.6%.

When there  are two successive Profit of x % and y % then the resultant profit  per cent is given by 

If there is a Profit of  x% and loss of  y %  in a transaction, then the  resultant profit or loss% is given by 

Note-  For profit use sign + in previous formula and for loss use – sign.

if resultant come + then there will be overall profit, if it come – then  there will be overall loss.

Example 2:

If two articles are sold at same selling price one at 30% profit another at 30% loss then what is his overall percentage profit or loss?

FALSE WEIGHT PROBLEMS

Shown or indicate weight is always equivalent to selling price, and actual/true weight is equivalent to cost price.

If a trader professes to sell his goods at cost price, but uses false weights, then 

Example 3:

A shopkeeper takes 20%, extra quantity while purchasing the milk, and gives 25% less than the indicated weight while selling the milk. Find the profit percentage of he sells at the cost price only. 

Solution: 

Suppose the price of milk = 1 Rs per ml shopkeeper takes 120 ml, and pays only Rs. 100

While selling he gives only 75 ml and shows 100 ml.

Total selling price of 120 ml

 100/75×120 = 160, hence percentage profit = 60% 

16 September, 2016

Reasoning Study Notes: Machine Input-Output

In machine input-output, the basic problem is the problem of time. You can obviously get the logic if you are not bound by time, but the challenge is to get the logic as quickly as possible and get done with the questions to score more.

Here you can learn how to solve machine input by saving your time

The general instruction of what a machine input-output question says are:

“When a word and number arrangement machine is given an input line words and number, it arranges them following a particular rule. The following is the illustration of input and re-arrangement”

 

Example: Input: name 37 11 is his 42 Khan 28

Step I:is name 37 11 his 42 Khan 28

Step II:is 42 name 37 11 his Khan 28

Step III:is 42 his name 37 11 Khan 28

Step IV:is 42 his 37 name 11 Khan 28

Step V: is 42 his 37 Khan name 11 28

Step VI:is 42 his 37 Khan 28 name 11

VIth step is the last step.

The last step is the final output the machine.

So what has happened in this example of VI Steps?

Input is given to you and it is simplified in the subsequent steps. By simplified we mean they’ve applied a certain logic, if you know questions of series, the series in made on a certain logic and by analysing that logic you get to know the next value in that series. Similarly, in questions of Machine Input Output, you have to analyse the given Input and its subsequent steps and understand or find out the logic behind it. It means your job is to identify the logic through which the input-output machine has transformed the input to output and you have to apply the same logic in the subsequent step of questions asked. And the last step is the final output.

FIND THE LOGIC!!!

Compare quickly the Input and the final step and try to deduce the logic through which the machine has produced the output.

Example: Input: 96 amber cola 84 new 6

Step I: 6 96 cola 84 new amber

Step II: 6 84 96 new cola amber

Step II is the final output

Here we can see that the logic applied is arranging the numbers in ascending order (right to left) and arranging the alphabets in alphabetical order (left to right).

Observe the happenings in the subsequent step. Is the machine shifting only 1 item at a time or is it shifting two or more?

Example: Input: 96 amber cola 84 new 6

Step I: 6 96 cola 84 new amber

Step II: 6 84 96 new cola amber

Step II is the final output

In this example, the machine is shifting two items at a time, i.e. a number and a word in each step.

Observe the direction in which shifting has taken place- left to right, right to left.

Example: Input: 96 amber cola 84 new 6

Step I: 6 96 cola 84 new amber

Step II: 6 84 96 new cola amber

Step II is the final output

Here the number are arranged from left to right direction and words are arranged in right to left direction

Quickly and carefully analyse and try to discriminate according to the first letter of given words is they are in alphabetical sequence or is there any particular arrangement related to vowels and consonants and analyse the numbers too.

Example: Input: assure 7 new 2 email 16 demand 3 quit 12 20 urban

Step I: assure 2 7 new email 16 demand 3 quit 12 20 urban

Step II: assure 2 7 new email 16 3 quit 20 urban demand 12

Step III: assure 2 email 3 7 new 16 quit 20 urban demand 12

Step IV: assure 2 email 3 7 quit 20 urban demand 12 new 16

Step V: assure 2 email 3 urban 7 quit 20 demand 12 new 16

Step VI: assure 2 email 3 urban 7 demand 12 new 16 quit 20

Step VI is the final output.

Logic: Here in each step a number and a word are arranged in pairs of Vowel+ Prime and Consonant +Composite. The words starting with vowels are arranged from left to right along with a prime number is ascending order. The words starting with a consonant are arranged from left to right (on the right end) in the next step along with a composite number is ascending order.

Quant Study Notes: Ratio and proportion

What is Ratio?

Ratio is a mathematical term used to compare two similar quantities expressed in the same units. The ratio of two terms ‘x’ and ‘y’ is denoted by x : y. In ratio x : y , we can say that x as the first term or antecedent and y, the second term or consequent.

In general, the ratio of a number x to a number y is defined as the quotient of the numbers x and y i.e. x/y. 

Example: The ratio of 25 km to 100 km is 25:100 or 25/100, which is 1:4 or 1/4, where 1 is called the antecedent and 4 the consequent.

Note that fractions and ratios are same; the only difference is that ratio is a unit less quantity while fraction is not. 

Compound Ratio

Ratios are compounded by multiplying together the fractions, which denote them; or by multiplying together the antecedents for a new antecedent, and the consequents for a new consequent. The compound of a : b and c : d is  i.e. ac : bd. 

Properties of Ratio:

a : b = ma : mb, where m is a constant

a : b : c = A : B : C is equivalent to a / A = b /B = c /C, this is an important property and has to be used in ratio of three things.

i.e. the inverse ratios of two equal ratios are equal. This property is called Invertendo

i.e. the ratio of antecedents and consequents of two equal ratios are equal. This property is called Alternendo.

This property is called Componendo.

  

This property is called Dividendo

This property is called Componendo - Dividendo


Dividing a Quantity Into a Ratio

Suppose any given quantity ‘a’ is to be divided in the ratio of m : n. 

Then, 

Proportion

When two ratios are equal, the four quantities composing them are said to be in proportion. 

If a/b=c/d, then a, b, c, d are in proportions. 

This is expressed by saying that ‘a’ is to ‘b’ is to ‘c’ is to ‘d’ and the proportion is written as 

a : b :: c : d or a : b = c : d 

(product of means = product of extremes)

If there is given three quantities like a, b, c of same kind then we can say it proportion of continued.

a : b = b : c the middle number b is called mean proportion. a and c are called extreme numbers.

So, b2 = ac. (middle number)2 = ( First number x Last number ).

Application: These properties have to be used with quick mental calculations; one has to see a ratio and quickly get to results with mental calculations.

Example: 

 should quickly tell us that 

Q. A certain amount was to be distributed among A, B and C in the ratio 2 : 3 : 4, but was erroneously distributed in the ratio 7 : 2 : 5. As a result of this, B received Rs. 40 less. What is the actual amount? 
(a) Rs. 210

(b) Rs. 270

(c) Rs. 230

(d) Rs. 280

(e) None of these

Q. Mixture of milk and water has been kept in two separate containers. Ratio of milk to water in one of the containers is 5 : 1 and that in the other container 7 : 2. In what ratio the mixtures of these two containers should be added together so that the quantity of milk in the new mixture may become 80%? 

(a) 2 : 3

(b) 3 : 2

(c) 4 : 5 

(d) 1 : 3 

(e) None of these

10 September, 2014

Top Recommended books for Quantitative aptitude and reasoning for placements and competitive exams

Hello Friends,


Today  i am providing you a list of best books for your preparation for placements and competitive exams like CAT,XAT,MAT,UPSC,Civil Services,IBPS,PO etc.

Below given serial is not the rating of books but just a normal sequence of books (all books are equally good and best for your preparation).