Download Aptitude training Material and more Study notes Training and Development in PDF only on Docsity!
Index
S.No Topic Page No
- 1 Number system Quantitative Aptitude - 2 Problems on Numbers - 3 Average - 4 Problems on Ages - 5 Ratio and Proportion - 6 Allegation and Mixture - 7 Percentage - 8 Profit and Loss - 9 Interest Calculation
- 10 Time Trains Speed Distance& Problems on
- 11 Boats and Streams
- 12 Time and Work
- 13 Pipes and Cistern
- 15 Permutation and Combination
- 16 Logarithms
- 17 Geometry and Mensuration
- 18 Data Interpretation.
- 19 Simplification
- 20 Elementary Statistics - 1 Number Series Reasoning Ability - 2 Odd Man Out & Analogy - 3 Coding & Decoding - 4 Direction - 5 Blood Relation - 6 Syllogism - 7 Image Analysis - 8 Puzzles - 9 Logical Sequence of Words
- 10 Cubes & Dice
- 11 Clocks
- 12 Calendar
- 13 Data Arrangement
- 14 Data Sufficiency
- 15 Statement and Argument
- 1 Parts of Speech Verbal Ability
- 2 Tense
- 3 Preposition & Article
- 4 Subject Verb Agreement
- 5 Active & Passive Voice
- 6 Direct Speech & Indirect Speech
- 7 Degrees of Comparison
- 8 Conditional Clause
- 9 Question Tag , Prefix & Suffix
- 10 Idioms and Phrases & Homonyms
- 11 Synonyms & Antonyms
- 12 Verbal Analogy
- 13 Sentence Completion
- 14 Spot the Error
- 15 Sentence Correction
- 16 Jumbled Sentence
- 17 Reading Analysis, Critical Reasoning & Cloze Test Comprehension, Passage
- 18 Statement & Assumption
- 1 Formulas / Rules Formulas
Example: 592482 is divisible by 3, since sum of its digits =(5+9+2+4+8+2)=30=(5+9+2+4+8+2)=30, which is divisible by 3
Divisibility by 4 - A number is divisible by 4 if number formed using its last two digits is completely divisible by 4.
Ex: 92648
Divisibility by 5 - A number is divisible by 5 if its unit digit is 0 or 5.
Ex: 34565 ,
Divisibility by 6 - A number is divisible by 6 if the number is divisible by both 2 and 3.
Ex: 4560
Divisibility by 8 - A number is divisible by 8 if number formed using its last three digits is completely divisible by 8.
Ex: 48512
Divisibility by 9 - A number is divisible by 9 if sum of its digits is completely divisible by 9.
Ex: 60732 is divisible by 9, since sum of digits =(6+0+7+3+2)=
Divisibility by 10 - A number is divisible by 10 if its unit digit is 0.
Ex: 86750
Divisibility by 11 - A number is divisible by 11 if difference between sum of digits at odd places and sum of digits at even places is either 0 or is divisible by 11.
Ex: The number 4832718 is divisible by 11, since :(sum of digits at odd places) - (sum of digits at even places)
=(8+7+3+4)−(1+2+8)=
Division Algorithm:
When a number is divided by another number then
Dividend = (Divisor x Quotient) + Reminder
Formulas of Number Series:
- 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/
- (1^2 + 2^2 + 3^2 + ..... + n^2 ) = n ( n + 1 ) (2n + 1) / 6
- (1^3 + 2^3 + 3^3 + ..... + n^3 ) = (n(n + 1)/ 2)^2
- Sum of first n odd numbers = n^2
- Sum of first n even numbers = n (n + 1)
Mathematical Formulas:
- (a + b)(a - b) = (a^2 - b^2 )
- (a + b)^2 = (a^2 + b^2 + 2ab)
- (a - b)^2 = (a^2 + b^2 - 2ab)
- (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
- (a^3 + b^3 ) = (a + b)(a^2 - ab + b^2 )
- (a^3 - b^3 ) = (a - b)(a^2 + ab + b^2 )
- (a^3 + b^3 + c^3 - 3abc) = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ac)
H.C.F : H.C.F stands for Highest Common Factor. The other names for H.C.F are Greatest Common Divisor (G.C.D) and Greatest Common Measure (G.C.M). The H.C.F. of two or more numbers is the greatest number that divides each one of them exactly.
L.C.M : The least number which is exactly divisible by each of the given numbers is called the least common multiple of those numbers.
LCM= (LCM of the numerators)/(HCF of the denominators)
HCF= (HCF of the numerators)/(LCM of the denominators)
Product of two numbers = Product of their H.C.F. and L.C.M.
Arithmetic progression:
It is a sequence of numbers in which the difference between any two successive numbers remains constant, e.g. the series 1,2,3,4.., 3,5,7,9.. and 5,10,15,20...
An Arithmetic progression with first term 'a' and common difference d is given by;
a, (a+d), (a+2d), (a+3d),….(a+nd)
The nth term of an arithmetic progression is given by;
Tn = a + (n-1)d
Where,
nth term is denoted by Tn.
The sum of first n terms of an arithmetic progression is given by:
Sn=n/2[2a+(n-1)d]
Sn=n/2[First term+Last term]
Geometric progression:
It is a sequence of numbers in which each number is found by multiplying the previous number by a fixed number or a common ratio.
A geometric progression with first term 'a' and common ratio 'r' is given by;
a, ar, ar^2,ar^3,…ar^n
The nth term of a geometric progression is given by;
Tn= ar^n-
The sum of first n terms in a geometric progression is given by;
12.There are three drums with 1653 litre, 2261 litre and 2527 litre of petrol.The greatest possible size of the measuring vessel with which we can measure up the petrol of any drum while every time the vessel must be completely filled.
A) 31 B) 27 C) 19 D) 41
13.The HCF of two numbers is 27 and their sum is 216.These numbers are:
A) 27,189 B) 81,189 C) 1 08,108 D) 154,
14.What is the least possible number which when by 24,32 and 42 in each case it leaves the remainder 5?
A) 677 B) 520 C) 540 D) 600
15.What is the least possible number which when divided by 13 leaves the remainder 3 and when it is divided by 5 it leaves the remainder 2.
A) 42 B) 54 C) 89 D) 102
16.Find the number of zeros at the end of 100!
A) 25 B) 50 C) 24 D) 100
17.Find the remainder when 5^123 is divided by 7?
A) 6 B) 5 C) 7 D) 8
18.Find the largest number of 4-digits divisible by 12, 15 and 18.
A) 9900 B) 9750 C) 9450 D) 9000
- Find the smallest number of five digits exactly divisible by 16, 24, 36 and 54
A) 10368 B) 10342 C) 12000 D) 12342
20.Find the total number of factors of 540.
A) 24 B) 20 C) 30 D) None of these
Exercise :
1.If 653x14y is divisible by 80 then the value of x+y is:
A) 2 B) 3 C) 4 D) 6
2.How many numbers are there between 200 and 800 which are divisible by both 5 and 7?
A) 35 B) 16 C) 17 D) Can’t be determined
3.Find the unit digit of 3^47+7^
A) 8 B) 6 C) 7 D) 4
- Find the remainder when 11^8 is divided by 7.
A) 2 B) 6 C) 5 D) 4
- 2+2^2+2^3+...+2^9=?
A) 2044 B) 1022 C) 1056 D) None of these
- A number which when divided by 32 leaves a remainder of 29.If this number divided by 8 the remainder will be:
A) 0 B) 1 C) 5 D) 3
- The ratio of two number is 15:11.If their HCF be 13 then these numbers will be:
A) 15:11 B) 75:55 C) 105:77 D) 195:
- The HCF and LCM of the two numbers is 12 and 600 respectively. If one of the number is 24, then the other number will be:
A) 300 B) 400 C) 1500 D) 650
- Find the least possible number which can be divided by 32,36 and 40
A) 1440 B) 1500 C) 1489 D) 1200
- What is the least possible number which when divided by 18,35 and 42 it leaves the 2, and 26 as the remainders respectively?
A) 614 B) 342 C) 600 D) 656
- The largest possible length of a tape which can measure 525 cm, 1050 cm and 1155 cm length of cloths in a minimum number of attempts without measuring the length of a cloth in a fraction of the tape’s length?
A) 25 B) 105 C) 75 D) None of these
- Find the smallest positive number which is exactly divisible by 1/3,1/2,3/7 and 4/11.
A) 12 B) 14 C) 18 D) 20
13.The least number which when divided by 2,3,4,5 and 6 leaves the remainder 1 in each case. If the same number is divided by 7 it leaves no remainder. The number is:
A) 231 B) 301 C) 371 D) 441
14.Three bells, toll at interval of 36 sec, 40 sec and 48 sec respectively. They start ringing together at particular time. They next toll together after:
A) 6mins B) 12 mins C) 18 mins D) 24 mins
15.The remainder when 8^1785 is divided by 7 is:
A) 5 B) 1 C) 6 D) Can’t be determined
16.The number of possible pairs of numbers, whose product is 5400 and HCF is 30:
A) 1 B) 2 C) 3 D) 4
PROBLEMS ON NUMBERS
A problem on numbers is one of the sub-categories in number system. Question is the form of a puzzle. To calculate problems on numbers: Analyze the given conditions Assume the unknown numbers and form equations. Finally, find the unknown numbers. Examples:
- A number is as much greater than 36 as is less than 86. Find the number? Solution: Let the number = x x – 36 = 86 – x = 86 + 36 = 122 2x = 122 122/2 = 61 x = 61
- Find a number such that when 15 is subtracted from 7 times the number, the result is 10 more than twice the number? Solution: Let the number = x 7x – 15 = 2x + 10 5x = 25 x = 25 / 5 = 5
- The sum of two numbers is 184. If one third of the one exceeds one-seventh of the other by 8, find the smaller number. Solution: Let the numbers be x and 184-x x/3 – 184-x / 7 = 8 7x – 3 (184-x) = 168 10x = 720 x = 72 The numbers are 72 and 112 Hence the smaller number is 72
- The difference of two numbers is 11 and one fifth of their sum is 9. Find the numbers. Solution: Let the numbers = x and y x – y =11 ……………………(1) 1/5 (x+y) = 9 x + y = 45 …………………..(2) Add (1) & (2) 2x = 56 x = 28 Numbers are 28 and 17
- The average of four consecutive even numbers is 27. Find the largest of these numbers? Solution: Let the numbers be = x, x+2, x+4, x+ Sum of these numbers = 27 * 4 = 108 or 4x = 96 or x = 24 Largest number = x + 6 = 30
SQUARES AND CUBES SQUARES: Tips:
(a+b)^2 = a^2 + 2ab + b^2
EXAMPLE:
1. (12)^2
By using a^2 2ab b^2 Ie., a = 1 ; b = 2 a^2 2ab b^2 (^12) 212 22
Hence, the square of 12 = 144
2. 26^ By using a^2 2ab b^2 Ie., a = 2 ; b = 6 a^2 2ab b 2
(^22) 226 62
(add 2 as the carry over) 4 + 2 = 6 (In the final calculation take the whole number to the solution)
(add 3 as the carry over) 24 + 3 = 27 (only take 7 and carry over 2)
(only take the unit digit number)
Hence, the square of 26 = 676
Factorization method:
- 676
26 2 676 4 46 276 276 0
Hence, 676 = 26
- 3364
58 5 3364 25 108 864 864 0
Hence, 3364 = 58
CUBE ROOT:
Tips:
Step 1: Group the number (from right to left) as three digit in one group ie., 17 and 576 Step 2: Group 1 576 = check the last digit ie., 6 6
Unit digit of number cubed Unit digit in cube roots 1 1 2 8 3 7 4 4 5 5 6 6 7 3 8 2 9 9 0 0
Step 3: Group 2 17 = comes in between 2^3 = 8 and 3^3 = 27, from this 2 options select the smaller number 2^3 = 8 2 Hence, the cube root of 14576 = 26
Step 1: Group the number (from right to left) as three digit in one group ie., 140 and 608 Step 2: Group 1 608 = check the last digit ie., 8 2 Step 3: Group 2 140 = comes in between 5^3 = 125 and 6^3 = 216, from this 2 options select the smaller number 5^3 = 125 5 Hence, the cube root of 140608 = 52
Workout:
- If the sum of 2 numbers is 42 and their product is 437, then find the absolute difference between the numbers? A) 2 B) 4 C) 6 D) 8
- Find the sum of the squares of three consecutive odd numbers is 2531. Find the numbers? A) 27, 29, 31 B) 28,31,29 C) 28,30,32 D) 26,28,
3.The sum of two numbers is 15 and the sum of their squares is 113. Find the numbers? A) 7,9 B) 8,9 C) 7,8 D) 8,
4.The sum of seven consecutive even natural numbers is 1617. How many of these numbers are prime? A) 227,224 B) 229,233 C) 232,227 D) 251,
- By how much is 3/4th^ of 568 lesser than 7/8th^ of 1008? A) 444 B) 448 C) 452 D) 456
- The difference between a number and its three-fifths is 50. What is the number? A) 75 B) 100 C) 125 D) None of these
- If a number is added to two-fifths of itself the value so obtained is 455. What is the number? A) 355 B) 350 C) 400 D) 420
- If a number is multiplied by two-thirds of itself the value so obtained is 864. What is the number? A) 34 B) 36 C) 38 D) 44
- If 1/3rd^ of 1/4th^ of a number is 15, then 3/10th^ of that number? A) 35 B) 36 C) 45 D) 54
- The difference between 3/5th^ of 2/3rd^ of a number and 2/5th^ of 1/4th^ of the same number is 288. What is the number? A) 850 B) 895 C) 955 D) 960
4.A number is doubled and 9 is added. If the resultant is trebled, it becomes 75. What is the number? A) 3.5 B) 6 C) 8 D) None of these
- Three-fourth of a number is 60 more than its one-third. The number is: A) 84 B) 108 C) 144 D) None of these
- A number whose fifth part increased by 4 is equal to its fourth part diminished by 10, is? A) 240 B) 260 C) 270 D) 280
- When 24 is subtracted from a number, it reduces to its four-seventh. What is the sum of the digits of that number? A) 1 B) 9 C) 11 D) Data inadequate
- Find the number which when multiplied by 15 is increased by 196? A) 14 B) 20 C) 26 D) 28
- If a number, when divided by 4, is reduced by 21, the number is? A) 18 B) 20 C) 28 D) 38
- A number whose fifth part increased by 4 equal to its fourth part diminished by 10, is? A) 240 B) 260 C) 270 D) 280
A) 202 B) 232 C) 242 D) 332
- The square root of 41209 is equal to A) 103 B) 203 C) 303 D) 403
- The square root of 123454321 is? A) 111111 B) 12341 C) 11111 D) 11211
- The number of digits in square root of 625685746009 is? A) 4 B) 5 C) 6 D) 7
15. √^ =?
A) 19 B) 155 C) 256 D) None of these
16.√^ =?
A) 17 B) 312 C) 441 D) None of these
- The value √^ √^ √^ √^ is
A) 4 B) 6 C) 8 D) 10
- Evaluate: √^ √^ √
A) 3 B) 5 C) 6 D) 6.
19.√^ is equal to A) 14 B) 15 C) 18 D) 24
A) 48 B) 54 C) 58 D) 84
- Columbus started his journey from Lucknow to Kolkata, which is 200km, at the speed of 40 km/hr then he went to Bangalore which at the speed is 300 km, at the speed of 20 km/hr. Further he went to Ahmedabad which is 500km, at the speed of 10 km/hr. The average speed of Columbus is:
A) 14 2/7 km/hr B) 14 5/7 km/hr C) 15.6 km/hr D) none of these
- The average weight of all the 11 players of Indian cricket team is 50kg. If the average of first six lightest weight player is 49 kg and that of the six heaviest players is 52 kg. The average weight of the player which lies in the sixth position in the list of players when all the 11 player are arranged in the order of increasing or decreasing weights:
A) 56 kg B) 52 kg C) 51kg D) none of the above
- The average presence of students in a class on Monday, Tuesday and Wednesday is 30 and on the Wednesday, Thursday, Friday and Saturday is 28 then number of students who attended the class on Wednesday is, if the average number of students on all the six day is 27:
A) 24 B) 25 C) 20 D) 40
- The average age of A, B, C, D and E is 40 years. The average age of A and B is 35 years and the average age of C and D is 42 years. The average age of E is:
A) 46 B) 48 C) 32 D) none of these
- The average temperature on Monday, Tuesday and Wednesday is 38 degree Celsius The average temperature on Tuesday, Wednesday and Thursday is 43 degree Celsius, if the average temperature on Monday and Thursday is 18.5 degree Celsius, The average temperature on Monday is:
A) 11 degree Celsius B) 21 degree Celsius C) 35 degree Celsius D) 27 degree Celsius
- The average weight of a class of 20 students is 45kgs. A new student whose weight is 40kgs replaces an old student of this class. Hence the average weight of the whole class decreases by 1 kg. The weight of the replaced student is:
A) 55 kgs B) 50kgs C) 60kgs D) none of these
- The average length of first 3 fingers is 3 inches and the average length of the other 2 fingers i.e., thumb and the index finger is 2.8 inches. If the length of the index fingers is 3 inches then the length of the thumb is:
a) 2 inches B) 2.6 inches C) 3 inches D) none of these
- Pankaj went to the post-office at the speed of 60 km/hr while returning for his home he covered the half of the distance at. the speed of 10 km/hr, but suddenly he realized that he was. getting late so he increased the speed and reached the home by. covering rest half of the distance at the speed of 30 km/hr. The average speed of the Pankaj in the whole length of journey is:
A) 5.67 km/hr B) 24 km/hr C) 22.88 km/hr D) 5.45 km/hr
- the average of 9 numbers is 11. if each of these 9 numbers is multiplied by 5and then 5 is added to each of these resultant numbers then the new average is?
A) 20 B) 30 C) 60 D) 50
- The average rain fall in the months of January and February is 6 cm and in the months of March to June is 5 cm and July to October is 10 cm and in the November and December, it is 6 cm. The average rainfall for the whole year is:
A) 7 B) 5.5 C) 7.5 D) None of these
- The average weight of 11 players of Indian cricket team is increased by 1 kg, when one player of the team weighing 55 kg replaced by a new player. The weight of the new player is
A) 55 kg B) 64 kg C) 66 kg D) none of these
- The average age of a family of 6 members 4 years ago was 25 years. Meanwhile a child was born in this family and still the average age of the whole family is same today. The present age of child is
A) 2 years B) 1 ½ years C) 1 year D) data insufficient
- The average age of 30 students of a class is 30 years. When the average age of class teacher is also included, the average age of the whole class increases by 1 year. The age of the class teacher is:
A) 31 years B) 60 years C) 61 yeas D) none of these
- The averages of A and B is 400 and the averages of C and D is 600 the averages of A, B, C and D is:
A) 500 B) 450 C) 525 D) 625
- The average weight of A, B, C and D is 40 kg. A new person E is also included in the group, then the average weight of the group is increased by 1 kg. Again a new person F replaces A, then the new average of 5 person becomes 42. The average weight of B, C, D, F is
A) 42 B) 41.25 C) 42.5 D) none of these
- The average income of A,B and C is Rs.12000 per month and the average income of B,C and D is Rs. 15000 per month. If the average salary of D is twice that of A , then the average salary of B and C is (in Rs.):
A) 8000 B) 18000 C) 13500 D) 9000
Exercise:
- The average price of 80 computers in an electronic shop is Rs. 30,000, if the highest and lowest price computers are sold out then the average price of the remaining 78 computers is Rs. 29,500. The cost of the highest price computer is Rs. 80,000. The cost of lowest price computer is:
A) 19000 B) 20000 C) 29000 D) can’t be determined
