P REPARATION 1 Prepared by : Rupal Patel TOPICS Numbers HCF and - - PowerPoint PPT Presentation

QUANTITATIVE APTITUDE PREPARATION

Prepared by : Rupal Patel

1

TOPICS

 Numbers  HCF and LCM  Simplifications  Square roots and cube roots  Problems on numbers  Surds and Indices  Ratio and Proportion  Chain Rule  Time and Work  Pipes and Cistern  Permutations and Combinations

2

HINDU ARABIC NUMBER SYSTEM

We have total 10 digits in Hindu Arabic

System.

Namely,

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Number is a group of digits called numerals. E.g. 83526

3

PLACE OF EACH DIGIT

E.g. 83526

8

3 5 2 6

4

Unit’s Place Ten’s Place Hundred’s Place Thousand’s Place Ten Thousand’s Place

HOW MANY ZEROS?

In 1 Thousand?

3 Zeros

In 10 Thousand?

4 Zeros

In 1 Lakh?

5 Zeros

In 1 Crore?

7 Zeros

In 10 Crore?

8 Zeros

5

WRITE THE GIVEN NUMBERS IN WORDS

1.

9,04,06,002 Nine crore four lakh six thousand two

2.

1,60,05,014 One crore sixty lakh five thousand fourteen

3.

5,04,080 Five lakh four thousand eighty

4.

2,07,09,207 Two crore seven lakh nine thousand two hundred seven

6

WRITE THE GIVEN NUMBERS IN FIGURES

1.

Six lakh thirty-eight thousand five hundred forty-nine 6,38,549

2.

Twenty-three lakh eighty thousand nine hundred seventeen 23,80,917

3.

Eight crore fifty-four lakh sixteen thousand eight 8,54,16,008

4.

Four lakh four thousand forty 4,04,040

7

FACE VALUE AND PLACE VALUE

The face value of a digit in a numeral is its own

value at whatever place it may be

E.g. 6872 Face value of 6 is 6 Face value of 8 is 8 Face value of 7 is 7 Face value of 2 is 2 Place value of 6 is 6000 Place value of 8 is 800 Place value of 7 is 70 Place value of 2 is 2

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EXAMPLES: FACE VALUE AND PLACE VALUE

1.

The difference between the place value and the face value of 6 in numeral 856973 is

a)

973

b)

6973

c)

5994

d)

None of these

2.

The difference between the place value of two seven’s in the numeral 69758472 is

a)

75142

b)

64851

c)

5149

d)

699930

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EVEN AND ODD NUMBERS

A number which is divisible by 2 is called even

number.

E.g. 0, 2, 4, 6, 8, 10, 12……….. A number which is not divisible by 2 is called

• dd number.

E.g. 1, 3, 5, 7, 9, 11, 13………...

10

PRIME NUMBERS

A number which is divisible by only two factors 1.

itself

2.

1

is called a prime number.

E.g. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,

43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

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SIMPLIFICATIONS

1.

35+15*1.5 = ?

a)

75

b)

51.5

c)

57.5

d)

5.25

2.

• 84*29+365 = ?

a) 2436

b)

2801

c)

• 2801

d)

• 2071

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HCF AND LCM

HCF – Highest Common Factor LCM – Least Common Multiple E.g. H.C.F of 36 and 84 is

36 = 6 * 6 = 2 * 3 * 2 * 3 = 22 * 32 84 = 12 * 7 = 2 * 2 * 3 * 7 = 22 * 3 * 7 HCF = 22 * 3 = 4 * 3 = 12

Find the HCF of 15, 25 and 75.

15 = 3 * 5 25 = 5 * 5 = 52

75 = 3 * 5 * 5 = 3 * 52

HCF = 5

13

HCF AND LCM

E.g. L.C.M of 16, 24, 36

16 = 2 * 2 * 2 * 2 = 24 24 = 2 * 2 * 2 * 3 = 23 * 3

36 = 2 * 2 * 3 * 3 = 22 * 32

LCM = 24 * 32 = 144

Find the LCM of 22, 54

22 = 2 * 11 54 = 2 * 3 * 3 * 3 = 2 * 33 LCM = 2 * 33 * 11 = 594

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EXAMPLES

1.

The traffic lights at three different road crossings change after every 48 sec., 72 sec. and 108 sec. respectively. If they all change simultaneously at 8 : 20 : 00 hours, then at what time will they again change simultaneously?

• Ans. LCM of (48, 72, 108) = 432 sec

So, the lights will again change simultaneously after every 432 sec. That is = 7 min and 12 sec. Next simultaneous change will take place at 8 : 27 : 12 hours.

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EXAMPLES

2.

The maximum number of students among them 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is.

• Ans. HCF of (1001, 910) = 91

So, 91 students will get the same number of pens and same number of pencils.

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EXERCISE

1.

One bell rings at an interval of 30 minutes and another at an interval of 25 minutes. If they both ring together at 10:00 am, the time when they will next ring together is

a) 12:30 am

b)

10:55 am

c)

12:30 pm

d)

11:30 pm

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EXERCISE

2.

What is the least number of students in a class, if they can be made to stand in rows of 8, 12, or 14 each?

a) 158

b)

168

c)

148

d)

178

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SIMPLIFICATIONS

“BDMAS” rule is used to decide the priority of

• perations.

B stands for Brackets D stands for Division M stands for Multiplication A stands for Addition S stands for Subtraction

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SIMPLIFY THE FOLLOWING EXAMPLES

1.

5005-5000/10 = ?

4505

2.

100+50*2 = ?

200

3.

2-[2-{2-2(2+2)}] = ?

• 6

4.

The sum of two integers is 25. One integer is

• 11. Find the other integer.

14

5.

A number divided by 2 is 5 less than that

• number. What is that number?

10

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SIMPLIFY THE FOLLOWING EXAMPLES

6.

Two pens and three pencils cost Rs. 86. Four pens and a pencil cost Rs. 112. Find the cost of a pen and that of a pencil.

pen=25 Rs. Pencil= 12 Rs.

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SIMPLIFY THE FOLLOWING EXAMPLES

1.

In a caravan, in addition to 50 hens there are 45 goats and 8 camels with some keepers. If the total number of feet be 224 more than the number of heads, find the number of keepers.

15

2.

A class starts at 10 a.m. and lasts till 1:27 p.m. Four periods are held during this

• interval. After every period, 5 minutes are

given free to the students. The exact duration

• f each period is :

48 minutes

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SIMPLIFY THE FOLLOWING EXAMPLES

3.

A long yard 225 meters long, 26 trees are planted at equal distances, one tree being at each end of the yard. What is the distance between two consecutive trees? 9 meters

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SQUARE ROOTS AND CUBE ROOTS

1.

Evaluate √6084 = ? 78

2.

Find the cube root of 2744. 14

3.

If x*y=x+y+ √xy, the value of 6*24 is :

42

4.

If y=5, then what is the value of 10y √y3 – y2

a)

50 √ 2

b)

100

c)

200 √ 5

d)

500

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PROBLEMS ON NUMBERS

1.

Find a number such that when 15 is subtracted from 7 times the number, the result is 10 more than twice the number.

5

2.

The sum of a rational number and its reciprocal is 13/6. Find the number.

2/3 or 3/2

3.

The difference of two numbers is 11 and one-fifth of their sum is 9. Find the numbers.

x=28 and y=17

4.

The difference between two digit number and the number obtained by interchanging the two digits is

• 36. What is the difference between the two digits of

that number? 4

25

RATIO AND PROPORTION AND CHAIN RULE

1.

If A:B = 5:7 and B:C = 6:11, then A:B:C is

30:42:77

2.

If 15 toys cost Rs. 234, what do 35 toys cost?

546 Rs.

3.

If 36 men can do a piece of work in 25 hours, in how many hours will 15 men do it?

60 hours

4.

36 men can complete a piece of work in 18 days. In how many days will 27 men complete the same work?

24 days

26

5.

If 20 men can build a wall 56 meters long in 6 days, what length of a similar wall can be built by 35 men in 3 days?

49 meters

6.

If 15 men, working 9 hours a day, can reap a field in 16 days, in how many days will 18 men reap the field, working 8 hours a day?

15 days

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TIME AND WORK

1.

A does a work in 10 days and B does the same work in 15 days. In how many days they together will do the same work?

6 days

2.

A man can do a job in 15 days. His father takes 20 days and his son finishes it in 25 days. How long will they take to complete the job if they all work together?

Approximately 6.4 days

28

PIPES AND CISTERNS

1.

Two pipes A and B can fill a tank in 36 hours and 45 hours respectively. If both the pipes are opened simultaneously, how much time will be taken to fill the tank?

20 hours

2.

Two pipes can fill a tank in 10 hours and 12 hours respectively while third pipe empties the full tank in 20 hours. If all the three pipes

• perates

simultaneously, in how much time will the tank be filled.

7 hours and 30 minutes

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PERMUTATIONS AND COMBINATIONS

n! = n*(n-1)*(n-2)*…..*3*2*1 Permutations = Different Arrangements

nPr = n! / (n-r)!

Combinations = Different groups or selections

nCr = n! / r!(n-r)!

Some Examples:- 1.

30! / 28!

870

2.

60P3

205320

3.

100C98

4950

4.

50C50

1

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PERMUTATIONS AND COMBINATIONS

1.

How many words can be formed by using all letters of the word “BIHAR”

120 words

2.

How many words can be formed by using all the letters of the word “DAUGHTER” so that the vowels always come together?

4320 ways

3.

In How many ways a committee of 5 members can be selected from 6 men and 5 ladies, consisting of 3 men and 2 ladies.

200 ways

4.

In how many ways can the letters of the word “Apple” be arranged?

60 ways

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PERMUTATIONS AND COMBINATIONS

5.

A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if atleast one black ball is to be included in the draw?

64 ways

6.

Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

25200 ways

7.

In how many different ways can the letters of the word “DETAIL” be arranged in such a way that the vowels occupy only the odd positions.

36 ways

32

PERMUTATIONS AND COMBINATIONS

8.

In a group of 6 boys and 4 girls, four children are to be selected, In how many different ways can they be selected such that at least one boy should be there?

209 ways

9.

In how many ways can a cricket eleven be chosen out of a batch of 15 players?

1365 ways

• 10. How many words can be formed from the

letters of the word “EXTRA”, so that the vowels are never together?

72 ways

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PROBABILITY

 Experiment – An operation which can produce some

well-defined outcomes is called an experiment.

 Random Experiment – An experiment in which all

possible outcomes are known but the exact output can not be predicted in advance, is called a random experiment.

 Examples: 1.

Tossing a fair coin

2.

Rolling an unbiased dice

3.

Drawing a card from a pack of well-shuffled cards

4.

Picking up a ball of certain colour from a bag contaning balls of different colours.

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PROBABILITY

 Sample Space – When we perform an experiment,

then the set S of all possible outcomes is called the sample space.

 Examples: 1.

In Tossing a fair coin, S = {H, T}

2.

In Rolling an unbiased dice, S = {1, 2, 3, 4, 5, 6}

3.

If two coins are tossed then, S = {HH, HT, TH, TT}

 Event – Any subset of a sample space is called an

event.

 Probability of Occurrence of an Event

P(A) = n(E)/n(S) Where E=Event, S=Sample space

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RESULTS ON PROBABILITY

 P(S)=1  Then the event is certain to occur.  E.g. The sun rises in the east.  0 ≤ P(A) ≤ 1  P(ø) = 0  Then the event is impossible.  For any events A and B,

P(A U B) = P(A)+P(B)-P(A∩B)

 Probability of not happening an event A’ is

P(A’) = 1 – P(A)

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PROBABILITY

1.

In a throw of a coin, find the probability of getting a head.

½ = 0.5

2.

Two unbiased coins are tossed. What is the probability of getting at most one head?

¾ = 0.75

3.

An unbiased dice is tossed. Find the probability of getting a multiple of 3.

1/3 = 0.33

4.

In a simultaneous throw of a pair of dice, find the probability of getting a total more than 7.

5/12

37

PROBABILITY

5.

A bag contains 6 white and 4 black balls. Two balls are drawn at random. Find the probability that they are of the same colour.

7/15

6.

Two dice are thrown together. What is the probability that the sum of the numbers on the two faces is divisible by 4 or 6?

7/18

7.

Two cards are drawn at random from a pack

• f 52 cards. What is the probability that either

both are black or both are queens?

55/221

38

PROBABILITY

8.

In a simultaneous throw of two coins, the probability of getting at least one head is:

¾

9.

Three unbiased coins are tossed. What is the probability of getting at least 2 heads.?

½

• 10. Tickets numbered 1 to 20 are mixed up and

then a ticket is drawn at random. What is the probability that the ticket drawn bears a number which is a multiple of 3?

3/10

39

PROBABILITY

• 11. In a lottery, there are 10 prizes and 25 blanks.

A lottery is drawn at random. What is the probability of getting a prize?

2/7

• 12. One card is drawn at random from a pack of

52 cards. What is the probability that the card drawn is a face card?

3/13

• 13. From a pack of 52 cards, two cards are drawn

together at random. What is the probability of both the cards being kings?

1/221

40

PROBABILITY

• 14. A box contains 5 green, 4 yellow and 3 white
• marbles. Three marbles are drawn at random.

What is the probability that they are not of the same colour.

41/44

• 15. In a class, 30% of the students offered

English, 20% offered Hindi and 10% offered

• both. If a student is selected at random, what

is the probability that he has offered English

• r Hindi?

2/5

• 16. Two dice are tossed. The probability that the

total score is a prime number is:

5/12

41

PROBABILITY

• 17. A man and his wife appear in an interview for

two vacancies in the same post. The probability of husband’s selection is 1/7 and the probability of wife’s selection is 1/5. What is the probability that only one of them is selected?

2/7

42

TIME AND DISTANCE

Speed = Distance / Time Time = Distance / Speed Distance = Speed * Time x km/hr = (x * 5/18) m/sec E.g. 1 kilo meter = 1000 meters 1 hour = 60 minutes 1 minute = 60 seconds 60*60 = 3600 1000/3600 = 5/18 Similarly, x m/sec = (x * 18/5) km/hr

43

TIME AND DISTANCE

1.

A cyclist covers a distance of 750 m in 2 min 30 sec. What is the speed in km/hr of the cyclist. 18 km/hr

2.

Peter can cover a certain distance in 1 hr. 24 min. by covering two kind of the distance at 4 kmph and the rest at 5 kmph. Find the total distance. 6 km

3.

If a man walks at the rate of 5 kmph, he misses a train by 7 minutes. However if he walks at the rate of 6 kmph, he reaches the station 5 minutes before the arrival of the train. Find the distance covered by him to reach the station. 6 km

4.

A car moves at the speed of 80 km/hr. What is the speed of the car in meters per second? 200/9 m/sec

44

TIME AND DISTANCE

5.

Which of the following trains is the fastest?

a)

25 m/sec

b)

1500 m/min

c)

90 km/hr

d)

None of these

6.

A person crosses a 600 m long street in 5 minutes. What is his speed in km per hour? 7.2 km/hr.

7.

How long will a boy take to run round a square field of side 35 meters, if he runs at the rate of 9 km/hr? 56 sec

8.

A car is running at a speed of 108 kmph. What distance will it cover in 15 seconds? 450 meters m/sec

45

TIME AND DISTANCE

9.

An express train travelled at an average speed

• f 100 kmph, stopping for 3 minutes after

every 75 km. How long did it take to reach its destination 600 km from the starting point?

6 hr and 21 min

• 10. A certain distance is covered by a cyclist at a

certain speed. If a jogger covers half the distance in double the time, The ratio of the speed of the jogger to that of the cyclist is:

1:4

46

BOATS AND STREAMS

In water, the direction along the stream is

called downstream.

And, the direction against the stream is called

upstream.

If the speed of a boat in still water is u km/hr

and the speed of the stream is v km/hr, then

Rate / Speed downstream = (u+v) km/hr Rate / Speed upstream = (u-v) km/hr

47

BOATS AND STREAMS

If the speed downstream is a km/hr and the

speed upstream is b km/hr, then

Speed in still water / Rate in still water

= ½(a+b) km/hr

Speed of current/ Rate of stream/current

= ½(a-b) km/hr

48

BOATS AND STREAMS

1.

In one hour, a boat goes 11 km along the stream and 5 km against the stream. The speed of the boat in still water (in km/hr) is:

8 km/hr

2.

A man can row upstream at 8 kmph and downstream at 13 kmph. The speed of the stream is:

2.5 kmph

3.

A man rows downstream 32 km and 14 km

• upstream. If he takes 6 hours to cover each

distance, then the velocity (in kmph) of the current is:

1.5 kmph

49

BOATS AND STREAMS

4.

A boat running downatream covers a distance

• f 16 km in 2 hours while for covering the

same distance upstream, it takes 4 hours. What is the speed of the boat in still water?

6 kmph

5.

A boatman goes 2 km aginst the current of the stream in 1 hour and goes 1 km along the current in 10 minutes. How long will it take to go 5 km in stationary water?

1 hr 15 min

50

51