Q UANTITATIVE A PTITUDE P REPARATION 1 Prepared by : Rupal Patel
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 2 Permutations and Combinations
H INDU A RABIC N UMBER S YSTEM 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
P LACE OF EACH DIGIT E.g. 83526 Ten Thousand’s Hundred’s Ten’s Unit’s Thousand’s Place Place Place Place Place 8 3 5 2 6 4
H OW 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
W RITE THE GIVEN NUMBERS IN WORDS 9,04,06,002 1. Nine crore four lakh six thousand two 1,60,05,014 2. One crore sixty lakh five thousand fourteen 5,04,080 3. Five lakh four thousand eighty 2,07,09,207 4. Two crore seven lakh nine thousand two hundred seven 6
W RITE THE GIVEN NUMBERS IN FIGURES Six lakh thirty-eight thousand five hundred 1. forty-nine 6,38,549 Twenty-three lakh eighty thousand nine 2. hundred seventeen 23,80,917 Eight crore fifty-four lakh sixteen thousand 3. eight 8,54,16,008 Four lakh four thousand forty 4. 4,04,040 7
F ACE V ALUE AND P LACE V ALUE 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 8
E XAMPLES : F ACE V ALUE AND P LACE V ALUE The difference between the place value and 1. the face value of 6 in numeral 856973 is 973 a) 6973 b) 5994 c) None of these d) The difference between the place value of two 2. seven’s in the numeral 69758472 is 75142 a) 64851 b) 5149 c) 9 699930 d)
E VEN 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 odd number . E.g. 1, 3, 5, 7, 9, 11, 13 ……… ... 10
P RIME NUMBERS A number which is divisible by only two factors itself 1. 1 2. 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 11
S IMPLIFICATIONS 35+15*1.5 = ? 1. a) 75 51.5 b) 57.5 c) 5.25 d) -84*29+365 = ? 2. a) 2436 2801 b) -2801 c) 12 -2071 d)
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 = 2 2 * 3 2 84 = 12 * 7 = 2 * 2 * 3 * 7 = 2 2 * 3 * 7 HCF = 2 2 * 3 = 4 * 3 = 12 Find the HCF of 15, 25 and 75. 15 = 3 * 5 25 = 5 * 5 = 5 2 75 = 3 * 5 * 5 = 3 * 5 2 13 HCF = 5
HCF AND LCM E.g. L.C.M of 16, 24, 36 16 = 2 * 2 * 2 * 2 = 2 4 24 = 2 * 2 * 2 * 3 = 2 3 * 3 36 = 2 * 2 * 3 * 3 = 2 2 * 3 2 LCM = 2 4 * 3 2 = 144 Find the LCM of 22, 54 22 = 2 * 11 54 = 2 * 3 * 3 * 3 = 2 * 3 3 LCM = 2 * 3 3 * 11 = 594 14
E XAMPLES The traffic lights at three different road 1. 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 15 8 : 27 : 12 hours.
E XAMPLES The maximum number of students among 2. 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. 16
E XERCISE One bell rings at an interval of 30 minutes 1. 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 10:55 am b) 12:30 pm c) 11:30 pm d) 17
E XERCISE What is the least number of students in a 2. class, if they can be made to stand in rows of 8, 12, or 14 each? a) 158 168 b) 148 c) 178 d) 18
S IMPLIFICATIONS “BDMAS” rule is used to decide the priority of operations. B stands for Brackets D stands for Division M stands for Multiplication A stands for Addition S stands for Subtraction 19
S IMPLIFY THE FOLLOWING EXAMPLES 5005-5000/10 = ? 1. 4505 100+50*2 = ? 2. 200 2-[2-{2-2(2+2)}] = ? 3. -6 The sum of two integers is 25. One integer is 4. 11. Find the other integer. 14 A number divided by 2 is 5 less than that 5. number. What is that number? 20 10
S IMPLIFY THE FOLLOWING EXAMPLES Two pens and three pencils cost Rs. 86. Four 6. pens and a pencil cost Rs. 112. Find the cost of a pen and that of a pencil. pen=25 Rs. Pencil= 12 Rs. 21
S IMPLIFY THE FOLLOWING EXAMPLES In a caravan, in addition to 50 hens there are 1. 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 A class starts at 10 a.m. and lasts till 1:27 2. p.m. Four periods are held during this interval. After every period, 5 minutes are given free to the students. The exact duration of each period is : 48 minutes 22
S IMPLIFY THE FOLLOWING EXAMPLES A long yard 225 meters long, 26 trees are 3. planted at equal distances, one tree being at each end of the yard. What is the distance between two consecutive trees? 9 meters 23
S QUARE ROOTS AND CUBE ROOTS Evaluate √ 6084 = ? 1. 78 Find the cube root of 2744. 2. 14 If x*y=x+y+ √xy, the value of 6*24 is : 3. 42 If y=5, then what is the value of 10y √y 3 – y 2 4. 50 √ 2 a) 100 b) 200 √ 5 c) 500 d) 24
P ROBLEMS ON NUMBERS Find a number such that when 15 is subtracted from 7 1. times the number, the result is 10 more than twice the number. 5 The sum of a rational number and its reciprocal is 2. 13/6. Find the number. 2/3 or 3/2 The difference of two numbers is 11 and one-fifth of 3. their sum is 9. Find the numbers. x=28 and y=17 The difference between two digit number and the 4. number obtained by interchanging the two digits is 25 36. What is the difference between the two digits of that number? 4
R ATIO AND P ROPORTION AND CHAIN RULE If A:B = 5:7 and B:C = 6:11, then A:B:C is 1. 30:42:77 If 15 toys cost Rs. 234, what do 35 toys cost? 2. 546 Rs. If 36 men can do a piece of work in 25 hours, in how 3. many hours will 15 men do it? 60 hours 36 men can complete a piece of work in 18 days. In 4. how many days will 27 men complete the same work? 24 days 26
If 20 men can build a wall 56 meters long in 6 days, 5. what length of a similar wall can be built by 35 men in 3 days? 49 meters If 15 men, working 9 hours a day, can reap a field in 6. 16 days, in how many days will 18 men reap the field, working 8 hours a day? 15 days 27
T IME AND WORK A does a work in 10 days and B does the same work 1. in 15 days. In how many days they together will do the same work? 6 days A man can do a job in 15 days. His father takes 20 2. 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 Two pipes A and B can fill a tank in 36 hours and 45 1. hours respectively. If both the pipes are opened simultaneously, how much time will be taken to fill the tank? 20 hours Two pipes can fill a tank in 10 hours and 12 hours 2. respectively while third pipe empties the full tank in 20 hours. If all the three pipes operates simultaneously, in how much time will the tank be filled. 7 hours and 30 minutes 29
P ERMUTATIONS AND COMBINATIONS n! = n*(n-1)*(n-2)* … ..*3*2*1 Permutations = Different Arrangements n P r = n! / (n-r)! Combinations = Different groups or selections n C r = n! / r!(n-r)! Some Examples:- 30! / 28! 1. 870 60 P 3 2. 205320 100 C 98 3. 4950 30 50 C 50 1 4.
P ERMUTATIONS AND COMBINATIONS How many words can be formed by using all 1. letters of the word “BIHAR” 120 words How many words can be formed by using all 2. the letters of the word “DAUGHTER” so that the vowels always come together? 4320 ways In How many ways a committee of 5 members 3. can be selected from 6 men and 5 ladies, consisting of 3 men and 2 ladies. 200 ways In how many ways can the letters of the word 4. 31 “Apple” be arranged? 60 ways
P ERMUTATIONS AND COMBINATIONS A box contains 2 white balls, 3 black balls and 5. 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 Out of 7 consonants and 4 vowels, how many 6. words of 3 consonants and 2 vowels can be formed? 25200 ways In how many different ways can the letters of 7. the word “DETAIL” be arranged in such a way that the vowels occupy only the odd positions. 32 36 ways
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