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48 PRACTICE BOOK ON QUICKER MATHS 5. The quotient arising from the division of 24446 by a certain number is 79 and the remainder is 35,what is the .divisor? a) 309 b)319 c)310 d) 379 6. A boy had to divide 49471 by 210. He made some mistake in copying the divisor and obtained as his quotient 246 with a remainder 25. What mistake did he make? a) He made no mistake b) He put down 120 for 21 0 c) He put down lO2fo~ 210 d) He put dwn 201 for 21 0 7. Ina division sum the dividend is 57324 and quotient 123. If the remainder is greater than the' quotient but less han twice the quotient. Find the divisor. a) 465 b) 475 c) 645 . d) 565 :. the least number to be added = 58. Ex. 2: Find the least number of 3 digits, which is exactly divisible by 14. Solo: The least number of3 digits = 100 On dividing 100 by 14, remainder = 2 To determine exactly divisible least number, the above method will be applied. :. The required number = Dividend + (Divisor...: Remainder) = 100+(14-2)= 112. Exercise 1. Answers I.b 2.a· 3.b 4.c 5.a 6.d 7.a 2. Rule 3 A number (Dividend) can be made completely divisible with the help of either of the following methods: Divisor) Dividend. (Quotient 3. 4. ,, 1" Iii[ I I. Remainder Method I: By subtracting remainder from dividend. For finding the greatest n-digit number completely divisible by a divisor, this rule is applicable. 5. Illustrative Examples 6. Ex. I: Find the greatest number of 3 digits, which is exactly divisible by 35. Solo: The greatest number of3 digit = 999 On dividing 999 by 35, remainder =-19. Now, applying the above method, the required number = dividend - remainder = 999 19=980 Ex.2: Find the least number that must be subtracted from 87375, to get a number exactly divisible by 698. Solo: On dividing 67375 by 698, the remainder is 125. By the above method, The least number to be subtracted is the remainder from dividend. :. the least number to be subtracted = 125. Method II: By adding (divisor- remainder) to dividend. For finding the least n-digit number completely divisible by a divisor, this rule is applicable. Illustrative Examples Ex. I: What least number must be added to 49123 to get a number exactly divisible by 263. Solo: On dividing 49123 by 263, the remainder is 205. By the above method, The least number to be added to the dividend = divisor - remainder =263-205=58. 7. 8. 9. 10. 11. 12. 13. 14. What least number must be subtracted from 5731625, to get a number exactly divisible by 3546? a) 1189 b) 1829 c) 1289 d) 1982 Find the least number of 5 digits which is exactly divisibleby456. a) 10456 b) 10424 c) 10032 d) 10023 Find the number which is nearest to 68624 and exactly divisible by 587 .. a) 68679 b)69156 c) 68569 d) 68689 Find the number nearest to 144759 and exactly divisible by 927. a) 144906 b) 144612 c) 144169 d) 144621 Find the greatest number of 5-digits, which is exactly divisible by 547. a) 99456 b) 99554 c) 10545 d) 99545 What least number must be added to 9541-31, to get a number exactly divisible by 548? a) 63 b)563 c) 485 d)611 What least number be subtracted from 650 I to get a number exactly divisible by 135? a)21 b) 12 c)35 d)53 What least number be added to 5200 to get a number exactly divisible by 180. a) 160 b) 60 c) 20 d) 180 Findthe number which is nearest to 6555 and exactly divisible by 21. a) 6558 b) 6576 c) 6552 d) 6534 Find the number which is nearest to 8845 and exactly divisible by 80. a) 8890 b) 8810 c) 8800 d) 8880 What least number must be subtracted from 13601 to get a number exactly divisible by 87. a)39 b)29 c)27 d)33 What least number must be added to 1056 to get a number exactly divisible by 23. a)21 b)23 c)2 d)4 The largest number offour digits exactly divisible by 88 is a) 9856 b) 9944 c) 9988 d) 9994 Find the greatest number of five digits exactly divisible by 279. Number System ~ . i i 3-digitnumber= 156 a) 99882 b) 99720 c) 99782 d) 99982 15. Find the nearest integer to 56100 ••• hich is exactly divisible by456. a) 56556 b) 56088 c) 56112 d) 56188 16. What is the nearest whole number to one million which is divisible by 537 without remainder? ta)999894 b) 999994 c)999~84 d) 999948 . 17. What least number must be added to 2716321 to make it . exactly divisible by 3456? a)3361 b)95 c) 105 d)3316 18:' What least number must be subtracted from 2716321 to make it exactly divisible by 3456? a)3361 b) 95 c) 85 d) 3613 19. Find'the least number of five digits which is exactly divisible by 654. a) 10190 b) 10654 c) 10464 d) 10644 20. Which least number should be subtracted from 427396 so that the remainder would be divisible by 15? (BSRB Delhi PO, 2000) a)6 b)1 c) 16 d)4 Answers I.c 8.c 15.b 2.c 9.c 16.a 3.a 1O.d 17.b 4.b 11. b 18.a 5.b 12.c 19.c 6.c 13.b 20.b 7.a 14.a Rule 4 Theorem: When two numbers, after being divided by a third number, leave the same remainder, the difference of those two numbers must be perfectly divisible by the third number. IIInstrative Examples I r Ex. I: 24j45 and 33334 are divided by a certain number of , three digits and the remainder is the same in both the cases. Find the divisor and the remainder. Soln: By the above theorem, the difference of 24345 and 33334 must be perfectly divisible by the divisor. We have the difference = 33334 - 24345 = 8989 = 101 x 89 Thus, the three-digit number is 101. The remainder can be obtained by dividing one of the numbers by 101. Ifwe divide 24345 by 101, the remainder is 4. Ex. 2: 451 and 607 are divided by a number and we get the same remainder in both the cases. Find all the possible divisors (other than 1). Soln: By the above theorem: 607 - 451 = 156 is perfectly divisible by those numbers (divisors). Now, 156=2x2x3 x 13 Thus, I-digit numbers = 2,3,2 x 2,2 x 3 = 2, 3, 4, 6 2-digit numbers = 12, 13,26,39,52,78 umber of foui digits and the remainder is the same 4 in both the 9 .. cases. Find the divisor. a) 1423 1432 c) 1422 . d) 1433 2. 31 593and 23456 are divided by a certain number of three digits and the remainder is the same in both the cases. Find the remainder . q) E x ~ rc is e 1.4 5 7 2 1 3 ~~ ~~ 0~ ~~ Answers 1. a 2. a Rule 5 .'\.. To find the prOduCt of the two numbers w!ten the sum and the difference of the two numbers are given. Product of the numbers (Sum + Difference )(Sum - Difference) 4 Illustrative Example a n d Ex. The sum of two numbers is 14 and their difference is 10. Find the product of the two numbers. Soln: Detail Method: Let the two numbers bex andy, then x+y= 14andx-y= 10 3 4 3 3 7 3 a r e d i v i d e d b y a c e r t a i n n Now, we have, (x+ yf = (x- yf +4xy or, (14)2 = (lO)i +4xy. :. xy = (14)2 -(lOf = 96 = 24 4 4 Quicker Method: Applying the above formula, we have Product = (i4+10~14-:10)_24 Note: The numbers can also be found by the direct formula x = Sum + Difference = 14 + 10 = 12 2 2 y = Sum-Difference = 14 -10 2 =2 2 Exercise 1. 2. 3. 4. The sum of two numbers is 20 and their difference is 10. Find the product of the two numbers. a) 60 b) 100 c) 80 d) 75 The sum of two numbers is 49 and their difference is 3. Find the product of the two numbers. a) 598 b) 958 c)589 d) 859 The sum of two numbers is 38 and their difference is 4. Find the produc~ of the two numbers. a) 537 b) 375 c)357 d) 753 The sum of two numbers is 24 and their difference is 18 .• 50 PRACTICE BOOK ON QUICKER MATHS Find the product of the two numbers. a) 54 b) 63 c)36 d) 64 5 The sum of two numbers is 33 and their difference is 21. Find the product of the two numbers. . a) 162 b) 126 c) 102 d)216 1 6. The difference of two numbers is .11 and "5 th of their sum is 9. The numbers are: [RRB Exam 1991 J a)31,20 b) 30, 19 Answers c) 29, 18 4.b d)28,17 5.a 1. d 2. a 3. c 6. d; Hint: See Note. Rule 6 Ex. If one-fifth of one-third of one-half of number is 15, find the number. Soln: Detail Method: Let the number be x. Then we have, . (*) The required number = 1 s( T)( T )( T ) I. ~% ~~ ~~ 435 10. If "9 of 10 of g of a number is 45, what is the number? [BSRBHyderabadPO 1999) a)450 b) 540 c) 560 d)650 11.Two-thirds of three-fifths of one-eighth of a certain number is 268.50. What is 30 per cent ofthat number? [NABARD 19991 a)1611.0 b) 716.0 c}1342.5 d) 596.60 124 12. Ifg of3 of5 of an umber is 12 then 30 per cent of the number will be ~~ [SBI Bank PO 2001) ~64 ~~ 1. c 2. b 8.c 9.d 3.a lO.b 4.c Il.a 3. 4. 5. 6. 7. 8. ~~ 5.a 12.c 6. b . 7.d The sum of the digits of a two-digit number is S./fthe digits are reversed, the number is decreased by N, then the numberisgivenby s[s+ :]+~[S- :] If one-third of one-sixth of two-third of number is 64, find or the number. ~ [ Decrease] ) [ Decrease] a) 1278 b) 1782 c) 1728 d)3456 5 Sum of digits + 9' + Sum of digits -9 If one-tenth of one-fourth of one-fifth of number is 10, find the number. Illustrative Example a) 200 b) 2000 c) 500 d) 1000 Ex. The sum ofthe digits ofa two-digit number is 8. Ifthe If three-fourth of two-third of two-fifth of one-half of digits are reversed, the numbe{ is decreased by 54. Find number is 60, find the number. the number. a) 600 b}400 c) 650 d) 575 Solo: Detail Method: Let the two-digit number be lOx + y. If two-fifth of one-third of two-third of number is 16, find Then, we have; x + y = 8 ... (1) and thenmber. 10y+ x= 10x+y-54 ~1~ ~~O ~IW ~1~ . = 54 =. 6 If one-fifth of two-third of one-half of number is 30, find or,x-y 9 .... (2) the number. From equations (1) and (2) a)450 b) 900 c) 950 d) 400 Three-fourth of one-fifth ofanumber is 60. The number 8+6 x = -- = 7 and y = 1 is: [Baok PO Exam, 1990] 2 a) 300 b) 400 c) 450 d) 1200 ... The required number = 7 x 10+ 1 =71 Four-fifths of three-eighths of a number is 24. What is Quicker Method: 250 per cent ofthat number? [BSRB Mumbai, 1998J The required number = a) 100 b) 160 c) 120 d) 200 DeCrease] I [ Decrease] Two-fifths ofthirty per cent of one-fourth ofa number is 15. What is 20 per cent of that number? [ [BSRB Mumbai 1998J 5 Sumofdi gits+-- 9- +"2 Sumofdigits--- 9- '2 2. ~~ Rule 7 =. 450 Note:(*) The resultant should be multiplied by the reverse of each fraction. Exercise a) 90 b) 150 c) 100 d) 120 Two-fifths of one-fourth of five eighths of a number is 6. What is 50 per cent ofthat number? [BSRB Calcutta PO 19991 Answers X(~X~)(~)=15 ;. x = 15x5x3><2 =450 Direct Formula: 9. - 7. 51 Number System Detail Method: Let the number = x. Then, x2 + x = 182 Exercise or, x2 +x-182 = 0 1. or, x2 + 14x-13x-182 The sum of the digits of a two-digit number is 12. lfthe digits are reversed, the number is decreased by 18. Find the number. ~~ ~~ ~M = 0 or, x(x+14)-13(x+14)=0 or, (x 13 )(x + 14) = 0 ~~ 2. The sum of the digits ofa two-digit-number is 9. If the digits are reversed, the number is decreased by 63. Find the number. a) 72 b)63 c) 54 d)81 3. The sum of the digits of a two-digit number is 10. 1f.the digits are reversed, the number is decreased by 72. Find the number. a)91 b) 82 c)73 d) 64 4. The sum ofthe digits ofa two-digit number is 13. lfthe digits are reversed, the number is decreased by 45. Find the number. a)85 b) 76 c)49 d)94 5. The sum of the digits of a two-digit number is 7. If the digits are reversed, the number is decreased by 45. Find the number. a) 52 b)43 c)61 d) 25 6. A certain number consists of two digits whose sum is 9. Ifthe order of digits is reversed; the new number is 9 less than the original number. The original number is a)45 . b)36 . c)54 d) 63 7. In a two-digit number the digit in the unit's place is more than the digit in the ten's place by 2. If the difference between the number and the number obtained by interchanging the digits is 18. What is the original number. [8m Associates PO 1999J c) 24 d) Data inadequate a) 46 b) 68 Answers 6.c 1. a 2. d 3. a 4. d 5.c ·7. d; Hint: Let the no. be lOx + y theny'=x+20r,y-x=2 .... (i) (IOy+x)-(IOx+y)= 18 CPL 6 3 6 2 3 6 or, 9y - 9x = 18 or, y-x =2 ..... (ii) From eqn (i) and (ii) we can't get any conclusion; \\\ \\\\\\\\\\\\\\\\\\\\\\\\\ Rule 8 If the sum of a number and its square is x, then the number ~-I~J [ is given by Example 2 . Illustrative Ex.: If the sum of a number and its square is 182, what is the number? or, x = 13 (negative value is neglected). Quicker Method: Applying the above rule, we have the required answer .J729 -1 27-1 =13 2 .J1+182x4 -1 2 2 Exercise 1. Ifthe sum ofa number and its square is 240, what is the number? a) 15 b) 18 c)25 d) 22 2. If the sum of a number and its square is 306, what is the number? a)16 b)18 c) 17 d)19 .3. If the sum of a number and its square is 702, what is the number? ~~ ~n ~~ ~~ 4 .. lfthe sum of a number and its square is 1560, what is the number? a)38 'b)37 c)36 d) 39 5. lfthe sum of a number and its square is 156, what is the number? a) 16 b) 14 c) 12 d) 13· 6. , If the sum of a number and its square is 210, what is the number? a) 12 b) 13 c) 14 d) 16 7. If the sum of a number and its square is 90, what is the number? a)7 b)8 c)9 d)8 8. Ifthe sum of a number and its square is 380, what is the number? a) 17 b) 18 c) 19 d)21 9. 'If the sum ofa number and its square is 342, what is the number? a) 14 b)28 c) 18 d) 23 10. lfthe sum ofa numper and its square is 552, what is the number? a)21 b)22 c)23 d) 24 Answers 1. it 2. c 8. c 9. c 3.a 10.c 4.d 5.c 6.c 7.c Rule 9 The sum of the digits of a two-digit number is S./fthe digits are reversed, the number is increased by N, then the num- PRACTICE BOOK ON QUICKER MATHS 52 ber is given by Illustrative Example 5[ S ~ '~]+~[ S + ~] or . Ex. Increase] 1 [. [ 5 Sum of digits - Increase + 2 Sum of digits + 9 Illustrative Example l If40% ofa number is 360, what will be 15%0f15%of that numb(lr? , . ~ Soln: Detail Method: Let the number bex.Then we have 40%ofx=360 ( 9j The sum ofthe digits ofa two-digit number is 8. If the digits are reversed, the number is increased by 54. Find the number. Soln: Detail Method: Letthe two digit number be lOx + y Then, we have, x + y = 8 ... (i) and lOy+x= lOx+y+54 or,y-x=6 .... (ii) From eqn (i) and (ii) x= 1 and y = 7. :. the required number = 1 x 10+7= 17 Quicker Method: Applying the above formula, we have :.x=360x100 =900 40 Ex.: Now, 15% ofx = ~x900 = 135 100 Again, 15%of135 =~x135=20.25 100 Quicker Method: Apply.ing the above rule, we have , the required answer = Exercise 1. . . [ +254J ReqUIred number = 5 88+91 [ 9 54] 2. = 10+7= 17 Exercise 1. 2. 3. f I , I i 4. 1i' '\ ' f 5. Ii: I 3. The sum of the digits of a two-digit number is 7, If the digits are reversed, the number is increased by 27. Find the number. a)25 b)34 c) 16 d) None of these The sum of the digits ofa two-digit number is 6. Ifthe digits are reversed, the number is increased by 36. Find the number. a) 24 b) 15 c)51 d) 42 The sum of the digits of a two-digit number is 9. If the digits are reversed, the number is increased by 63. Find the number. a) 27 b)36 c)45 d)18 The sum of the digits ofa two-digit number is 5. If the digits are reversed, the number is increased by 27. Find the number. a)23 b)32 c) 14 d)41 A number consists oftwo digits whose sum is 15. If9 is added to the number, then the digits change their places. The number is a) 69 b) 78 c) 87 d)96 15x15x360 = 20.25. 40xlOO ----. - 4. 5. If90% ofa number is 540, what will be 10% of5% ofthat number . a)30 b)3.5 c)3 d) 35 If35% ofa number is 385, what will be 5% of5% of that number. a) 11 b)5.5 <;:)2.5 d)2.75 If 17% of a number is 68, what will be 15% of25% ofthat number. a) 20 b}15 c)35 d)25 If 18% ofa number is 144, what will be 12% of25% of that number. a)8 b)12 c)16 d) 24 If39% of a number is 780, what will be 35% of 13% of that number. a)91 Answers 1. c 2. d b)52 3.b c)65 4.d d)78 5.a Rule 11 If the ratio of the sum and the difference aftwo numbers is J a+b\ a: b, then the ratio afthese two numbers is given by a _ b ( Answers 1. a Illustrative Example 2. b Ex. 3.d 4.c 5.b Rule 10 If x% of a number is n, then y% of z% of that number is yzn ] I I, \i .,~ , [ given by x x 100 . The ratio of the sum and the difference of two numbers is 7 : 1. Find the ratio of those two numbers. Soln: Detail Method: Let the two numbers be x and y. Then we have x+y='2x- y 1 =:::>x+y=7x-7y x84 or 6x = 8y . - == - = - = 4 : 3 , .. Y 6 3 • • Number System 5 3 . Quicker Method: Applying th.above rule, we have 7+1 8 4 the required ratio = -- = - = - = 4 : 3 7 -I 6 3 2. Exercise I. Ratio of the sum and the difference of the two numbers is 5: 3. Find the ratio of those two numbers. 'a)4:1 b)3:2 c)3:1 d)2:1. 2. Ratio of the sum and the difference of the two numbers is 9 : I. Find the ratio of those two numbers. a)5:3 b)5:4 c)4:1 d)5:2 3. Ratio of the sum and the difference of the two numbers is 7 : 3. Find the ratio of those two numbers. a)5:2 b)5:3 c)3:2 d)7:4' 4. Ratio of the sum and the difference ofthe two numbers /is 2 : 1. Find the ratio oflthose two numbers. a)l:2 b)3:2 c)4:3 d)3:1 5. Ratio of the sum and the difference of the two numbers is 13 : 3. Find the ratio ofthose two numbers. a)5:8 b)8:3 c)8:5 d)8:7 3. ~7 4. 5. 6. Answers I. a 2. b 4.d 3.a 5.c Rule 12 To find the difference of the two digits of a two-digit number, when the difference between two-digit number and the number obtained by interchanging the digits is given. Difference of two digits Diff. in original and interchanged number 9 Note: We cannot get the sum of two digits. Illustrative Example Ex. The difference between a two-digit number and the number obtained by irit~rchanging the digits is 27. What are the sum and the difference of the two digits of the number? Soln: Detail Method: Let the number be lOx + y. Then we have (lOx + y)-(10y+x)= 27 the sum of the two digits of the number? a)2 b) 1 c)9 d) Can't be determined The difference between two-digit number and the number obtained by interchanging the digits is 36. What is the difference ofthe two digits of the number? ~4 ~3 02 ~8 The difference between two-digit number and the number obtained by interchanging the digits is 63. What is the difference of the two digits of the number? 7. ~9 ~6 08 The difference between two-digit number and the number obtained by interchanging the digits is 9. What is the difference of the two digits of the number? ~2 ~5 03 ~I The difference between two-digit number and the number obtained by interchanging the digits is 72. What is the difference of the two digits of the number? a)7 b)9 c)8 d) Can't be determined The difference between two-digit number and the number obtained by interchanging the digits is 45. What is the difference of the two digits ofthe number? a)6 b)5 c)8 d) Can't be determined The difference between the digits of a two-digit number is one-ninth of the difference between the original number and the number obtained by interchanging the positions the digits. What definitely is the sum of the digits of that number? [BSRB Mumbai PO, 19981 a)5 b) 14 c) 12 d) Data inadequate of - - 1 8. The sUm of the digits of a two-digit number is - of the . 11 sum of the number and the number obtained by interchanging the position of the digits. What is the difference between the digits of that number? _________ _ [Bank of Baroda PO, 19991a)3 b)2 c)6 d) Data inadequate 9. The difference between a two-digit number and the number obtained by interchanging the position of the digits of that number is 54. What is the sum ofthe digits of that number? [BSRB Calcutta PO, 1999) a)6 b)9 c)15 d) Data inadequate I 10. The sum of the digits ofa two-digit number is "5 of the 27 or,9(x-y)=27 :.x-y=9=3 Thus, the difference is 3, but we cannot get the sum of two digits. Quicker Method: Applying the above rule, we have Required answer = 27 9 =3 Exercise I. The difference between a two-digit number and the number obtained by interchanging the digits is 18. What is difference between the number and the number obtained by interchanging the positions of the digits. What definitely is the difference between the digits of that number? [BSRB Chennai PO, 2000J a) 5 b) 9 c) 7 d) Data inadequate Answers . 1. d 2. a 7. d; Hint: 3.a 4.d 5.c 6.b x- y = ~{(lOx+ y)-(10y+x)}= ~(9X-9Y)= x- y r , . " 54 PRACTICE BOOK ON QUICKER MATHS 8. d; Hint: Let, the two no. be xy, ie lOx + y then, x +y = ~[(10x+ y)+(lOy + x )]= x+ y 11 Thus we see that the difference of x and y can't be determined. , Hence, the answer is,data inadequate. 9. d; Hint: See note. Let the two-digit no. be lOx + y According to question, (lOx +y)-(lOy+ x)= 54 9x-9y=54 :. x-y=6 10. a; Hint: Let the two-digit number be lOx + y Exercise 1. 2. 3. The average of 5 consecutive integers is 4. Find the average of the squares of these integers. a)22.5 b)45 c)l8 d) Can't be determined The average of 15 consecutive integers is 15. Find the average of the squares of these integers. a)243.66approx b) 300 c) 225.4 approx d) 394.26 approx The average of 9 con!>ecutive integers is 9. Find the average of the squares of these integers. a) 87 4. b) 87% c)88 d) 85+ Then, x + y = ~(IOX+ y-IOy-x) The average of 7 consecutive integers is 6. Find the average of the squares of these integers. 9 or, x+y= -(x- a) 46~ 3 , 5 y) b) 46L 3 or,4x-I4y=0::;.-=- x c)40 L d) 47 3 7 y2 Using componendo & dividendo, we have, x+y 7+2 9 x- y = 7-2 =5 iex-y=5K Here, K has the only possible value, K = 1. Because the difference of two single-digit numbers will always be of a single digit. '. II ! 11 Rule 13 Ex. I The average of7 consecutive integers is 7. Find the average of the squares of these integers. Soln: Use the formula: [for odd number of consecutive integers) Average of squares [. 'I II I i ~ I' , ~ = 1 'x[nJ(nl+lX2nJ+l) n2h+lX2n2+1)] No. of integers 6 6 Where, n] = Average + No. of integers -I 2 and n2 = Average _ No. of integers + 1 2 In the above case 7-1 n] =7+--=10 2 7 +1 n2 =7---=3 2 X :. Average of squares = L[10XllX21 3(4 7)] 7 6 -6 = L[385 -14]= ~ = 53 7 7 5. f the squares of these integers. T h e a v e r a g e o f 3 Rule 14 To find the least number which when divided by x], x2 and x3 leaves the remainders G], G2' and G3respectively. (XI - G]) = (Xl - G2) = (X3 - G3)' We have an established method that is given below. Required least number = (LCM of Xl' x2 and x3) (XI -al) or (X2 -a2) or (X3 -a3) c o n s e c u t i v e i n t e g e r s i s 3 . F i n d t h e a v e r a g e o Illustrative Example Ex.: Find the least number which, when divided by 13, 15 and 19, leaves the remainder 2,4 and 8 respectively. Soln: Applying the above rule, 13-2= 15~4= 19~ 18= II , Now, LCMof13, 15, 19=3705 :. therequiredleastnumber=3705-I1 =3694 Note: Find the least number which, when divided by 13, 15 and 19, leaves the remainders 1, 2, 3 respectively. Can we fmd the specific solution. No, because 13 - 1 '* 15-2,* 19-3 Exercise 1. 2. Find the least number which when divided by 24, 32 and 36 leaves the remainders 19,27, and 31 respectively. a) 288 b)283 c) 287 d) 285 Find the least number which when divided by 12,21 and 35 leaves the remainders 6,15, and 29 respectively. 5 5 Number System I 3. 4. 5. I I a)414 b)418 c) 420 d) 410 Find the least number which when'l!ivided by 48, 60 and 64 leaves the remainders38, 50, and 54 respectively. a) 860 b) 960 c) 950 d) 850 Find the least number which when divided by 5, 6, 8, 9 and 12 leaves the remainders 3, 4, 6, 7 and 10 respectively. a) 360 b)358· c) 362 d}258 Find the least number which when divided by 9, 10 and 15 leaves the remainders 4, 5, and 10 respectively. ~W ~~ ~~ ~W 3. 4. 5. Answers 1.b Find also the common remainder .. a)70,6 b)71,5 c) 75, Id)73,3 . The greatest number which when divides 99, 123 and 183 leaves the same remainder is a) 12 b) 24 c) 18 d) 26 Find the greatest number which divides 77, 112 and 287 and leaves.the same remainder in each case; a)35 b) 25 c)45 d)I5 Find the greatest number which divides 95, 195 and 175 and leaves the same remainder in each case. a)5 b) 10 c)20 d)25 Answers 2. a 3.c 4.b 1. a 5.c 2. c 3.a Rule 15 "To find the greatest number that will divide given numbers say XI' X2'''' Xll so as to leave the same remainder in each . case, wefind the HCF of the positive difference of numbers 4.a Rule 16 The ratio between a two-digit number and the sum of the digits of that number is a : b. If the digit in the unit's place is n more than the digit in the ten's place, then the number 9a) ie Ix) - x21, IX2 ~ x31, ... and so on. 5.c . is given by lIb _ 2a n and the digits in unit's place and Illustrative Example Ex. . Find the greatest number that will divide 55, 127 and 175 so as to leave the same remainder in eacQ case. Soln: Detail Method: Let x be the remainder, then the numbers (55 -x); (127 -x) and (175 -x) must be exactly divisible by the required number. Now, we know that if two numbers are divisible by a certain number, then their difference is also divisible by that number. Hence, the numbers (27-x)-(55-x1 (I75-x)-(127-x) and (I75-x)-(55-x) or, 72, 48 and 120 are also divisible by the required number. HCF of72, 48 and 120 is 24. Therefore, the required number is 24. Quicker Method: If you don't want to go into the details of the method, find the HCF of the positive differences of numbers. It wiII serve your purpose . quickly. For example, in the above case, positive difference of numbers are (127 55 = 72), (I 75 - 127 = 48) and (175-55 = 120). HCF of72, 48 and 120 is 24 :. required number = 24. Exercise ]. Find the greatest number which is such that when, 12288, 19139 and 28200 are divided by it, the remainders are al! the same. a)221 b)212 c) 122 d)321 2. Find the greatest number which is such that when 76, 151 and 226 are divided by it, the remainders are all alike. lOb - a ) .(d -b) ten ~ place are n lIb _ 2a and n lIb _ 2a respectively. Illustrative Example Ex.: The ratio between a two-digit number and the sum of the digits of that number is 4 : 1. If the digit in the unit's place is 3 more than the digit in the ten's place, what is the number? SoIn: Detail Method: Suppose the two-digit number = lOx + y lOx+ Then we have x+ y = y 4 1 or,lOx+y=4x+4y or,6x=3y or,2x=y or, x = y-x = 3 (given) andy = 6 :. the number is 36 . Quicker Method: Applying the above rule, we have Required number =(9x4 llxl-2x4 }x 3 = 9 x 4 x 3 = 36 3 ( ( Exercise 1. 2. The ratio between a two-digit number and the sum of the digits ofthat number is 5 : 1. If the digit in the unit's place is I more than the digit in the ten's place, what is the value of unit's place digit of that number? a)4 b)5 c)3 d)7 The ratio between a two-digit number and the sum of the digits of that number is 2: 1.Ifthe digit in the unit's place 56 3. 4. 5. PRACTICE BOOK ON QUICKER MATHS is 7 more than the digit in the ten's place. What is the va-Iue often's place digit of that number? a)1 b)2 c)3 d) 64 The ratio between a two-digit number and the sum of the digits of that number is 3 : 1. If the digit in the unit's place is 5 more than the digit in the ten's place. What is the value often's place digit of that number? ~4 ~3 02 ~I The ratio between a two-digit number and the sum ofthe digits of that number is 14: 5. If the digit in the unit's place is 6 more than the digit in the ten's place. What is the sum of the digits of that number? a) 10 b)12' c) 13 d)9 The ratio between a two-digit number and the sum ofthe digits of that number is 4 : 1. If the digit in the unit's place is 4 more than the digit in the ten's place. What is the sum of the digits of that number? a)9 b) 10 c) 15 d) 12 Answers Lb 2.a 3.c 4.a (xn + K) is divided byx-l. (i) Remainder = 1 + K; when K < x-1 (ii) Remainder = (1 + Remainder obtained when K is divided by x - 1); when K>x-1. Illustrative Example Ex.: Find the remainder when i 3 + I is divided by 6. Soln: Detail Method: See the following binomial expansion' (x+ YY' = x" + nC1x.II -1y+ nC'].x"-2y2 + nC3 x"-3y 3 + ... + IlC,,_lxyn-1 + I. Find the remainder when (919 + 6} is divided by 8. a)2 2. b)3 b)3 c)? d)5 Find the remainder when (523 + 3) is divided by 4. ~7 4. d)7 Find the remainder when (7 + 8) is divided by 6. a)2 3. c)5 13 ~4 03 Find the remainder when ~ i a)19 b)7 50 ~2 + 8) is divided by II. c)9 ")8 . 5. Find the remainder when (25625 + 241) is divided by 24. a) 23 b)2 c)1 d)Can'tbedetermined 3. b 4. c 5. b Answers I. d 2. b Rule 18 5.d Rule 17 To find the remainder when Exercise ytl We find that each of the terms except the last term 0" ) ~c~)Jltains x. It means each term except yn is perfectlyoivisible by x. Note: y" may be perfectly divisible by x but we cannot say without knowing the values of x and y. FojJowingthe same logic, i 3 = (6 + I Y 3 has each term except ]13 exactly 3 divisible by 6. Thus, when i is divided by 6 we have the . remainder 113 = I and hence, when (713 + I) is divided by 6 the remainder is I + I = 2. Quicker i\1ethod: Applying the above rule, we have K = I and x - I = 6 ie K <x-I. Therefore, we apply rule (i) :. required answer = I + I = 2. To find the all possible numbers, when the product of two numbers andth~ir HCF are given, we/ollow thefollowing method. Product Step I: Find the value of (HCF r. Step II: Find the possible pairs of value got in step I. Step III: Multiply the HCF with the pair of prime factors obtained in step II. Illustrative Example Ex.: The product of two numbers is 7168 and their HCF is 16. Find the numbers. 7168 =.28 Soln: Step I: (16)2 Stepn: (1,28),(2, 14),(4,7) StepID: (I x 16,28 x 16)and(4 x 16,7 x 16)or(l6,448) and (64, 112) Note: (2, 14), which are not prime to each other should be rejected. Exercise 1. The product of two numbers is 286 and their HCF is 12. Find the sum of the numbers. ·a) 12 b) 24 c)36 d)48 2. Th'e product of two numbers is 3125 and their HCF is 25. Find the sum of the numbers . a) 75 b) 100 c) 125 d)50 3. The product of two numbers is 20 16 and their HCF is 12. Find the number of all possible pairs of numbers. a)1 b)2 c)3 d) Can't be determined 4. The product of two numbers is 338 and their HCF is 13. Find the difference of the numbers. a)13 -b)26 c)39 d)52 Number System 57 Answers I. c 2. b 3.b 5. 4.a Rule 19 A number on being divided by d, and d2 successively leaves the 6. remainders r] and r2 respectively. If the number is divided by d l x d2, then the remainder is given by (d] xr2 H])' a)12 b) 10 c)14 d)16 A number on being divided by 10 and 11 successively leaves the remainders 5 and 7 respectively. Find the remainder when the same number is divided by 110. a) 70 b)98 c) 74 d)75 A number on being divided by 3 and 7 successively leaves the remainders 2 and 5 respectively. Find the sum ofdigits of the remainder when the same number is divided by 21. a)7 b) 17 c)S d)6 Illustrative Example Answers Ex. l. b A number on being divided by 5 and 7 successively leaves the remainders 2 and 4 respectively. Find the remainder when the same number is divided by 5 x 7 =35. Soln: Detail Method: Ifffi A 7 B· 2 .C4 In the above arrangement, A.is the number which, when divided by 5, gives B as a quotient and leaves 2 as a remainder. Again, when B is divided by 7, itgives C as a quotient and 4 as a remainder. For simplicity, we may take C = I. :.B=7xl+4=11 andA=5xIl+2=57 Now, when 57 is divided by 35, we get 22 as the remainder. Quicker Method: The required remainder = d] xr2 +r, = the first remainder = 2 r2 = the second remainder = 4 .'. the required remainder = 5 x 4 + 2 = 22. Exercise I. 2. 3. 4. A number on being divided by 12 and 15 successively leaves the remainders 4 and 6 respectively. Find the remainder when the same number is divided by 180. a)46 b) 76 c) 84 d) 18 A number on being divided by 5 and 7 successively leaves the remainders 3 and 6 respectively. Find the remainder when the same number is divided by 35. a) 33 b) 23 c) 32 d) Can't be determined A number on being divided by 8 and 9 successively leaves the remainders 5 and 7 respectively. Find the remainder when the same number is divided by 72. a)61 b)S c) 71 d)9 A number on being divided by 4 and 6 successively leaves the remainders 2 and 3 respectively. Find the remainder when the same number is divided by 24. 3:a 4.c 5.d 6.c Rule 20 To find the number of zeros at the end of the product. We know that zeros are produced only due to the following reasons. (i) If there is any zero at the end of (lny multiplicand. (ii) If 5 or multiple qf 5 are multiplied by any even number . To generalise the above two statements, we may say that (2)"' (5t has n zeros ifm > nor m zeros ifm <no Note: Always lesser value of the exponents of 5 and 2 will be . the required answer. Thus, write the product in the form bmx5 x ... ) n I Illustrative Example Ex.: Find the number of zeros at the end of the products. 12 x 18x 15 x40x25 x 16x55 x 105 Soln: 12 x 18 x 15 x40 x 25x 16x 55 x 105 = 12 x is x 16 x 40 x 15 x 25 x 55 x 105 Where, dl = the first divisor = 5 r] 2. a , = (22x3~ (2x9)x (2f x (2 3 x 5)x (5 x 3)" (5)2 x (5 x II)x (5 x 21) = 2'0 x56 x .... [Since numbers other than 2 and 5 are useless] Since 10> 6, there are 6 zerbsat the end of the product. Note: This is the easiest way to count the number of zeros in the chain of products. By this method, we can easily find that the product of 1 x 2 x 3 x ... x 100 contains 24 zeros. Exercise 1. 2. 3. 4. Find the number of zeros at the end of the product 15 x 16 x IS x 25 x 35 x 24 x 20 a) 10 b) S c) 5 d) Can't be determined Find the number of zeros at the end of the product 52 x20x28 xlOxl6xl25 a) 15 b) 22 c)7 d) 8 Find the numberof zeros at the end of the product 50 x 625 x 15 x lOx 30 a) 10 b)9 c) 12 d)3 Find the number of zeros at the end of the product PRACTICE BOOK ON QUICKER MATHS 58 150 x 250 x 625 x 125 x 75 x 20 x 16 a)9 b) 14 c)23 d)5 5. Find the number of zeros at the end of the product 70x80x 16x64x56x 13 x 18x3125 a) 16 b) 12 c)JO d)25 Answers 1. c 2. c 3.d To findthe number of different divisors. Find tlte prime factors of tlte number and increase tlte index of eaclt factor by 1. Tlte continued product of increased indices will give tlte result including unity and tlte number itself. Note: Also see Rule - 36. Illustrative Examples Ex.I: Find the number of different divisors of 50, besides unity and the number itself. Soln: If you solve this problem without knowing the rule, you will take the numbers in succession and check the divisibility. In dOIng so, y'ou may miss some numbers. It will a~o.take more time. Different divisors of 50 are: 1,2,5, 10,25,50 Ifwe exclude I and 50, the number of divisors will be 4. 2 By rule: 50 = 2x 5 x 5 = 2 x 5 ... the number oftotal divisors = (I + I) x (2 + I) =2x3=6 or, the number of divisors excluding 1 and 50 = 6 - 2 =4 Ex. 2: Rule 22 Ex. 1 : How many numbers up to 100 are divisible by 6? Soln: Divide 100 by 6. The quotient obtained is the required 'number of numbers. 100= lQ x 6+4 Thus, there are 16 numbers. Ex. 2: How many numbers up to 200 are divisible by 4 and 3 toaether? b ,Soln: LCM of 4 and 3 = 12 Now, divide 200 by 12 and the quotient obtained is the required number of numbers. 200 = 16 x 12 + 8 Thus, there are 16 numbers. Ex. 3: How many numbers between 100 and300 are divisibleby7? Soln: Upto-l00, there are 14 numbers which are divisible by 7 (since 100= 14 x 7 + 2). Up to 300, there are 42 numbers which are divisible by 7 (since 300=42 x 7 + 6) Hence, there are 42 - 14 = 28 numbers . 3'- Exercise 1. Find the different divisors of 3 7800, excluding unity. Soln: 2. 37800=2x2x2x3x3x3 x5x5x7 3. 23 x 4. 33 x 52 x i Total no. of divisors = (3 + 1)(3 + 1)(2 + 1) (I + 1)= 96 ... the number of divisors excluding unity =96-1 = 95. 5. Exercise I. Find the number of different divisors of307692. a)96 b) 12 e)6 d)48 ,2. Find the number of different divisors of 400, besides unity and the number itself. a) 15 b) 14 e) 13 d) 12 3. Find the number of divisors of999999, excluding unity. ~M 4. 6. ~~ ~~ Find the number of different divisors of 13231. ~M 5. ~~ ~4 ~~ 6.a Illustrative Examples Rule 21 1 5.c To find the number of numbers divisible by a certain integer. 5.b 4.a Answers l.a 2.e 3.e 4. b;Hint: 13231=131 x 101,131 and 101 are primes ~5 Find the no. of different divisors of30030, besides unity and the number itself. ~M ~~ ~~ ~~ Find the no. of different divisors of 4452. a) 24 b)32 e) 16 d)22 6. 7. 8. How many numbers up to 150 are divisible by 9? a) 16 b) 15 c) 10 d)6 How many numbers up to 200 are divisible by 7? a)26 b) 22 e) 18 d)28 How many numbers up to 532 are divisible by 15? a)25 b)26 e)36 d)35 How many numbers up to 300 are divisible by 5 and 7 together? a)9 b)8 e) 10 d)7 How many numbers up to 450 are divisible by 4, 6 and 8 together? ' a)19 b)18 e)17 d)16 How many numbers between 50 and 150 are divisible by 8? a) 24 b) 12 e)18 d)8 How many numbers between 100 and 200 are divisible by 2 and 8 together? a) 12 b) 13 e)9 d) 16 How many numbers between 100 and 300 are divisible by 9? a)11 b)13 e)19 d) 22 Answers l.a 2.d 3.d 4.b 5.b 6.b 7.b 8.d 1 5 9 Number System Rule 23 .The number which when multiplied by x is increased by y is Y) ( Illustrative Example 5. Answers Multiplier -1 . 1. a Ex. Find the number which when multiplied by 16 is increased by225. Soln: Detail Method: Let that number be x. Then Ex.: Exercise .. = 225 = 225 = IS the requ~~d number . 16 -lIS I. Find the sum of first 50 odd numbers. a) 6250 b) 2500 c) 2520 d) 2450 Find the value of (1 +3 + 5 + ... + 80thoddnumber)-(1 +3 +5 +7+ ... + 30th odd number) a) 5500 b) 6100 c) 5400 d) 7300 Find the value of 35 + 37 + ... + 25th odd number. a) 356 b) 336 c) 363 d)365 Find the value of I + 3 + 5 + ... + 199 a) 40000 b) 10000 c)39601 d)Can'tbedetermined Find the value of 15 + 17 + ... + 51 a) 627 b) 676 c) 725 d) None ofthese 2. Fmd the r umber which when~ultiplied by 36 is increased by a)30 b)28 c)32 d)35 rind t;lt: number which when multiplied by 9 is increased by 128. a) 12 b) IS _£) 16 d) 18 Find the number which when multiplied by 17 is increased by256. a) 12 b) 14 c)18 d) 16 Find the number which when multipliedby IS is increased by 378. a) 26 b) 16 c)27 d}28 Find the number which when multiplied by 26 is increased by625. a) 26 b)25 c) 24 d)27 Answers 1. a 2. c 3. 4. 5. 6. ~ 3.d 5.b 3. d) 1992 2. a tn = nth term of the series a = n(n+l) Find the value of] + 2 + 3 + ... + 105. Exercise 2. c) 1990 tn =a+(n-I)d 105(105 + I) Soln: Reuired sum = ---- = 5565 2 I. b) 1989 3. b; Hint: We have the following formula, first term of the series n = number of numbers d = common difference 2 Illustrative Example Ex.: is equal to AnsWers Rule 24 Theorem: Sum of all the firstn natural numbers = --; 1+3+5+ ... +3983 1992 a) 1988 1. b 4.c Find the value of I + 3 + 5 + ... + 20th odd number . Soln: 202 =400. 1050~ 5. 5.b Illustrative Example Exercise 4. 4.a 2 Quicker Method: Applying the above rule, we have 3. 3.a Theorem: Sum of first n odd numbers = n • . :. x = 225 = 15 . 15 2. 2. b Rule 25 16x-x = 225 I. Find the value of I + 2 + 3 + ... + 62. a) 1953 b) 1395 c) 1593 d) 1359 Find the value of (1 + 2 + 3 + 4 + ... + 80) - (1 + 2 + 3 + ... + 60) a) 1830 b) 1410 c)1140 d) 1380 (Increased Value) given by x _lor " 4. Find the sum of first 45 natural numbers. a) 1035 b) 1235 c) 1135 d) 1305 Find the sum of natural numbers between 20 and 100. a) 4480 b) 4840 c) 4800 d) 4850 Find the value ofl +2+3+ .... +210. a)22155 b)21255 c)22515 d) 22255 ( For the case of odd number a= l,d=2 :.tn =1+(n-I)2=2n~1 We apply this formula for solving this question. First we calculate 1+ 3 + 5 + ... + 33 and then 1+2 + 3 + ... +25th odd number. For getting required answer, we subtract first from second. How do we calculate first ie (I + 3 + 5 + ... + 33)? We have, PRACTICE BOOK ON QUICKER MATHS 60 .. 33=2h-l Eseeforrnu!a) :. n= 17 :. 1 +3 +5 + ... +33 =;01 +3+5 + ... + 17th odd number. Exercise = {17f =289 4. b .. 5. a 6. d Rule 26 Theorem: Sum of first n even numbers = n (n + 1) a) 5255 b) 5525 Find the value of2 +4 + 6 + 8 + ... + 1 00 (or 50th even number) Soln: 50 x (50 + 1) = 2550 Note: We have the following formula, t" = a+(n-1)d 3. 5. 5. Find the vahie of 1 + 2 + ... + (30th natural number? a) 9454 2 b) 9544 2 2 b) 440 2 2 4.d 5.b = n(I1~+IX2n+l) 6 )= 385, then the value of c)1155 d)(385x385) 2.a 3.d 4.a - -~----'-'---- , 2J 0 Rule 28 Theorem: Sum of cubes of first n natural numbers Illustrative Example Find the value of 13 + 23 + ... + 63 . . [6X(6+1)l2 =(21)2 =441 S I on. .. 2 ., J Exercise 2. 102 _ 1O(10+1X2xl0+1) 6 10x11x21 =----=385 6 6.a {Ix 2)2 +(2x2? +(2x3)2 + ... +(2xl0)2 Find the value of 13 +23 + ... +123. a) 6804 Find the value of 12 +22 +32 + ... +102 5.c 2 ·=2l1"+2"+3~+ .... +10 =4 x 385= 1540 1. Illustrative Example + + J + ... + d) 660 Z b) 1540 ', 2 Ex..: Rule 27 Solo: c)550 + ... +lO 7. b; Hint: 22+'42+ ... +20 2 22 ,,2 1 d)9555 Answers Theorem: Sumofsquarcs offirst n natural numbers Ex..: 2 If (1 +2 +3 l.b t" 3.a c) 9455 2 (1 +2 +3 + .... +10 )-(1+2+3+ ... +10) is equal to a) 770 Answers 2. a . d)4898 2 . (22 +42 +62 + ... +202) is Find the value of2 + 4 + 6 + .... + lOOth even number. a) 10000 b) 10100 c) 11000 d) 10101 Find the value of26 + 28 + ... + 28th even number. a)656 b) 665 c) 566 d)565 Find the value 01'2 +4 + 6 + .... + 1002. a)251502 b)250512 c)215502 d)255102 Find the value of68 + 70 + ... + 180 a) 7608 b) 7680 c) 6078 d) 7068 Find the value of2 + 4 + 6 ... + 56th even number. a)3912 b)3192 c)3219 d)3129 I. b c)4901 2 = 4. d) 1496 2 b) 4900 Exercise 3. 2 Find the value of 2 + 3 + ... + 24 . 7. = 2+2n-2 = 2n 2. d) 38205 2 c) 1469 2 a)330 tn = 2+(n-l)2 l. 2 b) 1649 a) 4899 n = no. of numbers d = common difference . For the case of even numbers 2 Find the value of 1 +2 +3 + ... +16 . 4. where, ~" = nth term d) 5252 2 2 6. a = first tertil 2 Find the value of 25 +26 + .... +50 . a) 38025 b)30825 c) 38250 a) 1946 Ex.: c )5552 2 2. Illustrative Example ro, n ,= 2 Find the value c.f 12 + 22 + ... + 252. 1. 3 d) 6408 3 b) 18495 c) 18497 3 3 3 d) 14895 Find the value of 8 + 9 + ... + 15 . a) 16316 4. c) 6048 3 Find the value of 2 + 3 + ... + 16 . a) 18496 3. b) 6084 b) 13661 3 c) 16361 d) 13616 3 Find the value of 1 +2 + ... +(10th natural numberY a) 3025 b) 3205 c) 3052 d) 3250 Number System 5. 6. 61 Find the value of 23 + 33 + 43 + ... + 93. a) 2024 b) 2025 c) 2225 Find the value of 33 + 43 + ... + 113. a) 4356 b)4348 c) 4347 2. d) 2205 3. d) 4374 4. Answers 1. b 2. b 4.a 3.d 5.a 5. 6.c From I to 31, how many are the odd numbers? a)15 b) 16 c)14 d)17 From I to 51, fmd the number of even and odd numbers. a) 26, 25 b) 25, 26 c)24,25 d)25,24 From 51 to 91, find the number of even and odd numbers. a)20,21 b)21,20 c)21,22 d) 19,20 From 51 to 90, find the number of even and· odd numbers. Rule 29 n 50 For exampIe,from 1 to 50, there are - = 25 odd numbers 2 )0 amI - = 75 even numbers. 2 Exercise 1. 2. 3. 4. 5. 6. In the first 62 counting numbers, fmd the number of even numbers. a)30 b)31 c)32 d) 34 From I to 78, how many are the odd numbers? a)20 b)38 c)39 d) 40 From I t028, find the number of even numbers. a) 14 b) 13 c)12 d) 15 From I to 100 find the number of even and the number of odd numbers. a) 50, 50 b)51,50 -.;)50,51 d) 49, 50 From 1 to 80 how many are the even numbers? a)41 b) 42 c)39 d) 40 From 50 90, find the number of odd and even numbers. a)20,21 b)20,20 c)21,22 d) 19,20 to' . c)20,21 d) 19,20 Answers 1. 3.b a 2. b 4.a 5.a Rule 31 The difference between the squares of two consecutive numbers is always an odd number and the difference between the squares of two consecutive numbers is the sum of the two consecutive numbers. For example, 16 and 25 are squares of 4 and 5 respectively (two consecutive numbers). :. Differimce = 25 - 16 = 9 (an odd number) and 52 - 42 (Difference) =5 + 4 = 9 2 2 Reasoning: a _b = (a-bXa+b)=a+b [: a-b = I] Exercise a) 24 1. 2. 3. b) 12 c) 18 d)8 c)8 d) 10 2 Find the value of 6 - 52 . a)ll b)9 2 2 Find the value of 35 -34 . a) 59 b)69 c)70 Find the value of d)71 102 _92 +82 _72 +62 _52 +42 _32 +22 _12 a) 50 b) Answers I. b b)21,20 a) 20, 20 n In the first n counting numbers, there are - odd and 2 . 2 even numbers provided n, the number of numbers, is even. 2. c 4.a 3.a 5.d 6.a 4. 292 +352 +332 +312 -342 -322 -302 -282. a) 250 b) 252 c)352 . d) 342 Rule 30 In the firstncounting numbers, if n, the number of num1 bers, is odd, then there are 2(n + 1) odd numbers and 65 c)45 d)55 Find the value of 5. I Find the value of 652 -642 a)129 b) 128 2(n -I) even numb~rs. AnsWers 51+1 For example, from I to 51 there are -- = 26 odd numbers 2 I. a 2. b c) 120 d) 125 5.a 3.d Rule 32 51-1 and -- = 25 even numbers. 2 Exercise 1. In the first 61 counting numbers, find the number of even numbers. a)30 b)31 c)32 d)29 To find the number in the unit-place for odd numbers. When there is an odd digit in tire unit place (except 5), '1lultipIy the number by itself· until you get 1 in the unit ?lace. (. .. 1r=(. .. 1) (. .. 3ft' = (. .. 1) (. .. 7/" = (. .. 1) ~I 9. 62 (. .. 9/" = (. .. 1) , where n = 1, 2, 3, .... Iilustrative Examples Ex. 1 : What is the number in the unit place in (729)59? Soln: When 729 is multiplied twice, the number in the unit place is I. In other. words, if729 is multiplied an even number oftimes, the number in the unit place will be I. Thus, the number in the unit place in (729 (729 )59 = (729 i 8 Y8 is I. ~'. x (729) = ( ... I)x (729) = 9 in the unit place Ex. 2: Find the number in the unit place in (6~ (623)38 and (623)39 . Solo: W~~~ 623 is multiplied twice, the number in the unit place is 9. When it is multiplied 4 times, th~ humber in the unit place is L Thus we say that if 623 is multiplied 4n number oftimes, the number in the unit place will be 1:80,' (623 )36 = (623 )4x9 == I in'the unit place (623)38 = (623)4x9 x (623)l = ( .. :I)x(. .. 9)= 9 in the ~3 ~6 ~9 What is the number in theunit place in (333 Y4 ? ~I ~9 ~6 02 P R A C T I C E B O O K O N Q U I C K E R unit place. (623 )39 = (623 )4x9 x (623)3 = (. .. ])x (...7) = 7 in the unit place. Exercise J. What is the number in the unit place in (659)56? a) I b) 9 c) 6 d) None of these ~ 2. What is the number in the unit place in (329 f ~I ~4 3. ~7 What is the number in the unit place in (147)48? ~7 4. ~9 ? ~6 ~9 What is the number in the unit place in (87 ~I ~7 ~9 6. to ? ~3 ~9 b)7 c)9 d)3 What is the number in the unit place in (6231)928? ~l 8. ~3 i s t h e What is the number in the unit place in (5427)641 ? a)] 7. ~7 10. W h a t ~I 5 .. What is the number in the unit place in (127)127? ~I M A T H S ~8 03 ~4 What is the numbe~ in the unit place in (543)12 ? n u m b e r •• in the unit place in (4673Y21 ? a) I b)6 c)3 d)9 II. What is the number, in the unit place in (5483)843 ? ~I ~7 ~9 12. What is the number in the unit place 111 (1243YC x (I 547YoO ? a) 1 b)2 c)3 13. What is the number in ~3 d)9 the unit place 61 Y x (12349l39 ? (24533 a) 7 b) I 14. What . is the number in the c)9 d)3 unit place in 59 (157)157 x(159Y ? a) 3 b}9 c)6 15. What is the number in the unit place d) I 11 1 38 (751Y51 X (263)l71 x(137Y x(339~39? a)7 b}9 c) I d)6 Answers 1. a 2. b 3. d; Hint: When 7 is multiplied 4 times, the number in the unit place is"l. ie if? is multiplied 4n number oftimes,the number in the unit place will be I. :. (147)48 = (147)4 x12 = I in the unit place. 4. c; Hint: (87)90 = (87)4x22 x 87 x87 =(. .. I)x( ... 9)=9 5.c 6.b 7.a 8.b 9.d 10. a Il.b 12.a;Hint: (1243Y6 = (I 243)4x19 =(. .. 1) intheunitplace. (1547)100 = (I 547)4x25 =(. .. 1) in the unit place. 13. a; Hint: (24533 Y61 = (24533 )4 190 x (24533) . X = (. .. I)x( .. .3)=(. .. 3) in the unit place (12349)839 = (12349)2X419 x(12349)= (. .. IX ... 9)= (. .. 9) in the unit place. 14.a 15. a; Hint: (751YS! =(. .. 1) in the unit place (263 f71 = (263 t67 x (263)3 = ( ... 1) x ( ... 7) f' ( ... 7) in the unit place . I (137)138 = (I 37)4X34 X (lp7)2 = (. .. I)x(. .. 9) = (. .. 9) in the unit place (339)339 = (339YX169 x(339)= (. .. I)x( ... 9)= ( ... 9) in the unit place. 6 3 Number System 3. ... required answer == ( .•. 1X .. 7X .. 9X···9) == ( ••• 7) Find the number in the unit place in (1602)602 in the unit a)2 4. place. b)4. a)2 5. b)4 ~6 ~8 7. ~6 a)6 Ex. 1 : Find the number in the unit place in (122 9. ~2 ~2 is 6. Therefore, X (J22)20 = (J22)4 S =( ... 6)0=6 in the unit place (J22?2 = (J22)4X S x (122Y = ( ... 6)x( . ..4) = 4 in the unit c)8 d)2 ~6 08 Find the number in the unit place in (9S8 y 17 and (122f . Soln: ( . 2) x (. .. 2) = .. .4 ( .. 2) x ( .. : 2) x ( ..... 2) = 8 ( . 2) x ( ... 2) x ( .... 2) x ( ... 2) == ...... 6 We know that( .... 6) x ( ......6) == .......6 Thus, when{ 122) is multiplied 4'n times, the last digit ~2 Find the number in the unit place ih (958 Y 16 . 8. yo, (122 Y2 ~4 08 b)4 ~4 Illustrative Examples 02 Find the number in the unit place in (216?\6. (. .. 6)" = ( 6) ( ... 8)4n =( ... 6); where n == 1,2,3, ... d) 8 Find the number in the unit place in (5924)429. ~4 ( .. .4yn =( .. 6) c)6 Find the number in the unit place in (194)64 . 6. ( ... 2 )4n == ( •.• 6) d)6 Find the number in the unit place in (1392)9\ . Rule 33 To find the number in the unit placefor even numbers. When there is an evell digit in the unit pletce, multiply the number by itself until you get 6 in the unit place. c)8 10. 06 ~4 ~8 Find the number in the unit place in (958 y 18. ~4 ~2 06 ~8 ~2 ~4 06 ~8 11.Find the number in the unitplacein (958 t9 . J2. Find the number in the unit place in 61 (1532Y62 x(34S4Y x (1236Y62 x{53 1 8)243 . a)2 b)4 c)6d) 8 place 13. Find the number in the unit place in (122)23 = (J22)4XS x (122Y ~ (. .. 6)x ( ... 8) = 8 in the unit place. 2 Ex. 2: Find the number in the unit place in (98)40, (98t {4152Yl X (3268)67 ;«S913)83 X (6217yo3 . a)4 Answers and (98)43 . 4.d c)6 5.a 10.a J1.a 12.a d)8 6:a 7.a l3.c 1. a 2. a Rule 34 8.d 9.d Ifthereis 1, 5 or 6 in the unit place oft/te given II u11lb er, then after (tny times of its J1tultiplicatioll, it will have the Soln: (98)4 = (. .. 6) ... (98)4n =( ... 6) 10 Thus, (98)40 = (98t b)2 3.b = (. .. 6)= 6 in the unit place same digit in the unit place ie X ( .. It = ( (98)42 =(98)4 IOx(98? =(. .. 6)><(. . .4)=4 intheunit place (98t 3 =(98)4XIOx(98Y = ( ... 6)x(. .. 2) = 2 in the unit ... 1) ( ... S}' = place ( .. Exercise 1. Find the number in the ur,it place in (542 to. a)6 2. b)2 c)3 Illustrative Example Ex.: d)9 Find the number in the unit place in (1542)541. a)2 b)4 c)6 .5) (. .. 6}' d)8 =Find (. ..the 6) number in the unit place in (621?40, (62Sfs, (636Y6 Soln: From the above rule, (621tO = ( ... 1 to = 1 in the unit place t! I ~ :I", 64 PRACTICE BOOK ON QUICKER MATHS Now, apply the above rule, Number of divisors = (7 + 1) (1 + 1)(2 + 1) = 84 (625)125 = ( .. .5)125 = 5 in the unit place (636)36 Exercise = ( ... 6)36 = 6 in the unit place' Exercise 1. 2. Find the number in the unit place in (1845 ~5 ~3 ~9 ~9 ~6 r . 1. Find the no. of divisors of225. r56 • 2. 'Find the no. of divisors of63504. a) 25 b)32 c)75 d) 56 Find the no. of divisors of 17640, besides unity and itself. a) 12 b) 60 c) 72 d) 70 Find the no. of divisors of25200, excluding unity. a) 90 b)89 c}88 d) 86 Find the no. of divisors of234. a) 12 b)6 c)2 d)8 Find the no. of divisors of9000. a)36 b)48 c) 54 d) 18 Find the no. of divisors of20570, besides unity and itself. ~1 Find the number in ,the unit place in (99026 ~3 3. 5 \ ~1 Find the number in the unit place in (441)441 X (495)126 X (1536)236 . ~1 4. ~5 ~6 ~4 3. 4. 5. ~O 6. Find the number in the unit place in (321)321 x (325 )326 . ~1 ~5 ~6 ~8 Answers La 2.c 7. ~9 a) 24 8. 3.d 4.b ~8 b) 22 ~6 c}21 d) 18 Find the no. of divisors of 1 0000, excluding itself. a) 24 ' b)25 c}16 d) 32 Answers Rule 35 Ex.: What is the number in the unit place when 781, 325, 497 and 243 are multiplied together? Soln: Multiply all the numbers in the unit place, ie 1 x 5 x 7 x 3, the result is a nU\TIber in which 5 is in the unit ,place. l.b 2.c 7. b 8.a 3.d Find the number in the unit place in 962 x <1,- 5 x 454 x 959. ~2 2. ~4 4. ~6 ~8 Find the number in the ~nit place in 954 x 9625 x 43216 x 15437 x 12343. ~O 3. 6.b ~l ~1 ~6 Find the number in the unit place in 14532 x 14531 x 243 x 245 x 247 x 249. ~3 ~6 ~4 ~O Find the number in the unit place in 1431 x 5343 x 9645 x 1489. ~3 ~6 ~O ~5 Let N = aP bq c" .... , then the sum of the divisors of a number' P 1 q a + -1 b + 1 -1 C"+1_1 --'-, x---x---x ... a-I b-l c-l 'Note: This includes unity and the number itself as divisors. Illustrative Example Ex.: Find the sum ofthe divisors of a number 8064. Soln: Factorize 8064 into its prime factors. 8064= 27 x31 x72 Now, apply the above rule Answers 27+1-1 31+ 1-1 72+ 1-1 1. a --,-x--x--- 2. a 3.d 2-1 3~1 7-1 256 -1 9 -1 343-1 4.d =-'-x--x-- 1 Rule 36 1. (p+ 1) (q+ 1) (r+ 1) ... 2. Find the no. of divisors of 8064. Soln: 8064= 27 x31 x72 6 Exercise Note: This includes unity and the number itself as divisors. Illustrative Example 2 =255 x4 x 57=58140. If N is a composite number and N = aP bq c" ... Where a, b, c, ... are different prime numbers and p, q, rare positive integers. Then the number of divisors are Ex.: 5.a Rule 37 Exercise 1. 4.b 3. 4. Find the sum of the divisors of a number 225. a)430 b) 403 c) 503 d) 303 Find the sum ofthe divisors of a number 63504. a) 213870 b)231807 c)213807 d)213708 Find the sum of the divisors of a number 17640. a) 66960 b) 66690 c) 96660 d) 69660 Find the sum of the divisors ofa number 180. a) 465 b) 546 c)564 d) 654 " , 65 Number System 5. Find the sum of the divisors ofa number 120. a) 360 b) 420 c)480 ~)630 6. Find the sum of the divisors ofa number 64. a) 128 b) 127 c) 63 d) 130 7. Find the sum of the divisors of a number 3 125. a) 3906 b) 3609 c) 3096 d) 3069 8. Find the number and the sum of the divisors of the number2460 excluding one and itself. a) 24, 7056 b) 42, 7056 c) 24,4594 d) 24,4595 Answers J.b 2.c 3.b 4.b 5.a 6.b 7.a 8. d; Hint: Sumofthe divisors excluding 1 and itself= 7056. :. sum of the divisors including I 'and itself =7056-(2460+ 1)=4595. Rule 38 es /' If theplac: of last two digits of a three-digit number are interchanged, a new number greater than the originalltumbel' by N is obtained, then the difference betWeen the last two digits of that numbefl is given by (~) or (Dif.fe~'ence i; two values). Illustrative Example Ex.: If the places of last two digits of a three digit number are interchanged, a new number greater than the original number by 54 is obtained. What is the difference between the last two digits of that number? [NABARD 1999] Soln: Detail Method: Let the three-digit number be 100x + 10 y + z . interchanged, a new number greater than the original number by 27 is obtained. What is the diffeience between the last two digits of that number? a)l b)2 <::)3 d)4 4. If the places oflast two-digits bf a three-digit number are interchanged, a new number greater than the original number by 36 is obtained. What is the difference between the last two digits of that number? ~l ~2 ~3 ~4 S. If the places oflast t\vocdigits of a tlu:ee-digit number are interchanged, a new number greater than the original number by 45 is obtained. What is the difference between the last two· digits of that number? a)3 b)4 c)5 d)6 6. Ifthe places oflast two-digits of a three-digit liumber are interchanged, a new nuniber greater than the original number by 63 is obtained. What is the difference between the last two digits of that number? ~7 ~S 06 ~8 7. If the places oflast two-digits of a three-digitnumber are interchanged, a new numb~r gi:eater than the original number by 72 is obtained. What is the difference between the last two digits of that number? ~7 ~5 ~4 ~8 8. If the places of last two-digitI! of a threecdigit number are interchanged, a new number greater than the original number by 8 I is obtained. What is the difference between the last two digits of that number? a)7 b}8 c)9 d) 1 Answers 3.c 1. b 2. a 7. d, 8. c Acconling to the question, (IOOx + IOz + y )-(lOOx + lOy + z)= 54 or, 4.d 5.c 6.a Rule 39 Exercise A Itumber is divided by a certain number NI and gives a remainder 'R'. If the same number is divided by altother number N z, then the new remainder is obtained by the following method. "Divide R by N z and the remainder obtained ilt this division will be the new remainder". (Note: Here Nj > N 2 and NI is divisible Nz.) 1. Illustrative Exampl~ 9z-9y=54 orz-y=6 QuiCker Method: Applying the above rule, we have 54 the required answer = 9" = 6 . 2. 3. If the places oflast two-digits of a three-digit number are ii1terchanged, a new number greater than the original number by 18 is obtained. What is the difference between the last two digits of that number? a) I b) 2 . c) 3 d) 4 If the places ofJast two-digits of a three-digit number are interchanged, a new number greater than the original number by 9 is obtained. What is the difference between the last two digits of that number? ~l ~3 ~4 ~6 If the places of last two-digits of a three-digit number are Ex.: A number when divided by 899 gives a remainder 63. What remainder will be obtained by dividing the same number by 29. So]n: Detail Method: Number == Divisor x Quotient + Remainder = 899 x Quotient + 63 = 29 x 3 I x Quotient + 2 x 29 + 5 Therefore, the remainder obtained by dividing the number by 29 is clearly 5. 66 PRACTICE BOOK ON QUICKER MATHS Quicker Method: Applying the above rule, we have, 63 + 29 i.e. 29) 63 (2 58 5 :. required answer = 5 Exercise 1. 2. 3. 5. 7. 8. 9. 5.d 4.c 6.a Suppose, the larger divisor is N J ' and the smaller divi- ~7 Where, N J = K N z and K = any integer> I. ~6 08' ~J 09 ~9 Now, when the number is divided by N z ' then remainder is Rz (say) and when the ~2 05 2N 2 +R 2 = R) In the given question, 357 _ 21 ~8 a)4 b) 18 c)9 K=-- Nz = 17and KN2 =357 d) 6 17 Here, K> 1 an integer. Now, we can apply the remainder rule. 2N2 +R2 = RJ or, 2 x 17 + 5 = R, ~4 10. A number when divided by 1404 gives a remainder 93. What remainder would be obtained by dividing the same number by 39? a)4 b) 13 c)19 d)15 II. A number when divided by 17, leaves a remainder 5. What remainder would be obtained by dividing the same number by 357? a)39 c)21 b)29 d)38 same number is divided by N\(=KNz),retnainderis R, (say). Then, by the remainder rule, we have the following formula, A number when divided by 1491 gives a remainder 83. What remainder would be obtained by dividing the same number by 21 ? a)21 b)2 c)20 d) 18 A number when divided by 1092 gives a remainder 60. What remainder would be obtained by dividing the same number by 28? A number when divided by 1156 gives a remainder 73. What remainder would be obtained by di" ,Jing the same numberby34? a)5 b) 17 c) 13 d)4 A number when divided by 1836 gives a remainder 79. What remainder would be obtained by dividing the same number by 36? a)7 b)9 c) 19 d) 14 A number when divided by 1207 gives a remainder 85. What remainder would be obtained by dividing the same number by 17? a)7 b)2 c)O d)6 A number when divided by 2470 gives a remainder 80. What remainder would be obtained by dividing the same numher by 38? 7.a sor is Nz . A number when divided by 609 gives a remainder 65. What remainder would be obtained by dividing the same number by 29? a)6 b)5 c)6 d)7 A number when divided by 738 gives a remairider 92. What remainder would be obtained by dividing the same number by 18? ~6 6. 3.a 8:c 9.a 10.d 11. a; Hint: Here we apply "Remainder Rule". This rule is applicable when the same number (dividend) is divided by two different divisors which are multiples of each other. A number when divided by 221 gives a remainder 43, what remainder will be obtained by dividing the same number by 17? ~2 4. Answers I.d 2.d :. RJ =39 Hence, the required remainder = 39. Note: All the other questions can also be solved by this rule. Rule 40 If the sum of two numbers is x and their difference is y, then the difference of their squares is xy. Illustrative Example Ex.: The sum oftwo numbers is 75 and their difference is 20. Find the difference of their squares. Soln: Detail Method: Let the numbers bex andy. According to the question, x+y= 75 .... (i) and x-y=20 ... (ii) Now, multiplying eqn (i) and (ii), we get 2 x -l = Difference of the squares of numbers =75 x20= 1500 Quicker Method: Applying the above rule, we have, required answer = 75 x 20 = 1500 Exercise I. The sum of two numbers is 100 and their difference is 37. The difference of their squares is. [Clerk's Grade Exam, 1991] i I -.II 8. Number System 6 7 a)37 b) 100 c) 63 d)3700 . The sum of two numbers is 50 and their ditzerence is 6. The difference of their squares is a) 400 b)500 c)350 d) 300 3. The sum of two numbers is 75 and their difference is 9. The difference of their squares is a) 685 b) 625 c) 675 d) 775 4. The sum of two numbers is 160 and their difference is 39. The difference of their squares is a) 6420 b) 4620 c) 8420 d) 6240 5. The sum of two numbers is 175 and their difference is 75. The difference of their squares is a) 13025 b) 13125 c) 13215 d) 13152 3.c 5.b 4.d 2. Answers I. d Answers 1. a If the two consecutive numbers arex andy, then the difference of their squares is given by x + y. Illustrative Example Ex.:· Two consecutive numbers are 8 and 9. Find the difference of their squares. Soln: Detail Method: Required answer = 92 - 82 = 81- 64 = 17 Quicker Method: Applying the above rule, we have the required answer = 8 + 9 = 17 Exercise 2. 3. 4. 5. or, x2 +1+2x-x2 =37 or, 2x = 37 - 1 = 36 :.x=18 and x+I=19 :. numbers are 18, and 19 Quicker Method: Applying the above rule, we have the required an.swer = 37-1 37+1 -- and -- = 18 and 19 22· Two consecutive numbers are 17 and 18. Find the difference of their squares. a)36 b)25 c)35 d)34 Two consecutive numbers are 75 and 76. Find.the difference of their squares. a) 141 b)151 c)l3l d)115 Two consecutive numbers are 79 and 80. Find the differ- . ence of their squares. a) 159 b) 169 c) 149 d)158 Two consecutive numbers are 15 and 16. Find the difference of their squares. a)31 b)32 c)30 d)21 Two consecutive numbers are 26 and 27. Find the difference of their squares. a)53 b)52 c) 43 d) 63 Answers I.c 2.b 3.a 4.a 2. 3. 4. 5. The difference between the squares of two consecutive numbers is 39. Find the numbers. a) 19,20 b)20,21 c) 18, 19 d) 17, 18 The difference between the squares oEtwo consecutive numbers is 27. Find the numbers. a)14,15 b)13,14 c)15,16 d)16,7 The difference between the squares of two consecutive numbers is 35. Find the numbers. a) 14, 15 b) 15, 16 c) 17,18 d) 18, 19 The difference between the squares of two consecutive numbers is 59. Find the numbers. a)29,30 b)30,31 c)28,29 d)27,28 The difference between the squares of two consecutive numbers is 77. Find the numbers. a)3839 b)39,40 c)40,41 d)37,38 5.a Rule 43 . If the sum of two numbers is x and sum of their squres is y, then the Exercise I. 5.a 4.a 3.c Rule 42 1. 2. d 2. b (i) product of numbers is given by (X2 - YJ l-2- and· X-~2y-x2] [ (ii) tlte numbers are. 2 [x+~] and 2 Illustrative Example The sum of two numbers is 13 and the sum of their squares is 85. Find the numbers. Soln:- Detail Method: Let the numbers be x and y. According to the question, Ex.: x+y= 13 .... (i) and x2 +.l = 85 .... (ii) Now, from eqn (i) and eqn (ii), we have (x+ y)2 = 169 68 PRACTICE BOOK ON QUICKER MATHS 2 Soln: Detail Method: Let the numbers be x and y. According to the question, J or, x + y + 2xy = 169 or, 2xy= 169-85 = 84 :. xy;= 42 [xy = product of two numbers] Again, (x- x2 + y2 = 90 ..... (i) and (x- yf = 46 .... (ii) yf = (x+y)2 -4xy From eqn (ii) =169-4x42=1 :. x-y= 1 .... (iii) From eqn (i) and eqn (iii) we have, x= 7 andy=6 :. Numbers are 7 and 6 Quicker Method: Applying the above rule, we have, '(x-yY ~ 46 2 or, x + y2 - 2xy = 46 or, 90 - 2xy = 46 [Puttingthe value of eqn (i)] 90-46 or, ,xy= ---=22 2 . n-.J170-169 reqUired answers = ----- and 2 :. product of two numbers = 22 Quicker Method: Applying the above rule, we have 90-46 the required answer = --, - = 22 2 13+.J170-169 =6and7 2 Exercise 1. The sum of two numbers is 15 and sum of their squares is 113. The numbers are: [CDS Exam, 1991] a)4,11 b) 5, 10 c)6,9 d)7,8 2. The sum of two numbers is 25 and sum of their squares is 313. The numbers are: a)12,13 b)20,25 c) 9, 16 d)21,4 3. The sum of two numbers is 26 and sum of their squares is 340. The numbers are: a) 12, 14 b)11,15 c) 9, 17 d)8,18 4. The sum of two numbers is 30 and sum oftheir squares is 458. The numbers are: a)14,16 b) 12; 18 c) 13, 17 d)II,15 5. The sum of two numbers is 14 and sum of their square.§ is 100. The numbers are: a) 6, 8 b)5,9 c)4,10 d)3,11 6. The sum of two numbers is 13 and sum of their squares 89. Find the product of the two numbers. a) 40 b)36 c)22 d)30 7. ,The sum of two numbers is 32 and sum of their squares 514. Findthe product of the two numbers. a)510 b)225 c}255 d) 355 Answers 1. d 2: a Exercise 1. 2. 3. 4. 5. The sum of squares of two numbers is 80 and the square of their difference is 36. The product of the two numbers is [Clerks' Grade Exam, t 9911 a) 22 b) 44 c)58 d)116 The sum of squares of two numbers is 40 and the square of their difference is 20. The product ofthe two numbers is a) 10 b)20 c)15 d)16 The sum of squares of two numbers is 95 and the square of their difference is 37. The product of the two numbers is a) 18 b) 19 c)29 d) 27 The sum of squares of two numbers is 94 and the square oftheir difference is 24. The product ofthe two numbers is a)36 b)40 c)30 d)35 The sum of squares oftV;'o numbers is 87 and the square of their difference is 25. The product of the two. numbers is b)35 a)31 Answers 3. a 4. c 5. a 6. a 7. c 1. a 2. a 3.c c)32 4.c d)30 5.a Rule 44 Rule 45 If the sum of squares of two numbers is x and the square of their difference is y, then the product of the tlllO numbers is If the product of two !lumbers is x ami the sum of their squa}'es is y, then (i) the sum of the two /lumbers is given by (x;y). ~ Y + 2x ami (if) tlte difference is ~ y - Illustrative Example Illustrative Example Ex.: 2; . The sum of squares of two numbers is 90 and the square of their difference is 46. The product of the two numbers is Ex.: The product of two numbers is 143. The sum of their squares is 290. Find the sum of the two numbers and also find the difference of the two numbers. Soln: Detail Method: Let the numbers be x and y. .. ...- .. 69 Number System Rule 46 According to the question, xy= 2 143 and x + l The denominator of a rational number is 'D' more than its numerator. lfthe numerator is increased by x and the denominator is decreased by y,we obtain p, then the rational = 290 Now, 2 (x+yf =x +y2+2xy =290+2 x 143 =576 X-P(D-y)] or,x+y= ...)576 =24 numberis given by . x + (yP - D) . .'. Sum of the numbers = 24 Again, Illustrative Example (x - y [ Ex.: y = x + y2 - 2xy 2 =290-286=4 or,x-y=2 ... difference of the numbers = 2 Quicker Method: Applying the above rule, we have the sum ofthe numbers The denominator ofa rational number is 3 more than its numerator. If the numerator is increased by 7 and the denominator is decreased by 2, we obtain 2. The rational number is ______ _ Soln: Detail Method: Let the numerator be x and the denominator = x + 3. According to the question, x+7 x+3-2 = ...)290+2xI43 =...)576 =24 and the difference of the numbers 2 or, x + 7 = 2x + 2 ... x = 5 = ...)290-2xI43 =14 =2 ... Numerator = 5 and the denominator = 5 + 3 = 8 Exercise 1.The product of two numbers is 12.0. The sum of their squares is 289. The sum of the two numbers is __ '_. [Clerks' Grade Exam, 1991] a) 20 b}23 c) 169 d)33 2. The product of two numbers is 48. The sum of their squares is 100. The sum ofthe two numbers is __ . ~14 ~12 ~18 ~24 3. The product of two numbers is 168. The sum of their squares is 340. The sum of the two numbers· is __ . a)36 b)24 c)26 d) 28 4. The product of two numbers is 36. The sum of their squares is 97. The sum of the two numbers is __ . a) 13 . b)12 c}15 d}l1 5. The product of two numbers is 35. The sum of their squares is 74. The sum of the two numbers is __ . a) 13 b) 12 c) 14 d) 17 6. The productof two numbers is 120. The sum of their squares is 289. The difference of the two numbers is a)7 b)9 c)8 d) 23 7. The product of two numbers is 180. The sum of their squares is 369. The,difference of the two numbers is 8. b)2 d) 15 c)4 Answers 2.a ~. b 3.c 4.a 5.b 6.a 5 Quicker Method: Applying the above rule, we have . 7-2(3-2)_ 5 ReqUIred answer = 7 + (2 x 2 - 3) '8 Exercise 1. The numerator of a rational number is 4 less than its denominator. If the numerator is increased by 8 and the denominator is decreased by 2, we obtain 3. Find the rational number. 3 b) 7" 2. 3. 1 c) "5 5 d) "9 The denominator of a rational number is 6 more than its numerator. If the numerator is increased by 9 and the denominator is decreased by 5, we obtain 5. Find the rational number. 1 2 b) . 8 a) 7 " The denominator of a rational number is 3 more than its numerator. If the numerator is increased by 6 and the denominator is decreased by 2, we obtain 2. Find the rational number. a)3 b) 27 c)5 d) 17 The product of two numbers is 224. The sum of their squares is 452. The difference of the two numbers is -a)30 l.b 7.a ... rational number = '8 1 4. 5 4 d) 7" b) '8 a) ' 3 The denominator of a rational number is 8 more than its numerator. If the numerator is increased by 7 and the denominator is decreased by 8, we obtain 8. Find the 70 PRACTICE BOOK ON QUICKER MATHS rational number. 123 a) 5. 6. "9 b) 10 c) IT The denominator of a rational number is 2 more than its numerator. If the numerator is increased by 9 and the denominator is decreased by 5, we obtain 7. Find the rational number. 5 7 9 3 a)- b) c} d)7 9 II 5 The denominator of a fraction is 2 more than thrice its numerator. If the numerator as well as denominator is 1 increased by one, the fraction becomes "3' What was the original fraction. [SBI PO, 1999] 435 a) 5 13 b) 11 c) 13 Answers I.c 2.a d) IT x 1 y :. x:y= 1:2 2 Quicker Method: Applying the above ruie, we have 150-100 1 . the required ratio = 100 - =]:2 :2 Note: In case tHe total ie (A + B) becomes P% of the number 100 ) ( A, the r'atio between A and B is given py P -1 00 . Exercise 1. When a number added another totalis the becomes 333~isper centto ofthe first number number.the What 3 ratio between the first and the second number? a) 3 : 7 b) 7: 4 c) 7: 3 d) Data inadequate When a number is 2. added to another number the total becomes 333 ~ per cent afthe second number. What is 3 - .. the ratio between the first and the second number? 3.d . 4.~ 5.a ISBI PO 20001 6. b; Hint: This type of question may be solved by hit and trial a)3:7 b)7:4 c)1:3 d)4:7 method. When a number is added to another number the total becomes 3 First divide the question in different parts. Then start from the . 250 per cent ofthe second number. What is the answer-choices one-by-one. The choice, which' satisfies all ratio between the first and the second number? a)3:2 the parts of the given question, will be required answer. For b)2:3 c)4:3 d)3:4 example, in the above question we have two parts. 4. When a number is added to another number the total becomes (I) The denominator of a fraction is 2 more than thrice its 175 per cent of the first number. What is the ratio between the numerator. first and the second number? (II) If the num.erator as well as denominator is increased ~ 5. by 1, a)4:3 b)3:4 c)5:3 d)3:5 the fractIOn becomes 1/3. When a number is added to another number the total becomes . Both parts will be satisfied by the answer choice (b), hence 275 per cent of the first number. What is the (b) is the required answer. ratio between the first and the second number? a)4:7 b}7:4 c)3:8 d)8:3 6. When a number is added to another number the total becomes Rule 47 125 per cent of the second number. What is the ratio between When a lIumber 'A' is added to another number 'B' alltl the total ie the first and the second number? (A + B) becomes P% of the lIumber B, thell the ratio a) 1:4 b)4:1 c)1:2 d)2:1 7 When a number is added to another number the total becomes between A and B is given by 375 per cent of the second number. What is the . ratio between the first and the second number? a)4:11 Illustrative Example b)11:4 c)4:7 d)7:4 Ex.: When a number is added to another number the total When a number is added to another number the total becomes becomes 150 per cent of the second number. What is the 8 375 per cent of the first number. What is the . ratio between the first and the second number? ratio between the first and the second num ber? Solo: Detail Method: a)4:11 b}II:4 c)4:7 d)7:4 Let the numbers be x and y.' When a number is added to another number the total becomes According to the question, 150 9. 225 per cent of the first numbel. What is the x+y= 150%ofy= lOOy ratio between the first and the second number? a)5:4 b)4:5 c)3:4 d)4:3 ]0. When a number is added to another number the total becomes 3 I or x + y = --'-y or x = - y 225 per cent of the second number. What is the , 2 ' 2 . (P-I00). wo 7 1 Number System Answers ratio between the first and the second number? a)3:4 b)4:3 c)5:4 "p)4:5 4.d 3.b 1. a 2. a 5.c Rule 49 Answers I.a 2.c 3.a 8.a 9.b 10.c 4.a 5.a 6.a 7.b Rule 48 The sum of three consecutive even or odd immbers is Pless Two different numbers when divided by the Sllme divisor, leaves remainders x and y respectively, and when their sum is divided by the same divisor, remainder is z, then the divisor is given by (x + Y - z). Or, Divisor = (sum of remainders) - (Remainder when sum is divided) a or more than b ofQ. Then the middle number is given by Illustrative Example Ex: Two different numbers when divided by the same divisor, left remainders II and 21 respectively, and when their sum was divided by the same divisor,remainder was 4. What is the divisor? Soln: Applying the above rule, we have the required answer= II +21-4=28 Note: +ve and -ve sign indicate more and less respectively Exercise Illustrative Example Ex: The sum of three consecutive even numbers is 15 less than three-fourth of60. What is . the middle number? Soln: DetailMethod: Let the middile number be x According to the question, 60x3 ' x-2+x+x+2= ---15 4 ' or, 3x= 30 :.x= 10 ... required answer = 10 Quicker Method: Since we have less type of question, the above formula will be like . Q(~)-p Middle number = ----= 3 60x~-15 4 3 10. Exercise 1. The sum of three consecutive even numbers is 14 less than one-fourth of 176. What is the middle number. [BSRB Mumbai PO, 19981 a)IO b)8 c)6 d)4 2. The sum of three consecutive odd numbers is 15 more than one fourth of 120. What is the middle number? a)15 b)13 c) 17 d)21 3. The sUm of three consecutive even numbers is 24 less than one-sixth of324. What is the middle number? a) 12 b) 10 c) 14 d)20 , 4. The sum of three consecutive even numbers is 8 less than two-third of 66. What is the middle number? a) 10 b)18 c)16 d)12 5. The sum of three consecutive odd numbers is 25 more than two-fifth of 65. What is the middle number? a)]5 b) 19 c) 17 d)21 - ,I. Two different numbers when divided by the same divisor, left remainders 10 and 15 respectively, and when their sum was divided by the same divisor, remainder was 3. What is the divisor? a) 22 b)25 c)23 d)21 2. Two differ.ent numbers when divided by the same divisor, left remainders 5 and 7 respectively, and when their sum was divided by 'the same divisor, remainder was 2. What is the divisor? a)11 b)12 c) 10 d)9 3. Two different numbers when divided by the same divisor, left remainders 13 and 23 respectively, and when their sum was divided by the same divisor, remainder was 5. What is the divisor? a) 32 b)36 <:)30 d)31 4. Two different numbers when divided by the same divisor, left remainders 12 and 21 respectively, and when their sum was divided by the same divisor, remainder was 4. What is the divisor? a)28 b) 27 c)31 d)29 5. Two different numbers when divided by the same divisor, left remainders 15 and 17 respectively, and when their sum was divided by the same divisor, remainder was 8. What is the divisor? ~~ ~~ Answers 3.d 1. a 2. c 0TI 4.d ~~ 5.a Rule 50 If the product of two numbers is x and the sum o.lthese two . [ --]j ; y+ y -4x numbers isy, then the 1lumbers a~egive1l by ~---- 2 , 72 PRACTICE BOOK ON QUICKER MATHS an" [y-~~' Rille 51 -4x ). If the product of two numbes is x and the difference between these two numbers is y, then the numbers are Illustrative Example The product of two numbers is 192 and the sum of these two numbers is 28. What is the smaller ofthese two numbers? [BSRB Calcutta PO 1999] Soln: Detail Method: Let the numbers be x and y. :. xy= 192,x+y=28 ..... (i) ~y2+4X+Y] 2 Ex: ,'. (x_y)2=(x+y)2_4xy Illustrative Example . The product of two numbers is 192 and the difference of these two numbers is 4. What is the greater of these two numbers? Soln: Detail Method: Let the numbers is x and y. xy= 192 andx-y=4 ...• (i} (x+y? = (x- yf +4xy = (4? +4x]92=784 x+y=28 .... (ii) , Solving eqn (i) and eqn (ii) we have x= ]6andy= 12 :. Greater number = 16 Quicker Method: Applying the above rule, we have req.uired answer 28+4 ~y2 +4x + y = 2 _ 32 = 16 2 2 2 Note: ~/+4x+y ~y2+4x-y --'->----- 2 Exercise I I : 1.. The product of two numbers is ] 54 and the sum of these two numbers is 25. Find the difference between the numbers. a)3 b)4 c)5 d)8 2. The product of two numbers is 252 and the sum of these two numbers is 33. Find the greater number. a)2] b)]2 c)]3 d)23 3. The product of two numbers is 255 and the sum of these two numbers is 32. Find the smaller number. f; I ,I 11 a)]7 4. ~ I I ~ 5. b)]6 c)]5 d)]3 The product of two numbers is ] 68 and the sum ofthese two numbers is 26. Find the smaller number. a)12 b)]4 c) 16 d) 18 The product of two numbers is 486 and the sum of these two numbers is 45. Find the smaller number. a) 12 b) 18 c)26 d) 34 Answers I. a 2. a .fi84 +4 _ 28+4 222 16 and 28-·h82 -4xl92 _ 28-4 = 24 = 12. 2 :. smaller number = 12. . Ex: .. 28+~282 -4x192 the reqUIred numbers = ---2 = 2 [ , =784-768 = 16 :. x - y = 4 .... (ii) Combining eqn (i) and eqn (ii) x= 16,andy= 12 :.smallernumber= 12. Quicker Method: Applying the above rule, we have 28+ .,}784 - 768 2 [~y2+4X-Y] and 2 Exericse 1. The product of two numbers. is 22] and the difference of these two numbers is 4. Find the smaller number. a) 13 b)14 c)16 d)17 2. The product oftwo numbers is ] 98 and the difference of these two numbers is 7. Find the greater number. a) 18 b) 15 c) 13 d) I] 3. . The product of two numbers is ] 80 and the difference of these two numbers is 3. Find the sum of the numbers. a)26 b)25 c)28 d) 27 4. The product of two numbers is 594 and the difference of these tw'J numbers is 5. Find the sum of the numbers. a)46 b)39 c)40 d)49 5. The product of two numbers is 468 and the difference of these two numbers is 8. Find the sum oHhe numbers. a) 42 b)44 c)48 d)34 Answers 3.c 4.a 5.b I. a 2. a 3.d 4.d S.b Number System 73 Miscellaneous J. a) 125 d)25 •• If a fi'action's numerator is increased by 1 and the de2 "3' But when the numerator is increased by 5 and the denominator is increased by 1 then the fraction becomes ~ .What is the value of the original fraction? 4 [Bank of Baroda PO, 1999] 3 5 5 ~- 6 ~- ~~ 7 8 7 7 2. If the numerator of a fraction is increased by 2 and denominator is increased by 3, the fraction becomes 7/9; and if numerator as well as denominator are" decreased by 1 the fraction becomes 4/5. What is the original fi'action? [SBI Associates PO, 19991 13 a) 1 6 -5 9 c) 6' b)I l 17 d) 21 3. e) None of these If the numerator of a fraction is increased by 2 and the denominator is increased by 1, the fraction becomes "8 5 and if the numerator ofthe same fraction is increased by 3 and the denominator is increased by I the fraction 3 . becomes -. Whatis the original fraction? 4 [Guwahati PO Exam, 1999) 2 a) Data inadequate b) 3 d)7 4. 5. 4 "7 c) "7 e) None of these In a two-digit number, the digit at unit place is I more than twice of the digit at tens place. If the digit at unit and tens place be interchanged, then the difference between the new number and original number is less than I to that of original number. What is the original number? [BSRB Hydcrabad PO, 1999] a) 52 b) 73 c)25 d)49 e)37 I c) 40 e) None of these 6. riominator is increased by 2 then the fraction becomes ~- b)70 5 "5 ofa number is equal to "8 of the second number.lf35 is added to the first number then it becomes 4 times of second number. What is the value of the second number? [BSRB Hyderabad PO, 19991 The ratio of two numbers is 3 : 2. If! 0 and the sum of the two numbers are added to their product, square of sixteen is obtained. What could be the smaller number? [NABARD,19991 a) 14 b) 12 c) 16 d) 18 e) None of these 7. The numbers x, y, z are such that xy = 96050 and xz="· 95625 and y is greater than z by one. Find out the numberz. [NABARD,1999j a) 425 b) 220 c) 525 d) 226 e) 225 8 If the sum of one-half, one-third and one-fourth of a number exceeds the number itself by 4, what could be . the number? [NABARD,19991 a) 24 b)36 c) 72 d) 84 e) None of these When any number is divided by 12 then dividend bel 9. comes - of the other number. By how much per cent is 4 first number greater than the second number? [BSRB Chennai PO, 2000) a)200 b) 150 c)300 d) Data inadequate e) None of these 10. A number gets reduced to its one-third when 48 is substracted from it. What is two-third of that number? [BSRB Bhopal PO, 2000J c)36 a) 24 b) 72 d)46 e) None of these II. The sum of three consecutive numbers is given. What is he difference between first and third number? IBSRB Bhopal PO, 20001 a)-One 2) Three c) Either one or three e) d) Two None of these 12. If the two digits of the age of Mr Manoj are reversed thcn the new age so obtained is the age of his wife. - I 11 of the sum of their ages is equal to the difference between their ages. IfMr Manoj is elder than his wife then find the difference between their ages. [BSRB Bangalore PO, 20001 a) Cannot be detennined b) 10 years c) 8 years d) 7 years e) 9 years 13. A number is greater than the square of 44 but smaller than the square of 45. If one part of the number is the square of 6 and the number is a multiple of 5, then find the number. [BSRB Bangalore PO, 20001 a) 1940 b)2080 c) 1980 d) Cannot be determined e) None of these ]4. Ifa number is decreased by 4 and divided by 6 the result I I ! i f 74 PRACTICE BOOK ON QUICKER MATHS i~ 9. What would be the result in is subtracted from the nllmber and then it is divided by 5? [BSRB Delhi PO, 2000] 2 I' 2 a)9- b) 10c) 11.55 5 d) 11 e) None of these ]5. A two-digit number is seven times the sum of its digits. Ifeach digit is increased by 2, the number thus obtained is 4 more than six times the sum of its digits. Find the number. [BSRB Patna PO, 2001] a)42 b)24 c)48 d) Data inadequate e) None of these ]6. The digit in the units place of a number is equal to the digit in the tens place of half ofthat number and the digit in the tens place of that number is less than the digit in units place of half of the number by 1. ]fthe sum of the digits of the number is seven, then what is the number? ISBI BankPO, 2001] , a)52 b) 16 c)34 d) Data inadequate e) None of these ] 7. A fraction becomes 4 when 1 is added to both the numerator and denominator, and it becomes 7 when I is subtraced from both the numerator and denominator. The numerator of the given fraction is: a)2 b)3 c)7 d) ]5 (NDAExam 1990) 18. If I is added to the denominator of a fraction, the fraction becomes (1/2). If I is added to the numerator, the fraction becomes 1. The fraction is: 4 a) -: :; 5 b) "9 2 10 d)- c) 3' 11 [-CDS Exam 1991 I 19. The sum of two numbers is twice their difference. If one 6fthe numbers is 10, the other number is: .., 1 1 .., 1 a).J3 c)300r - - d)300r.Jb)30 3 3 3 [RRB Exam 1991] , l ' 4 20. 5 of a certain number is 64. Half ofthat number is: a) 32 b)40 c)80 d) 16 IBSRB Exam 1991] I 1 21. - ofa number subtracted from :;- of the number gives 4 .) 12. The number is: c) 72 d)63 a)]44 b)]20 IHotel Management, 19911 22. If one fifth of a number decreased by 5 is 5, then the number is: c)60 d) 75 a)25 b)50 [Clerks' Grade Exam 19911 23. I I times a number gives 132. The number is a)]] b)]2 c) 13.2 d)Noneofthese [Clerks' Grade Exam 1991/ 2 24. A number is25 more than its 5th. The number is: [Clerk.s' Grade E~am, 19911 125 a) -3" 125 b)7 c)60 d) 80 25. 24 is divided into two paJ1S such that 7 times the first part added to 5 times the second part makes ] 46. The first part is: b) 13 c) 16 d) 17 a) 11 [RRB Exam 19911 4 26. 2 5 of a number exceeds its '3 by 8. The number is: a)30 b) 60 c) 90 d)Noneofthese IRRB Exam 19891 27. The difference between squares of two numbers is 256000 and sum of the numbers is 1000. The numbers are: a) 628, 372 b) 600, 400 c) 640, 630 d) None of these [GICAAO Exam, 19881 28. Three numbers are in the ratio 3 : 4: 5. The sum of the largest and the sm~lIlest equals the sum of the third and 52. The smallest number is: a) 20 b)27 c)39 d) 52 [Accountants' Exam 19861 29. A positive number when decreased by 4, is equal to 2 I times the reciprocal of the number. The number is: a)3 b)5 c)7 -(~I)~ --[NDAExam 19871 30. The sum on Immb(;}FS'r~n58. If the ratio between first and second be 2 : 3 and that between second and third be 5 : 3, then the second number is: d)48 a)30 b)20 c)58 [SSC Exam 1986/ 31. Two numbers are such that the ratio between them is 3 : 5; but if each is il)creased by 10, the ratio between them becomes 5: 7. The numbers are: a)3,5 b)7,9 c) 13,22 d)]5,25 [RRB Exam. 19891 32. Divide 50 into twq parts so that the sum of their recipro cals is (1/12): a)20,30 b) 24, 26 c)28,22 d)36,14 [RRB Exam 19881 33. The sum of seven numbers is 235. The average of the first three is 23 and that of the last three is 42. The fourth number is: c) 69 d) ]95 a) 40 b)126 [Clerks' Grade Exam. 1991/ 34. How many figures (digits) are required to number a book 9. Number System 7 5 cont~jning 200 pages? a) 200 b) 600 c) 492 2 35. In a question, divisor is 3" of the dividend and 2 times the remainder. rfthe remainder is 5, find the dividend. a) 15 b)25 c) 18 ---:;:J = "8 or, 8x •. 5y = -II Y . x ]. c; Let the fraction be - then, y . .... (i) 5 Also,.we have y + I ="4 or,4x+20=5y+5 or,4x=5y-]5 From (i) and (ii), we get 2y+1 5y-]5 Y = 2x+ ] .... (i) imd(lOy+x)-(IOx+y)= lOx + y-] or, 9y - 9x = ] Ox + Y - ] or, ] 9x - 8y = ] Puttingthe value of (i) in equation (ii) we get, ] 9x - 8(2x + ]) = ] or, ] 9x - ] 6x - 8 = ] or,3x= 9 or, x= 3 So, y= 2 x 3 + ] =7 ... original number = ] a x 3 + 7 = 37 ] 5 . -=- (I') 5 · -1=-11 C . 2y+] 2x7+I or, 7y=49 ..... y=7and x=--=---=53 3 x 5 ... y =7 .. 11 8 .... 25 or, -11 +35 = 411 8 x+2 7 tively. Then y + 3 = 9 x-I 4 .... (ii) Solving (i) and (ii), we get x = 5, Y = 6 Reqd fraction = 5/6 x 3. d; Let the original fraction be - . y (But -ve value cannot be accepted) So, x = 6. Hence, smaller number = 2x = ]2 7. e; xy=96050 ..(i) and xz= 95625 .... (ii) andy-z=] ... (iii) .: ............. ............................ y 96050 3842 . Dlvldmg (I) by (ll), we get -; = 95625 = 3825 .... (IV) 8. e; Let the number be x. . (~+~+~)x=(6+4+3)x=~x or, 9(x + 2) = 7(y + 3) or; 9x- 7y = 3 I 6 Combining (iii) and (iv), we get z = 225 2. c; Let the numerator and denominator be x and y respec- = 1 25 -41 , or 4· , Y.=J = 5" or, Sx - 4y . fraction = - 7 ... 4. e; Let the original number be ] Ox + Y .... {ii) --. =-- or 8y+4= ]5y-45 required original fraction =. 3 x = 3 and y = T ... 1l=40 6. b; Let the two numbers be 3x and 2x. According to the question, ]0 + (3x+2x)+(3x x 2x)=(]6)2 or, 6x2 + 5x - 246 = 0 or, 6x2 +4]x - 36x -246 =0 or, 6x(6x +4 I) - 6(6x +4]) = 0 or, (6x +4])(x -6) =0 ... x=6 y+2 = 3" or, 3x +3 =2y+4 x+5 .... (ii) solving eqn (i) and (ii), we get 1+35=411 2 or, 3x = 2y + 1 .... (i) x+3 3 Again, y +] == 4' or, 4x - 3y = -9 . '5 .8 Answers 3 Then d) 24 [SSe 94) 36. A number when divided by 5 leaves a remainder 3. What is the remainder when the square ofthe same number is divided by 5? a)9 b)3 c) ] d)4 [MBA 19901 37.Assuming that A, Band C are different single-digit numerical values other than what is already used in the following equation, what number C definitely cannot be? 8A2+3B5+C4- ]27] b) 9 c) Either 7 or 9 e) None of these a)7 d) 6 x+] x+2 5 d) 372 . '" [MBA 1980] .... (i) .. 2 3 4 ]2 According to the question, 12 ]3 -x-x = 4 . x=48 12 .. 9. d; Here neither the remainder nor the dividend nor the second number is given, so can't be determined. ] O. e; Let the number be x. x 2 then ,x--=48 3 . -x=48 .. 3 76 PRACTICE BOOK ON QUICKER MATHS 11. d; Let the three consecutive numbers be x, x + I and x + 2 respectively. :. Diff. between first and third numbers =x+2-x=2 12. e; Let the age ofMr Manoj be (lOx + y)yrs. :. His wife's age = (1 Oy + x) years 1 Solving (i) and (ii), we get x = ·15, Y = 3. x 18. c; Let the required fraction = xI :. y+l ='2 ~ 2x-y=] .. ~ x+1 And, ,-- = 1 ~ x - y= -1 Y Solving (i}and (ii), we getx = 2,y= 3. Then, (lOx+y+ lOy+x) - = 10x+y -lOy-x" II x 5 or, x + y = 9x - 9y or, 8x = I Oy or, y = '4 :. x = 5 and y = 4 (because any other multiple of5 will make x 6ftwo digits) :~ Diff=IOx +y -lOy-x =9x-9y=9(x -y) =9(5 4)= 9 yrs 13.c; Let the number be x. 2 44 <x<452 ~ 1936<x<2025 .... (i) From equation (i), the required number will be any numberbetweenl1936 and 2025 . Since one part of the number is the square of6 means one factor is 36. :. LCMof36and5= 180 :. Number will be multiple ofl80 ie 180x II = 1980 the only value which satisfies the equation (i) 14. d;Let the number be x :. The fraction is '3 lO+x =2(x-1O) ~ x=30. 4 20 b·-xx=64 - x=--=80 ·, 5 64x5 --- 4 1 I . -xx= -x80 = 40 .. 2 2 1) 1 1 ( 21 a' -xx--xx =12 - -x=12~x=144 ·,3 4 --- 12' . 2 '(~OfX)~5= 5~~=10=>x=50 ·,5 5 . 23. b; l1x= 132 ~ x= 12 2 3x ' 125 2 ·.,x--x=25~-=25~x=-' 5 '5 3. ~ 25. b; Let these parts be x and (24 - x). Then, 7x+5(24-x)= 146 ~ x= 13 . So the first part is 13. 26. b;Let the number be x. Then, 4 2 2 -x--x =8~-x =8~ x =60 5 3 15 27. a; Let the numbers be x and y. Then, x2 '- y2 = 256000 and x + y = 1000. 2 , x _ y2 256000 x - y = -- = -- = 256 .. x+ y 1000 Solving x +y= 1000, x-y =256, we get x =628,y= 372. 28. c; Let the numbers be 3x, 4x and 5x. 5x+3x=Ax+52 ~ x= 13 . :. smallest number = 3x = 39. 29. c; Let the number be x. Then, x+i Then, y-:;:-T = 4 ~ x - 4y = 3 x-I 2 . 19. b;Let the other number = x x-4 x-3 58-3 . --=9 -x=58Again --=--=11 6 --.,5 5 15. a; Let the two-digit number be lOx + y. IOx+y=7(x+y) ~ x=2y .... (i) 10(x +2)+y+2 = 6(x + y+4)+4 or, lOx+y+22=6x+6y+28 ~ 4x-5y=6 .. : .. (ii) Solving equations (i) and (ii), We get x = 4 and y = 2 16. a; Let 1/2 of the no. = lOx + y and the no. = 10V + W From the given conditions, W = x and V = Y -I Thus the no, = lO(y - I) + x .... (A) :. 2(10x+y)= lO(y-I)+x ~.8y-19x= 10 V+W,\=7 ~ y-l +x=;7 :. x+y=8 .... Solving equations (i) and (ii), we get (i) x = 2 and y = 6 :. From equation (A), .... Number = 1O(y-I)+x=52 (ii) x 17 d' Let the required . , fraction be.- . Y And -- = 7 ~ x - 7y = -6 , y-] y. 4 21 24 2<' x- =-~x - x- l=U~X= x 7 :. required number = 7 ------_._._ .. _~ _. - --... 77 . Number System Number oftwo digit pages from 10 to 99 == 90 Number of three digit pages froni 100 to 200 == 101 :. total number of required figures = 9 x I + 90 x 2 + 10 1 x 3 = 492 35. a; According to the question 30. a; Let the numbers be x, y, z. Then, x_2y_5 ;-3'-;-3 3 .. ., => 3x=2yand5z=3y. 3 33 9 . y=_x,z=-y=-x-:-x=-x ., 2· 5 52 10 39· ~ 34x=680 -. x=20 ., x+-x+-,--x=98 2 10 -;" 2 Divisor = "3 x dividend and 2 x remainder -;'" 3 2'(3 ",30) So, second number = '2 x == . x 20 31. d; Let the numbers be 3x and 5x. . 36. d; The number is of the form (5x + 3), where x is an integer 3x+IO =~=>x=5 ., 5x+ 10 7 25x2 +30x+9 25x2 30x 5+4 ---- --+-+-- Hence, the numbers are 15,25. 32. a; Let the numbers be x and (50 - x). Then, I I I 50-x+x => x(50.~x) ;+ 2 or - x dividend = 2 x 5 , 3 2x5x3 .. Dividend = --2-· - = 15. 50-x = 12 I 12 => x2-50x+600=0 =:>x=30or20. 33. a; (23 x 3 +x +42 x 3)=235 => x =40: :. fourth number = 40. 34. c; Number of one digit pages from I to 9 = 9 5 555 :. the remainder is 4. 37. e; SinceA+ B +C= 16 (Possible values of A, Band C are 0, 6, 7 & 9). Also kt: B, B;f: C, A ;f: C . IfC = 6, A + B should be 10, which is not possible. IfC = 9, A+ B should be 7, which is also not possible. IfC = 0, A + B should be 16 which is also not possible.