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1.Decimal fractions: Fraction in which denominations are powers of 10 are decimal fractions. Example:1 /10 = 0.1, 1 / 100 = 0.01 2.Convertion of Decimal into fraction:- Example: 0.25 = 25/100 = ¼ 3.i) If numerator and denominator contain same number of decimal places, then we remove decimal sign. Thus, 1.84/2.99 =184/299 Problems 1.0.75 =75/100 =3/4 2.Find products= 6.3204*100 = 632.04 3.2.61*1.3=261*13=3393 some of decimal places 2 +1 =3 sol: 3.393 4.If 1/3.718 =0.2689,then find value of 1/0.0003718 ? Sol: 10000/3.718 =10000*1/3.718 =10000*0.2689 = 2689 5.Find fractions : i) 0.37 = 37/99 ii)3.142857 =3+0.142857 =3 +142857/999999 = 3 142857/ 999999 iii) 0.17=17-1/90 =16/90=8/45 iv)0.1254 =1254 -12/9900 =1242/9900=69/550 6.Fraction 101 27/100000 Sol: 101+27/100000 =101+0.00027 =101.00027 7.If 47.2506 =4A + 7/B +2C + 5/D + 6E then 40+7+0.2+0.05+0.0006 Sol: compairing terms 4A= 40 => A=10 7/B = 7 => B=1 2C= 0.2=> C=0.1 5/D= 0.05=>D=5/0.05 =>5*100/5 =100 6E= 0.0006=> E= 0.0001 5A + 3B+6C+ D+ 3E = 5*10+ 3*1+ 6*0.1 + 100+ 3*0.0001 =50+3+0.6+100+0.0003 =153.6003 8.4.036 dividedby 0.04 Sol: 4.036/0.04 =4036/4 =100.9 9.[ 0.05/0.25 + 0.25/ 0.05]3 Sol: =>[5/25 + 25/5] = [1/5+ 5]3 =26/53 =5.23 = 140.603 10.The least among the following :- a. 0.2 b.1/0.2 c. 0.2 d. 0.22 sol:10/2 =5 0.2222 0.04 0.04 < 0.2 < 0.22 --------<5 Since 0.04 is least (0.2)2 is least. 11.Let F= 0.84181 Sol: when F is written as a fraction in lowest terms, denominator exceeds numerator by 84181 -841 /99000 = 83340/99000 =463/550 Required distance = (550 – 463) = 87 12.2 .75 + 3.78 Sol: [-2+0.75]+[-3+0.78] =-5+[0.75+0.78] = -5+1.53 =-5+1+0.53 = -4+0.53 = 4.53 13.the sum of first 20 terms of series is 1/5*6 +1/6*7+1/7*8—– Sol: [1/5 -1/6]+[1/6-1/7]+[1/7-1/8]+———————— = [1/5-1/25] =4/25=0.16 14.13 +23+ ————+93 =2025 Sol: value of (0.11) 3+ (0.22) 3+———(0.99)3 => (0.11) [1+2+--------+9] =0.001331*2025 =2.695275 15.(0.96)3 – (0.1)3/ (0.96)2 +0.096 +(0.1)2 Sol: formula => a3 -b3/a2 +ab +b2 =a -b (0.96-0.1)=0.86 16.3.6*0.48*2.50 / 0.12*0.09*0.5 Sol: 36*48*250/12*9*5=800 17.find x/y = 0.04/1.5 = 4/150 =2/75 find y-x/y+x (1- x/y) / (1+ x/y) 1 - 2/75 /1 +2/75 =73/77 18.0.3467+0.1333 Sol: 3467 -34/9900 + 1333-13/9900 = 3433 +1320/9900 = 4753/9900 = 4801 -48/9900 =0.4301 SQUARE AND CUBE ROOTS Formula: The Product of two same numbers in easiest way as follow. Example:let us calculate the product of 96*96 Solution: Here every number must be compare with the 100. See here the given number 96 which is 4 difference with the 100. so subtract 4 from the 96 we get 92 ,then the square of the number 4 it is 16 place the 16 beside the 92 we get answer as 9216. 9 6 - 4 ————– 9 2 ————– 4*4=16 9 2 1 6 therefore square of the two numbers 96*96=9216. Example: Calculate product for 98*98 Solution: Here the number 98 is having 2 difference when compare to 100 subtract 2 from the number then we get 96 square the number 2 it is 4 now place beside the 96 as 9604 9 8 - 2 ————- 9 6 ————- 2*2=4 9 6 0 4. so, we get the product of 98*98=9604. Example: Calculate product for 88*88 Solution: Here the number 88 is having 12 difference when compare to 100 subtract 12 from the 88 then we get 76 the square of the number 12 is 144 (which is three digit number but should place only two digit beside the 76) therefore in such case add one to 6 then it becomes 77 now place 44 beside the number 77 we will get 7744. 88 -12 ———— 76 ———– 12*12=144 76 + 144 ——————– 7744 ——————– Example: Find the product of the numbers 46 *46? Solution:consider the number 50=100/2. Now again go comparision with the number which gets when division with 100.here consider the number 50 which is nearer to the number given. 46 when compared with the number 50 we get the difference of 4. Now subtract the number 4 from the 46, we get 42. As 50 got when 100 get divided by 2. so, divided the number by 2 after subtraction. 42/2=21 now square the the number 4 i.e, 4*4=16 just place the number 16 beside the number 21 we get 2116. 4 6 4 —————- 4 2 as 50 = 100/2 42/2=21 now place 4*4=16 beside 21 2 1 1 6 Example: Find the product of the numbers 37*37 Solution: consider the number 50=100/2 now again go comparision with the number which gets when division with 100. here consider the number 50 which is nearer to the number given. 37 when compared with the number 50 we get the difference of 13. now subtract the number 13 from the 37, we get 24. as 50 got when 100 get divided by 2. so, divided the number by 2 after subtraction. 24/2=12 now square the the number 13 i.e, 13*13=169 just place the number 169 beside the number 21 now as 169 is three digit number then add 1 to 2 we get 1t as 13 then place 69 beside the 13 we get 1369. 3 7 1 3 —————– 2 4 as 50 = 100/2 24/2=12 square 13* 13=169 1 2 + 1 6 9 ———————– 1 3 6 9 ————————- Example: Find the product of 106*106 Solution: now compare it with 100 , The given number is more then 100 then add the extra number to the given number. That is 106+6=112 then square the number 6 that is 6*6=36 just place beside the number 36 beside the 112,then we get 11236. 1 0 6 + 6 ——————— 1 1 2 ——————– now 6* 6=36 place this beside the number 112, we get 1 1 2 3 6 Square root: If x2=y ,we say that the square root of y is x and we write ,√y=x. Cube root: The cube root of a given number x is the number whose cube is x. we denote the cube root of x by x1/3 . Examples: 1.Evaluate 60841/2 by factorization method. Solution: Express the given number as the product of prime factors. Now, take the product of these prime factors choosing one out of every pair of the same primes. This product gives the square root of the given number. Thus resolving 6084 in the prime factors ,we get 6084 2 6024 2 3042 3 1521 3 507 13 169 13 6084=21/2 *31/2 *131/2 60841/2=2*3*13=78. Answer is 78. 2.what will come in place of question mark in each of the following questions? i)(32.4/?)1/2 = 2 ii)86.491/2 + (5+?1/2)2 =12.3 Solution: 1) (32.4/x)1/2=2 Squaring on both sides we get 32.4/x=4 =>4x=32.4 =>x=8.1 Answer is 8.1 ii)86.491/2 + √(5+x2)=12.3 solution:86.491/2 + (5+x1/2 )=12.3 9.3+ √(5+x1/2 )=12.3 => √(5+x1/2 ) =12.3-9.3 => √(5+x1/2 )=3 Squaring on both sides we get (5+x1/2 )=9 x1/2 =9-5 x1/2 =4 x=2. Answer is 2. 3.√ 0.00004761 equals: Solution: √ (4761/108) √4761/√ 108 . 69/10000 0.0069. Answer is 0.0069 4.If √18225=135,then the value of √182.25 + √1.8225 + √ 0.018225 + √0.00018225. Solution: √(18225/100) +√(18225/10000) + √(18225/1000000) +√(18225/100000000) =√(18225)/10 + (18225)1/2/100 + √(18225)/1000 + √(18225)/10000 =135/10 + 135/100 + 135/1000 + 135/10000 =13.5+1.35+0.135+0.0135=14.9985. Answer is 14.9985. 5.what should come in place of both the question marks in the equation (?/ 1281/2= (162)1/2/?) ? Solution: x/ 1281/2= (162)1/2/x =>x1/2= (128*162)1/2 => x1/2= (64*2*18*9)1/2 =>x2= (82*62*32) =>x2=8*6*3 =>x2=144 =>x=12. 6.If 0.13 / p1/2=13 then p equals Solution: 0.13/p2=13 =>p2=0.13/13 =1/100 p2=√(1/100) =>p=1/10 therefore p=0.1 Answer is 0.1 7.If 13691/2+(0.0615+x)1/2=37.25 then x is equals to: Solution:37+(0.0615+x)1/2=37.25(since 37*37=1369) =>(0.0615+x)1/2=0.25 Squaring on both sides (0.0615+x)=0.0625 x=0.001 x=10-3. Answer is 10-3. 8.If √(x-1)(y+2)=7 x& y being positive whole numbers then values of x& y are? Solution: √(x-1)(y+2)=7 Squaring on both sides we get (x-1)(y+2)=72 x-1=7 and y+2=7 therefore x=8 , y=5. Answer x=8 ,y=5. 9.If 3*51/2+1251/2=17.88.then what will be the value of 801/2+6*51/2? Solution: 3*51/2+1251/2=17.88 3*51/2+(25*5)1/2=17.88 3*51/2+5*51/2=17.88 8*51/2=17.88 51/2=2.235 therefore 801/2+6 51/2=(16*51/2)+6*1/25 =4 51/2+6 51/2 =10*2.235 =22.35 Answer is 22.35 10.If 3a=4b=6c and a+b+c=27*√29 then Find c value is: Solution: 4b=6c =>b=3/2*c 3a=4b =>a=4/3b =>a=4/3(3/2c)=2c therefore a+b+c=27*291/2 2c+3/2c+c=27*291/2 =>4c+3c+2c/2=27*291/2 =>9/2c=27*291/2 c=27*291/2*2/9 c=6*291/2 11.If 2*3=131/2 and 3*4=5 then value of 5*12 is Solution:Here a*b=(a2+b2)1/2 therefore 5*12=(52+122)1/2 =(25+144)1/2 =1691/2 =13 Answer is 13. 12.The smallest number added to 680621 to make the sum a perfect square is Solution: Find the square root number which is nearest to this number 8 680621 824 64 162 406 324 1644 8221 6576 1645 therefore 824 is the number ,to get the nearest square root number take (825*825)-680621 therefore 680625-680621=4 hence 4 is the number added to 680621 to make it perfect square. 13.The greatest four digit perfect square number is Solution: The greatest four digit number is 9999. now find the square root of 9999. 9 9999 99 81 189 1819 1701 198 therefore 9999-198=9801 which is required number. Answer is 9801. 14.A man plants 15376 apples trees in his garden and arranges them so, that there are as many rows as there are apples trees in each row .The number of rows is. Solution: Here find the square root of 15376. 1 15376 124 1 22 53 44 244 976 976 0 therefore the number of rows are 124. 15.A group of students decided to collect as many paise from each member of the group as is the number of members. If the total collection amounts to Rs 59.29.The number of members in the group is: Solution: Here convert Money into paise. 59.29*100=5929 paise. To know the number of member ,calculate the square root of 5929.7 5929 77 49 147 1029 1029 0 Therefore number of members are 77. 16.A general wishes to draw up his 36581 soldiers in the form of a solid square ,after arranging them ,he found that some of them are left over .How many are left? Solution: Here he asked about the left man ,So find the square root of given number the remainder will be the left man 1 36581 191 1 29 265 261 381 481 381 100(since remaining) Therefore the left men are 100. 17.By what least number 4320 be multiplied to obtain number which is a perfect cube? Solution: find l.c.m for 4320. 2 4320 2 2160 2 1080 2 540 2 270 3 135 3 45 3 15 5 4320=25 * 33 * 5 =23 * 33 * 22 *5 so make it a perfect cube ,it should be multiplied by 2*5*5=50 Answer is 50. 18.3(4*12/125)1/2=? Solution: 3(512/125)1/2 3(8*8*8)1/2/(5*5*5) 3(83)1/2/(53) ((83)/(53))1/3 =>8/5 or 1 3/5. | |
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