SIMPLIFICATION

Simplification Techniques and Tricks

Classification:-

Natural Numbers (प्राकृत संख्या):

all counting numbers ( 1,2,3,4,5….∞)

Whole Numbers (पूर्ण संख्या):
natural number + zero( 0,1,2,3,4,5…∞)
Integers:

All whole numbers including Negative number + Positive number(∞……-4,-3,-2,-1,0,1,2,3,4,5….∞)

Even & Odd Numbers:

All whole number divisible by 2 is Even (0,2,4,6,8,10,12…..∞) and which does not divide by 2 are Odd (1,3,5,7,9,11,13,15,17,19….∞)
Prime Numbers:

It can be positive or negative except 1, if the number is not divisible by any number except the number itself.
(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61….∞)

Composite Numbers:

Natural numbers which are not prime
Co-Prime:
Two natural number a and b are said to be co-prime if their HCF is 1.

Divisibility

Divisible by 2 End with 0,2,4,6,8 are divisible by 2 254,326,3546,4718 all are divisible by 2
Divisible by 3 Sum of its digits is divisible by 3 375,4251,78123 all are divisible by 3. [549=5+4+9][5+4+9=18]18 is divisible by 3 hence 549 is divisible by 3.
Divisible by 4 Last two digit divisible by 4 5648 here last 2 digits are 48 which is divisible by 4 hence 5648 is also divisible by 4.
Divisible by 5 Ends with 0 or 5 225 or 330 here last digit digit is 0 or 5 that mean both the numbers are divisible by 5.
Divisible by 6 Divides by Both 2 & 3 4536 here last digit is 6 so it divisible by 2 & sum of its digit (like 4+5+3+6=18) is 18 which is divisible by 3.Hence 4536 is divisible by 6.
Divisible by 8 Last 3 digit divide by 8 746848 here last 3 digit 848 is divisible by 8 hence 746848 is also divisible by 8.
Divisible by 10 End with 0 220,450,1450,8450 all numbers has a last digit zero it means all are divisible by 10.
Divisible by 11 [Sum of its digit in
odd places-Sum of its digits
in even places]= 0 or multiple of 11
Consider the number 39798847
(Sum of its digits at odd places)-(Sum of its digits at even places)(7+8+9+9)-(4+8+7+3)
(23-12)
23-12=11, which is divisible by 11. So 39798847 is divisible by 11.

Division and Remainder Rules

Assume we separate 45 by 6
consequently ,speak to it as:
profit = ( divisor✘quotient ) + leftover portion
on the other hand
divisior= [(dividend)- (remainder]/remainder
could be compose it as
x = kq + r where (x = dividend,k = divisor,q = quotient,r = leftover portion)

Case:
On separating a specific number by 342, we get 47 as leftover portion. In the event that the same number is separated by 18, what will be the rest of
Number = 342k + 47
( 18 ✘19k ) + ( 18 ✘2 ) + 11
18 ✘( 19k + 2 ) +11.
Leftover portion = 11

Entirety Rules

(1+2+3+………+n) = 1/2 n(n+1)
(1²+2²+3²+………+n²) = 1/6 n (n+1) (2n+1)
(1³+2³+3³+………+n³) = 1/4 n2 (n+1)2
Number juggling Progression (A.P.)
an, a + d, a + 2d, a + 3d, ….are said to be in A.P. in which first term = an and basic distinction = d.
Give the nth term a chance to be tn and keep going term = l, then
a) nth term = a + ( n – 1 ) d
b) Sum of n terms = n/2 [2a + (n-1)d]
c) Sum of n terms = n/2 (a+l) where l is the last term

THIS IS SHORT TRICK TO SOLVE SIMPLIFICATION

Question Without detail

54789 – 8462 + 282 – 742 =?
9 – 2 + 2 -2 = 7 (takes the last digit right to left)
8 – 6 + 8 – 4 = 6
7 – 4 + 2 – 7 = -2 Answer is 8 (10 – 2 = 8) (10 nearest number of 2 in the form of 2 digit value )
carry 1 4 – 1 =3 (54789)
3 – 8 = -5 (10 – 5 = 5)
5 – 1 = 4 used carry 1 rule (54789)
Answer is 45867

carry 1 rule : carry the 1 digit to next digit
For example in case 7 – 4 + 2 – 7 = -2 Answer ( in case 1 carry 4 to 5 that is next digit 54789 )
10 – 2 = 8
4 – 1 = 3 in next step used the 3 not 4
With detail
54789 – 8462 + 282 – 742 =?
Step first takes the last digit right to left of the all values (54789 = 9)
9 – 2 + 2 -2 = 7 (9 + 2 -2 -2 = 11 – 4 = 7)
Step In second step takes second last digit right to left of the all values (54789 = 8)
8 – 6 + 8 – 4 = 6 (8 + 8 – 6 – 4 = 6)
Step same takes next digit right to left (54789 = 7 )
7 – 4 + 2 – 7 = 5 – 7 = -2 in case Answer is 8 (10 – 2 = 8) (Now in case -2 (remaining – ) so used the 2 digit nearest number like that in case 10 so used 10 – 2 = 8 and carry 1 number to next digit
( 4 – 1= 3 ) 54789
Step (54789 = 4 ) (4- 1 = 3) used due to takes 1 carry (54789)
3 – 8 = -5 (10 – 5 = 5) same case 3rd step

Step 5

5 – 1 = 4 (used due to takes 1 carry )

Answer is 45867

Question : 45.89-2.769+142.8=?
So convert in same form and add extra zero 45.890-002.769+142.800 or write in this type
045.890 – 002.769+142.800
0-9+0 = used carry rule 10-9-0=1
9-1-6+0=2
8-7+8 = 1+ 8 = 9
5-2+2=5
4-0+4=8
0-0+1=1
185.921
Example 2 = 245 + 412 -121 =?
245 + 412 -121 =?
5 + 2 -1 = 6
4 + 1 – 2 = 3
2+4-1=5
Answer = 536

What should come in place of the question mark (?) in the following questions?

1. 52 x 7 x ? = 4004
(a) 17 (b) 13 (c) 11 (d) 9 (e) 15

2. 555.55+55.55+15+0.55+0.05=?
(a) 626.65 (b) 626.50 (c) 625.70 (d) 626.70 (e) None of these

3. 361? = ?289
(a) 357 (b) 399 (c) 323 (d) 327 (e) None of these

4. 12.50 x 14 ÷ 8.75 + 12 = 20 + ?
(a) 10 (b) 12 (c) 14 (d) 16 (e) 18

5. 1805 ÷ 19 + 65 = 200 + ?
(a) 70 (b)-40 (c) 30 (d) -30 (e) 40

6. ?529 = 324?
(a) 404 (b) 408 (c) 410 (d) 414 (e) 416

7. 61.25 x 1.08 x 1.8=?
(a) 119.07 (b) 117.09 (c) 97.07 (d) 107.12 (e) None of these

8. (3080 + 6160) ÷ ?=330
(a) 26 (b) 22 (c) 28 (d) 29 (e) 18

9. ? x (523.5 + 687.5)=24220
(a) 32 (b) 22 (c) 28 (d) 20 (e) 30

10. ( 5 𝑋 5 𝑋 5 𝑋 52 + 2 + 2 + 2 ) =?
(a) 78.125 (b) 76.125 (c) 68.125 (d) 72.125 (e) 74

11. 264 x ? x 6 = 28512
(a) 18 (b) 13 (c) 16 (d) 17 (e) 20

12. {2.002+7.9(2.8-1.4)} =?
(a) 11.312 (b) 12.204 (c) 13.062 (d) 14.442 (e) 11.006

13. 32 x 15 - 1850 ÷37=?
(a) 295 (b) 326 (c) 410 (d) 430 (e) None of these

14. 125 + 265 x 9=?
(a) 2510 (b)(c) 399 (d) 298.125 (e) None of these

15.16.23 x 12.9 + 17.32 =?
(a) 294.0036 (b) 490.4706 (c) 226.687 (d) 432.795 (e) None of these

16. 2.5 x 4.8 ÷ ? =30
(a) 1.4 (b) 0.2 (c) 1.2 (d) 0.4 (e) None of these

17. 18 𝑋 15−20 3.2+9.4 −7.6=?
(a) 75 (b) 100 (c) 150 (d) 50 (e) None of these

18. 963 x 874 =?
(a) 840788 (b) 843588 (c) 83972 (d) 842536 (e) None of these

19. 444 ÷ (16 x 1.5) =?
(a) 41.625 (b) 222 (c) 18.5 (d) 185 (e) None of these

20. 725 ÷ 25 -13 =?
(a) 16 (b) 29 (c) 12 (d) 18 (e) None of these

21. 558 𝑋 4518 𝑋 4.5=?
(a) 314 (b) 313 (c) 312 (d) 311 (e) None of these

22. 559+965=? X 16 (a) 92.05 (b) 95.25 (c) 93.15 (d) 94.35 (e) None of these

23. {(337 + 146) x 8} =?
(a) 3884 (b) 1515 (c) 3864 (d) 1505 (e) 3846

24. 126 ÷ 7 ÷ 0.6 =?
(a) 3 (b) 30 (c) 300 (d) 6 (e) 18

25. 8.75 x ? – 17.5 =350
(a) 32 (b) 36 (c) 40 (d) 42 (e) 44

26. 36 x 1.75 + 24 x 3.25 =?
(a) 137.5 (b) 141 (c) 143.75 (d) 145 (e) 147.25

27. 4.8 x 8.75 x 6.4 =?
(a) 262.2 (b) 264.4 (c) 266.6 (d) 268.8 (e) 270.2

28. 1772 x 34 -578=?
(a) 57226 (b) 27368 (c) 59670 (d) 29565 (e) 28486

29. 12.7 x 3.9 x11.4=?
(a) 432.314 (b) 486.624 (c) 564.642 (d) 584.27 (e) 593.310

30. 284 ÷ 8 ÷ 0.4=?
(a) 82.25 (b) 84.45 (c) 86.65 (d) 88.75 (e) 90.25

31. (4270÷122)+ ?=200
(a) 135 (b) 145 (c) 155 (d) 165 (e) 175

32. 32 x 15 -1850 ÷ 37=?
(a) 295 (b) 326 (c) 410 (d) 430 (e) None of these

33. 11.02 x 8 + 1.5 x 0.6=?
(a) 89.06 (b) 89.60 (c) 89.96 (d) 89.26 (e) None of these

34. 20 x 168 ÷ 14 – 40 =? + 110
(a) 90 (b) 80 (c) 200 (d) 240 (e) None of these

35. (2222.5 – 1680 x 0.75) x 1.2=?
(a) 1015 (b) 1115 (c) 1155 (d) 1555 (e) 1515

36. (0.1x0.1x0.1x0.01) + (0.2x0.01x0.1)= (?)
(a) 0.00003 (b) 0.00012 (c) 0.0003 (d) 0.00021 (e) None of these

37. 13.2 x 11.7 x 8.6 =?
(a) 1328.184 (b) 1416.916 (c) 1448.642 (d) 1581.124 (e) None of these

38. 11.5 x ?=193.2
(a) 15.4 (b) 16.8(c) 14.2 (d) 17.6(e) 15.8

39. 184 ÷ 46 ÷ 0.008 ÷ 12.5=?
(a) 20 (b) 40 (c) 80 (d) 400 (e) 500

40. 85943 – 71206 + 37517-3512+119=?
(a) 48861 (b) 47641 (c) 47929 (d) 48781 (e) 49101

41. 2522÷97+34 x 43 -96+69 =?
(a) 1641 (b) 4161 (c) 1461 (d) 6411 (e) 1146

42. 1131÷87+2-8 x16 =?
(a) -113 (b) 113 (c) 114 (d) -114 (e) -115

43. 11+19+17 x 65 –1064 ÷ 19 +288 =?
(a) 1201 (b) 1367 (c) 1368 (d) 1907 (e) 2253

44. 86+34 x 79 -1512÷63+13 x 10 =?
(a) 8782 (b) 8728 (c) 2887 (d) 2878 (e) 2788

45. 73+19-2385÷53+89 x 98 =?
(a) 8769 (b) 7896 (c) 8967 (d) 6987 (e) 9678

46. 20+17-5757÷101+64 x 98 =?
(a) 2625 (b) 2265 (c) 6252 (d) 5622 (e) 2256

47. 24 x 16 − 736÷23 (42 x 17) ÷102 =?
(a) 51 (b) 51.28 (c) 50.28 (d) 53 (e) 54

48. 29.92 x 2.4+21.28 x 4.5 =?
(a) 167.568 (b) 176.568 (c) 176.658 (d) 167.658 (e) None of these

49. 5610 ÷ 35.0625 × 2.5 =?
(a) 400 (b) 320 (c) 415 (d) 360 (e) 435

50. 12.4 X ? X 16.5= 2905.32
(a) 13.4 (b) 15.2 (c) 16.2 (d) 14.2 (e) None of these

51. 3772 ÷ 164 x 29.5 =?
(a) 699.5 (b) 715.0 (c) 6875 (d) 678.5 (e) 6895

52. 4+4 x 18−6−8123 x 6−146 x 5 =?
(a) 1 (b) 2 (c) 6.65 (d) 7.75 (e) None of these

53. 180 x 15−12 x 20140 x 8+2 x 55 =?
(a) 17 (b) 45 (c) 2 (d) 4 (e) None of these

54. The value of 25-5[2+3{2-2(5-3) + 5}-10]÷4 is:
(a) 5 (b) 23.25 (c) 23.75 (d) 25 (e) None of these

55. 11.6 + 9.28 ÷ 0.464 -0.2828 ÷ 0.07 =? (a) 9.2 (b) 9.56 (c) 27.2 (d) 27.(e) None of these

56. 5.8 x 2.5 + 0.6 x 6.75 + 139.25 =?
(a) 157.30 (b) 157.80 (c) 158.40 (d) 160.30 (e) None of these

57. 2.375 x 5.22 ÷ 0.87 – 1.425 x 0.02 =?
(a) 0.142215 (b) 1.42215 (c) 14.2215 (d) 1.042215 (e) None of these

58. 2.002+7.9{2.8-6.3(3.6 – 1.5) + 15.6} = ?
(a) 2.002 (b) 4.2845 (c) 40.843 (d) 42.845 (e) None of these

59. 14.6 x 12.8 + 3.9 x 4.5 =?
(a) 217.23 (b) 204.43 (c) 200.53 (d) 1097.19 (e) None of these

60. 74.6−38.9−5.726.4−18.9 =?
(a) 0.5 (b) 2.0 (c) 1.5 (d) 3.0 (e) None of these

ANS KEY:-

1. (c) 2. (d) 3. (c) 4. (b) 5. (b) 6. (d) 7. (a) 8. (c) 9. (d) 10. (a) 11. (a) 12. (c) 13. (d) 14. (a) 15. (c) 16. (d) 17. (d) 18. (e) 19. (c) 20. (a) 21. (e) 22. (b) 23. (c) 24. (b) 25. (d) 26. (b) 27. (d) 28. (c) 29. (c) 30. (d) 31. (d) 32. (d) 33. (a) 34. (a) 35. (c) 36. (d) 37. (a) 38. (b) 39. (b) 40. (a) 41. (c) 42. (a) 43. (d) 44. (d) 45. (a) 46. (c) 47. (c) 48. (a) 49. (a) 50. (d) 51. (d) 52. (d) 53. (c) 54. (c) 55. (d) 56. (b) 57. (c) 58. (d) 59. (b) 60. (e)