# SIMPLIFICATION

It usually includes simpler expression of mathematical problems following the rule BODMAS { B=bracket O= of; D= division; M= multiplication; A= addition; S= subtraction). It comprises square root, cube root, surds and indices, percentage.  SIMPLIFICATION Answers are given just after each question, explanations are given after

### SIMPLIFICATION NUMARICAL PROBLEMS:- (find the correct option)

1)108 X 106 – 94 X 94 = ? in simplification

Options : i)2400 ii)2000 iii)1904 iv)1906

simplification ans:- (i)

2)8796 X 223 + 8796 X 77 = ? in simplification

Option in simplification :(i)2736900 (ii)2638800 (iii)2658560 (iv)2716740

simplification ans:- (ii)

3)8988 ÷8÷4 = ? simplification

simplification option : (i)4494 (ii)561.75 (iii)2247 (iv)280.875

#### 4)287 X 287 + 269 X 269 – 2 X 287 X 269 =?

(i)534 (ii)446 (iii)354 (iv)324

ans:- (iv)

5)What is the unit digit in the product 365X 659X 771

(i)1 (ii)2 (iii)4 (iv)6

ans:- (iii)

6)666 ÷ 6 ÷ 3 =?

(i)37  (ii) 333 (iii) 111 (iv)84

ans :- (i)

7)2056 X 987 =?

(i)1936372 (ii) 2029272 (iii) 189172 (iv)1926172

ans:- (ii)

8)587 X 999 =?

(i)586413 (ii)587523 (iii)614823 (iv)615173

Simplification ans:-(i)

9)9548 + 7314 = 8362 + ?

(i)8230 (ii)8410 (iii)8500 (iv)8600

Simplification ans:-(iii)

10)1904 X 1904 =?

(i)3654316 (ii)3632646 (iii)3625216 (iv)3623436

ans:- (iii)

11)There are two numbers whose total is 12 and when multiplied result is 35, sum of the reciprocals is:

(i)12/35 (ii) 1/35 (iii) 35/8 (iv) 7/32

ans:- (i)

12)(800 ÷64) X (1296 ÷36) =?

(i)420 (ii) 460 (iii)500 (iv)none of these

ans:- (iv)

#### 13)9 3/4 + 7 2/17 –(9 1/15)= ?

(i) correct fraction {7 719/1020 }(ii) correct fraction { 9 817/1020 }(iii) correct fraction { 9 719/1020 }(iv) correct fraction { 7 817 /1020}

ans:- (iv)

14)8597 -? = 7429 – 4358

(i)5426 (ii) 5706 (iii)5526 (iv)5476

Simplification ans:- (iii)

15)There is a proper fraction and reciprocal whose difference is 9/20, The fraction is

(i)3/5 (ii)3 /10 (iii) 4/5 (iv) 4/3

ans:- (iii)

16)?- 19657 – 33994 = 9999

(i)63650 (ii)53760 (iii)59640 (iv)61560

ans :- (i)

#### 17)112 X 54 =?

(i)67000 (ii)70000 (iii) 76500 (iv)77200

ans:- (ii)

18)Which of the following fractions is the largest   ( SIMPLIFICATION)

(i)7/8 (ii) 13/16 (iii) 31/40 (iv) 63/80

ans:- (i)

19) The fractional form of 6.46 is

(i)646/99   (ii) 64640/1000 (iii)  640/1000 (iv)646/100

Simplification ans :- (iv)

20)The fraction 10127/100000 in decimal form is

(i)0.10127 (ii) 1012.7 (iii)101.00027 (iv) 101.000027

ans :- (iii)

#### 21)  0.0203 X 2.92  =?

0.0073 X 14.5 X 0.7

(i) 0.8 (ii) 1.45 (iii) 2.40 (iv) 3.25

ans :- (i)

22)4.036 is divided by 0.04 it gives

(i)009 (ii)10.09 (iii) 100.9 (iv) none of the above

ans :- (iii)

23)34.95 + 240.016 + 23.98 =?

(i)0946 (ii) 298.111 (iii) 298.946 (iv) 299.09

Simplification ans:- (iii)

#### 24)The least among the following is  ( SIMPLIFICATION)

(i)2 (ii) 1÷0.2 (iii)0.4 (iv) (0.2)2

ans:- (iv)

25)   5 X 1.6 – 2 X 1.4

1.3

(i) 0.4 (ii) 1.2 (iii) 14 (iv) 4

ans :- (iv)

26)In the product of 95.75 and 0.02554 how many numbers are there in the right of decimal ( SIMPLIFICATION)

(i)5 (ii) 6 (iii)7 (iv)none of the above

ans :- (ii)

27)Which fractions lies between  3/4  and  5/6 ?

(i) 1/2 (ii) 2/3  (iii) 4/5  (iv) 9/10

ans:- (iii)

#### 28)O.04 X 0.0162 is equal to

(i)6.48 X 10-3 (ii) 6.48 X 10-4 (iii) 6.48 X 10-5 (iv) 6.48 X 10-6

ans:- (ii)

29)If 144/0.144 = 14.4 /x then value of x is

(i)0.0144 (ii) 1.44 (iii) 14.4 (iv) 144

ans:- (i)

30)0.009 /? = 0.01

(i)0.0009 (ii) 0.09 (iii) 0.009 (iv) 0.9

ans :- (iv)

31){(0.1667) (0.8333) (0.3333)}/ {(0.2222) (0.667) (0.1250)} is approximately equals to

(i)2 (ii) 2.40 (iii) 2.43 (iv) 2.50

ans:- (iv)

32)3889 + 12.952 – ? = 3854.002i)47.095 (ii) 47.752 (iii) 47.932 (iv) 47.95

ans:- (iv)

33)(1000)9 ÷ (10)24 =?

(i)10,000 (ii) 1000 (iii) 100  (iv) 10

ans:- (ii)

34)(476 + 424)2 – 4 X 476 X 424 = ?

(i)2906 (ii) 3116 (iii) 2704 (iv) 2904

ans :- (iii)

35) Which one of numbers is not a perfect square of a natural number?

(i) 30976 (ii) 75625 (iii) 28561 (iv) 43642

ans :- (iv)

36){(489 + 375)2 – (489 – 375)2}/(489 X 375)

(i)144 (ii) 864 (iii) 2 (iv)4

ans:- (iv)

37))397 X 397 + 104 + 104 + 2 X 397 X 104

(i)250001 (ii) 251001 (iii) 260101 (iv) 261001

ans :- (ii)

38)(12)3 X (6)4 ÷ 432 = ?

(i)5184 (ii) 5060 (iii) 5148 (iv) 5048

ans :- (i)

39)0.0169 X ? =(1.3)2

(i)101 (ii) 102 (iii) 103 (iv) 104

ans :- (ii)

#### 40)The square root of 64009 is

(i)253 (ii) 347 (iii) 363 (iv) 803

ans:- (i)

41)If a = 0.1039 then the value of √(4a + 4a +1) + 3a is

(i)0.1039  (ii) 0.2078 (iii) 1.1039 (iv) 2.1039

ans :- (iii)

42)The square root of (7 + 3√5) (7 + 3√5) is

(i)5 (ii) 2 (iii) 4 (iv) 35

ans :- (ii)

43)Cube root of 0.000216 is

(i)0.6 (ii) 0.06 (iii) 0.77 (iv) 0.87

ans:- (ii)

44)(243)n/5 X 32n+1  =?

9nX 3n-1

(i)1 (ii)2 (iii)9 (iv)3n

ans :- (iii)

45)(17)3.5 X (1.7)? = (1.7)8

(i)2.29 (ii) 2.75 (iii) 4.25 (iv) 4.5

Ans:- (iv) ( 400x12x5)

#### 46)If (a/b)x-1 = (b/a)x-3 then the value of x  is

(i)½ (ii) 1 (iii) 2 (iv) 7/2

ans:- (iii)

47)(256)0.16 X (256)0.09= ?

(i)4 (ii) 16 (iii) 64 (iv)256.25

ans:- (i)

#### 48) what should come in place of both x in the equation x/√128 = √162/ x

(i)12 (ii)14 (iii) 144 (iv)196

ans:- (i)

49) 1164 X 128 ÷8.008 + 969.007 = ?

(i)18800 (ii) 19000 (iii)19600 (iv) 19200

ans:-  (iii)

50) 12.25 X ? X 21.6 = 3545.64

(i) 14.8 (ii) 12.6 (iii) 15.8 (iv) 13.4

ans:- (iv)

400 X 12 X 5 = ?

Ans: 24000

#### EXPLANATION OF NUMERICAL PROBLEMS

1)Let 106 be a and 94 be b; so it will be a X a – b X b = a2 – b2 = (a+b) (a-b)

Putting value of a and b (106 + 94) (106 – 94) = 200 X 12 = correct: 2400 (i)) ans. ( 400x12x5)

2)Taking 8796 common we get 8796( 223 + 77) = 8796 X 300 = correct:2638800 (ii))

#### 3)Doing the division one by one

(8988 /8)÷4  = 8988 ÷4 = 2247 ÷4 = 2247

8                 2             2 X 4

Correct: 280.875 (iv) ans ( 256^0.16 x (256)^0.09 is equal to)

4) Let 287 as a and 269 as b

So, a X a + b X b -2 ab = a2+ b2– 2ab = (a-b)2; putting the value of a and b

(287 – 269)2 = (18)2 = correct:324 (iv) ans.

#### 5)Unit digit in 34 = 1 => (34)16 = 1 ; so, 365 = (34)16 X 31 = 1 X 3 = 3;

(62)28 X 63 = 6

74 = 1 => (74)17 X 73 = 1 X 3 => 3

So, unit digit will be  (3 X 6 X 3 ) =correct: 4 (iii)ans.

#### 6)Doing the division one by one

666 ÷3   =111 ÷3 =correct: 37 (i))ans.

6

7)It is the multiplication between 4-digit and 3- digit ; 2056 X 0987

Step 1: 6 X 7 = 42                                                                               (2)

=>Step 2 : 5 X 7 + 6 X 8 + 4 (carry) = 87                                              (7)

Step 3 : 7 X 0 + 6 X 9 + 5 X 8  + 8 (carry) = 102                             (2)

=>Step 4 : 6 x 0 + 7 x 2 + 5 x 9 + 8 X 0 + 10 (carry) = 69                  (9)

Step 5 : 5 X 0 + 2 X 8 + 9 X 0 + 6 (carry) = 22                               (2)

=>Step 6 : 0 X 0 + 2 X 9 + 2 (carry) = 20                                            (0)

Step 7 : 2 X 0 + 2 (carry) = 2                                                           (2)

#### ( what is the unit digit in the product (365 x 659 x 771))

8)587 X (1000 – 1) = 587000 – 587 =correct: 586413(i)

9)9548 + 7314 – 8368 = ?

16862 – 8362 = correct:8500 (iii) ans.

#### 10)(1904 ) X (1904 ) = (1900 + 4)2 = (1900)2 + (4)2 + 2 X 1900 X 4 ( SIMPLIFICATION)

= 3610000 + 16 + 15200 = correct :3625216(iii) ans.

11)Let the number be a and b, so according to the question: a + b = 12 and ab = 35;

(a+b)/ab = a/ab + b /ab = 1/a + 1/b = 12/35

So sum of reciprocal of given number is  correct: 12/35 (i)

#### ans. ( 256^0.16 x (256)^0.09 is equal to)

12)Given expression = 800/64 X 1296/30 = correct : 450 (iv) ( 144/0.144=14.4/x)

13).9 3/4 + 7 2/17  – 9 1/15 = 9 + 7 -9  + (3/4 + 2/17 – 1/15) = 7 + (765 + 120 -68)/1020

#### = correct:  7  817 /1020  (iv)ans.

14)Let 8597 – x = 3071 => x = 8597 -3071 = correct : 5526 (iii)

15).The required fraction is x/1 and reciprocal is 1/x ; so, (1/x) – (x/1) = 9/20 => (1-x)2/x = 9/20

Or,20 -20x2 = 9x

=> 20x2+ 25x -16x -20 = 0

Or, 5(4x +5) – 4 (4x+ 5) = 0

=> (4x + 5) (5x -4) = 0

Or, x = – 5/4 (not possible) or 4/5

#### So, correct : 4/5 (iii) ans.

16)Let x = 9999 + 19657 + 33994 => 9999 + 53651 =correct :63650 (i)

17) (112 X 54) = 112 X (10/2)4 = 112 X 104 = 1120000/16 = correct :70,000 (ii)ans

24

#### 18)LCM of 8, 16,40 and 80 = 80; so, 7/8 = 70/80 ; 13/16 = 65/80 ; 31/ 40 = 62/80 and 63/80 =63/80

so, 70/80 > 65/80 > 63/80 > 62/80

therefore, correct : 7/8 is the greatest (i)ans.

19) 4 = 6 + 0.46 ; or 6 + 46/100 => (600 + 46)/100 = so, correct :646/100(iv) ans

#### 20)101 27/ 100000 = 101 + 27 / 10,0000 = 101 + 0 +0.00027 =correct: 000027(i) ans.

21)  0.0203 X 2.92=

0.0073 X 14.5 X 0.7

203 X 292 X 104 X 10 X10                                   =correct: 4/5  OR 0.8 (i)ans.

73 X 445 X 7 X 104 X 102

22).036 ÷0.04 = 4036 X 100 = 1009 / 10 =  correct:  100.9 (iii)

ans.

23) 95

240.016

023.98

298.946

#### Correct ans is 298.946 (iii) ans.  ( (?)-19657-33994=9999)

24).1 ÷ 0.2 = 1/0.2 = 10/2 = 5

0.2 = 2/10 = 0.2

0.4 = 4/10 = 0.4

(0.2)2 = 0.04

#### Correct: Least number is (0.2)2(iv)ans. ( (?)-19657-33994=9999)

25)6 X 5 = 8

1.4 X 2 = 2.8

(8 -2.8)÷ 1.3 = 5.2 ÷  1.3 = 4

#### Correct :4 (iv)ans.

26).As the last two digits are 5 and 4 so by multiplying we get ZERO (0) which will be cancelled after decimal; so after decimal there will be 6 digits not seven.

Correct : 6 (ii) ans.

27) Let’s convert all fractions into decimal like: 3÷4 = 0.75 ; 5÷6 = 0.83 ;

1÷2 = 0.5; 2÷3 =  0.66; 4÷5 = 0.8 ; 9÷10 = 0.9

So 0.8 is between 0.75 and 0.83

#### Correct : 4/5 (iii)ans.

28) 4 X 162 = 648, sum of decimal place is 6;

So, ans is 0.000648

Correct : 6.48 X 10-4 (ii)ans.

#### 29)144/0.144 = 14.4 /x => x = (14.4 X 0.144)/144 => (144 X 144)/(144 X 10000) = 0.0144

Correct : 0.0144 (i)ans. (find the value of (256)0.16 x (256)0.09)

30).(0.009 /x) = 0.01 => x = (9 X 100)/ (1 X 1000) = 9/10 = 0.9

Correct :0.9 (iv)ans.

31) 0.3333/0.2222 = 3/2

=>0.1667/0.6667 = 1/4

0.8333/0.1250 = 6/10

So, 3/2 X 1/4 X 66/10 = 99/40 = 100/40 (approx)= 5/2 = 2.5

#### Correct : 2.50 (iv)ans.

32)x=(3889 + 12.952) – 3854.002 => 3901.952 -3854.002 = 47.95

Correct :  47.95 (iv) ans

33) (1000)9/ (10)24 = {(10)3}9/ (10)24 = (10)27/(10)24 = 1027-24 = 103

#### Correct : 1000(ii) ans

34) let  take a = 476 ; b = 424

Taking formula of (a +b)2 – 4ab

=> (900)2 – 807296 = 810000 – 807296 = 2704

#### Correct : 2704 (iii) ans.

35) Last digit of a square number cannot be ended with  2 ; so 143642 cannot be a square of natural number; where others are square numbers.

Correct :143642 (iv) ans.

36) {(a + b)2 – (a –b)2}/ab = 4ab /ab => 4 ( SIMPLIFICATION)

#### Correct : 4 (iv) ans.

37) (397)2 + (104)2 + 2 X 397 X 104 = (397 + 104)2= (501)2 = (500 + 1)= 250000 + 2 X 500 X 1 + 1

= 251001

#### Correct : 251001 (ii) ans

38) {(12)3X 64}/432 = {(12)3 X (6)4}/ 12 X 62 = (123- 1) X (64-2) = 122X 62= (72)2

Correct : 5184 (i)ans

39) { (13) X (13) X x}/10000 = 1.69 => x = 100 ans (ii)

40) √64009 =     |6  40  09| 253

– 4

45   | 240 |

-225

503     | 1509 |

-1509

0

#### Correct : 253 (i)ans

41)  √{(1)2 + (2a)2 – 2 X 2a X 1} + 3a = √ (1 – 2a)2 + 3a = 1 – 2a +3a = 1 +a

(putting the value of a = 0.1039) so,  1 + 0.1039

Correct : 1.1039 (iii)ans

#### 42)  √{ (7 + 3√5)  X ( 7 + 3√5)} = √ (7)2 – (3√5)2 = √ 49 – 45 = √4 = 2

Correct : 2 (ii)ans

43) (0.000216)1/3 = (216/1000000)1/3 = {(6 3)/106}1/3 = (63×1/3)/106×1/3 = 6/102 = 6/100 ( SIMPLIFICATION)

Correct : 0.06 (ii) ans

44) {(243)n/5 X 32n+1} / {9n X 3n-1} = {(35)n/5 X (3)2n+1} / { (32)n X (3)n-1}

=> [3nX 32n+1]/ [32n X 3n-1] => { 3n+2n 1}/{ 32n+n-1} = (33n+1)/ (33n-1) = 33n+1-3n+1= 32

#### Correct: 9 (iii)ans ( find the value of (256)0.16 x (256)0.09)

45) (17)3.5 X 17 x = 178=> (17)3.5+ x = 178 so, 3.5 + x = 8 => x = 8 – 3.5; Correct: 4.5(iv) ans

46) (a/b)x-1= (a/b)-(x-3) => so, x-1 = -x+3

Or, x + x = 3 +1

=> 2x = 4

Or x = 4/2

#### Correct : 2 (iii) ans ( what is the unit digit in the product (365 x 659 x 771))

47) (256)0.16 X (256)0.09 = (256)0.16 + 0.09= (256)25/100 = (256)1/4= (44)1/4

Correct :4 (i) ans

48) x2= √162 X √128 => √ 2 X 9 X 9 X √2 X 8 X 8 = √ 2 X 2 X 9 X 9 X 8 X 8

X2 = 2 X 9 X 8 => 144 so, x = √144

#### Correct: 12 (i) ans

49) 1164 X 128 X 1000   + 969.007= 1164 X 16 X 1000  + 969.007=>(let 1001 be taken 1000 approx)

8008                                 1001

So, 1164 X 16 X 1000 +969.007 = (18624 + 969.007) = 19593.007

1000

#### Correct : 19600 approx (iii) ans

50) let (?) be x

x = (3545.64)/ (12.25 X 21.6) = 3283 / 205 = 16.01 ≡ 16.2 (approx)

Correct : 16.2(iv) ans

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