# Number System Practice Questions (Level - 1) - 2

41.The largest natural number by which the product of three consecutive even natural number is always divisible, is:
a. 16
b. 24
c. 48
d. 96
Correct Option: C
Explanation: Take the numbers as 2n, 2n + 2, and 2n + 4. By taking 2 common, their product is 8 x n(n+1)(n+2).  Any threee consecutive numbers product is divisible by 6 so the given number is a product of 8 x 6 = 48

42.If n is any positive integer, then $({3^{4n}} - {4^{3n}})$ is always divisible by
a. 7
b. 17
c. 112
d. 145
Correct Option: B
Explanation: ${a^n} - {b^n}$ is always divisible by a - b.  So $({3^{4n}} - {4^{3n}}) = {\left( {{3^4}} \right)^n} - {\left( {{4^3}} \right)^n} = {81^n} - {64^n}$ is divisible by 81 - 64 = 17

43.If $- 1 \le x \le 2$ and ${\rm{1}} \le {\rm{y}} \le {\rm{3}}$ , then least possible value of (2y-3x) is :
a. 0
b.-3
c.-4
d.-5
Correct Option: C
Explanation:
For (2y-3x) to be minimum, the condition is that y must be substituted with least value and x must be with large value.
$(2 \times 1 - 3 \times 2) = - 4$

44. 8756 x 99999=?
a. 815491244
b. 796491244
c. 875591244
d. None of these
Correct Option: C
Explanation: Write 99999 as (10000 - 1)

45. 9787 x 123 + 9787 x 77=?
a. 1867400
b. 1957400
c. 1967600
d. 1887400
Correct Option: B

46. $\left[ {\displaystyle\frac{{973 \times 973 \times 973 + 127 \times 127 \times 127}}{{973 \times 973 + 127 \times 127 - 973 \times 127}}} \right]$ is equal to :
a. 1000
b. 1100
c. 846
d. None of these
Correct Option: B
Explanation: $\displaystyle\frac{{{a^3} + {b^3}}}{{{a^2} + {b^2} - ab}} = a + b$
So the given equation is equal to 973 + 127 = 1100

47.$\displaystyle\frac{{{{(856 + 167)}^2} + {{(856 - 167)}^2}}}{{856 \times 856 + 167 \times 167}}$ is equal to :
a. 1
b. 2
c. 689
d. 1023
Correct Option: B
Explanation:
${(a + b)^2} + {(a - b)^2} = 2({a^2} + {b^2})$

48. $\displaystyle\frac{{{{(469 + 174)}^2} - {{(469 - 174)}^2}}}{{469 \times 174}}$ is equal to :
a. 2
b. 4
c. 643
d. 295
Correct Option: B
Explanation:
${(a + b)^2} - {(a - b)^2} = 4ab$

49.$\left[ {\displaystyle\frac{{147 \times 147 + 147 \times 143 + 143 \times 143}}{{47 \times 147 \times 147 - 143 \times 143 \times 143}}} \right]$ is equal to
a. 4
b. $\displaystyle\frac{1}{4}$
c. 290
d. $\displaystyle\frac{1}{{290}}$
Correct Option: B
Explanation: ${a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})$
$\left[ {\displaystyle\frac{{{a^2} + a \times b + {b^2}}}{{{a^3} - {b^3}}}} \right] = \displaystyle\frac{1}{{a - b}}$

50.The value of $\left[ {\displaystyle\frac{{{2^n} + {2^{n - 1}}}}{{{2^{n + 1}} - {2^n}}}} \right]$ is :
a. $\displaystyle\frac{1}{2}$
b. $\displaystyle\frac{3}{2}$
c. ${2^{\displaystyle\frac{{n - 1}}{{n + 2}}}}$
d. None of these
Correct Option: B
Explanation:
The expression can be written as $\displaystyle\frac{{{2^{n - 1}}(2 + 1)}}{{{2^n}(2 - 1)}}$
$\displaystyle\frac{{{2^{n - 1}}(2 + 1)}}{{{2^{n - 1}}{{.2}^1}(2 - 1)}}$
$\Rightarrow \displaystyle\frac{{(2 + 1)}}{2} = 3/2$

51. $\displaystyle\frac{4}{5}$ of a number exceeds its $\displaystyle\frac{2}{3}$ by 8. The number is
a. 30
b. 60
c. 90
d. None of these
Correct Option : B
Explanation:
$\displaystyle\frac{4}{5}a - \frac{2}{3}a = 8$
$\Rightarrow \displaystyle\frac{{12a - 10a}}{{15}} = 8$
$\Rightarrow 2a = 120 \Rightarrow a = 60$

52. If 1 is added to the denominator of fraction, the fraction becomes $\displaystyle\frac{1}{2}$.  If 1 is added to the numerator, the fraction becomes 1.  The fraction is
a. $\displaystyle\frac{4}{7}$
b. $\displaystyle\frac{5}{9}{\rm{ }}$
c. $\displaystyle\frac{2}{3}$
d. $\displaystyle\frac{{10}}{{11}}$
Correct Option: C
Explanation:
Let the required fraction be $\displaystyle\frac{a}{b}$. Then
$\displaystyle\frac{a}{{b + 1}} = \displaystyle\frac{1}{2} \Rightarrow 2a - b = 1$ ---- (1)
$\Rightarrow \displaystyle\frac{{a + 1}}{b} = 1$  $\Rightarrow a - b = - 1$ ------- (2)
Solving (1) & (2) we get a = 2, b=3
Fraction = $\displaystyle\frac{a}{b}{\rm{ = }}\displaystyle\frac{2}{3}$

53. The sum of two numbers is twice their difference.  If one of the numbers is 10, the other number is
a. ${\rm{3}}\displaystyle\frac{1}{3}$
b. 30
c. ${\rm{ - 3}}\displaystyle\frac{1}{3}$
d. $4\displaystyle\frac{1}{4}$
Correct Option:  B
Explanation:
Let the other number be a
${\rm{10 + a = 2(a - 10)}} \Rightarrow {\rm{a = 30}}$

54. The sum of squares of two numbers is 80 and the square of their difference is 36.  The product of the two numbers is.
a. 22
b. 44
c. 58
d. 116
Correct Option: A
Explanation :
Let the numbers be x and y . Then
${{\rm{x}}^2} + {y^2} = 80$ and ${({\rm{x - y}})^2} = 36$
${(x - y)^2} = 36 \Rightarrow {x^2} + {y^2} - 2xy = 36$
$\Rightarrow 2xy = ({{\rm{x}}^2} + {y^2}) - 36 = 80 - 36 = 44$
$\Rightarrow xy = 22$

55. 11 times a number gives 132.  The number is
a. 11
b. 12
c. 13.2
d. None
Correct Option: B
Explanation:
Let the number be 'N'
$11 \times {\rm{N = 132}} \Rightarrow N = 12$

56. If 1/5th of a number is decreased by 5 is 5, then the number is
a. 25
b. 50
c. 60
d. 75
Correct Option: B
Explanation :
Let the number be a, then
$\displaystyle\frac{a}{5} - 5 = 5 \Rightarrow \frac{a}{5} = 10 \Rightarrow a = 50$

57. The difference of two numbers is 2 and the difference of their squares is 12.  The sum of the number is
a. 6
b. 8
c. 10
d. 22
Correct Option: A
Explanation:
Let the numbers be a, b; $\Rightarrow$ a-b=2 and
${a^2} - {b^2} = 12$
$a + b = \displaystyle\frac{{{a^2} - {b^2}}}{{a - b}} = \displaystyle\frac{{12}}{2} = 6$

58. The sum of two numbers is 29 and the difference of their squares is 145.  The difference between the number is
a. 13
b. 5
c. 8
d. 11
Correct Option: B
Explanation:
Let the numbers be x and y, then
$(x - y) = \displaystyle\frac{{{x^2} - {y^2}}}{{(x + y)}}$
$= \displaystyle\frac{{145}}{{29}} = 5$

59. The difference between the squares of two consecutive numbers is 35.  The numbers are
a. 14,15
b. 15,16
c. 17,18
d. 18,19
Correct Option: C
Explanation:
Let the numbers be a and (a+1)
${(a + 1)^2} - {a^2} = 35$
$\Rightarrow {a^2} + 2a + 1 - {a^2} = 35$
$\Rightarrow 2a = 34$ or a = 17
The numbers are 17 & 18.

60.$\displaystyle\frac{{{3^{th}}}}{4}{\rm{ of }}\displaystyle\frac{{{1^{th}}}}{5}$
of a number is 60.  The number is
a. 300
b. 400
c. 450
d. 1200
Correct Option: B
Explanation:
Let the number be N. Then
$\displaystyle\frac{3}{4} \times \frac{1}{5} \times N = 60 \Rightarrow 3N = 1200 \Rightarrow N = 400$

61. 24 is divided into two parts such that 7 times the first part added to 5 times the second part gives 146. The first part is
a. 11
b. 13
c. 16
d. 17
Correct Option: B
Explanation:
Let the first and second parts be a and 24-a, then
${\rm{7a + 5(24 - a) = 146}}$
$\Rightarrow {\rm{7a + 120 - 5a = 146}}$
$\Rightarrow {\rm{2a = 26}}$ or a = 13

62. The product of two numbers is 120.  The sum of their squares is 289.  The sum of the two numbers is :
a. 20
b. 23
c. 169
d. None
Correct Option: B
Explanation:
Let the number be x and y . Then
${(x + y)^2} = ({x^2} + {y^2}) + 2xy = 289 + 2x120$
$= 289 + 240 = 529 \Rightarrow x + y = \sqrt {529} = 23$

63. The sum of squares of two numbers is 68 and the square of their difference is 36.  The product of the two numbers is
a. 16
b. 32
c. 58
d. 104
Correct Option:a
Explanation:
Let the numbers be x and y. Then
${x^2} + {y^2} = 68{\rm{ \& (x - y}}{{\rm{)}}^2} = 36$
But ${(x - y)^2} = 36 \Rightarrow {x^2} + {y^2} - 2xy = 36$
$\Rightarrow 68 - 2xy = 36 \Rightarrow 2xy = 32$
$\Rightarrow xy = 16$

64. 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
a. 40
b. 126
c. 69
d. 195
Correct Option: A
Explanation:
$(23 \times 3 + a + 42 \times 3) = 235 \Rightarrow a = 40$

65. Two numbers are such that the ratio between them is 3:5 but if each is increased by 10, the ratio between them becomes 5 : 7, the numbers are
a. 3, 5
b. 7, 9
c. 13, 22
d. 15, 25
Correct Option: D
Explanation:
Let the numbers be 3a and 5a
Then $\displaystyle\frac{{3a + 10}}{{5a + 10}} = \displaystyle\frac{5}{7}$
$\Rightarrow 7(3a + 10) = 5(5a + 10) \Rightarrow a = 5$
The numbers are 15 & 25

66.  A fraction becomes 4 when 1 is added to both the numerator and denominator, and it becomes 7 when 1 is subtracted from both the numerator and denominator.  The numerator of the given fraction is :
a. 2
b. 3
c. 7
d. 15
Correct Option: D
Explanation:
Let the required fraction be $\displaystyle\frac{a}{b}$
Then $\displaystyle\frac{{a + 1}}{{b + 1}} = 4 \Rightarrow a - 4b = 3$
and $\displaystyle\frac{{a - 1}}{{b - 1}} = 7 \Rightarrow a - 7b = - 6$
Solving these equations we get,
a=15
b=3

67. A number exceeds 20% of itself by 40.  The number is
a. 50
b. 60
c. 80
d. 320
Correct Option: A
Explanation:
Then $a - \displaystyle\frac{{20}}{{100}}a = 40 \Rightarrow 5a - a = 200$
$\Rightarrow a = 50$

68. Three numbers are in the ratio 3:4:5.  The sum of the largest and the smallest equals the sum of the third and 52.The smallest number is :
a. 20
b. 27
c. 39
d. 52
Correct Option: C
Explanation:
Let the numbers be 3 N, 4 N and 5 N
Then 5N+3N = 4 N +52
$\Rightarrow 4N = 52 \Rightarrow N = 13$
The smallest number =3N=3 x 13=39

69. If 16% of 40% of a number is 8, then the number is
a. 200
b. 225
c. 125
d. 320
Correct Option: C
Explanation:
Let $\displaystyle\frac{{16}}{{100}} \times \displaystyle\frac{{40}}{{100}} \times a = 8$

a = $\displaystyle\frac{{8 \times 100 \times 100}}{{16 \times 40}} = 125$

20. If 3 is added to the denominator of a fraction, it becomes $\displaystyle\frac{1}{3}$ and if 4 be added to its numerator, it becomes $\displaystyle\frac{3}{4}$ ; the fraction is :
a. $\displaystyle\frac{4}{9}$
b. $\displaystyle\frac{3}{{20}}$
c. $\displaystyle\frac{7}{{24}}$
d. $\displaystyle\frac{5}{{12}}$
Correct Option: D
Explanation:
Let the required fraction be $\displaystyle\frac{a}{b}$
Then $\displaystyle\frac{a}{{b + 3}} = \displaystyle\frac{1}{3}$
$\Rightarrow 3a - b = 3$ and
$\displaystyle\frac{{a + 4}}{b} = \displaystyle\frac{3}{4} \Rightarrow 4a - 3b = -16$ solving,we get
a=5, b=12; required answer = 5/12

71.Of the three numbers, the first is twice the second and is half of the third.  If the average of three numbers is 56, then the smallest number is
a. 24
b. 36
c. 40
d. 48
Correct Option: A
Explanation:
Let the second number is a. Then the first number is 2a and third number is 4a.
$\displaystyle\frac{{2a + a + 4a}}{3} = 56 \Rightarrow 7a = 3 \times 56$ or
a= $\displaystyle\frac{{3 \times 56}}{7} = 24$
Smallest number is 24

72.The difference of two numbers is 8 and $\displaystyle\frac{{{1^{th}}}}{{12}}$ of the sum is 1.  The numbers are
a. 10, 2
b. 18, 26
c. 10, 18
d. 26, 34
Correct Option: A
Explanation :
Let the numbers be a and (a+8). Then
$\displaystyle\frac{1}{{12}}\left[ {a + (a + 8)} \right] = 1 \Rightarrow 2a + 8 = 12$
$\Rightarrow a = 2,a + 8 = 10$

73.A number is 25 more than its $\displaystyle\frac{2}{5}{\rm{th}}$. The number is
a. 60
b. 80
c. $\displaystyle\frac{{125}}{3}$
d. $\displaystyle\frac{{125}}{7}$
Correct Option: C
Explanation:
Let the number be N. Then
$N - 25 = \displaystyle\frac{2}{5}{\rm{ }}N{\rm{ or 5N - 125 = 2N}}$
or N=125/3

74.The sum of three numbers is 68.  If the ratio between first and second be 2 : 3 and that between second and third be 5 : 3, then the second number is
a. 30
b. 20
c. 58
d. 48
Correct Option: A
Explanation:
Let the numbers be x,y,z.  Then
$\displaystyle\frac{x}{y} = \displaystyle\frac{2}{3},\displaystyle\frac{y}{z} = \displaystyle\frac{5}{3} \Rightarrow \displaystyle\frac{x}{y} = \displaystyle\frac{{2 \times 5}}{{3 \times 5}} = \displaystyle\frac{{10}}{{15}}$
and $\displaystyle\frac{y}{z} = \frac{{5 \times 3}}{{3 \times 3}} = \frac{{15}}{9}$
$\Rightarrow x:y:z = 10:15:9$

75. The sum of two numbers is 100 and their difference is 37. The difference of their squares is
a. 37
b. 100
c. 63
d. 3700
Correct Option: D
Explanation:
Let the numbers be x and y
Then x+y=100 & x-y=37
${x^2} - {y^2} = (x - y)(x + y) = 100 \times 37 = 3700$