# Problems on Numbers for Exam

Q51. The number (5n^{2} + 5n) for natural number n is always divisible by

(a) 5 only

(b) 10 only

(c) 5 and 10 only

(d) None of the above

**Answer:** (c) 5 and 10 only

5n^{2} + 5n = 5n (n + 1) ∀ n ∈ N

putting n = 1, 2, 3, ⋅ ⋅ ⋅ ⋅ ⋅ all numbers are divisible by 5 an 10 only

Q52. The numbers 1, 3, 5 ⋅ ⋅ ⋅ ⋅ ⋅ 25 are multiplied together. The number of zeros at the right end of the product is

(a) 0

(b) 1

(c) 2

(d) 3

**Answer:** (a) 0

Clearly, the product will contain 5 at the unit place. So number of zeros at the right end of the product is 0.

Q53. Find the number such that its third is greater than its 4th by 12

(a) 99

(b) 105

(c) 123

(d) 144

**Answer:** (d) 144

Let the number be x

A/Q, x X | 1 | = x X | 1 | + 12 |

3 | 4 |

⇒ | x | - | x | = 12 |

3 | 4 |

⇒ | 4x - 3x | = 12 |

12 |

⇒ x = 12 X 12 = 144

∴ The number is 144

Q54. The sum of one half, one-third and one-fourth of a number exceeds the number by 10. The number is

(a) 103

(b) 111

(c) 120

(d) 135

**Answer:** (c) 120

Let the number be x

A/Q, x X | 1 | + x X | 1 | + x X | 1 | = x + 10 |

2 | 3 | 4 |

⇒ | x | + | x | + | x | = x + 10 |

2 | 3 | 4 |

⇒ | 6x + 4x + 3x | = x + 10 |

12 |

⇒ 13x = 12x + 120

⇒ 13x - 12x = 120

⇒ x = 120

Q55. If the sum of three consecutive numbers is more than the middle number by 120, the middle number is

(a) 60

(b) 65

(c) 75

(d) 80

**Answer:** (a) 60

Let the consecutive numbers be x, x+1, x+2

A/q, x + (x+1) + (x+2) = (x+1) + 120

⇒ 3x + 3 = x + 121

⇒ 2x = 118

⇒ x = 59

∴ Middle number = x + 1 = 59 + 1 = 60

Q56. The sum of two numbers is 2490. If 5.5% of one number is equal to 9.5% of the other, the smaller number is

(a) 536

(b) 725

(c) 825

(d) 913

**Answer:** (d) 913

Let, the number's be x and (2490 - x)

Now, x X 5.5% = (2490 - x) X 9.5%

⇒ x X (5.5/100) = (2490 -x) X (9.5/100)

⇒ 5.5x = 23655 - 9.5x

⇒ 15x = 23655

⇒ x = 1577

One Number is 1577

And other number is = 2490 - 1577 = 913

∴ Smaller number = 913

Q57. The product of two successive number is 1980. Which is the greater number ?

(a) 41

(b) 45

(c) 52

(d) 63

**Answer:** (b) 45

Let, the numbers be x, x+1

A/Q, x(x+1) = 1980

⇒ x^{2} + x - 1980 = 0

⇒ x^{2} + 45x - 44x - 1980 = 0

⇒ x(x + 45) - 44(x + 45) = 0

⇒ (x + 45) (x - 44) = 0

∴ x = 44

∴ Greater number = 44 + 1 = 45

Q58. If the difference of two numbers is 4 and the difference of their square is 56, then the larger number is

(a) 7

(b) 8

(c) 9

(d) 10

**Answer:** (c) 9

Let, the numbers be x and y

Now x - y = 4 ---------> (1)

And x^{2} - y^{2} = 56

⇒ (x + y)(x - y) = 56

⇒ (x + y). 4 = 56

⇒ x + y = 14 -------> (2)

(1) + (2) ⇒ 2x = 18

⇒ x = 9

(2)⇒ y = 14 - 9 = 5

∴ Larger number = 9

Q59. If the product of two number is 6 and one of the numbers is 3/2, then the sum of the two numbers is

(a) 3½

(b) 3¾

(c) 5½

(d) 7¾

**Answer:** (c) 5½

Let, the numbers be x and y

Now x X y = 6 y = 3/2

⇒ x X (3/2) = 6

⇒ x = 12/3 = 4

∴ x + y = 4 + (3/2) = 11/2 = 5½

Q60. The product of two fractions is 14/15 and their quotient is 35/24. The smaller of the fraction is

(a) 1/3

(b) 4/5

(c) 3/7

(d) 5/11

**Answer:** (b) 4/5

Let the fractions be x and y

Now x X Y = 14/15 ------> (1)

And x/y = 35/24 -------> (2)

(1) X (2)⇒ xy X | x | = | 14 | X | 35 |

y | 15 | 24 |

⇒ x^{2} = |
49 | = ( | 7 | )^{2} |

36 | 6 |

⇒ x = 7/6

(1)⇒ y = | 14 | X | 1 | = | 14 | X | 6 |

15 | x | 15 | 7 |

= 4/5

∴ x > y

∴ Smaller fraction = 4/5