9. 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

Answer: 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.

10. $\displaystyle\frac{{{3^{th}}}}{4}{\rm{ of }}\displaystyle\frac{{{1^{th}}}}{5}$ of a number is 60. The number is
a. 300
b. 400
c. 450
d. 1200

Answer: B

Explanation:
Let the number be N. Then
$\displaystyle\frac{3}{4} \times \frac{1}{5} \times N = 60 \Rightarrow 3N = 1200 \Rightarrow N = 400$.

11. 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

Answer: 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

12. 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

Answer: B

Explanation:
Let the number be x and y . We know that,
${(x + y)^2} = ({x^2} + {y^2}) + 2xy = 289 + 2\times120$
$ = 289 + 240 = 529 \Rightarrow x + y = \sqrt {529} = 23$

13. 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

Answer: A

Explanation:
Let the numbers be x and y. Then
${x^2} + {y^2} = 68$
But ${(x − y)^2} = 36 \Rightarrow {x^2} + {y^2} − 2xy = 36$
$ \Rightarrow 68 − 2xy = 36 \Rightarrow 2xy = 32$
$ \Rightarrow xy = 16$

14. 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

Answer: A

Explanation:
Average of the first three is 23. Therefore their sum = 23 x 3
Average of the last three is 42. Therefore their sum = 42 x 3
Sum of all number = Sum of first three + Fourth number + Sum of last three.
$(23 \times 3 + a + 42 \times 3) = 235 \Rightarrow a = 40$

15. 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

Answer: 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.

16. 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

Answer: 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