## Problems on ages

1. Raghu's age after 15 years will be 5 times his age 5 years back. What is the present age of Raghu ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;Raghu's&space;present&space;age&space;be&space;x&space;years.&space;Then,$
$\par&space;Raghu's&space;age&space;after&space;15&space;years&space;=&space;(x&space;+&space;15)&space;years.$
$\par&space;Raghu's&space;age&space;5&space;years&space;back&space;=&space;(x&space;-&space;5)&space;years.$
$x&space;+&space;15&space;=&space;5(x&space;-&space;5)$
$x&space;+&space;15&space;=&space;5x&space;-&space;25$
$4x&space;=&space;40$
$x&space;=&space;10$			
2. The product of the ages of Ankit and Nikita is 240. If twice the age of Nikita is more than Ankit's age by 4 years, what is Nikita's age ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;Ankit's&space;age&space;be&space;x&space;years.&space;Then,&space;Nikita's&space;age&space;=$  $\frac{240}{x}&space;years.$

$2&space;\times&space;\frac{240}{x}&space;-&space;x&space;=&space;4$

$480&space;-&space;x^{2}&space;=&space;4x$

$x^{2}&space;+&space;4x&space;-&space;480&space;=&space;0$

$(x&space;+&space;24)&space;(x&space;-&space;20)&space;=&space;0$

$x&space;=&space;20$

$\par&space;Hence,&space;Nikita's&space;age&space;=$ $(\frac{240}{20})$  $\par&space;years&space;=&space;12&space;years$			
3. The present age of a father is 3 years more than three times the age of his son. Three years hence, father's age will be 10 years more than twice the age of the son. Find the present age of the father ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;the&space;son's&space;present&space;age&space;be&space;x&space;years.$

$\par&space;Then&space;father's&space;present&space;age&space;=&space;(3x&space;+&space;3)&space;years.$

$(3x&space;+&space;3&space;+&space;3)&space;=&space;2(x&space;+&space;3)&space;+&space;10$

$3x&space;+&space;6&space;=&space;2x&space;+&space;16$

$x&space;=&space;10$

$\par&space;Hence&space;father's&space;present&space;age&space;=&space;(3x&space;+&space;3)&space;=&space;(3\times&space;10&space;+&space;3)&space;=&space;33&space;years$			
4. One year ago, the ratio of Gaurav's and Sachin's age was 6 : 7 respectively. Four years hence, this ratio would become 7 : 8. How old is Sachin ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;Gaurav's&space;and&space;Sachin's&space;ages&space;one&space;year&space;ago&space;be&space;6x&space;and&space;7x&space;years&space;respectively.$

$\par&space;Gaurav's&space;age&space;4&space;years&space;hence&space;=&space;(6x&space;+&space;1)&space;+&space;4&space;=&space;(6x&space;+&space;5)&space;years$

$\par&space;Sachin's&space;age&space;4&space;years&space;hence&space;=&space;(7x&space;+&space;1)&space;+&space;4&space;=&space;(7x&space;+&space;5)&space;years$

$\frac{6x&space;+&space;5}{7x&space;+&space;5}&space;=&space;\frac{7}{8}$

$8(6x&space;+&space;5)&space;=&space;7(7x&space;+&space;5)$

$48x&space;+&space;40&space;=&space;49x&space;+&space;35$

$x&space;=&space;5$

$\par&space;Hence&space;Sachin's&space;present&space;age&space;=&space;(7x&space;+&space;1)&space;=&space;36&space;years$			
5. Suneel's age after six years will be three-seventh of his father's age. Ten years ago, the ratio of their ages was 1 : 5. What is Suneel father's age at present ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;the&space;ages&space;of&space;Suneel&space;and&space;his&space;father&space;10&space;years&space;ago&space;be&space;x&space;and&space;5x&space;years&space;respectively.$

$\par&space;Suneel's&space;age&space;after&space;6&space;years&space;=&space;(x&space;+&space;10)&space;+&space;6&space;=&space;(x&space;+&space;16)&space;years$

$\par&space;Father's&space;age&space;after&space;6&space;years&space;=&space;(5x&space;+&space;10)&space;+&space;6&space;=&space;(5x&space;+&space;16)&space;years$

$(x&space;+&space;16)&space;=&space;\frac{3}{7}(5x&space;+&space;16)$

$7(x&space;+&space;16)&space;=&space;3(5x&space;+&space;16)$

$7x&space;+&space;112&space;=&space;15x&space;+&space;48$

$8x&space;=&space;64$

$x&space;=&space;8$

$\par&space;Hence,&space;Suneel&space;father's&space;present&space;age&space;=&space;(5x&space;+&space;10)&space;=&space;50&space;years.$			
6. Sachin is younger than Rahul by 4 years. If their ages are in the respective ratio of 7 : 9, how old is Sachin ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;Rahul's&space;age&space;be&space;x&space;years.&space;Then,&space;Sachin's&space;age&space;=&space;(x&space;-7)&space;years$

$\frac{x-7}{x}&space;=&space;\frac{7}{9}$

$9x&space;-&space;63&space;=&space;7x$

$2x&space;=&space;63$

$x&space;=&space;31.5$

$\par&space;Hence,&space;Sachin's&space;age&space;=&space;(x&space;-&space;7)&space;=&space;24.5&space;years$			
7. The ratio between the present ages of P and Q is 6 : 7. If Q is 4 years old than P, what will be the ratio of the ages of P and Q after 4 years ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;P's&space;age&space;and&space;Q's&space;age&space;be&space;6x&space;years&space;and&space;7x&space;years&space;respectively.$

$\par&space;Then,&space;7x&space;-&space;6x&space;=&space;4$

$x&space;=&space;4$

$\par&space;Required&space;ratio&space;=&space;(6x&space;+&space;4)&space;:&space;(7x&space;+&space;4)&space;=&space;28&space;:&space;32&space;=&space;7&space;:&space;8$			
8. The ratio of the present ages of P and Q is 5 : 7 respectively. If the difference between Qs present age and P’s age after 6 years is 2, what is the total of P’s and Q’s present ages ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;the&space;present&space;ages&space;of&space;P&space;and&space;Q&space;be&space;5x&space;years&space;and&space;7x&space;years&space;respectively.$

$\par&space;Then,&space;7x&space;-&space;(5x&space;+&space;6)&space;=&space;2$

$2x&space;=&space;8$

$x&space;=&space;4$

$\par&space;Required&space;sum&space;=&space;5x&space;+&space;7x&space;=&space;12x&space;=&space;48&space;years$			
9. At present, the ratio between the ages of Arun and Deepak is 4 : 3. After 6 years, Arun’s age will be 26 years. What is the age of Deepak at present ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;the&space;present&space;ages&space;of&space;Arun&space;and&space;Deepak&space;be&space;4x&space;years&space;and&space;3x&space;years&space;respectively.&space;Then,$

$4x&space;+&space;6&space;=&space;26$

$4x&space;=&space;20$

$x&space;=&space;5$

$\par&space;Deepak's&space;age&space;=&space;3x&space;=&space;15&space;years$			
10. Present ages of X and Y are in the ratio 5 : 6 respectively. Seven years hence this ratio will become 6 : 7 respectively. What is X’s present age in years ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;the&space;present&space;ages&space;of&space;X&space;and&space;Y&space;be&space;5x&space;years&space;and&space;6x&space;years&space;respectively.$

$\frac{5x&space;+&space;7}{6x&space;+&space;7}&space;=&space;\frac{6}{7}$

$7(5x&space;+&space;7)&space;=&space;6(6x&space;+&space;7)$

$x&space;=&space;7$

$\par&space;X's&space;present&space;age&space;=&space;5x&space;=&space;35&space;years.$"			
11. Present ages of David and Balu are in the ratio of 5 : 4 respectively. Three years hence, the ratio of their ages will become 11 : 9 respectively. What is Balu’s present age in years ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;the&space;present&space;ages&space;of&space;David&space;and&space;Balu&space;be&space;5x&space;years&space;and&space;4x&space;years&space;respectively.$

$\frac{5x+3}{4x+3}&space;=&space;\frac{11}{9}$

$9(5x+3)&space;=&space;11(4x+3)$

$x&space;=&space;6$

$\par&space;Balu's&space;present&space;age&space;=&space;4x&space;=&space;24&space;years$			
12. Six years ago, the ratio of the ages of Ram and Sai was 6 : 5. Four years hence, the ratio of their ages will be 11 : 10. What is Sai’s age at present ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;the&space;ages&space;of&space;Ram&space;and&space;Sai&space;6&space;years&space;ago&space;be&space;6x&space;and&space;5x&space;years&space;respectively.$
$\frac{(6x&space;+&space;6)&space;+&space;4}{(5x&space;+&space;6)&space;+&space;4}&space;=&space;\frac{11}{10}$
$10(6x+10)&space;=&space;11(5x+10)$
$5x&space;=&space;10$
$x&space;=&space;2$
$\par&space;Sai's&space;present&space;age&space;=&space;(5x+6)&space;=&space;16&space;years.$			
13. The ratio of present ages of two brothers is 1 : 2 and 5 years back, the ratio was 1 : 3. What will be the ratio of their ages after 5 years ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;the&space;present&space;ages&space;of&space;the&space;two&space;brothers&space;be&space;x&space;years&space;and&space;2x&space;years&space;respectively.$
$\frac{x&space;-&space;5}{2x&space;-&space;5}&space;=&space;\frac{1}{3}$
$3(x&space;-&space;5)&space;=&space;(2x&space;-&space;5)$
$x&space;=&space;10$
$\par&space;Required&space;ratio&space;=&space;(x&space;+&space;5)&space;:&space;(2x&space;+&space;5)&space;=&space;15&space;:&space;25&space;=&space;3&space;:&space;5$			
14. Hitesh is 40 years old and Ronnie is 60 years old. How many years ago was the ratio of their ages 3 : 5

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Suppose,&space;the&space;ratio&space;was&space;3&space;:&space;5,&space;x&space;years&space;ago.$
$\frac{40&space;-&space;x}{60&space;-&space;x}&space;=&space;\frac{3}{5}$
$5(40&space;-&space;x)&space;=&space;3(60&space;-&space;x)$
$2x&space;=&space;20$
$x&space;=&space;10$			
15. The of the father's age to his son's age 7 : 3. The product of their ages are 756. The ratio of their ages after 6 years will be

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;the&space;present&space;ages&space;of&space;the&space;father&space;and&space;son&space;be&space;7x&space;years&space;and&space;3x&space;years&space;respectively.$
$7x\times&space;3x&space;=&space;756$
$21x^{2}&space;=&space;756$
$x^{2}&space;=&space;36$
$x&space;=&space;6$
$\par&space;Required&space;ratio&space;=&space;(7x&space;+&space;6)&space;:&space;(3x&space;+&space;6)&space;=&space;48&space;:&space;24&space;=&space;2&space;:&space;1$			
16. The present ages of three persons are in proportions 4 : 7 : 9. Eight years ago, the sum of their ages was 56. Find their present ages (in years).

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;their&space;present&space;ages&space;be&space;4x,&space;7x&space;and&space;9x&space;years&space;respectively.$
$\par&space;(4x&space;-&space;8)&space;+&space;(7x&space;-&space;8)&space;+&space;(9x&space;-&space;8)&space;=&space;56$
$20x&space;=&space;80$
$x&space;=&space;4$
$\par&space;So,&space;their&space;present&space;ages&space;are&space;16&space;years,&space;28&space;years&space;and&space;36&space;years.$			
17. The ratio between the school ages of Neelam and Shaan is 5 : 6 respectively. If the ratio between the one-third age of Neelam and half of Shaan’s age is 5 : 9, then what is the school age of Shaan ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;the&space;school&space;ages&space;of&space;Neelam&space;and&space;Shaan&space;be&space;5x&space;and&space;6x&space;years&space;respectively.$
$\frac{\frac{1}{3}&space;\times&space;5x}{\frac{1}{2}&space;\times&space;6x}&space;=&space;\frac{5}{9}$
$(\frac{1}{3}\times&space;9\times&space;5x)&space;=&space;(\frac{5}{2}\times&space;6x)$
$15&space;=&space;15$
$\par&space;Thus,&space;Shaan's&space;age&space;cannot&space;be&space;determined.$			
18. The ratio between the present ages of A and B is 5 : 3 respectively. The ratio between A’s age 4 years ago and B’s age 4 years hence is 1 : 1. What is the ratio between A’s age 4 years hence and B’s age 4 years ago ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;the&space;present&space;ages&space;of&space;A&space;and&space;B&space;be&space;5x&space;and&space;3x&space;years&space;respectively.$
$\frac{5x&space;-&space;4}{3x&space;+&space;4}&space;=&space;\frac{1}{1}$
$5x&space;-&space;4&space;=&space;3x&space;+&space;4$
$2x&space;=&space;8$
$x&space;=&space;4$
$\par&space;Required&space;ratio&space;=&space;(5x&space;+&space;4)&space;:&space;(3x&space;-&space;4)&space;=&space;24&space;:&space;8&space;=&space;3&space;:&space;1$			
19. Ten years ago, A was half of B in age. If the ratio of their present ages is 3 : 4, what will be the total of their present ages ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

Explanation:

			$\par&space;Let&space;the&space;ages&space;of&space;A&space;and&space;B&space;10&space;years&space;ago&space;be&space;x&space;and&space;2x&space;years&space;respectively.$
$\frac{x+10}{2x+10}&space;=&space;\frac{3}{4}$
$4(x&space;+&space;10)&space;=&space;3(2x&space;+&space;10)$
$2x&space;=&space;10$
$x&space;=&space;5$
$\par&space;Sum&space;of&space;their&space;present&space;ages&space;=&space;(x&space;+&space;10)&space;+&space;(2x&space;+&space;10)&space;=&space;(3x&space;+&space;20)&space;=&space;35&space;years$			
20. A is two years older than B who is twice as old as C. If the total of the ages of A, B and C be 27, then how old is B ?

A. 3 : 4

B. 3 : 5

C. 4 : 3

D. None of these

			$\par&space;Let&space;C's&space;age&space;be&space;x&space;years.&space;then,$
$\par&space;B's&space;age&space;=&space;2x&space;years$
$\par&space;A's&space;age&space;=&space;(2x&space;+&space;2)&space;years$
$(2x&space;+&space;2)&space;+&space;2x&space;+&space;x&space;=&space;27$
$5x&space;=&space;25$
$x&space;=&space;5$
$\par&space;B's&space;age&space;=&space;2x&space;=&space;10&space;years.$			`