1. The ratio between the parents ages of A and B is 5 : 3 respectively. The ratio between A's age 4 years ago and B's 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) 1 : 3

b) 3 : 1

c) 2 : 1

d) 4 : 1

Explanation: Let A's age be 5x years, then B's age = 3x years

$$\eqalign{ & \frac{{5x - 4}}{{3x + 4}} = \frac{1}{1} \cr & 5x - 4 = 3x + 4 \cr & 2x = 8 \cr & x = 4 \cr & \frac{{{\text{A's age 4 years hence}}}}{{{\text{B's age 4 years ago }}}} \cr & = \frac{{5x + 4}}{{3x - 4}} \cr & = \frac{{5 \times 4 + 4}}{{3 \times 4 - 4}} \cr & = \frac{{24}}{8} \cr & = \frac{3}{1} \cr & = 3:1 \cr} $$

2. The ratio between the ages of Neelam and Shiny is 5 : 6 respectively. If the ratio between the one-third age of Neelam and half of Shiny's age is 5 : 9, then what is Shiny's age = ?

a) 25 years

b) 30 years

c) 36 years

d) Cannot be determined

Explanation: Let Neelam's age be 5x years and Shiny's age be 6x years

$$\eqalign{ & \left( {\frac{1}{3} \times 5x} \right):\left( {\frac{1}{2} \times 6x} \right) = 5:9 \cr & \frac{{5x}}{{3 \times 3x}} = \frac{5}{9} \cr} $$

Shiny's age cannot be determined

3. 18 years ago, a man was three times as old as his son. Now, the man is twice as old as his son. The sum of the present ages of the man and his son is =

a) 54 year

b) 72 years

c) 105 years

d) 108 years

Explanation: Let the son's age 18 years ago be x years,

Then man's age 18 years ago = 3x years

$$\eqalign{ & \left( {3x + 18} \right) = 2\left( {x + 18} \right) \cr & 3x + 18 = 2x + 36 \cr & x = 18 \cr} $$

Sum of their present ages

$$\eqalign{ & \left( {3x + 18 + x + 18} \right){\text{years}} \cr & \left( {4x + 36} \right){\text{years}} \cr & \left( {4 \times 18 + 36} \right){\text{years}} \cr & {\text{ 108 years}} \cr} $$

4. The age of a man 10 years ago was thrice the age of his son. 10 years hence, the man's age will be twice the age of his son. The ratio of their present ages is = ?

a) 5 : 2

b) 7 : 3

c) 9 : 2

d) 13 : 4

Explanation: Let son's age 10 years ago be x years,

Then man's age 10 years ago = 3x years

Son's present age = (x + 10) years

Man's present age = (3x + 10) years

$$\eqalign{ & \left( {3x + 10} \right) + 10 = 2\left( {x + 10 + 10} \right) \cr & 3x + 20 = 2\left( {x + 20} \right) \cr & 3x + 20 = 2x + 40 \cr & x = 20 \cr} $$

Ratio of present ages of man and the son

$$\eqalign{ & {\text{ = }}\frac{{3x + 10}}{{x + 10}} \cr & = \frac{{3 \times 20 + 10}}{{20 + 10}} \cr & = \frac{{70}}{{30}} \cr & = 7:3 \cr} $$

5. Tanya's grandfather was 8 times older to her 16 years ago. He would be 3 times of her age 8 years from now. 8 years ago, what was the ratio of Tanya's age to that of her grandfather ?

a) 1 : 2

b) 1 : 5

c) 3 : 8

d) None of these

Explanation: 16 years ago, let T = x years and G = 8x years

After 8 years from now,

T = (x + 16 + 8) years and G = (8x + 16 + 8) years

$$\eqalign{ & {\text{8x + 24 = 3}}\left( {x + 24} \right) \cr & 8x - 3x = 72 - 24 \cr & 5x = 48 \cr & 8{\text{years ago,}} \cr & {\text{ }}\frac{{\text{T}}}{{\text{G}}} \cr & = \frac{{x + 8}}{{8x + 8}} \cr & = \frac{{\frac{{48}}{5} + 8}}{{8 \times \frac{{48}}{5} + 8}} \cr & = \frac{{48 + 40}}{{384 + 40}} \cr & = \frac{{88}}{{424}} \cr & = \frac{{11}}{{53}} \cr} $$

6. Rajan got married 8 years ago. His present age is $$\frac{6}{5}$$ times his age at the time of his marriage. Rajan's sister was 10 years younger to him at the time of his marrige. The age of Rajan's sister is =

a) 32 years

b) 36 years

c) 38 years

d) 40 years

Explanation: Let Rajan's age 8 years ago be x years,

Present age = (x + 8) years

$$\eqalign{ & x + 8 = \frac{6}{5}x \cr & 5x + 40 = 6x \cr & x = 40 \cr} $$

Rajan's sister's age 8 years ago

= (40 - 10) years

= 30 years

His sister's age now

= (30+8) years

= 38 years

7. A couple has a son and a daughter. The age of the father is four times that of the son and the age of the daughter is one-third of that of her mother. The wife is 6 years younger to her husband and the sister is 3 years older then her brother. The mother's age is = ?

a) 42 years

b) 48 years

c) 54 years

d) 63 years

Explanation: M → Mother, F → Father, S → Son and D → Daughter

F = 4S, D = $$\frac{1}{3}$$M, M = F - 6 and S = D - 3

$$\eqalign{ & M = 3D = 3\left( {S + 3} \right) \cr & = 3S + 9 = \frac{3}{4}F + 9 = \frac{3}{4}\left( {M + 6} \right) + 9 \cr & = \frac{3}{4}M + \frac{3}{4} \times 6 + 9 \cr & \Rightarrow \left( {M - \frac{3}{4}M} \right) = \left( {\frac{9}{2} + 9} \right) \cr & \Rightarrow \frac{1}{4}M = \frac{{27}}{2} \cr & \Rightarrow M = \left( {\frac{{27}}{2} \times 4} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 54\,{\text{years}} \cr} $$

8. Reenu's father was 38 years of age when she was born while her mother was 36 years old when her brother 4 years younger to her was born. What is the difference between the ages of her parents ?

a) 2 Years

b) 4 Years

c) 6 Years

d) 8 years

Explanation: Mother's age when Reenu's brother was born = 36 year

Father's age when Reenu's brother was born = (38 + 4) years = 42 year

Required difference = (42 - 36) years = 6 years

9. A man was asked to state his age in years. His reply was, "Take my age 3 years hence, multiply it by 3 and then subtract 3 times my age 3 years ago and you will know how old I am". What is the age of the man ?

a) 18 years

b) 20 years

c) 24 years

d) 32 years

Explanation: Let the present age of the man be x years

$$\eqalign{ & {\text{3}}\left( {x + 3} \right) - 3\left( {x - 3} \right) = x \cr & \left( {3x + 9} \right) - \left( {3x - 9} \right) = x \cr & x = 18 \cr} $$

The present age of the man is 18 years

10. The ratio of a man's age and his son's age is 7 : 3 and the product of their ages is 756. The ratio of their ages after 6 years will be ?

a) 5 : 2

b) 2 : 1

c) 11 : 7

d) 13 : 9

Explanation: Let the man's age be 7x years

Then son's age = 3x years

$$\eqalign{ & {\text{7}}x \times 3x = 756 \cr & 21{x^2} = 756 \cr & {x^2} = 756 \cr & {x^2} = 36 \cr & x = 6 \cr} $$

The ratio of their ages after 6 years

$$\eqalign{ & {\text{ = }}\left( {7x + 6} \right):\left( {3x + 6} \right) \cr & = \left( {7 \times 6 + 6} \right):\left( {3 \times 6 + 6} \right) \cr & = 48:24 \cr & = 2:1 \cr} $$