Problems On Ages — Part 2 MCQs with Explanations

Let the ages of Kunal and Sagar 6 years ago be 6x and 5x years Then, $$\eqalign{ & \frac{{\left( {6x + 6} \right) + 4}}{{\left( {5x + 6} \right) + 4}} = \frac{{11}}{{10}} \cr & \Rightarrow \frac{{6x + 10}}{{5x + 10}} = \frac{{11}}{{10}} \cr & \Rightarrow 10\left( {6x + 10} \right) = 11\left( {5x + 10} \right) \cr & \Rightarrow 60x + 100 = 55x + 100 \cr & \Rightarrow 5x = 10 \cr & \Rightarrow x = 2 \cr} $$ Sagar's present age = (5x + 6) years = (5 × 2 + 6) years = 16 years

Vimal's present age = (8 + 2) years = 10 years Sneh's father's age = 2(10 + 10) years = 40 years Sneh's age $$\eqalign{ & {\text{ = }}\left( {\frac{1}{6} \times 40} \right){\text{ years }} \cr & {\text{ = }}\frac{{20}}{3}{\text{years }} \cr & {\text{ = 6}}\frac{2}{3}{\text{years}} \cr} $$

Let Sulekha age be 9 year, Then Arunima's age = 8x years $$\eqalign{ & \frac{{9x + 5}}{{8x + 5}} = \frac{{10}}{9} \cr & \Rightarrow 9\left( {9x + 5} \right) = 10\left( {8x + 5} \right) \cr & \Rightarrow 81x + 45 = 80x + 50 \cr & \Rightarrow x = 5 \cr} $$ Difference in their ages = (9x - 8x) years = x years = 5 years

Let the present age of Y be a years Three years ago X's age = 3a years Then, present age of X is (3a + 3) Z's present age = 2a According to question Now, (3a + 3) - 2a = 12 a = 9 year ∴ Present age of Z = 2a = 2 × 9 = 18 years

Let the age of the son and the daughter of Poorvi be 6a years and 7a years respectively. 5 years hence, present age of son = 6a - 5 and present age of daughter = 7a - 5 According to the question, Eight years ago, the age of Poorvi = 6a - 5 + 7a - 5 = 13a - 10 So, present age of Poorvi = 13a - 10 + 8 = 13a - 2 Since, present age of Poorvi husband = 3 (6a - 5) The difference of present age of Poorvi husband and Poorvi = 7 (Given) $$\eqalign{ & {\text{3}}\left( {6a - 5} \right) - \left( {13a - 2} \right) = 7 \cr & \Rightarrow 18a - 15 - 13a + 2 = 7 \cr & \Rightarrow 5a = 20 \cr & \Rightarrow a = 4 \cr} $$ The present age of daughter = (7a - 5) = 7 × 4 - 5 = 23 years

Let present age of father , mother and son be x, y and z respectively Sum of present ages of father and son = (Mother's present age + 8 years) x + z = y + 8 years.................(i) Mother's present age = Son's present age + 22 years) ⇒ y = z + 22 years..............(ii) Put the value of y in equation (i) we get x + z = z + 22 + 8 x + z = z + 30 x = 30 years ∴ Father's present age = 30 years Age of father after four years = 30 + 4 = 34 years ∴ Required age of father = 34 years

Let the age of Rahul, Sagar and Purav be x, y and z respectively According to the given information Age of Sagar - Age of Rahul = Age of Rahul - Age of Purav $$\eqalign{ & \Rightarrow y - x = x - z \cr & \Rightarrow 2x = y + z......(i) \cr} $$ Also y + z = 66 years From (i) x = 33 years Also as per equation (i) we have Purav's age + Sagar age = 66 years by going through option (A) given Purav = 18, and Rahul = 33 years, Sagar = 48 years Difference between Rahul's and Purav's age = 33 - 18 = 15 years

Let present ages of Simmi and Niti be a and b years respectively. Ten years hence, the ratio between Simmi's age and Niti's age = 7 : 9 According to question $$\frac{{a + 10}}{{b + 10}} = \frac{7}{9}$$ By cross multiplying we get $$\eqalign{ & \Rightarrow 9a + 90 = 7b + 70 \cr & \Rightarrow 7b - 9a = 20......(i) \cr & {\text{Also}},\frac{{a - 2}}{{b - 2}} = \frac{1}{3} \cr} $$ By cross multiplying we get $$\eqalign{ & \Rightarrow 3a - 6 = b - 2 \cr & \Rightarrow 3a - b = 4.....(ii) \cr} $$ Multiplying equation (ii) by 3 $$9a - 3b = 12.....(iii)$$ Adding equation (ii) and (iii) we get $$\eqalign{ & - 9a + 7b = 20 \cr & 4b = 32 \cr & b = 8{\text{ year}} \cr} $$ From equation (ii) we get $$\eqalign{ & a = \frac{{4 + b}}{3} \cr & \,\,\,\,\, = \frac{{4 + 8}}{3} \cr & \,\,\,\,\,\, = \frac{{12}}{3} \cr & \,\,\,\,\,\, = 4\,{\text{year}} \cr} $$ Since, Abhay is 4 years older to Niti So, Abhay's present age = 8 + 4 = 12 years

\[\begin{array}{*{20}{c}} {{\text{After four years}} \to }&{{\text{Father}}}&:&{{\text{Son}}} \\ {}&4&:&1 \end{array}\] Four years ago → 4x - 4, x - 4 4x - 4 + x - 4 = 52 5x - 8 = 52 x = 12 \[\begin{array}{*{20}{c}} {{\text{Present age}} \to }&{{\text{Father}}}&:&{{\text{Son}}} \\ {}&{44}&:&8 \\ {{\text{After 10 years}} \to }&{{\text{Father}}}&:&{{\text{Son}}} \\ {}&{54}&:&{18} \\ {}&3&:&1 \end{array}\]

Let the income of A and B be 8 years ago = 5x, 4x $$\eqalign{ & \frac{{\text{A}}}{{\text{B}}} = \frac{{5x + 8}}{{4x + 8}} = \frac{6}{5} \cr & 25x + 40 = 24x + 48 \cr & x = 8 \cr & \frac{{\text{A}}}{{\text{B}}} = \frac{{5x + 8 + 7}}{{4x + 8 + 7}} \cr & = \frac{{5 \times 8 + 15}}{{4 \times 8 + 15}} \cr & = \frac{{55}}{{47}} \cr} $$ Sum of ages of A and B = A + B = 55 + 47 = 102

$$\eqalign{ & 1{\text{ unit}} \to 4{\text{ years}} \cr & {\text{So current age}} \cr & {\text{Man}}:{\text{Father}}:{\text{Son}} \cr & \,\,\,{\text{48}}\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{60}}\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{8}} \cr & {\text{Required answer}} = \frac{{60 + 12}}{{8 + 12}} \cr & = \frac{{72}}{{20}} \cr & = \frac{{18}}{5}{\text{ or }}18:5 \cr} $$

Present age of B = 35 years