Problems On Ages — Part 1 MCQs with Explanations

$$\eqalign{ & {\text{Let}}\,{\text{Ronit's}}\,{\text{present}}\,{\text{age}}\,{\text{be}}\,x\,{\text{years}}. \cr & {\text{Then,}}\,{\text{father's}}\,{\text{present}}\,{\text{age}}\, \cr & = \left( {x + 3x} \right)\,{\text{years}} \cr & = 4x\,{\text{years}} \cr & \therefore \left( {4x + 8} \right) = \frac{5}{2}\left( {x + 8} \right) \cr & \Rightarrow 8x + 16 = 5x + 40 \cr & \Rightarrow 3x = 24 \cr & \Rightarrow x = 8 \cr & {\text{Hence,}}\,{\text{required}}\,{\text{times}} \cr & = \frac{{ {4x + 16} }}{{ {x + 16} }} \cr & = \frac{{48}}{{24}} \cr & = 2 \cr} $$

Let the ages of children be x , ( x + 3), ( x + 6), ( x + 9) and ( x + 12) years. Then, x + ( x + 3) + ( x + 6) + ( x + 9) + ( x + 12) = 50 ⇒ 5 x = 20 ⇒ x = 4. ∴ Age of the youngest child = x = 4 years.

Let the son's present age be x years. Then, (38 - x ) = x ⇒ 2 x = 38. ⇒ x = 19. ∴ Son's age 5 years back (19 - 5) = 14 years.

Let C's age be x years. Then, B's age = 2 x years. A's age = (2 x + 2) years. ∴ (2 x + 2) + 2 x + x = 27 ⇒ 5 x = 25 ⇒ x = 5. Hence, B's age = 2 x = 10 years.

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{present}}\,{\text{ages}}\,{\text{of}}\,{\text{Sameer}}\,{\text{and}}\,{\text{Anand}}\,{\text{be}}\, \cr & 5x\,{\text{years}}\,{\text{and}}\,4x\,{\text{years}}\,{\text{respectively}} \cr & {\text{Then,}}\,\frac{{5x + 3}}{{4x + 3}} = \frac{{11}}{9} \cr & \Rightarrow 9\left( {5x + 3} \right) = 11\left( {4x + 3} \right) \cr & \Rightarrow 45x + 27 = 44x + 33 \cr & \Rightarrow 45x - 44x = 33 - 27 \cr & \Rightarrow x = 6 \cr & \therefore {\text{Anand's}}\,{\text{present}}\,{\text{age}} \cr & = 4x \cr & = 24\,{\text{years}}\, \cr} $$

Let the son's present age be x years. Then, man's present age = ( x + 24) years. ∴ ( x + 24) + 2 = 2( x + 2) ⇒ x + 26 = 2 x + 4 ⇒ x = 22

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

Let the present ages of son and father be x and (60 - x ) years respectively. Then, (60 - x ) - 6 = 5( x - 6) ⇒ 54 - x = 5 x - 30 ⇒ 6 x = 84 ⇒ x = 14 ∴ Son's age after 6 years = ( x + 6) = 20 years.

Let the present ages of Arun and Deepak be 4 x years and 3x years respectively. Then, 4x + 6 = 26 ⇒ 4x = 20 ⇒ x = 5 ∴ Deepak's age = 3x = 15 years.

$$\eqalign{ & {\text{Let}}\,{\text{Rahul's}}\,{\text{age}}\,{\text{be}}\,x\,{\text{years}}. \cr & {\text{Then,}}\,{\text{Sachin's}}\,{\text{age}} = \left( {x - 7} \right)\,{\text{years}}. \cr & \therefore \frac{{x - 7}}{x} = \frac{7}{9} \cr & \Rightarrow 9x - 63 = 7x \cr & \Rightarrow 2x = 63 \cr & \Rightarrow x = 31.5 \cr & {\text{Hence,}}\,{\text{Sachin's}}\,{\text{age}} \cr & = \left( {x - 7} \right)\,{\text{years}} \cr & = 24.5\,{\text{years}} \cr} $$

Let their present ages be 4 x , 7 x and 9 x years respectively. Then, (4 x - 8) + (7 x - 8) + (9 x - 8) = 56 ⇒ 20 x = 80 ⇒ x = 4 ∴ Their present ages are 4 x = 16 years, 7 x = 28 years and 9 x = 36 years respectively.

Mother's age when Ayesha's brother was born = 36 years. Father's age when Ayesha's brother was born = (38 + 4) years = 42 years. ∴ Required difference = (42 - 36) years = 6 years.

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{mother's}}\,{\text{present}}\,{\text{age}}\,{\text{be}}\,x\,{\text{years}}{\text{.}} \cr & {\text{Then,}}\,{\text{the}}\,{\text{person's}}\,{\text{present}}\,{\text{age}} \cr & = {\frac{2}{5}x} \,\,{\text{years}} \cr & \therefore {\frac{2}{5}x + 8} = \frac{1}{2}\left( {x + 8} \right) \cr & \Rightarrow 2\left( {2x + 40} \right) = 5\left( {x + 8} \right) \cr & \Rightarrow x = 40 \cr} $$

Given that: 1. The difference of age b/w R and Q = The difference of age b/w Q and T. 2. Sum of age of R and T is 50 i.e. (R + T) = 50. Question: R - Q = ?. Explanation: R - Q = Q - T (R + T) = 2Q Now given that, (R + T) = 50 So, 50 = 2Q and therefore Q = 25. Question is (R - Q) = ? Here we know the value(age) of Q (25), but we don't know the age of R. Therefore, (R-Q) cannot be determined.

Let the ages of father and son 10 years ago be 3x and x years respectively. Then, (3x + 10) + 10 = 2[(x + 10) + 10] ⇒ 3x + 20 = 2x + 40 ⇒ x = 20 ∴ Required ratio = (3x + 10) : (x + 10) = 70 : 30 = 7 : 3

Let Nishi's age be 6x years, Then Vinnee's age = 5x years $$\eqalign{ & \therefore \frac{{6x + 9}}{{5x + 9}} = \frac{9}{8} \cr & \Rightarrow 8\left( {6x + 9} \right) = 9\left( {5x + 9} \right) \cr & \Rightarrow 48x - 45x = 81 - 72 \cr & \Rightarrow 3x = 9 \cr & \Rightarrow x = 3 \cr & {\text{Difference in their ages}} \cr & {\text{ = }}\left( {6x - 5x} \right) \cr & = x {\text{ years}} \cr & = {\text{3 years}} \cr} $$

Let Shakti's age be 8$$x$$ years, Then Kanti's age = 7$$x$$ years $$\eqalign{ & \therefore \frac{{8x + 10}}{{7x + 10}} = \frac{{13}}{{12}} \cr & \Rightarrow 12\left( {8x + 10} \right) = 13\left( {7x + 10} \right) \cr & \Rightarrow 96x + 120 = 91x + 130 \cr & \Rightarrow 5x = 10 \cr & \Rightarrow x = 2 \cr} $$ Difference between their ages $$\eqalign{ & = \left( {8x - 7x} \right) \cr & = {\text{ }}x\,{\text{years}} \cr & {\text{ = 2}}\,{\text{years}} \cr} $$

$$\eqalign{ & {\text{A's age = }}\left( {44 \times \frac{6}{{11}}} \right){\text{years}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = 24 years}} \cr & {\text{And B's age = }}\left( {44 - 24} \right){\text{years}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = 20 years}} \cr} $$ Ratio of their ages after 8 years $$\eqalign{ & {\text{ = }}\frac{{\left( {24 + 8} \right)}}{{\left( {20 + 8} \right)}} \cr & = \frac{{32}}{{28}} \cr & = \frac{8}{7} \cr & = 8:7 \cr} $$

Let Farah's age 8 years ago be x years. Then , her present age = (x + 8) $$\eqalign{ & \therefore x + 8 = \frac{9}{7}x \cr & \Rightarrow 7x + 56 = 9x \cr & \Rightarrow 2x = 56 \cr & \Rightarrow x = 28 \cr} $$ ∴ Farah's age now = (x + 8) years = (28 + 8) years = 36 Her daughter's age now =$$\left( {\frac{1}{6} \times 36} \right)$$ years = 6 years Her daughter's age 3 years ago = (6 - 3) years = 3 years

Let the daughter's age be x years. Then, mother age = 3x years 3x + 12 = 2 (x + 12) ⇒ 3x + 12 = 2x + 24 ⇒ x = 12 Present age of daughter = 12 years

Let the son's age be $$x$$ years , Then Mr. Sanyal's age = 3$$x$$ years $$\eqalign{ & \therefore \frac{{3x + 6}}{{x + 6}} = \frac{5}{2} \cr & \Rightarrow 2\left( {3x + 6} \right) = 5\left( {x + 6} \right) \cr & \Rightarrow 6x + 12 = 5x + 30 \cr & \Rightarrow x = 18 \cr} $$ ∴ Present age of Mr. Sanyal = 3x years = (3 × 18) years = 54 years

Average age of man and his daughter = 34 years Their total age = (34 × 2) years = 68 years Let man's age be x years, Then daughter age = (68 - x) years $$\eqalign{ & \therefore \frac{{x + 4}}{{68 - x + 4}} = \frac{{14}}{5} \cr & \Rightarrow 5\left( {x + 4} \right) = 14\left( {72 - x} \right) \cr & \Rightarrow 5x + 20 = 1008 - 14x \cr & \Rightarrow 19x = 988 \cr & \Rightarrow x = 52 \cr} $$ ∴ Daughter's present age = (68 - 52) years = 16 years Alternate Solution : According to question, After 4 years, the total age of man & daughter is = [(34 × 2) + 4 + 4] = 76 years After 4 years their age ratio is 14 : 5 (given) So, 4 years after the daughter age will be = 76 × $$\frac{5}{19}$$ = 20 years ∴ Daughter present age = 20 - 4 = 16 years

Let son's age 10 years ago be $$x$$ years Then, father age 10 years ago = 3$$x$$ years Son's age now = ($$x$$ + 10) years, Father age now = (3$$x$$ + 10) years $$\eqalign{ & \left( {3x + 10} \right) + 10 = 2\left[ {\left( {x + 10} \right) + 10} \right] \cr & \Rightarrow 3x + 20 = 2\left( {x + 20} \right) \cr & \Rightarrow 3x + 20 = 2x + 40 \cr & \Rightarrow x = 20 \cr} $$ Ratio of present ages of father and son $$\eqalign{ & {\text{ = }}\frac{{3x + 10}}{{x + 10}} \cr & = \frac{{3 \times 20 + 10}}{{20 + 10}} \cr & = \frac{{70}}{{30}} \cr & = \frac{7}{3} \cr & = 7:3 \cr} $$

Let daughter's age be x years. Suresh's age = 2x years $$\eqalign{ & \therefore \frac{{2x + 6}}{{x + 6}} = \frac{{23}}{{13}} \cr & \Rightarrow 13\left( {2x + 6} \right) = 23\left( {x + 6} \right) \cr & \Rightarrow 26x + 78 = 23x + 138 \cr & \Rightarrow 3x = 60 \cr & \Rightarrow x = 20 \cr} $$ Present age of Suresh = 2x years $$\eqalign{ & {\text{ = }}\left( {2 \times 20} \right){\text{ years }} \cr & {\text{ = 40 years}} \cr} $$

Let son's age 10 years ago be x years. Then, man's age 10 years ago = 7x years Son's present age = (x + 10) years, Man's present age = (7x + 10) years $$\eqalign{ & \therefore {\text{2}}\left[ {\left( {7x + 10} \right) + 2} \right]{\text{ = 5}}\left[ {\left( {x + 10} \right) + 2} \right] \cr & \Rightarrow 2\left( {7x + 12} \right) = 5\left( {x + 12} \right) \cr & \Rightarrow 14x + 24 = 5x + 60 \cr & \Rightarrow 9x = 36 \cr & \Rightarrow x = 4 \cr} $$ Son's present age = (x + 10) years = (4 + 10) years = 14 years

Let Samina's age be 7x years. Then, Suhana's age = 3x years $$\eqalign{ & \therefore \frac{{7x + 6}}{{3x + 6}} = \frac{5}{3} \cr & \Rightarrow 3\left( {7x + 6} \right) = 5\left( {3x + 6} \right) \cr & \Rightarrow 21x + 18 = 15x + 30 \cr & \Rightarrow 6x + 12 \cr & \Rightarrow x = 2 \cr} $$ Difference in their ages = (7x - 3x) years = 4x years = (4 × 2) years = 8 years

Let A's age be 5x years. Then, B's age = 6x years $$\eqalign{ & \therefore \frac{{5x + 6}}{{6x + 6}} = \frac{6}{7} \cr & \Rightarrow 7\left( {5x + 6} \right) = 6\left( {6x + 6} \right) \cr & \Rightarrow 35x + 42 = 36x + 36 \cr & \Rightarrow x = 6 \cr} $$ B's age 5 years ago = (6x - 5) years = (6 × 6 - 5) years = 31 years

Let daughter's age be $$x$$ years. Then, mother's age = (56 $$-\, x\,$$) years $$\eqalign{ & \left( {56 - x} \right)+4 = 3\left( {x + 4} \right) \cr & \Rightarrow 60 - x = 3x + 12 \cr & \Rightarrow 4x = 48 \cr & \Rightarrow x = 12 \cr} $$ ∴ Daughter's age = 12 years And the mother's age = (56 $$-$$ 12) = 44 years

Let mother's age be 2x years, Then, son's age = x years $$\eqalign{ & \therefore \left( {2x - 10} \right) = 3\left( {x - 10} \right) \cr & \Rightarrow 2x - 10 = 3x - 30 \cr & \Rightarrow x = 20 \cr} $$ Son's age = 20 years

$$\eqalign{ & {\text{Ram's father's age}} \cr & {\text{ = 50 years, }} \cr & {\text{Ram's age}} \cr & {\text{ = }}\left( {\frac{3}{5} \times 50} \right){\text{years}} \cr & {\text{ = 30 years}} \cr & {\text{Ram's wife's age}} \cr & {\text{ = }}\left( {\frac{4}{5} \times 30} \right){\text{years}} \cr & {\text{ = 24 years}} \cr & {\text{Ram's son's age}} \cr & {\text{ = }}\left( {\frac{1}{3} \times 24} \right){\text{years}} \cr & {\text{ = 8 years}} \cr} $$

Let Rajan's age 8 years ago be x years, Then, present age = (x + 8) years $$\eqalign{ & \therefore x + 8 = \frac{6}{5}x \cr & \Rightarrow 5x + 40 = 6x \cr & \Rightarrow x = 40 \cr} $$ Rajan's sister's age 8 years ago = (40 - 10) years = 30 years ∴ His sister's age now = (30+8) years = 38 years

M → Mother, F → Father, S → Son and D → Daughter F = 4S, D = $$\frac{1}{3}$$M, M = F - 6 and S = D - 3 $$\eqalign{ & \therefore M = 3D = 3\left( {S + 3} \right) \cr & = 3S + 9 = \frac{3}{4}F + 9 = \frac{3}{4}\left( {M + 6} \right) + 9 \cr & = \frac{3}{4}M + \frac{3}{4} \times 6 + 9 \cr & \Rightarrow \left( {M - \frac{3}{4}M} \right) = \left( {\frac{9}{2} + 9} \right) \cr & \Rightarrow \frac{1}{4}M = \frac{{27}}{2} \cr & \Rightarrow M = \left( {\frac{{27}}{2} \times 4} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 54\,{\text{years}} \cr & \therefore {\text{ The mother is 54 years old}}{\text{.}} \cr} $$

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

Let the present age of the man be x years Then, $$\eqalign{ & \Rightarrow {\text{3}}\left( {x + 3} \right) - 3\left( {x - 3} \right) = x \cr & \Rightarrow \left( {3x + 9} \right) - \left( {3x - 9} \right) = x \cr & \Rightarrow x = 18 \cr} $$ ∴ The present age of the man is 18 years

Let the man's age be 7x years , Then son's age = 3x years $$\eqalign{ & \therefore {\text{7}}x \times 3x = 756 \cr & \Rightarrow 21{x^2} = 756 \cr & \Rightarrow {x^2} = 756 \cr & \Rightarrow {x^2} = 36 \cr & \Rightarrow x = 6 \cr} $$ The ratio of their ages after 6 years $$\eqalign{ & {\text{ = }}\left( {7x + 6} \right):\left( {3x + 6} \right) \cr & = \left( {7 \times 6 + 6} \right):\left( {3 \times 6 + 6} \right) \cr & = 48:24 \cr & = 2:1 \cr} $$

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

Let Neelam's age be 5x years and Shiny's age be 6x years $$\eqalign{ & \left( {\frac{1}{3} \times 5x} \right):\left( {\frac{1}{2} \times 6x} \right) = 5:9 \cr & \Rightarrow \frac{{5x}}{{3 \times 3x}} = \frac{5}{9} \cr} $$ Thus, Shiny's age cannot be determined

Let the son's age 18 years ago be x years, Then man's age 18 years ago = 3x years $$\eqalign{ & \left( {3x + 18} \right) = 2\left( {x + 18} \right) \cr & \Rightarrow 3x + 18 = 2x + 36 \cr & \Rightarrow x = 18 \cr} $$ Sum of their present ages $$\eqalign{ & \Rightarrow \left( {3x + 18 + x + 18} \right){\text{years}} \cr & \Rightarrow \left( {4x + 36} \right){\text{years}} \cr & \Rightarrow \left( {4 \times 18 + 36} \right){\text{years}} \cr & \Rightarrow {\text{ 108 years}} \cr} $$

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 $$\eqalign{ & \left( {3x + 10} \right) + 10 = 2\left( {x + 10 + 10} \right) \cr & \Rightarrow 3x + 20 = 2\left( {x + 20} \right) \cr & \Rightarrow 3x + 20 = 2x + 40 \cr & \Rightarrow x = 20 \cr} $$ Ratio of present ages of man and the son $$\eqalign{ & {\text{ = }}\frac{{3x + 10}}{{x + 10}} \cr & = \frac{{3 \times 20 + 10}}{{20 + 10}} \cr & = \frac{{70}}{{30}} \cr & = 7:3 \cr} $$

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