\displaystyle \textbf{Question 1: }\text{Ten years later, }A\text{ will be twice as old as }B\text{ and}
\displaystyle \text{five years ago, }A\text{ was three times as old as }B.\text{ What are the present}
\displaystyle \text{ages of }A\text{ and }B\text{?}
\displaystyle \text{Answer:}
\displaystyle \text{Let the present ages of }A\text{ and }B\text{ be }x\text{ and }y\text{ years, respectively.}
\displaystyle \text{Ten years later, their ages will be }(x+10)\text{ and }(y+10)\text{ years.}
\displaystyle \therefore x+10=2(y+10)
\displaystyle \Rightarrow x-2y=10\qquad\ldots\text{(i)}
\displaystyle \text{Five years ago, their ages were }(x-5)\text{ and }(y-5)\text{ years.}
\displaystyle \therefore x-5=3(y-5)
\displaystyle \Rightarrow x-3y=-10\qquad\ldots\text{(ii)}
\displaystyle \text{Subtracting equation (ii) from equation (i),}
\displaystyle y=20
\displaystyle \text{Substituting }y=20\text{ in equation (i),}
\displaystyle x-2(20)=10
\displaystyle \Rightarrow x=50
\displaystyle \therefore \text{The present ages of }A\text{ and }B\text{ are }50\text{ years and }20\text{ years.}
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Five years ago, Nuri was thrice as old as Sonu.}
\displaystyle \text{Ten years later, Nuri will be twice as old as Sonu. How old are Nuri}
\displaystyle \text{and Sonu?} 
\displaystyle \text{Answer:}
\displaystyle \text{Let the present ages of Nuri and Sonu be }x\text{ and }y\text{ years, respectively.}
\displaystyle \text{Five years ago, their ages were }(x-5)\text{ and }(y-5)\text{ years.}
\displaystyle \therefore x-5=3(y-5)
\displaystyle \Rightarrow x-3y=-10\qquad\ldots\text{(i)}
\displaystyle \text{Ten years later, their ages will be }(x+10)\text{ and }(y+10)\text{ years.}
\displaystyle \therefore x+10=2(y+10)
\displaystyle \Rightarrow x-2y=10\qquad\ldots\text{(ii)}
\displaystyle \text{Subtracting equation (i) from equation (ii),}
\displaystyle y=20
\displaystyle \text{Substituting }y=20\text{ in equation (ii),}
\displaystyle x-2(20)=10
\displaystyle \Rightarrow x=50
\displaystyle \therefore \text{Nuri is }50\text{ years old and Sonu is }20\text{ years old.}
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Six years hence a man's age will be three times}
\displaystyle \text{the age of his son and three years ago he was nine times as old as his son.}
\displaystyle \text{Find their present ages.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the present ages of the man and his son be }x\text{ and }y\text{ years, respectively.}
\displaystyle \text{Six years hence, their ages will be }(x+6)\text{ and }(y+6)\text{ years.}
\displaystyle \therefore x+6=3(y+6)
\displaystyle \Rightarrow x-3y=12\qquad\ldots\text{(i)}
\displaystyle \text{Three years ago, their ages were }(x-3)\text{ and }(y-3)\text{ years.}
\displaystyle \therefore x-3=9(y-3)
\displaystyle \Rightarrow x-9y=-24\qquad\ldots\text{(ii)}
\displaystyle \text{Subtracting equation (ii) from equation (i),}
\displaystyle 6y=36
\displaystyle \Rightarrow y=6
\displaystyle \text{Substituting }y=6\text{ in equation (i),}
\displaystyle x-3(6)=12
\displaystyle \Rightarrow x=30
\displaystyle \therefore \text{The present ages of the man and his son are }30\text{ years and }6\text{ years.}
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Ten years ago, a father was twelve times as old as}
\displaystyle \text{his son and ten years hence, he will be twice as old as his son will be then.}
\displaystyle \text{Find their present ages.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the present ages of the father and son be }x\text{ and }y\text{ years, respectively.}
\displaystyle \text{Ten years ago, their ages were }(x-10)\text{ and }(y-10)\text{ years.}
\displaystyle \therefore x-10=12(y-10)
\displaystyle \Rightarrow x-12y=-110\qquad\ldots\text{(i)}
\displaystyle \text{Ten years hence, their ages will be }(x+10)\text{ and }(y+10)\text{ years.}
\displaystyle \therefore x+10=2(y+10)
\displaystyle \Rightarrow x-2y=10\qquad\ldots\text{(ii)}
\displaystyle \text{Subtracting equation (i) from equation (ii),}
\displaystyle 10y=120
\displaystyle \Rightarrow y=12
\displaystyle \text{Substituting }y=12\text{ in equation (ii),}
\displaystyle x-2(12)=10
\displaystyle \Rightarrow x=34
\displaystyle \therefore \text{The present ages of the father and son are }34\text{ years and }12\text{ years.}
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Father's age is three times the sum of ages of his}
\displaystyle \text{two children. After 5 years his age will be twice the sum of ages of the}
\displaystyle \text{two children. Find the age of father.}\hfill\text{[CBSE 2003, 2019]}
\displaystyle \text{Answer:}
\displaystyle \text{Let the father's present age be }x\text{ years.}
\displaystyle \text{Let the sum of the present ages of his two children be }y\text{ years.}
\displaystyle \text{Father's age is three times the sum of ages of his two children.}
\displaystyle \therefore x=3y\qquad\ldots\text{(i)}
\displaystyle \text{After 5 years, the father's age will be }(x+5)\text{ years.}
\displaystyle \text{The sum of ages of the two children after 5 years will be }(y+10)\text{ years.}
\displaystyle \therefore x+5=2(y+10)
\displaystyle \Rightarrow x-2y=15\qquad\ldots\text{(ii)}
\displaystyle \text{Substituting }x=3y\text{ in equation (ii),}
\displaystyle 3y-2y=15
\displaystyle \Rightarrow y=15
\displaystyle \text{From equation (i),}
\displaystyle x=3(15)=45
\displaystyle \therefore \text{The present age of the father is }45\text{ years.}
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Two years ago, a father was five times as old as}
\displaystyle \text{his son. Two years later, his age will be 8 more than three times the age}
\displaystyle \text{of the son. Find the present ages of father and son.}\hfill\text{[CBSE 2004]}
\displaystyle \text{Answer:}
\displaystyle \text{Let the present ages of the father and son be }x\text{ and }y\text{ years, respectively.}
\displaystyle \text{Two years ago, their ages were }(x-2)\text{ and }(y-2)\text{ years.}
\displaystyle \therefore x-2=5(y-2)
\displaystyle \Rightarrow x-5y=-8\qquad\ldots\text{(i)}
\displaystyle \text{Two years later, their ages will be }(x+2)\text{ and }(y+2)\text{ years.}
\displaystyle \therefore x+2=3(y+2)+8
\displaystyle \Rightarrow x-3y=12\qquad\ldots\text{(ii)}
\displaystyle \text{Subtracting equation (i) from equation (ii),}
\displaystyle 2y=20
\displaystyle \Rightarrow y=10
\displaystyle \text{Substituting }y=10\text{ in equation (ii),}
\displaystyle x-3(10)=12
\displaystyle \Rightarrow x=42
\displaystyle \therefore \text{The present ages of the father and son are }42\text{ years and }10\text{ years.}
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{The ages of two friends Ani and Biju differ by 3 years.}
\displaystyle \text{Ani's father Dharam is twice as old as Ani and Biju is twice as old as his}
\displaystyle \text{sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the}
\displaystyle \text{ages of Ani and Biju.} 
\displaystyle \text{Answer:}
\displaystyle \text{Let the ages of Ani and Biju be }x\text{ and }y\text{ years, respectively.}
\displaystyle \text{Since their ages differ by }3\text{ years, }x-y=\pm3.
\displaystyle \text{Dharam's age}=2x\text{ years.}
\displaystyle \text{Since Biju is twice as old as Cathy, Cathy's age}=\frac{y}{2}\text{ years.}
\displaystyle \text{Dharam is older than Cathy by }30\text{ years.}
\displaystyle \therefore 2x-\frac{y}{2}=30
\displaystyle \Rightarrow 4x-y=60\qquad\ldots\text{(i)}

\displaystyle \text{Case I: }x-y=3\qquad\ldots\text{(ii)}
\displaystyle \text{Subtracting equation (ii) from equation (i),}
\displaystyle 3x=57
\displaystyle \Rightarrow x=19
\displaystyle \text{Substituting }x=19\text{ in equation (ii),}
\displaystyle 19-y=3
\displaystyle \Rightarrow y=16
\displaystyle \therefore \text{Ani is }19\text{ years old and Biju is }16\text{ years old.}

\displaystyle \text{Case II: }x-y=-3\qquad\ldots\text{(iii)}
\displaystyle \text{Subtracting equation (iii) from equation (i),}
\displaystyle 3x=63
\displaystyle \Rightarrow x=21
\displaystyle \text{Substituting }x=21\text{ in equation (iii),}
\displaystyle 21-y=-3
\displaystyle \Rightarrow y=24
\displaystyle \therefore \text{Ani is }21\text{ years old and Biju is }24\text{ years old.}
\displaystyle \therefore \text{The possible ages are }(19,16)\text{ years or }(21,24)\text{ years.}
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Two years ago, Salim was thrice as old as his}
\displaystyle \text{daughter and six years later, he will be four years older than twice her}
\displaystyle \text{age. How old are they now?}
\displaystyle \text{Answer:}
\displaystyle \text{Let the present ages of Salim and his daughter be }x\text{ and }y\text{ years.}
\displaystyle \text{Two years ago, their ages were }(x-2)\text{ and }(y-2)\text{ years.}
\displaystyle \therefore x-2=3(y-2)
\displaystyle \Rightarrow x-3y=-4\qquad\ldots\text{(i)}
\displaystyle \text{Six years later, their ages will be }(x+6)\text{ and }(y+6)\text{ years.}
\displaystyle \therefore x+6=2(y+6)+4
\displaystyle \Rightarrow x-2y=10\qquad\ldots\text{(ii)}
\displaystyle \text{Subtracting equation (i) from equation (ii),}
\displaystyle y=14
\displaystyle \text{Substituting }y=14\text{ in equation (ii),}
\displaystyle x-2(14)=10
\displaystyle \Rightarrow x=38
\displaystyle \therefore \text{Salim is }38\text{ years old and his daughter is }14\text{ years old.}
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{The age of the father is twice the sum of the ages}
\displaystyle \text{of his two children. After 20 years, his age will be equal to the sum of the}
\displaystyle \text{ages of his children. Find the age of the father.} 
\displaystyle \text{Answer:}
\displaystyle \text{Let the present age of the father be }x\text{ years.}
\displaystyle \text{Let the sum of the present ages of his two children be }y\text{ years.}
\displaystyle \text{The father's age is twice the sum of the ages of his two children.}
\displaystyle \therefore x=2y\qquad\ldots\text{(i)}
\displaystyle \text{After 20 years, the father's age will be }(x+20)\text{ years.}
\displaystyle \text{The sum of the children's ages will then be }(y+40)\text{ years.}
\displaystyle \therefore x+20=y+40
\displaystyle \Rightarrow x-y=20\qquad\ldots\text{(ii)}
\displaystyle \text{Substituting }x=2y\text{ in equation (ii),}
\displaystyle 2y-y=20
\displaystyle \Rightarrow y=20
\displaystyle \text{From equation (i),}
\displaystyle x=2(20)=40
\displaystyle \therefore \text{The present age of the father is }40\text{ years.}
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Three years ago, Rashmi was thrice as old}
\displaystyle \text{as Nazma. Ten years later, Rashmi will be twice as old as Nazma. How old}
\displaystyle \text{are Rashmi and Nazma now?}\hfill\text{[CBSE 2024]}
\displaystyle \text{Answer:}
\displaystyle \text{Let the present ages of Rashmi and Nazma be }x\text{ and }y\text{ years.}
\displaystyle \text{Three years ago, their ages were }(x-3)\text{ and }(y-3)\text{ years.}
\displaystyle \therefore x-3=3(y-3)
\displaystyle \Rightarrow x-3y=-6\qquad\ldots\text{(i)}
\displaystyle \text{Ten years later, their ages will be }(x+10)\text{ and }(y+10)\text{ years.}
\displaystyle \therefore x+10=2(y+10)
\displaystyle \Rightarrow x-2y=10\qquad\ldots\text{(ii)}
\displaystyle \text{Subtracting equation (i) from equation (ii),}
\displaystyle y=16
\displaystyle \text{Substituting }y=16\text{ in equation (ii),}
\displaystyle x-2(16)=10
\displaystyle \Rightarrow x=42
\displaystyle \therefore \text{Rashmi is }42\text{ years old and Nazma is }16\text{ years old.}
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{16 years ago, at the time of marriage, Ajay was}
\displaystyle \text{5 years elder to his wife. The present ages of the wife and Ajay are in the}
\displaystyle \text{ratio }8:9.\text{ Find their ages at the time of their marriage.}\hfill\text{[CBSE 2024]}
\displaystyle \text{Answer:}
\displaystyle \text{Let the present ages of Ajay's wife and Ajay be }x\text{ and }y\text{ years.}
\displaystyle \text{Their present ages are in the ratio }8:9.
\displaystyle \therefore \frac{x}{y}=\frac{8}{9}
\displaystyle \Rightarrow 9x-8y=0\qquad\ldots\text{(i)}
\displaystyle \text{At the time of marriage, their ages were }(x-16)\text{ and }(y-16)\text{ years.}
\displaystyle \text{Ajay was }5\text{ years older than his wife.}
\displaystyle \therefore (y-16)-(x-16)=5
\displaystyle \Rightarrow y-x=5\qquad\ldots\text{(ii)}
\displaystyle \Rightarrow y=x+5
\displaystyle \text{Substituting }y=x+5\text{ in equation (i),}
\displaystyle 9x-8(x+5)=0
\displaystyle \Rightarrow x=40
\displaystyle \therefore y=40+5=45
\displaystyle \text{Age of Ajay's wife at the time of marriage}=40-16=24\text{ years.}
\displaystyle \text{Ajay's age at the time of marriage}=45-16=29\text{ years.}
\displaystyle \therefore \text{At marriage, Ajay's wife was }24\text{ years and Ajay was }29\text{ years old.}
\displaystyle \\

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