  Question

Seven years ago Varun's age was five times the square of Swati's age. Three years hence Swati's age will be two fifth of Varun's age. Find their present ages.

Solution

Seven years ago. let Swati's age be $$x$$ years. Then, seven years ago Varun's age was $$5{x}^{2}$$ years.Swati's present age $$=(x+7)$$ years, Varun's present age$$=\left( 5{ x }^{ 2 }+7 \right)$$ yearsThree years hence, we haveSwati's age $$(x+7+3)$$ years $$=(x+10)$$ yearsVarun's age $$=\left( 5{ x }^{ 2 }+7+3 \right) =\left( 5{ x }^{ 2 }+10 \right)$$It is given that three years hence Swati's age wil be $$\cfrac{2}{5}$$ of Varun's age$$\quad \therefore x+10=\cfrac { 2 }{ 5 } \left( 5{ x }^{ 2 }+10 \right)$$$$\Rightarrow x+10=2{ x }^{ 2 }+4$$$$\Rightarrow 2{ x }^{ 2 }-x-6=0$$$$\Rightarrow 2{ x }^{ 2 }-4x+3x-6=0$$$$\Rightarrow (2x+3)(x-2)=0$$$$\Rightarrow x-2=0\Rightarrow x=2$$Hence Swati's present age $$=(2+7)$$ years $$=9$$ yearsVarun's present age $$=\left( 5\times { x }^{ 2 }+7 \right)$$ years $$=27$$ yearsMaths

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