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.

Answer 1

$\underline{Let \: seven \: years \: ago , }$

Age of Swati = x years

Age of Varun = 5x² years [ that is 5 times the square of swati's age]





$\underline{After \: 3 \: years, }$

Age of Swati = age of 7 years ago + 7 years + 3 years = x years + 7 years + 3 years = ( x + 10 ) years

Age of Varun = age of 7 years ago + 7 years + 3 years = 5x² years + 7 years + 3 years = ( 5x² + 10 ) years




$\underline{According \: to \: the \: question: }$


= > Age of Swati after 3 years = two - fifth of age of Varun


= > ( x + 10 ) = 2 / 5 of ( 5x² + 10 )


$= > x + 10 = \dfrac{2}{5} (5 {x}^{2} + 10)$


$= > 5(x + 10 ) = 2( 5{x}^{2} + 10)$


= > 5x + 50 = 10x² + 20


= > 0 = 10x² - 5x + 20 - 50


= > 0 = 10x² - 5x - 30


= > 0 = 2x² - x - 6


= > 0 = 2x² - ( 4 - 3 ) x - 6


= > 0 = 2x² - 4x + 3x - 6


= > 0 = 2x( x - 2 ) + 3( x - 2 )


= > 0 = ( x - 2 ) ( 2x + 3 )




$\: \: \: \: \: \: \: \: \: \: \: \: \: \underline{By \: Zero \: Product \: Rule , }$



= > x = 2 or x = - 3 / 2




As we x assumed x is the age, x cant be a -ve value. Taking the +ve value that is 2.






So,
Present age of Swati = Age of 7 years ago + 7 years = ( x + 7 ) years = ( 2 + 7 ) years = 9 years


Present age of Varun = age of 7 years ago + 7 years = ( 5x² + 7 ) = { 5( 2 )² + 7 } years = { 20 +7 } years = 27 years






Therefore,
Present age of Swati = 9 years
Present age of Varun = 27 years

Answer 2

Let seven years, age of Swati was x years and age of Varun was 5x² years

Present age of Swati = ( x + 7 ) years

Present age of Varun = ( 5x² + 7 ) years



Given, after 3 years swati's age will be 2 / 5 of Varun's age.


Age of Swati after 3 years = ( x + 7 + 3 ) years = ( x + 10 ) years

Age of Varun after 3 years = ( 5x² + 7 + 3 ) years = ( 5x² + 10 ) years





Given, age of Swati after 3 years = age of varun after 3 years

( x + 10 ) = 2 / 5 × ( 5x² + 10 )

( x + 10 ) = 2( x² + 2 )

x + 10 = 2x² + 4

2x² - x + 4 - 10 = 0

2x² - x - 6 = 0

2x² - 4x + 3x - 6 = 0

2x( x - 2 ) + 3( x - 2 ) = 0


x = 2 [ Age can't be negative ]





Therefore,

present age of Swati = ( 2 + 7 ) = 9 years

present age of Varun = 5(2)²+7=27 years


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