Question

A girl is twice as old as her sister. Four years hence, the product of their ages (in years) will be 160. Find their present ages.

Answer 1

$\huge\star{\green{\underline{\mathfrak{Answer :-}}}}$

Let the age of the girl be x.

According to the question,

$\mapsto\tt{\pink{Sister's\:age=2x}}$

After 4 years,

The age of the girl = x + 4.

The age of her sister = 2x + 4.

According to the question,

$\mapsto\tt{\pink{(x+4)(2x+4)=160}}$

Solving the equation,

$\mapsto\tt{\pink{2{x}^{2}+4x+8x+16=160}}$

$\mapsto\tt{\pink{2{x}^{2}+12x+16=160}}$

$\mapsto\tt{\pink{2{x}^{2}+12x=160-16}}$

$\mapsto\tt{\pink{2{x}^{2}+12x=144}}$

$\mapsto\tt{\pink{2{x}^{2}+12x-144=0}}$

$\mapsto\tt{\pink{2({x}^{2}+6x-72=0)}}$

$\mapsto\tt{\pink{{x}^{2}+6x-72=0}}$

By splitting the middle term,

$\mapsto\tt{\pink{{x}^{2}+-6x+12x-72=0}}$

$\mapsto\tt{\pink{x(x-6)+12(x-6)=0}}$

$\mapsto\tt{\pink{(x-6)(x+12)=0}}$

So, age of the girl can be 6 years old or - 12 years old.

But we know that age can never be negative.

So, The age of the girl is 6 years old and her sister is 2x = 2 * 6 = 12 years old.

$\rule{200}4$

Answer 2

$\huge\orange{\frak{\underline{\underline{\:QUESTION}}}}$

A girl is twice as old as her sister. Four years hence, the product of their ages (in years) will be 160. Find their present ages .

$\huge\orange{\frak{\underline{\underline{\:ANSWER}}}}$

Let,

• Age of Girl = x years .
• Age of her sister = 2x years .

$\:\:\:\:\:\pink{\sf{\:(According\:to\:question)}}$

• After Four years, Age of girl = (x+4)
• After Four Years, Age of her sister = (2x+4)

$\pink{\sf{\:(Four\:years\:hence\:the\:product\:of\:their\:ages\:(in\:years)\:will\:be\:160)}}$

$\mapsto\sf{\:(x+4)(2x+4)\:=\:160}$

$\mapsto\sf{\:(2x^2+4x+8x+16)\:=\:160}$

$\mapsto\sf{\:(2x^2+12x+144)\:=\:0}$

$\:\:\:\:\:\pink{\sf{\:(Taking\:2\:Common)}}$

$\mapsto\sf{\:2(x^2+6x-72)\:=\:0}$

$\mapsto\sf{\:(x^2+6x-72)\:=\:0}$

$\mapsto\sf{\:(x^2+12x-6x-72)\:=\:0}$

$\mapsto\sf{\:x(x+12)-6(x+12)\:=\:0}$

$\mapsto\sf{\:(x-6)(x+12)\:=\:0}$

$\mapsto\sf{\:(x-6)\:=\:0\:\:Or\:\:\:(x+12)\:=\:0}$

$\mapsto\sf{\:(x)\:=\:6\:\:Or\:\:\:(x)\:=\:-12}$

But, Age is always in positive,

Then, X = -12 is negligible

So, We take X = 6

$\pink{\sf{\underbrace{AnsWeR}}}$

$\mapsto\green{\sf{\:Age\:Of\:Girl\:=\:6\:years}}$

$\mapsto\green{\sf{\:Age\:of\:her\:Sister\:=\:2\times12\:=\:24\:years}}$

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