Question

State true or false:
A girl is twice as old as her sister. Four years hence, the product of their ages (in years) will be 160. Their present ages are 6 years and 12 years.
(A) True
(B) False

Answer 1

Hint: Assume the ages of the girl and her sister to be some variable as per the condition given in the question. Use the second given condition to establish a quadratic equation in the assumed variable. Solve that equation to verify the result.

Complete step-by-step solution:
According to the question, a girl is twice as old as her sister.
Let the age of the girl is $2x$ years. Then the age of her sister will be $x$ years.
Further it is given that four year hence, the product of their ages will be 160.
After four years, the age of the girl will be $2x + 4$ and the age of her sister will be $x + 4$. Thus from the given condition, we’ll get:
$ \Rightarrow \left( {2x + 4} \right)\left( {x + 4} \right) = 160$
Now we have to solve this quadratic equation. We will solve it by expanding first and then factoring.
So on expanding it, we will get:
$
   \Rightarrow 2{x^2} + 8x + 4x + 16 = 160 \\
   \Rightarrow 2{x^2} + 12x + 16 = 160 \\
 $
We can cancel out the common factor 2 from it. We have:
$
   \Rightarrow {x^2} + 6x + 8 = 80 \\
   \Rightarrow {x^2} + 6x - 72 = 0 \\
 $
Now we will factorize the quadratic equation by splitting the middle term into two terms. This is shown below:
$
   \Rightarrow {x^2} + 12x - 6x - 72 = 0 \\
   \Rightarrow x\left( {x + 12} \right) - 6\left( {x + 12} \right) = 0 \\
   \Rightarrow \left( {x - 6} \right)\left( {x + 12} \right) = 0 \\
 $
Putting both the factors to zero, we’ll get:
$
   \Rightarrow x - 6 = 0{\text{ and }}x + 12 = 0 \\
   \Rightarrow x = 6{\text{ and }}x = - 12 \\
 $
We will ignore the negative value of $x$ because age can never be negative. Thus $x = 6$ is the only valid solution.
Hence the age of the sister is $x$ i.e. 6 years old and the age of the girl is $2x$ i.e. 12 years old.

Therefore the given statement is true. (A) is the correct option.

Note: If we are facing any difficulty solving a quadratic equation using factorization method, we can also use a direct formula to find its roots. Let the quadratic equation be:
$ \Rightarrow y = a{x^2} + bx + c$
The formula to determine its roots is:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Using this formula we can also find the roots if they are imaginary, complex or irrational numbers.