Question

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

Answer 1

Hint: According to the question given in the question we have to find the present ages of the girl and her sister. When a girl is twice as old as her sister and four year hence, the product of their ages (in ages) will be 160. So, first of all we have to let the age of sister = $x$ years and as given that girl is twice as old as her sister so, we can obtain the age of the girl.
Now, as mentioned in the question, the product of their ages (in years) will be 160 so after substituting the values we will obtain a quadratic equation and by solving the obtained quadratic equation we can obtain present ages of both of them.

Complete step-by-step answer:
Step 1: First of all we have to assume the ages of both of them, so the age of the sister is $ = x$years.
Now, as given in the question, the girl is twice as old as her sister hence the age of the girl $ = 2x$years.
Step 2: Now, as we know that after four years the product of a girl and her sister ages (in years) will be 160. Hence,
Age of sister after four years $ = (x + 4)$ years and,
Age of girl after four years$ = (2x + 4)$ years
Step 3: Now, on multiplying the obtained ages of girl and her sister as obtained in step 2 and as we know the product of girl and her sister ages (in years) will be 160. Hence,
$ \Rightarrow (2x + 4)(x + 4) = 160$
On, solving the expression obtained just above by multiplying each term of the expression with each other.
$
   \Rightarrow 2x(x + 4) + 4(x + 4) = 160 \\
   \Rightarrow 2{x^2} + 8x + 4x + 16 = 160 \\
   \Rightarrow 2{x^2} + 12x + 16 - 160 = 0............(1)
 $
Step 4: On, dividing by 2 in the both sides of the expression as obtained in the step 3.
$
   \Rightarrow \dfrac{{2{x^2} + 12x + 16 - 160}}{2} = \dfrac{0}{2} \\
   \Rightarrow {x^2} + 6x + 8 - 80 = 0 \\
   \Rightarrow {x^2} + 6x - 72 = 0.............(2)
 $
Step 5: Now, to solve the quadratic expression (2) as obtained in step 4 we have to find the factors of 72.
$
   \Rightarrow {x^2} + (12 - 6)x - 72 = 0 \\
   \Rightarrow {x^2} + 12x - 6x - 72 = 0 \\
   \Rightarrow x(x + 12) - 6(x + 12) = 0 \\
   \Rightarrow (x + 12)(x - 6) = 0
 $
Step 6: Now, to obtain the value of x we have to solve both of the roots obtained as in the step 5.
$
   \Rightarrow (x + 12) = 0 \\
   \Rightarrow x = - 12
 $
As we can see that the value of x is a negative number but the age of any person can’t be negative.
$
   \Rightarrow (x - 6) = 0 \\
   \Rightarrow x = 6 {\text{years}}
 $
So, the age of the sister is 6 years.
Step 7: Now, to find the age of the girl we have to substitute the value of x as obtained in the step 6 into 2x hence,
$ \Rightarrow 2 \times 6 = 12$ $\text years$.

Hence, when a girl is twice as old as her sister and Four year hence, the product of their ages (in years) will be 160. Find the present ages are 6 years and 12 years.

Note: If it is given in the question that the age of a person is four year hence then we have to add the 4 years to the given age of the person and if it is given that the age of a person was four year ago then we have to subtract the 4 years to the given age of the person.