Question

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.
True
False

Answer 1

Hint: Equation problems of age are part of the quantitative aptitude section. Equations are a convenient way to represent conditions or relations between two or more quantities. An equation could have one, two or more unknowns.
If the age of a person is ‘x’, then ‘n’ years after today, the age = x+n. Similarly, n years in the past, the age of this would have been x-n years.

Complete step-by-step answer:
It is given that girl is twice as old as her sister.
Let the present age of girl be 2x
Then, the age of her sister be x
Four years hence, the age of girl will be (2x+4) and (x+4)
According to the question,
$\begin{gathered}
  \left( {2x + 4} \right)\left( {x + 4} \right) = 160 \\
  2{x^2} + 8x + 4x + 16 = 160 \\
  2{x^2} + 12x + 16 - 160 = 0 \\
  2{x^2} + 12x - 144 = 0 \\
  2\left( {{x^2} + 6x - 72} \right) = 0 \\
  {x^2} + 6x - 72 = 0 \\
  {x^2} + 12x - 6x - 72 = 0 \\
  x\left( {x + 12} \right) - 6\left( {x + 12} \right) = 0 \\
  \left( {x + 12} \right)\left( {x - 6} \right) = 0 \\
  \therefore x = - 12{\text{ and }}x = 6 \\
\end{gathered} $
As age cannot be negative
∴The age of her sister is 6yrs
And the age of girl = $2x = 2 \times 6 = 12yrs$
Hence, it is true that the present ages of girls are 6 years and 12 years.

Note: The basic rule is that if the number of unknowns is equal to the number of conditions, then these equations are solvable, otherwise not. The most important step is to recheck the answer obtained by placing it in the equation formed to ensure that no error has been made while calculating the value.
If the age is given in the form of a ratio, p:q, then the age shall be considered as qx and px.
If the current age is x, then 1/n of the age is x/n.
If the current age is x, then n times the age is nx.