Question

The ages of Kavya and Karthik are 11 years and 14 years. In how many years will the product of their ages become 304?

Answer 1

Hint: We can take a variable x equal to the number of years after the product of their ages becomes 304. And then we can form a quadratic equation by multiplying their ages after x years and equating that with 304.

Complete step-by-step answer:
Let after x years the product of the ages of Kavya and Karthik become equal to 304.
So, after x years the age of Kavya will be (11 + x) years.
And after x years the age of Karthik will be (14 + x) years.
So, according to the question
\[\left( {11 + x} \right)\left( {14 + x} \right) = 304\]
Now we had to solve the above equation to find the value of x.
So, expanding the LHS of the above equation to find the value of x. We get,
\[
  154 + 11x + 14x + {x^2} = 304 \\
  {x^2} + 25x + 154 - 304 = 0 \\
  {x^2} + 25x - 150 = 0 \\
 \]
Now solve the above quadratic equation by making factors to find the value of x.
Splitting the middle term of the above equation and then taking common terms we get,
\[
  {x^2} + 30x - 5x - 150 = 0 \\
  x\left( {x + 30} \right) - 5\left( {x + 30} \right) = 0 \\
\]
Taking (x + 30) common from the above equation. We get,
(x + 30)(x – 5) = 0
So, x = – 30 years or x = 5 years
Now as we know that years cannot be negative. So, the value of x must be equal to 5 years.
Hence, after 5 years the product of the ages of Kavya and Karthik must be equal to 304.


Note:- Whenever we come up with this type of problem first, we have to make a quadratic equation using given conditions. Now after that there is also an alternate method to solve the quadratic equation and that is if \[a{x^2} + bx + c = 0\] is any quadratic equation then the value of x must be equal to \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] where a, b and c are the coefficient of \[{x^2}\], x and the constant term.