Question

The difference of squares, of two numbers is 88. If the larger number is 5 less than twice the smaller number, then the product of two numbers is ___.

Answer 1

Hint: In order to solve this question, we need to assume the larger number as x and smaller as y and then form the linear equations using given information. Then we can find out both the numbers by solving linear equations.

Complete step-by-step answer:
The given numbers are x and y where x > y. Then as we are given that the difference of squares of the numbers is 88.
$ \Rightarrow {x^2} - {y^2} = 88$ …………….. (1)
Now, we are also given at the larger number is 5 less than twice the smaller number
$ \Rightarrow x + 5 = 2y$
$ \Rightarrow x = 2y - 5$ ……………… (2)
Using value of x from Equation (2) in Equation (1), we get,
${\left( {2y - 5} \right)^2} - {y^2} = 88$
$ \Rightarrow 3{y^2} + 25 - 20y = 88$
$ \Rightarrow 3{y^2} - 20y - 63 = 0$
Factorising the Equation,
$ \Rightarrow 3{y^2} - 27y + 7y - 63 = 0$
$ \Rightarrow 3y\left( {y - 9} \right) + 7\left( {y - 9} \right) = 0$
$ \Rightarrow \left( {y - 9} \right)\left( {3y + 7} \right) = 0$
$ \Rightarrow y - 9 = 0{\text{ and }}3y + 7 = 0$
$ \Rightarrow y = 9{\text{ and y}} = - \dfrac{7}{3}$
Using these values in Equation (2), we get,
$x = 13{\text{ and x}} = - \dfrac{{29}}{3}$
But as x > y
$\therefore x = 13,y = 9$
Therefore, the product of x and y is $ = 13 \times 9 = 117$.

Note: Whenever we face such types of problems the key point to remember is that we need to have a good grasp over quadratic equations and linear equations in two variables. In these types of questions, we should always find one variable in terms of the other and then use it for simplification. This will help us to find out the answer.