Question

The sum of the two numbers is given to be $27$ and their product is $180$. What will be the two numbers?

• A. $9,18$
• B. $12,15$
• C. $13,14$
• D. $11,16$

Answer 1

The correct option is B $12,15$

Let the two numbers be $x$ and $y$.

Given: $x+y=27$ and $xy=180$

The equation $x+y=27$ can be written as $x=27-y$.

Substitute this value of $x$ in $xy=180$, we get

$\begin{array}{cc}(27-y)y& =180\\ 27y-y^2& =180\\ y^2-27y+180=0& \end{array}$

Write $27y$ as the sum of $12y$ and $15y$, we get

$\begin{array}{cc}{y}^2-27y+180& =0\\ y^2-12y-15y+180& =0\\ y(y-12)-15(y-12)& =0\\ (y-12)(y-15)& =0\\ y& =12\text{or}15\end{array}$

Thus, if $y=12$, then $x=27-12=15$

and if $y=15$, then $x=27-15=12$.

Thus, the two numbers are $12,15$.