Question

The product of two whole numbers is 560 and their sum is 51 . What are the two numbers ?

Answer 1

35 and 16.

Step-by-step explanation:

Let the two whole numbers be x and y respectively.

• The product of the two whole numbers is 560: xy=560

,

• Their sum is 51: x+y=51

`x+y=51\implies x=51-y`

Substitute x=51-y into xy=560.

`\begin{gathered} xy=560 \\ y(51-y)=560 \\ 51y-y^2=560 \\ \implies y^2-51y+560=0 \end{gathered}`

Next, solve the quadratic equation for y:

`\begin{gathered} y^2-16y-35y+560=0 \\ y(y-16)-35(y-16)=0 \\ (y-35)(y-16)=0 \\ \implies y-35=0\text{ or }y-16=0 \\ y=35\text{ or }y=16 \end{gathered}`

Finally, solve for x:

`\begin{gathered} \text{When }y=35,x=51-35=16 \\ \text{When }y=16,x=51-16=35 \end{gathered}`

Thus, the two numbers are 35 and 16.