# The average of two numbers is 41.125, and their product is 1683. What are the numbers?

Feb 23, 2017

The two numbers are $38.25$ and $44$

#### Explanation:

Let the numbers be $a$ and $b$.

As their average is $\frac{a + b}{2}$, we have $\frac{a + b}{2} = 41.125$

or $a + b = 41.125 \times 2 = 82.25$

or $a = 82.25 - b$ i.e. the numbers are $\left(82.25 - b\right)$ and $b$

As the product of numbers is $1683$, therefore

$b \left(82.25 - b\right) = 1683$

or $82.25 b - {b}^{2} = 1683$

or $329 b - 4 {b}^{2} = 6732$ - multiplying each term by $4$

i.e. $4 {b}^{2} - 329 b + 6732 = 0$

and using quadratic formula $b = \frac{329 \pm \sqrt{{329}^{2} - 4 \times 4 \times 6732}}{8}$

= $\frac{329 \pm \sqrt{108241 - 107712}}{8} = \frac{329 \pm \sqrt{529}}{8}$

= $\frac{329 \pm 23}{8}$

i.e. $b = \frac{352}{8} = 44$ or $b = \frac{306}{8} = \frac{153}{4} = 38.25$

ans $a = 82.25 - 44 = 38.25$ or $a = 82.25 - 38.25 = 44$

Hence the two numbers are $38.25$ and $44$