Question

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

Answer 1

The two numbers are `38.25` and `44`

Explanation:

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

As their average is `(a+b)/2`, we have `(a+b)/2=41.125`

or `a+b=41.125xx2=82.25`

or `a=82.25-b` i.e. the numbers are `(82.25-b)` and `b`

As the product of numbers is `1683`, therefore

`b(82.25-b)=1683`

or `82.25b-b^2=1683`

or `329b-4b^2=6732` - multiplying each term by `4`

i.e. `4b^2-329b+6732=0`

and using quadratic formula `b=(329+-sqrt(329^2-4xx4xx6732))/8`

= `(329+-sqrt(108241-107712))/8=(329+-sqrt529)/8`

= `(329+-23)/8`

i.e. `b=352/8=44` or `b=306/8=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`