## Forty-Five

The sum of two numbers is 45. The sum of their quotient and the quotient’s reciprocal is 2.05. What is the product of the two numbers?
Source: mathcontest.olemiss.edu 9/19/2011

SOLUTION
Let $x$ and $y$ be the two numbers.
$x+y=45$        (1)
$\frac{x}{y}+\frac{y}{x}=2.05$
(2)

Simplify Eq. (2)
$\frac{x^2+y^2}{xy}=2.05$
$x^2+y^2=2.05xy$        (3)
$\left (x+y\right )^2=x^2+2xy+y^2$        (4)

Substitute the value of $x+y$ from Eq. (1) and the value of $x^2+y^2$ from Eq. (3) into Eq. (4)
$45^2=2xy+2.05xy$
$45^2=xy\left (2+2.05\right )$
$45^2=4.05xy$
$xy=\frac{45^2}{4.05}$
$xy=500$

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , . Bookmark the permalink.

### 2 Responses to Forty-Five

1. Mariano,Jimel Ryan D. says:

There are 7 boxes each written with letter A to G (A for the first box, B for the 2nd and so on). You have 7 balls each with letter A to G (A for the first ball, B for the 2nd and so on) You randomly put one ball in each box (blindfoldedly). Find the probability that you put 3 balls correctly (that is, you put the ball in the box with the same letter)?

solution here plssss
this is my email address…. jimelryanmariano@yahoo.com
thank you

2. mvtrinh says:

Sorry for the late response. Please see post “Seven Boxes”.