How many distinct rectangles are there with integer side lengths such that the numerical value of area of the rectangle in square units is equal to $5$ times the numerical value of the perimeter in units? (Two rectangles are considered to be distinct if they are not congruent.)
We have the following :
xy = 5 [ 2 (x + y) ] where x, y are the dimensions of each rectangle
xy = 10x + 10y
Solving for y, we have that
y = [ 10x ] / [ x - 10 ]
It's clear that x > 10 and as x→ infinity, y → 10
And because y decreases as x increases....the smallest integer value possible for y is when y = 11 and x = 110
And y will have an integer value of 60 whenever x = 12
And when x = 14, y = 35
And when x = 15, y = 30
And when x= 20, y = 20
And no x integer values from 21 - 29 result in y being an integer
And when x = 30, y = 15...but these are just the same dimensions as when x = 15 and y = 30
So....there are 5 distinct rectangles possible which fit the criteria :
20 x 20
15 x 30
14 x 35
12 x 60
11 x 110
How many distinct rectangles are there with integer side lengths such that the numerical value of area of the rectangle in square units is equal to 5 times the numerical value of the perimeter in units?
Formula:
\(\begin{array}{rcll} \text{area of the rectangle } &=& xy \\ 5 \times \text{ the numerical value of the perimeter } &=& 5\times [ 2(x+y) ] \\ xy &=& 5\times [ 2(x+y) ] \\ xy &=& 10\times (x+y) \\ xy &=& 10x+ 10y \\ xy - 10x - 10y &=& 0 \quad & | \quad + 100 \\ xy - 10x - 10y +100 &=& 100 \\ (x-10)\times (y-10) &=& 100 \\\\ \mathbf{(x-10)\times (y-10)} & \mathbf{=} & \mathbf{100} \\ \end{array} \)
So x-10 and y-10 are integers, whose product is 100
How many divisors does 100 have?
Divisors:
1 | 2 | 4 | 5 | 10 | 20 | 25 | 50 | 100 (9 divisors)
\(\begin{array}{|rrcll|} \hline \text{or } & 1\times 100 &=& 100 \\ \text{or } & 2\times 50 &=& 100 \\ \text{or } & 4\times 25 &=& 100 \\ \text{or } & 5\times 20 &=& 100 \\ \text{or } & 10\times 10 &=& 100 \\ \hline \end{array}\)
Solution:
\(\begin{array}{|rclcl|} \hline \underbrace{(x-10)}_{=1} &\times& \underbrace{(y-10)}_{=100} & = & \mathbf{1\times 100} \\\\ x-10 = 1 && y-10 = 100 \\ x = 11 && y = 110 \\ &\mathbf{(11\times 110)} \\ \hline \underbrace{(x-10)}_{=2} &\times& \underbrace{(y-10)}_{=50} & = & \mathbf{2\times 50} \\\\ x-10 = 2 && y-10 = 50 \\ x = 12 && y = 60 \\ &\mathbf{(12\times 60)} \\ \hline \underbrace{(x-10)}_{=4} &\times& \underbrace{(y-10)}_{=25} & = & \mathbf{4\times 25} \\\\ x-10 = 4 && y-10 = 25 \\ x = 14 && y = 35 \\ &\mathbf{(14\times 35)} \\ \hline \underbrace{(x-10)}_{=5} &\times& \underbrace{(y-10)}_{=20} & = & \mathbf{5\times 20} \\\\ x-10 = 5 && y-10 = 20 \\ x = 15 && y = 30 \\ &\mathbf{(15\times 30)} \\ \hline \underbrace{(x-10)}_{=10} &\times& \underbrace{(y-10)}_{=10} & = & \mathbf{10\times 10} \\\\ x-10 = 10 && y-10 = 10 \\ x = 20 && y = 20 \\ &\mathbf{(20\times 20)} \\ \hline \end{array}\)
There are 5 distinct rectangles:
\(\mathbf{(11\times 110)} \\ \mathbf{(12\times 60)} \\ \mathbf{(14\times 35)} \\ \mathbf{(15\times 30)} \\ \mathbf{(20\times 20)} \)