# The length of a rectangular garden is 5 less than two times the width. There is a 5 foot wide sidewalk on 2 sides that has an area of 225 sq ft. How do you find the dimensions of the garden?

Sep 23, 2015

Dimensions of a garden are $25$x$15$

#### Explanation:

Let $x$ be the length of a rectangle and $y$ is the width.
The first equation that can be derived from a condition "The length of a rectangular garden is 5 less than two times the width" is
$x = 2 y - 5$

The story with a sidewalk needs clarification.
First question: is sidewalk inside the garden or outside?
Let's assume its outside because it seems more natural (a sidewalk for people going around the garden enjoying the beautiful flowers growing inside).
Second question: is sidewalk on two opposite sides of the garden or on two adjacent?
We should assume, the sidewalk goes along two adjacent sides, along the length and the width of the garden. It cannot be along opposite two sides because sides are different and the problem would not be properly defined.

So, a sidewalk of 5 foot wide goes along two adjacent sides of a rectangle, turning at ${90}^{0}$ around the corner. Its area consists of the part going along the length of a rectangle (area is $5 \cdot x$), along its width (area is $5 \cdot y$) and includes the $5$x$5$ square at the corner (area is $5 \cdot 5$).
This is sufficient to derive the second equation:
$5 \cdot x + 5 \cdot y + 5 \cdot 5 = 225$
or
$x + y = 40$

Now we have to solve a system of two equations with two unknown:
$x = 2 y - 5$
$x + y = 40$

Substituting $2 y - 5$ from the first equation into the second for $x$:
$2 y - 5 + y = 40$
or
$3 y = 45$
or
$y = 15$
from which
$x = 2 \cdot 15 - 5 = 25$
So, the garden has dimensions $25$x$15$.