Work Step by Step
To find the area of the garden, we need to know the dimensions of the garden and plug them into the following formula to calculate area: $A = lw$ We are given that the length is $15$ times the width. Let us define variables: $w$ = width of the garden $l = 15w$ We are also given that $160$ feet of fence surrounds the garden. We can think of the fence as being the perimeter of the garden. If we have the expressions for the dimensions, then we can use the formula for the perimeter of a rectangle to figure out the actual values of the width and length. The formula for calculating the perimeter of a rectangle is given by: $P = 2l + 2w$, where $P$ is the perimeter, $l$ is the length, and $w$ is the width. Let's plug in what we have: $160 = 2(15w) + 2w$ Multiply to simplify: $160 = 30w + 2w$ Combine like terms: $160 = 32w$ Divide both sides by $32$ to solve for $w$: $w = 5$ Now that we have the width, we can find the length: $l = 15(5)$ Multiply to find the length of the garden: $l = 75$ We now have the dimensions. Let's plug those values into the formula for the area of a rectangle: $A = 75(5)$ Multiply to simplify: $A = 375$ The area of the garden is $375$ square feet. This corresponds to option $G$.