word problem

**mojojojo** · May 24th 2010, 04:00 PM

if the length and the width of a rectangle are increased by two inches, the area of the rectangle is 120 square inches. if the length and the width of a rectangle are decreased by two inches, the area of the rectangle is 48 square inches. find the length of the rectangle.

**Soroban** · May 24th 2010, 05:39 PM

Hello, mojojojo!

If the length and the width of a rectangle are increased by two inches,
the area of the rectangle is 120 square inches.
If the length and the width of a rectangle are decreased by two inches,
the area of the rectangle is 48 square inches.
Find the length of the rectangle.

Let $L$ = length of the rectangle.
Let $W$ = width of the rectangle.

Then we have: . $\begin{array}{ccccc}(L+2)(W+2) &=& 120 & [1] \\<br /> (L-2)(W-2) &=& 48 & [2]\end{array}$

$\begin{array}{cccccc}<br /> \text{From [1] we have:} & LW + 2L + 2W + 4 &=& 120 \\<br /> \text{From [2] we have:} & LW - 2L - 2W + 4 &=& 48 \end{array}$

. . Subtract: . $4L + 4W \:=\:72 \quad\Rightarrow\quad W \:=\:18-L$

Substitute into [1]: . $(L+2)(20-L) \:=\:120 \quad\Rightarrow\quad L^2 - 18L + 80 \:=\:0$

. . $(L-10)(L-8) \:=\:0 \quad\Rightarrow\quad L \;=\;10,\:8$

Assuming $L > W$ we have: . $L \:=\:10\text{ inches.}$

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