The area of a rectangle remains unchanged when it is made 212 inches longer and 23 inch narrower, or when it is made 212 inches shorter and 43 inch wider. Its area, in square inches, is:

1967 AMC 12 Problem 19

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19.

The area of a rectangle remains unchanged when it is made

2\dfrac12

inches longer and

\dfrac23

inch narrower, or when it is made

2\dfrac12

inches shorter and

\dfrac43

inch wider. Its area, in square inches, is:

30

\dfrac{80}{3}

24

\dfrac{45}{2}

20

Let the original length and width be

l

and

w

Expand both equal-area equations; the

lw

terms cancel

The two conditions are

\begin{aligned} lw&=\left(l+\frac52\right)\left(w-\frac23\right),\\ lw&=\left(l-\frac52\right)\left(w+\frac43\right). \end{aligned}

After expanding, these give

-4l+15w=10

and

8l-15w=20.

Hence

l=\frac{15}{2},

w=\frac{8}{3},

and

lw=20.

Therefore, the correct answer is E.