Problem Solving

GMAT Paper Tests

A square playground has the same area as a rectangular playground that is 30 meters longer but 20 meters narrower. What is the length, in meters, of a side of the square playground?
$$A.\ 10\sqrt{5}$$
$$B.\ 10\sqrt{6}$$
$$C.\ 25$$
$$D.\ 50$$
$$E.\ 60$$
OA E

AAPL wrote:GMAT Paper Tests

A square playground has the same area as a rectangular playground that is 30 meters longer but 20 meters narrower. What is the length, in meters, of a side of the square playground?
$$A.\ 10\sqrt{5}$$
$$B.\ 10\sqrt{6}$$
$$C.\ 25$$
$$D.\ 50$$
$$E.\ 60$$

We can PLUG IN THE ANSWERS, which represent a side of the square.
Since the dimension of the rectangular playground are INTEGER values, the correct answer is probably also an integer.
When the correct answer is plugged in, the square and the rectangle will have the same area.

D: 50, implying that the area of the square = 50² = 2500
Since the rectangular playground is 30 meters longer but 20 meters narrower, the area of the playground = (50+30)(50-20) = 80*30 = 2400.
The areas are not equal.
Eliminate D.

E: 80, implying that the area of the square = 60² = 3600
Since the rectangular playground is 30 meters longer but 20 meters narrower, the area of the playground = (60+30)(60-20) = 90*40 = 3600.
Success!
The areas are equal.

The correct answer is E.

Algebra:
Let s = a side of the square, implying that the area of the square = s².
Since the rectangular playground is 30 meters longer but 20 meters narrower, the area of the playground = (s+30)(s-20) = s² + 10s - 600.
Since the two areas are equal, we get:
s² = s² + 10s - 600
0 = 10s - 600
600 = 10s
60 = s.

Let the side of the square be 'a' and that of rectangle be l (length) and b (breadth)
$$\begin{align*} l&=a+30 \\ b&=a-20 \\ a^2&=(a+30)(a-20) \\ a^2&=a^2+10a-600a \\ a&=60\end{align*}$$
Regards!