# If the length of the rectangle is increased by 50%, then by what part should its width

**If the length of the rectangle is increased by 50%, then by what part should its width be reduced so that the area of the rectangle does not change?**

Let x be the length of this rectangle and y the width of this rectangle.

Then the area S of this rectangle will be:

S = x * y.

If the length of this rectangle is increased by 50%, then the length of the resulting rectangle will be:

x + (50/100) * x = x + (5/10) * x = x + 0.5 * x = 1.5x.

Let us denote by a the part by which it is necessary to reduce the width of the new rectangle so that the area of the rectangle does not change.

After changing the length and width, the area of the resulting rectangle will be:

1.5 * x * a * y.

Let’s express this area in terms of the area of the original rectangle:

1.5 * x * a * y = 1.5 * a * x * y = 1.5 * a * S.

In order for the area not to change, it is necessary that the equality is fulfilled:

1.5 * a * S = S.

From this ratio it follows:

1.5 * a = 1;

a = 1 / 1.5;

a = 10/15:

a = 2/3.

Answer: The correct answer is 2/3.