# 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.