Let the sides of the rectangular paper initially be $3x$ and $4x$. The ratio of the sides is $3:4$, and the longer side is $4x$.

The process is to fold the paper in half across the longer side.

Folding Process

Fold 1

The paper is folded across the longer side, $4x$. This side is halved.

The new side lengths are $3x$ and $\frac{4x}{2} = 2x$.

The ratio of the sides is now $2x : 3x$, or $2:3$. The longer side is $3x$.

Fold 2

The paper is folded across the current longer side, $3x$. This side is halved.

The new side lengths are $2x$ and $\frac{3x}{2}$.

Let's check the ratio. The smaller side is $\frac{3x}{2}$ (since $3/2 = 1.5$, which is less than 2) and the larger side is $2x$.

The ratio of the sides is $\frac{3x}{2} : 2x$, which simplifies to $\frac{3}{2} : 2$, or $3 : 4$.

The ratio of the sides is $3:4$ after the second fold.

Fold 3

The paper is folded across the current longer side, $2x$. This side is halved.

The new side lengths are $\frac{3x}{2}$ and $\frac{2x}{2} = x$.

The ratio of the sides is now $x : \frac{3x}{2}$, or $1 : \frac{3}{2}$, which is $2:3$. The longer side is $\frac{3x}{2}$.

Fold 4

The paper is folded across the current longer side, $\frac{3x}{2}$. This side is halved.

The new side lengths are $x$ and $\frac{(3x/2)}{2} = \frac{3x}{4}$.

The ratio of the sides is now $\frac{3x}{4} : x$, or $\frac{3}{4} : 1$, which is $3:4$.

The ratio of the sides is $3:4$ after the fourth fold.

Summarizing Side Ratios After Folding

Let's list the ratio of the smaller side to the larger side after each fold:

Identifying the Pattern

From the table, we can see that the ratio of the sides becomes $3:4$ after 2 folds, and again after 4 folds.

The pattern shows that the original ratio of $3:4$ is restored after every two folds.

Conclusion

The ratio of the sides will be $3:4$ after every second fold.