A rectangular paper of sides in the ratio 3 ∶ 4 is folded in half across the longer side. The process is repeated several times. After how many foldings will the ratio of the sides be 3 ∶ 4? (Ignore the finite thickness of the paper.)
Every second fold
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:
| Number of Folds | Side Lengths | Ratio (smaller : larger) |
|---|---|---|
| 0 (Initial) | $3x, 4x$ | $3 : 4$ |
| 1 | $2x, 3x$ | $2 : 3$ |
| 2 | $\frac{3}{2}x, 2x$ | $3 : 4$ |
| 3 | $x, \frac{3}{2}x$ | $2 : 3$ |
| 4 | $\frac{3}{4}x, x$ | $3 : 4$ |
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.
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