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.
Important Questions from Miscellaneous
A stone is thrown horizontally from the top of a 20 m high building with a speed of 12 m/s. It hits the ground at a distance R from the building. Taking g = 10 m/s2 and neglecting air resistance will give :
A sphere of volume V is made of a material with lower density than water. While on Earth, it floats on water with its volume f1V (f1 < 1) submerged. On the other hand, on a spaceship accelerating with acceleration a < g (g is the acceleration due to gravity on Earth) in outer space, its submerged volume in water is f2V. Then:
A railway wagon (open at the top) of mass M1 is moving with speed v1 along a straight track. As a result of rain, after some time it gets partially filled with water so that the mass of the wagon becomes M2 and speed becomes v2. Taking the rain to be falling vertically and the water stationery inside the wagon, the relation between the two speeds v1 and v2 is :
Consider the following statements:
1. Distance between the longitudes becomes zero on North Pole and South Pole.
2. Distance between the longitudes is maximum on the Equator.
3. Number of longitudes is more than number of latitudes.
Which of the statements given above is/are correct?
One block of 2⋅0 kg mass is placed on top of another block of 3⋅0 kg mass. The coefficient of static friction between the two blocks is 0⋅2. The bottom block is pulled with a horizontal force F such that both the blocks move together without slipping. Taking acceleration due to gravity as 10 m/s2, the maximum value of the frictional force is :
