Question

Consider two rectangular sheets, Sheet M and Sheet N of dimensions 6 cm x 4 cm each.
Folding operation 1: The sheet is folded into half by joining the short edges of the current shape.
Folding operation 2: The sheet is folded into half by joining the long edges of the current shape.
Folding operation 1 is carried out on Sheet M three times.
Folding operation 2 is carried out on Sheet N three times.
The ratio of perimeters of the final folded shape of Sheet M to the final folded shape of Sheet N is \(\underline{\hspace{2cm}}\).

• A. 13 : 7
• B. 3 : 2
• C. 7 : 5
• D. 5 : 13

Hint:

When folding sheets, remember that the dimensions of the sheet are halved in each operation, and the perimeter is calculated based on the final dimensions.

Answer 1

The Correct Option is A

We are given two rectangular sheets, each with dimensions \( 6 \, \text{cm} \times 4 \, \text{cm} \). There are two folding operations described:
- Folding operation 1: The sheet is folded in half by joining the short edges of the current shape. - Folding operation 2: The sheet is folded in half by joining the long edges of the current shape. Let's analyze the two sheets separately: Step 1: Analyze Sheet M Initially, Sheet M has dimensions 6 cm (length) and 4 cm (width). - First fold (operation 1): When we fold the sheet in half by joining the short edges (4 cm), the width becomes halved, so the new dimensions are 6 cm (length) and \( \frac{4}{2} = 2 \) cm (width).
- Second fold (operation 1): We fold it again along the short edge (now 2 cm). The new dimensions become 6 cm (length) and \( \frac{2}{2} = 1 \) cm (width).
- Third fold (operation 1): Folding once more along the short edge (now 1 cm), we get the final dimensions of Sheet M as 6 cm (length) and \( \frac{1}{2} = 0.5 \) cm (width).
Thus, the final dimensions of Sheet M are 6 cm by 0.5 cm. The perimeter of this folded shape is: \[ P_M = 2 \times (6 + 0.5) = 2 \times 6.5 = 13 \, \text{cm}. \] Step 2: Analyze Sheet N Initially, Sheet N also has dimensions 6 cm (length) and 4 cm (width). - First fold (operation 2): We fold the sheet in half by joining the long edges (6 cm). The new dimensions are \( \frac{6}{2} = 3 \) cm (length) and 4 cm (width).
- Second fold (operation 2): We fold it again along the long edge (now 3 cm), so the new dimensions are \( \frac{3}{2} = 1.5 \) cm (length) and 4 cm (width).
- Third fold (operation 2): After another fold along the long edge (now 1.5 cm), the final dimensions are \( \frac{1.5}{2} = 0.75 \) cm (length) and 4 cm (width).
Thus, the final dimensions of Sheet N are 0.75 cm by 4 cm. The perimeter of this folded shape is: \[ P_N = 2 \times (0.75 + 4) = 2 \times 4.75 = 9.5 \, \text{cm}. \] Step 3: Calculate the Ratio of Perimeters Now, we can find the ratio of the perimeters of the final folded shapes of Sheet M and Sheet N: \[ \text{Ratio} = \frac{P_M}{P_N} = \frac{13}{9.5} \approx 1.368 \approx 3 : 2. \] Thus, the ratio of the perimeters of the final folded shape of Sheet M to the final folded shape of Sheet N is \( 3 : 2 \), which corresponds to option (B). Final Answer: 3 : 2