A rectangle and a parallelogram have equal areas. The base of the parallelogram is 20 cm and the altitude is 6 cm. Which one of the following cannot be the ratio of dimensions of the rectangle ?

Options

A. `7 : 5`
B. `40 : 3`
C. `15 : 2`
D. `30 : 1`

Answer

To solve the problem, we need to find the area of the parallelogram and then determine which ratio of dimensions for the rectangle cannot be possible given that both shapes have equal areas.

### Step-by-Step Solution:

1. **Calculate the Area of the Parallelogram:**
The area \( A \) of a parallelogram can be calculated using the formula:
\[ A = \text{base} \times \text{height} \]
Given:
- Base = 20 cm
- Height = 6 cm

Substitute the values:
\[ A = 20 \, \text{cm} \times 6 \, \text{cm} = 120 \, \text{cm}^2 \]

2. **Set the Area of the Rectangle Equal to the Area of the Parallelogram:**
Since the rectangle and the parallelogram have equal areas, the area of the rectangle \( A_r \) is also 120 cm².
\[ A_r = \text{length} \times \text{breadth} = L \times B = 120 \, \text{cm}^2 \]

3. **Express Length and Breadth in Terms of a Ratio:**
Let’s denote the length as \( L = x \) and the breadth as \( B = y \). Therefore, we have:
\[ x \times y = 120 \]

4. **Consider the Ratio of Length to Breadth:**
The ratio of the dimensions can be expressed as:
\[ \frac{x}{y} = k \quad \text{(where \( k \) is some ratio)} \]
This implies:
\[ x = ky \]

5. **Substituting into the Area Equation:**
Substitute \( x \) in the area equation:
\[ (ky) \times y = 120 \]
This simplifies to:
\[ ky^2 = 120 \]

6. **Finding Possible Values for \( k \):**
Rearranging gives:
\[ y^2 = \frac{120}{k} \]
For \( y^2 \) to be a valid positive number, \( k \) must be a positive factor of 120. The factors of 120 are:
- 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

7. **Identify Which Ratio Cannot Be a Dimension:**
The question asks for a ratio that cannot be represented as \( k \). If \( k \) is not a rational number or not a factor of 120, it cannot represent the ratio of dimensions of the rectangle.

For example, if we consider the ratio \( \frac{7}{5} \):
\[ k = \frac{7}{5} \quad \text{(which is not a factor of 120)} \]
Therefore, this ratio cannot represent the dimensions of the rectangle.

### Conclusion:
The ratio that cannot be the dimensions of the rectangle is \( \frac{7}{5} \).