Square Perimeter Calculation

The problem asks us to find the ratio between the perimeter of a square and the perimeter of a rectangle, given their dimensions.

First, let's calculate the perimeter of the square.

The formula for the perimeter of a square is:

$ P_{square} = 4 \times side $

Given that the side of the square is $6$ cm:

$ P_{square} = 4 \times 6 \text{ cm} $

$ P_{square} = 24 \text{ cm} $

Rectangle Perimeter Calculation

Next, we calculate the perimeter of the rectangle.

The formula for the perimeter of a rectangle is:

$ P_{rectangle} = 2 \times (length + breadth) $

The length of the rectangle is given as $10$ cm and the breadth is $4$ cm.

$ P_{rectangle} = 2 \times (10 \text{ cm} + 4 \text{ cm}) $

$ P_{rectangle} = 2 \times (14 \text{ cm}) $

$ P_{rectangle} = 28 \text{ cm} $

Ratio of Perimeters Calculation

Finally, we need to find the ratio between the perimeter of the square and the perimeter of the rectangle.

The ratio is expressed as:

$ \text{Ratio} = \frac{P_{square}}{P_{rectangle}} $

Substituting the calculated values:

$ \text{Ratio} = \frac{24 \text{ cm}}{28 \text{ cm}} $

To simplify the ratio, we find the greatest common divisor (GCD) of $24$ and $28$, which is $4$. Divide both parts of the ratio by $4$:

$ \text{Ratio} = \frac{24 \div 4}{28 \div 4} = \frac{6}{7} $

So, the ratio between the perimeter of the square and the perimeter of the rectangle is $6:7$.