The side of a square and the diameter of a circle are equal in length. If the side of the square is taken equal to the diameter of the circle and π is taken as 22/7, what is the ratio between the area of the square and the area of the circle?

Introduction / Context:
This problem compares the areas of two basic plane figures, a square and a circle, when their key dimensions are linked. The key relationship is that the side of the square is equal to the diameter of the circle. Such questions check understanding of area formulas and the ability to manipulate algebraic expressions and ratios, especially when a specific value for π is provided for easy calculation.

Given Data / Assumptions:

• Side of the square = s.
• Diameter of the circle = s (same as the square side).
• Radius of the circle = s / 2.
• Area of square = s^2.
• Area of circle = π * (radius)^2.
• Use π = 22/7 as instructed.

Concept / Approach:
The idea is to express the area of both figures in terms of the single variable s, since side of the square and diameter of the circle are equal. Once both areas are expressed in the same variable, we can form a ratio of square area to circle area and simplify that ratio. Because the variable s appears in both numerator and denominator, it cancels out, leaving a pure numerical ratio that can be simplified using the given value of π.

Step-by-Step Solution:
Let the side of the square be s.Then the diameter of the circle is also s, so the radius r = s / 2.Area of the square = s^2.Area of the circle = π * r^2 = π * (s / 2)^2 = π * s^2 / 4.Ratio of areas (square : circle) = s^2 : (π * s^2 / 4).Cancel s^2 from both sides to get 1 : (π / 4).This is equivalent to 4 : π.Using π = 22/7, the ratio becomes 4 : 22/7.Multiply both terms by 7 to clear the fraction: 4 * 7 : 22 = 28 : 22.Simplify 28 : 22 by dividing both by 2 to get 14 : 11.

Verification / Alternative check:
We can verify with a numerical example. Let s = 14 units. Then radius r = 7 units. Area of square = 14^2 = 196 square units. Area of circle = (22/7) * 7^2 = (22/7) * 49 = 22 * 7 = 154 square units. Ratio 196 : 154 simplifies on dividing by 14 to 14 : 11, which confirms the derived ratio is correct and consistent with the formula-based solution.

Why Other Options Are Wrong:
28 : 11 is too large and results from not simplifying the intermediate ratio properly. 7 : 22 and 22 : 7 invert or distort the correct relationship between square and circle area. 11 : 14 is the reverse of the correct ratio and would correspond to area of circle : area of square, not the other way around. Therefore, only 14 : 11 matches the correct derivation.

Common Pitfalls:
One mistake is to forget that the diameter of the circle equals the side of the square, and incorrectly take the radius as s instead of s / 2. Another common error is to leave the ratio in terms of π without substituting π = 22/7 or to fail to simplify the fraction completely. Students may also accidentally calculate the ratio of circle area to square area instead of square to circle. Careful attention to the wording of the problem prevents such errors.

Final Answer:
The ratio of the area of the square to the area of the circle is 14 : 11.