The area of a square is 1024 sq cm. Find the ratio of the length to the breadth of a rectangle, where the rectangle's length is twice the side of that square and the rectangle's breadth is 12 cm less than the same square's side.

Introduction / Context:
This question tests basic mensuration and proportional reasoning. You first convert a square's area into its side length, then form the rectangle's dimensions using the given relationships, and finally write the ratio length:breadth in simplest form. The key skill is correctly translating “twice the side” and “12 cm less than the side” into algebraic expressions and simplifying the resulting ratio.

Given Data / Assumptions:

• Area of square = 1024 sq cm
• Let side of square = s cm
• Rectangle length = 2*s cm
• Rectangle breadth = (s - 12) cm

Concept / Approach:
For a square: Area = s*s, so s = sqrt(Area). Once s is known, compute rectangle length and breadth from the given relations. Then ratio = length:breadth and reduce by dividing by the greatest common factor.

Step-by-Step Solution:
Area = s^2 = 1024s = sqrt(1024) = 32 cmRectangle length = 2*s = 2*32 = 64 cmRectangle breadth = s - 12 = 32 - 12 = 20 cmRequired ratio (length:breadth) = 64:20Simplify by dividing both terms by 4: 64/4 : 20/4 = 16:5

Verification / Alternative check:
Quick check: if breadth is 20 and length is 64, length is clearly a bit more than 3 times breadth (3.2 times). The simplified ratio 16:5 = 3.2 confirms this consistency. Also, since length = 2*s and breadth = s-12, with s=32, both values are positive and reasonable.

Why Other Options Are Wrong:
8:5 corresponds to 64:40, which would mean breadth 40 (not 20). 5:16 reverses the ratio (breadth:length). 13:12 and 17:13 do not match 64:20 after reduction, so they come from incorrect side calculation or incorrect subtraction of 12.

Common Pitfalls:
Mistaking sqrt(1024) (it is 32, not 16). Using 12 less than the length instead of 12 less than the square's side. Writing breadth:length instead of length:breadth. Forgetting to simplify the ratio to lowest terms.

Final Answer:
16:5