Two regular polygons are such that the ratio between their numbers of sides is 1 : 2 and the ratio between the measures of one interior angle of each polygon is 3 : 4. How many sides does each polygon have?

Introduction / Context:
This question tests properties of regular polygons, specifically the formula for the measure of an interior angle. You are told how the number of sides of two polygons compare and how their interior angles compare. From this you need to deduce the exact number of sides of each polygon.

Given Data / Assumptions:
• Let the first polygon have n sides and the second polygon have 2n sides (ratio 1 : 2).
• The ratio of one interior angle of the first polygon to one interior angle of the second polygon is 3 : 4.
• Both polygons are regular, meaning all sides and all interior angles are equal within each polygon.

Concept / Approach:
The measure of each interior angle of a regular n sided polygon is given by
Interior angle = [(n − 2) × 180] / n.
For the first polygon, interior angle A1 = 180(n − 2) / n.
For the second polygon with 2n sides, interior angle A2 = 180(2n − 2) / (2n) = 180(n − 1) / n.
The given ratio A1 : A2 = 3 : 4 allows setting up an equation to solve for n.

Step-by-Step Solution:
Step 1: First polygon interior angle A1 = 180(n − 2) / n. Step 2: Second polygon interior angle A2 = 180(n − 1) / n. Step 3: Given A1 : A2 = 3 : 4, so [180(n − 2) / n] / [180(n − 1) / n] = 3 / 4. Step 4: Simplify. The factors 180 and n cancel, leaving (n − 2) / (n − 1) = 3 / 4. Step 5: Cross multiply: 4(n − 2) = 3(n − 1). Step 6: Expand: 4n − 8 = 3n − 3. This gives n = 5. Step 7: Thus the first polygon has 5 sides and the second polygon has 2n = 10 sides.

Verification / Alternative check:
Compute interior angles explicitly. For n = 5, first polygon angle A1 = 180(5 − 2)/5 = 180 × 3 / 5 = 108 degrees. For 2n = 10, second polygon angle A2 = 180(10 − 2)/10 = 180 × 8 / 10 = 144 degrees. The ratio 108 : 144 simplifies to 3 : 4, confirming the correctness.

Why Other Options Are Wrong:
• 10, 20 and 4, 8 or 3, 6 do not satisfy the given angle ratio when you apply the interior angle formula.
• Only 5, 10 yields the required 3 : 4 interior angle ratio along with the given 1 : 2 side ratio.

Common Pitfalls:
Learners sometimes confuse the formula for interior angles with the exterior angle formula. Another pitfall is failing to simplify the fraction properly when eliminating the common factor of 180 and n. Carefully writing the ratio as a fraction and canceling common terms avoids these algebraic slips.

Final Answer:
The two polygons have 5 sides and 10 sides respectively.