Rectangle — If the base increases by 10% but area stays the same, by what percentage must the altitude (height) decrease?

Introduction / Context:
Area of a rectangle is base * height. If area remains constant, any multiplicative change in one side must be balanced by the reciprocal change in the other side. We convert the reciprocal factor into a percentage decrease.

Given Data / Assumptions:

• Base scales by 1.10.
• Area unchanged.

Concept / Approach:
Let original height be H. New height H' must satisfy (1.10 * base) * H' = base * H ⇒ H' = H / 1.10. The decrease = (1 − 1/1.10) * 100%.

Step-by-Step Solution:
1/1.10 = 10/11 ≈ 0.909090…Decrease = (1 − 10/11) * 100% = (1/11) * 100% = 100/11 % = 9 1/11 %

Verification / Alternative check:
Example: base 100 → 110; height 55 → 50. Height reduction is 5/55 ≈ 9.09% and area remains 5,500.

Why Other Options Are Wrong:
10% and 11% are approximations; 12 3/11% and 8 3/11% do not match the exact reciprocal relation.

Common Pitfalls:
Subtracting 10 from 100 directly or using additive logic instead of reciprocal scaling.

Final Answer:
9 1/11 %