A broker's income stays constant while commission rate rises from 4% to 5%. By what percentage did the business volume (turnover) slump?

Introduction / Context:
This problem links a broker’s income to commission rate and business volume. Income = rate * business volume. If income remains unchanged while the rate increases, volume must fall proportionally. We quantify that fall (slump) as a percentage.

Given Data / Assumptions:

• Initial commission rate = 4%.
• New commission rate = 5%.
• Income before = Income after.
• Business volume is directly proportional to income/rate.

Concept / Approach:
Let the original business volume be B. Original income = 0.04B. With the new rate 5%, to keep income same we need 0.05B’ = 0.04B → B’ = (0.04/0.05)B = 0.8B. The slump = (B − B’)/B = 0.2 = 20% decrease.

Step-by-Step Solution:

Verification / Alternative check:
Assume B = 100 units. Old income = 4. New rate = 5%. To earn income 4, required volume = 4/0.05 = 80. Fall from 100 to 80 is 20%.

Why Other Options Are Wrong:

• 8% or 1%: Underestimate; ignores proportionality.
• 80%: Misinterpretation; that would imply business nearly collapsed.

Common Pitfalls:
Confusing percentage points with percentage change, or comparing new and old rates instead of equating incomes.

Final Answer: