Commission changes and profit impact: A seller allows a commission on the written (marked) price. With a 10% commission, the seller makes a 20% gain on cost. If the commission is increased to 20% (same written price), what is the new gain percentage on cost?

Introduction / Context:
Commissions are reductions from the written (marked) price that reduce the seller’s realized revenue. Here, we compare two scenarios with different commission rates but the same written price, and measure the resulting profit percentage on cost in the second scenario.

Given Data / Assumptions:

• Commission on written price: case 1 = 10%, case 2 = 20%.
• In case 1, gain on cost = 20%.
• Written price (W) is unchanged between cases.

Concept / Approach:
Let CP denote cost price and W the written price. In case 1, realized SP₁ = 0.90 W = 1.20 CP ⇒ CP = 0.75 W. In case 2, realized SP₂ = 0.80 W. Compute profit % in case 2 as (SP₂ − CP)/CP * 100 using CP = 0.75 W determined from case 1.

Step-by-Step Solution:

Verification / Alternative check:
Take W = 100 as a check. Then CP = 75; with 20% commission, SP₂ = 80; profit = 5 on cost 75 ⇒ 6.666…%.

Why Other Options Are Wrong:

• 5%, 8%, 5 1/3%, 10%: Each corresponds to incorrect bases or arithmetic slips (e.g., using W instead of CP for profit base).

Common Pitfalls:
Confusing commission with discount or mixing bases. Profit is always calculated over cost price.

Final Answer:
6 2/3%