When a producer allows 36% commission on the retail price of his product, he earns a profit of 8.8%. What would be his profit percent, if the commission is reduced to 25%

Answer

To solve the problem step by step, we will first determine the cost price (CP) of the product based on the initial conditions provided and then calculate the new profit percentage when the commission is reduced.

### Step 1: Understand the initial conditions
Given:
- Commission = 36% of Retail Price (RP)
- Profit = 8.8%

### Step 2: Calculate the Selling Price (SP) based on the commission
If the commission is 36%, then the producer receives 64% of the retail price as the selling price.

\[ SP = RP \times (1 - \text{Commission}) = RP \times (1 - 0.36) = RP \times 0.64 \]

### Step 3: Relate Selling Price to Cost Price
The profit percentage is given as 8.8%. This means:

\[ SP = CP \times \left(1 + \frac{\text{Profit Percentage}}{100}\right) = CP \times \left(1 + 0.088\right) = CP \times 1.088 \]

### Step 4: Set up the equation
From the above two equations, we can equate SP:

\[ RP \times 0.64 = CP \times 1.088 \]

### Step 5: Express CP in terms of RP
Rearranging the equation gives us:

\[ CP = \frac{RP \times 0.64}{1.088} \]

### Step 6: Calculate the new Selling Price with reduced commission
Now, if the commission is reduced to 25%, the new Selling Price will be:

\[ SP' = RP \times (1 - 0.25) = RP \times 0.75 \]

### Step 7: Calculate the new profit percentage
Using the new Selling Price, we can find the new profit percentage. The new profit can be calculated as:

\[ \text{New Profit} = SP' - CP \]

Substituting the values we have:

\[ \text{New Profit} = (RP \times 0.75) - \left(\frac{RP \times 0.64}{1.088}\right) \]

### Step 8: Calculate the new profit percentage
The new profit percentage will be:

\[ \text{Profit Percentage} = \frac{\text{New Profit}}{CP} \times 100 \]

Substituting CP from Step 5 into the equation:

\[ \text{Profit Percentage} = \frac{(RP \times 0.75) - \left(\frac{RP \times 0.64}{1.088}\right)}{\frac{RP \times 0.64}{1.088}} \times 100 \]

### Step 9: Simplify the expression
After simplifying the above expression, we can calculate the exact profit percentage.

### Final Calculation
After performing the calculations, we find that the new profit percentage is approximately **24.1%**.

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