If a commission of 10% is given on the marked price of a book, the publisher gains 20%. If the commission is increased to 15%, what is the gain percent?
Understand the Problem
The question involves calculating the new gain percentage for a publisher when the commission on the marked price of a book is increased from 10% to 15%. The initial scenario provides a gain of 20% with a 10% commission, and we need to determine how this changes when the commission increases.
Answer
The new gain percentage is approximately $13.33\%$.
Answer for screen readers
The new gain percentage is approximately $13.33%$.
Steps to Solve
- Define the Variables
Let:
- ( CP ) = Cost Price of the book
- ( MP ) = Marked Price of the book
- ( SP ) = Selling Price of the book
- Establish Initial Conditions
Given the initial commission is 10% and leads to a 20% gain, we can express these relationships. The Selling Price ( SP ) can be represented as:
$$ SP = CP + 0.2 \cdot CP = 1.2 \cdot CP $$
- Express the Selling Price with the Commission
With a commission of 10%, the Selling Price can also be expressed in terms of the Marked Price:
$$ SP = MP - 0.1 \cdot MP = 0.9 \cdot MP $$
- Set Up the Equation
Now we can set the two expressions for ( SP ) equal to each other:
$$ 1.2 \cdot CP = 0.9 \cdot MP $$
- Solve for Marked Price in terms of Cost Price
Rearranging the above equation gives us:
$$ MP = \frac{1.2 \cdot CP}{0.9} = \frac{4}{3} CP $$
- Calculate New Selling Price with Increased Commission
Now, with the new commission at 15%, the Selling Price is:
$$ SP = MP - 0.15 \cdot MP = 0.85 \cdot MP $$
- Substitute the Marked Price
Substituting ( MP ) into the new Selling Price equation:
$$ SP = 0.85 \cdot \left(\frac{4}{3} CP\right) = \frac{3.4}{3} CP $$
- Determine New Gain Percentage
The new gain can be calculated as:
$$ Gain% = \frac{SP - CP}{CP} \cdot 100 = \frac{\frac{3.4}{3} CP - CP}{CP} \cdot 100 $$
- Simplify the Gain Percentage
Simplifying gives:
$$ Gain% = \left(\frac{3.4 - 3}{3}\right) \cdot 100 = \frac{0.4}{3} \cdot 100 \approx 13.33% $$
The new gain percentage is approximately $13.33%$.
More Information
This problem illustrates how changes in commission rates can significantly impact a publisher's profit margin. With increased commissions, the overall gain decreases, reflecting the importance of understanding the impact of pricing strategies in business.
Tips
- Not converting percentage increases into decimal form when calculating. Always remember to divide by 100.
- Forgetting to set the Selling Price expressions equal to each other, which is crucial for solving the problem accurately.
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