Hike in Selling Price to Maintain Profit After Cost Price Increase
In the world of electronic gadgets, costs can often fluctuate. Understanding how these changes impact the selling price is crucial. For instance, if a part of an electronic gadget costs one-third of the gadget's selling price, and the cost of this part increases by 10, how much should the selling price be adjusted to maintain a 25% profit margin? Let's delve into a step-by-step solution to answer this question and ensure businesses maintain their profit margins even amidst cost fluctuations.
Step-by-Step Solution
To solve this problem, we need to define a few key variables and break down the calculations:
Define Variables
Let S be the selling price of the gadget. The cost price of the part is S/3. The profit margin is 25%. This means the cost price of the gadget (excluding the part) is 75% of the selling price: 0.75S.Total Cost Price of the Gadget
The total cost price of the gadget includes the cost of the part and other costs, represented as x: [ CP_{text{gadget}} CP_{text{part}} x frac{S}{3} x ]
Cost Price of the Part Increase
If the cost price of the part is increased by 10, the new cost price of the part becomes: [ CP_{text{part new}} CP_{text{part}} times 1.1 frac{S}{3} times 1.1 frac{1.1}{3}S ]
New Total Cost Price of the Gadget
The new total cost price of the gadget is now: [ CP_{text{gadget new}} CP_{text{part new}} x frac{1.1}{3}S x ]
Maintain Profit at 25
To maintain a 25% profit margin, the new selling price (S_{text{new}}) must satisfy: [ 0.75S_{text{new}} CP_{text{gadget new}} frac{1.1}{3}S x ]
Express (x)
From the profit margin calculation, we can express (x) in terms of (S): [ x 0.75S - frac{S}{3} 0.75S - 0.3333S 0.4167S ]
Substitute (x) Back into the Equation
Substituting (x) back into the equation for (CP_{text{gadget new}}): [ CP_{text{gadget new}} frac{1.1}{3}S 0.4167S ]
Converting (frac{1.1}{3}) to a decimal approximation: [ frac{1.1}{3} approx 0.3667 ]
Therefore: [ 0.3667S 0.4167S 0.75S_{text{new}} ]
Simplifying the left side: [ 0.7834S 0.75S_{text{new}} ]
Solving for (S_{text{new}}): [ S_{text{new}} frac{0.7834S}{0.75} approx 1.0445S ]
Calculating the Hike
The increase in the selling price is: [ text{Hike} S_{text{new}} - S 1.0445S - S 0.0445S ]
This hike is approximately (4.45%) of the original selling price.
Conclusion
To maintain a 25% profit margin after a 10 increase in the cost price of the part, the selling price of the gadget needs to be increased by approximately (4.45%).
Understanding the impact of cost changes on selling prices is crucial for businesses to maintain profitability. This problem highlights the importance of adjusting prices appropriately to keep profit margins intact.