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Zorluk: Çok zorPercentage Calculations

Calculate the base production cost and the net percentage increase based on the successive price changes described below.

Cevap:A technology firm sets the retail price of an industrial 3D printer by applying a 125%125\% markup to its base production cost. During a trade show, the retail price is discounted by 40%40\%. Partners who place orders at the show receive an additional 20%20\% discount on the trade show price.

If a partner pays $5,400\$5,400 for the printer, the base production cost is ${{blank1}}\$\{\{blank_1\}\}, and the firm's final selling price represents an overall increase of {{blank2}}%\{\{blank_2\}\}\% over the base production cost.

Cevap

The base production cost is $5,000, and the final selling price represents an 8% overall increase.
To solve problems with successive percentage changes, express each change as a multiplier. A 125% markup is a multiplier of 2.25. A 40% discount is a multiplier of 0.60. A 20% discount is a multiplier of 0.80. Multiplying these sequentially (2.25 × 0.60 × 0.80) yields a final compound multiplier of 1.08. This means the final price is 1.08 times the base cost, which represents an 8% overall increase. Setting 1.08C equal to the final price of 5,400allowsustosolveforCbydividing5,400by1.08,givingabasecostof5,400 allows us to solve for C by dividing 5,400 by 1.08, giving a base cost of 5,000.

Adım Adım Çözüm

1
Express the initial retail price in terms of the base production cost (C).
Retail Price = C + 1.25C = 2.25C
A 125% markup means adding 125% of the cost to the original 100% cost, resulting in a multiplier of 2.25.
2
Apply the first discount (40%) to the retail price.
Trade Show Price = 2.25C × (1 - 0.40) = 2.25C × 0.60 = 1.35C
A 40% discount means the buyer pays 60% of the current price. Discounts are multiplicative.
3
Apply the second discount (20%) to the trade show price.
Final Partner Price = 1.35C × (1 - 0.20) = 1.35C × 0.80 = 1.08C
Successive discounts apply to the most recently discounted amount, not the original price.
4
Set up an equation for the final price and solve for the base cost (C).
1.08C = 5400 → C = 5400 / 1.08 = 5000
This reverse-calculates the initial unknown variable using the final compound multiplier and the given final amount.
5
Determine the net percentage increase from the final compound multiplier.
Multiplier is 1.08, which represents 100% + 8%. Thus, the net increase is 8%.
The coefficient of C directly indicates what percentage the final price is of the original cost.

Anahtar Kavram

Successive Percentage Adjustments and Reverse Calculation
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Percentage Calculations

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