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Zorluk: ZorPercentages, Percent Change, and Interest

In a given fiscal quarter, a retail company increased the unit price of its primary product by p%p\% while the total quantity of the product sold decreased by p%p\%, where p>0p > 0. As a result of these two concurrent changes, the total revenue generated from the product experienced a net decrease of exactly 4%4\%. Which of the following statements must be true? Select all that apply.

  1. The value of pp is 2020.Cevap
  2. The final revenue generated from the product is equal to 96%96\% of the initial revenue.Cevap
  3. If the unit price had instead decreased by p%p\% and the quantity sold had increased by p%p\%, total revenue would still have decreased by 4%4\%.Cevap
  4. D
    The net percentage change in total revenue depends directly on the initial unit price of the product.
  5. E
    The sequential percentage changes of +p%+p\% and −p%-p\% offset each other, resulting in zero net change in total revenue.

Cevap

The correct statements are that the value of p is 20, the final revenue is 96% of the initial revenue, and reversing the direction of the percentage changes would still result in a 4% decrease.
The correct options accurately identify that p=20p = 20, that the final revenue is 96%96\% of the initial value, and that reversing the roles of increase and decrease produces the exact same net multiplier of 0.960.96. Using the difference of squares identity (1+x)(1−x)=1−x2(1 + x)(1 - x) = 1 - x^2, the combined multiplier for price and quantity shifts is 1−(p/100)2=0.961 - (p/100)^2 = 0.96. Solving gives p2=400p^2 = 400, so p=20p = 20. Additionally, (1−0.20)(1+0.20)=0.96(1 - 0.20)(1 + 0.20) = 0.96, showing symmetry in net outcome.

Adım Adım Çözüm

1
Express the final revenue as a function of initial revenue, price change, and quantity change.
Final Revenue = Initial Revenue ×(1+p100)×(1−p100)\times (1 + \frac{p}{100}) \times (1 - \frac{p}{100})
Total revenue is Price ×\times Quantity, so independent percentage changes multiply.
2
Apply the difference of squares formula and set equal to the given net revenue decrease.
(1+p100)(1−p100)=1−p210000=1−0.04=0.96(1 + \frac{p}{100})(1 - \frac{p}{100}) = 1 - \frac{p^2}{10000} = 1 - 0.04 = 0.96
A 4%4\% decrease leaves 96%96\% or 0.960.96 of the original value.
3
Solve the algebraic equation for pp.
p210000=0.04  ⟹  p2=400  ⟹  p=20\frac{p^2}{10000} = 0.04 \implies p^2 = 400 \implies p = 20
Isolating pp demonstrates that p=20p = 20.
4
Evaluate alternative scenario with price decrease and quantity increase.
(1−20100)(1+20100)=(0.80)(1.20)=0.96(1 - \frac{20}{100})(1 + \frac{20}{100}) = (0.80)(1.20) = 0.96
Multiplication is commutative, yielding the exact same 4%4\% net decrease.

Anahtar Kavram

Successive Percent Changes and Base Shift Multipliers
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Konular

Percentages, Percent Change, and Interest

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