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Zorluk: ZorPercents and Percent Change

At the beginning of an experiment, the number of bacteria in Population X is 50%50\% greater than the number of bacteria in Population Y. During the experiment, Population X increases by 10%10\% and Population Y increases by r%r\%. At the end of the experiment, the number of bacteria in Population X is 25%25\% greater than the number of bacteria in Population Y. What is the value of rr?

  1. A
    24
  2. 32Cevap
  3. C
    35
  4. D
    40

Cevap

32
The correct answer is 3232. If the initial population of Y is assumed to be 100100, then the initial population of X is 150150 because it is 50%50\% greater. A 10%10\% increase of X results in a final population of 150×1.10=165150 \times 1.10 = 165. Since the final population of X is 25%25\% greater than the final population of Y, we have 165=1.25×Yfinal165 = 1.25 \times Y_{\text{final}}, which gives Yfinal=132Y_{\text{final}} = 132. The growth from 100100 to 132132 represents a 32%32\% increase, so r=32r = 32.

Adım Adım Çözüm

1
Set up initial values for both populations.
Let the initial size of Population Y be 100100. Since Population X is 50%50\% greater than Y, the initial size of Population X is 150150.
Choosing a concrete initial value like 100100 simplifies the percentage calculations.
2
Calculate the final population of X.
The final population of X is 150×(1+0.10)=165150 \times (1 + 0.10) = 165.
Population X increased by 10%10\% during the experiment.
3
Calculate the final population of Y.
Since the final population of X is 25%25\% greater than the final population of Y, we write 165=1.25×Yfinal165 = 1.25 \times Y_{\text{final}}, which yields Yfinal=132Y_{\text{final}} = 132.
We must divide the final population of X by 1.251.25 to find the final population of Y, because X is 25%25\% greater than Y.
4
Find the percentage increase of Population Y.
The percent increase from 100100 to 132132 is 132−100100×100%=32%\frac{132 - 100}{100} \times 100\% = 32\%. Therefore, r=32r = 32.
This determines the value of rr representing the percentage growth of Population Y.

Anahtar Kavram

Multi-step percentage change and identifying the correct baseline value.
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Konular

Percents and Percent Change

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