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Zorluk: OrtaPercentages, Percent Change, and Interest

In Year 1, a non-profit organization received all of its funding from two sources: private donations and government grants. Private donations accounted for 60%60\% of the total funding, and government grants accounted for the remaining 40%40\%. In Year 2, private donations increased by 25%25\% compared to Year 1, while government grants decreased by 15%15\% compared to Year 1. Which of the following statements must be true? Select all such statements.

  1. The total funding received by the organization in Year 2 was 9%9\% greater than the total funding received in Year 1.Cevap
  2. In Year 2, private donations accounted for more than 68%68\% of the total funding received in Year 2.Cevap
  3. C
    In Year 2, government grants accounted for 34%34\% of the total funding received in Year 2.
  4. D
    In Year 2, private donations accounted for 75%75\% of the total funding received in Year 2.
  5. E
    The ratio of private donations to government grants in Year 2 was 25%25\% greater than the ratio of private donations to government grants in Year 1.

Cevap

The correct statements are that the total funding in Year 2 was 9%9\% greater than in Year 1, and that private donations accounted for more than 68%68\% of the total funding in Year 2.
Total funding in Year 2 is 0.60(1.25)+0.40(0.85)=0.75+0.34=1.090.60(1.25) + 0.40(0.85) = 0.75 + 0.34 = 1.09 times the Year 1 total, which represents a 9%9\% increase. Furthermore, private donations in Year 2 represent 0.751.09≈68.81%\frac{0.75}{1.09} \approx 68.81\% of the Year 2 total funding, which is greater than 68%68\%.

Adım Adım Çözüm

1
Define variables for Year 1 funding components relative to total Year 1 funding (T1T_1).
Private donations in Year 1: P1=0.60T1P_1 = 0.60 T_1. Government grants in Year 1: G1=0.40T1G_1 = 0.40 T_1.
Establishing Year 1 amounts in terms of total funding T1T_1 provides the baseline for percentage changes.
2
Calculate Year 2 funding amounts after applying the respective percent changes.
P2=1.25×0.60T1=0.75T1P_2 = 1.25 \times 0.60 T_1 = 0.75 T_1. G2=0.85×0.40T1=0.34T1G_2 = 0.85 \times 0.40 T_1 = 0.34 T_1.
Increasing private donations by 25%25\% multiplies P1P_1 by 1.251.25; decreasing government grants by 15%15\% multiplies G1G_1 by 0.850.85.
3
Compute total Year 2 funding (T2T_2) and compare it with T1T_1.
T2=P2+G2=0.75T1+0.34T1=1.09T1T_2 = P_2 + G_2 = 0.75 T_1 + 0.34 T_1 = 1.09 T_1. Net change is a 9%9\% increase.
Adding the components yields total Year 2 funding, confirming that T2T_2 is 1.091.09 times T1T_1.
4
Calculate component shares as percentages of total Year 2 funding (T2T_2).
Share of private donations in Year 2 = 0.75T11.09T1≈68.81%\frac{0.75 T_1}{1.09 T_1} \approx 68.81\%. Share of government grants in Year 2 = 0.34T11.09T1≈31.19%\frac{0.34 T_1}{1.09 T_1} \approx 31.19\%.
Percentage share of a total requires dividing each component by the new total funding (1.09T11.09 T_1), not the original baseline (T1T_1).

Anahtar Kavram

Weighted percentage change and percentage base shift
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Konular

Percentages, Percent Change, and Interest

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