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Zorluk: OrtaRates, Ratios, Percentages, and Applied Word Problems

An eco-friendly power facility generates electricity exclusively using solar panels and wind turbines. Last month, energy from solar panels was produced at a constant cost of 0.04perkilowatt−hour(kWh),andenergyfromwindturbineswasproducedataconstantcostof0.04 per kilowatt-hour (kWh), and energy from wind turbines was produced at a constant cost of 0.06 per kWh. What was the average cost per kWh of the total energy generated by the facility last month?

(1) Last month, the facility generated 300,000 kWh of energy using solar panels.
(2) Last month, the ratio of the energy generated by wind turbines to the energy generated by solar panels was 2 to 3.

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Cevap
  3. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
The question asks for the weighted average cost per kWh of electricity produced by two sources with given unit costs (0.04and0.04 and 0.06). A weighted average of two component rates depends solely on the ratio of the quantities involved, not on the total absolute quantities. Statement (1) gives only the quantity of solar energy without providing any information about wind energy, making it insufficient. Statement (2) gives the exact ratio of wind energy to solar energy (2:32:3), which allows us to find the exact weighted average cost ($0.048 per kWh) without knowing the total energy produced. Therefore, Statement (2) alone is sufficient.

Adım Adım Çözüm

1
Rephrase the question stem target using weighted averages.
Let SS be the kWh generated from solar panels and WW be the kWh generated from wind turbines. The average cost per kWh is given by 0.04S+0.06WS+W=0.04(SS+W)+0.06(WS+W)\frac{0.04S + 0.06W}{S + W} = 0.04\left(\frac{S}{S+W}\right) + 0.06\left(\frac{W}{S+W}\right).
Simplifying the formula shows that the average cost depends strictly on the relative ratio of WW to SS (or the proportion of each source to the total), rather than absolute kWh quantities.
2
Evaluate Statement (1): Last month, the facility generated 300,000 kWh of energy using solar panels.
S=300,000S = 300,000, but WW is unknown.
Since the amount of wind energy WW is completely unknown, we cannot calculate the weighted average cost. Statement (1) is NOT sufficient.
3
Evaluate Statement (2): The ratio of energy generated by wind turbines to solar panels was 2 to 3.
WS=23⇒W=23S\frac{W}{S} = \frac{2}{3} \Rightarrow W = \frac{2}{3}S. Substituting into the average cost expression yields 0.04S+0.06(23S)S+23S=0.04S+0.04S53S=0.085/3=$0.048\frac{0.04S + 0.06(\frac{2}{3}S)}{S + \frac{2}{3}S} = \frac{0.04S + 0.04S}{\frac{5}{3}S} = \frac{0.08}{5/3} = \$0.048 per kWh.
Because SS cancels out completely, knowing only the ratio gives a unique numerical value for the average cost per kWh. Statement (2) is ALONE sufficient.

Anahtar Kavram

Weighted Average Ratios in Data Sufficiency
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Konular

Rates, Ratios, Percentages, and Applied Word Problems

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