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Difficulty: MediumPercentages, Percent Change, and Interest

In Month 1, a data center's two facilities, Facility A and Facility B, consumed a combined total of 120,000 kWh120,000\text{ kWh} of electricity. In Month 2, Facility A decreased its electricity consumption by 15%15\%, while Facility B increased its electricity consumption by 20%20\%. If the combined electricity consumption of the two facilities increased by 6%6\% overall in Month 2, how many kilowatt-hours (kWh\text{kWh}) of electricity did Facility A consume in Month 1?

Answer: 48000 kWh

Answer

Facility A consumed 48,000 kWh of electricity in Month 1.
Let AA represent the electricity consumed by Facility A in Month 1. The remaining electricity consumed by Facility B in Month 1 is 120,000−A120,000 - A. In Month 2, the decrease in Facility A's consumption is 15%15\% of AA, or 0.15A0.15A, and the increase in Facility B's consumption is 20%20\% of (120,000−A)(120,000 - A), or 0.20(120,000−A)0.20(120,000 - A). The net increase overall is 6%6\% of 120,000120,000, which is 7,200 kWh7,200\text{ kWh}. Setting up the change equation −0.15A+0.20(120,000−A)=7,200-0.15A + 0.20(120,000 - A) = 7,200 leads to −0.35A+24,000=7,200-0.35A + 24,000 = 7,200, which simplifies to 0.35A=16,8000.35A = 16,800. Solving for AA yields A=48,000 kWhA = 48,000\text{ kWh}.

Step-by-Step Solution

1
Define variables for the initial Month 1 consumption of each facility.
Let AA = Month 1 consumption of Facility A. Month 1 consumption of Facility B = 120,000−A120,000 - A.
Expressing one unknown in terms of the total reduces the system to a single variable equation.
2
Determine the net kilowatt-hour change for Month 2.
Net change = 0.06×120,000=7,200 kWh0.06 \times 120,000 = 7,200\text{ kWh}.
An overall increase of 6% applies to the total starting value of 120,000 kWh.
3
Express the individual percentage changes in terms of AA and equate to the net change.
−0.15A+0.20(120,000−A)=7,200-0.15A + 0.20(120,000 - A) = 7,200.
Facility A's 15% decrease contributes −0.15A-0.15A and Facility B's 20% increase contributes +0.20(120,000−A)+0.20(120,000 - A).
4
Solve the linear equation for AA.
−0.35A+24,000=7,200⇒0.35A=16,800⇒A=48,000-0.35A + 24,000 = 7,200 \Rightarrow 0.35A = 16,800 \Rightarrow A = 48,000.
Dividing 16,80016,800 by 0.350.35 yields the original consumption of Facility A.

Key Concept

Weighted Percent Change and Systems of Percent Equations
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Topics

Percentages, Percent Change, and Interest

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