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Difficulty: HardFractions, Decimals, and Percentages

A smart thermostat is programmed to reduce a home's heating energy usage. In January, the heating energy usage is reduced by 15%15\% compared to the baseline usage. In February, the usage is reduced by an additional 18\frac{1}{8} of January's usage level. In March, the usage is reduced by 20%20\% of February's usage level. If the baseline heating energy usage was 400400 kilowatt-hours (kWh), what is the total reduction in energy usage from the baseline to the end of March, in kWh?

Answer: 162 kWh

Answer

The total reduction in energy usage from the baseline to the end of March is 162 kWh.
The baseline usage of 400400 kWh is reduced by 15%15\% in January, leaving 340340 kWh. In February, a reduction of 18\frac{1}{8} of 340340 kWh reduces usage by 42.542.5 kWh, leaving 297.5297.5 kWh. In March, a reduction of 20%20\% of 297.5297.5 kWh reduces usage by 59.559.5 kWh, leaving a final usage of 238238 kWh. The total reduction is the difference between the baseline and final usage levels, which is 162162 kWh.

Step-by-Step Solution

1
Calculate January's energy reduction and January's usage level.
January reduction is 6060 kWh; January usage level is 340340 kWh.
January's reduction is 15%15\% of the 400400 kWh baseline (0.15×400=600.15 \times 400 = 60 kWh). Subtracting this reduction from the baseline gives January's usage level (400−60=340400 - 60 = 340 kWh).
2
Calculate February's energy reduction and February's usage level.
February reduction is 42.542.5 kWh; February usage level is 297.5297.5 kWh.
February's reduction is 18\frac{1}{8} of January's usage level (340340 kWh), which is 0.125×340=42.50.125 \times 340 = 42.5 kWh. Subtracting this from January's level gives February's usage level (340−42.5=297.5340 - 42.5 = 297.5 kWh).
3
Calculate March's energy reduction and March's usage level.
March reduction is 59.559.5 kWh; March usage level is 238238 kWh.
March's reduction is 20%20\% of February's usage level (297.5297.5 kWh), which is 0.20×297.5=59.50.20 \times 297.5 = 59.5 kWh. Subtracting this from February's level gives March's usage level (297.5−59.5=238297.5 - 59.5 = 238 kWh).
4
Calculate the total energy reduction from the baseline to the end of March.
The total reduction is 162162 kWh.
The total reduction can be found by adding the reductions from each of the three months (60+42.5+59.5=16260 + 42.5 + 59.5 = 162 kWh) or by subtracting the final usage level from the baseline (400−238=162400 - 238 = 162 kWh).

Key Concept

Applying sequential percentage and fraction reductions to changing base values.
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Topics

Fractions, Decimals, and Percentages

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