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Difficulty: MediumRatios, Rates, and Proportions

A community solar farm distributes electrical energy to three types of facilities: residential homes, small businesses, and municipal buildings. The ratio of energy distributed to residential homes to energy distributed to small businesses is 5:35:3. The ratio of energy distributed to small businesses to energy distributed to municipal buildings is 4:14:1. If the municipal buildings receive a total of 4545 megawatt-hours (MWh) of energy, what is the total amount of energy, in megawatt-hours, distributed by the solar farm?

  1. A
    255
  2. B
    300
  3. C
    450
  4. D
    480
  5. 525Answer

Answer

The total energy distributed by the solar farm is 525 megawatt-hours.
To solve this problem, express the two ratios with a unified scale. Residential to Small Businesses is 5:35:3 and Small Businesses to Municipal is 4:14:1. Multiplying the first ratio by 44 gives 20:1220:12 and the second ratio by 33 gives 12:312:3. This yields a three-part ratio of 20:12:320:12:3 for Residential : Small Businesses : Municipal. Since Municipal buildings account for 33 units out of the total 3535 units (20+12+3=3520 + 12 + 3 = 35) and receive 4545 MWh, each ratio unit corresponds to 453=15\frac{45}{3} = 15 MWh. Multiplying the total 3535 units by 1515 MWh gives 525525 MWh.

Step-by-Step Solution

1
Express both ratios using a common term for small businesses.
Residential to Small Businesses =5:3=20:12= 5:3 = 20:12. Small Businesses to Municipal =4:1=12:3= 4:1 = 12:3. Combining these gives a ratio of Residential to Small Businesses to Municipal of 20:12:320:12:3.
To combine two ratios sharing a common quantity, scale the ratios so the common quantity has the same numerical value in both.
2
Determine the value of one ratio unit.
Municipal buildings equal 33 ratio units, which corresponds to 4545 MWh. Therefore, 11 ratio unit =453=15= \frac{45}{3} = 15 MWh.
Knowing the actual quantity associated with one component allows us to find the constant multiplier for the ratio.
3
Calculate the total energy distributed.
Total ratio units =20+12+3=35= 20 + 12 + 3 = 35 units. Total energy =35×15=525= 35 \times 15 = 525 MWh.
Multiply the sum of all ratio parts by the value of a single ratio unit to find the grand total.

Key Concept

Combining Compound Ratios and Solving Proportions
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Topics

Ratios, Rates, and Proportions

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