A municipal solar power facility distributes its total generating capacity among three sec… — ACT

A municipal solar power facility distributes its total generating capacity among three sectors: residential, commercial, and municipal. The residential sector is allocated $\frac{7}{15}$ of the total capacity, and the commercial sector is allocated $36\%$ of the total capacity. The remaining $26$ megawatts (MW) of capacity is allocated to municipal buildings. What is the total generating capacity, in megawatts, of the facility?

Answer: 150 megawatts

Answer

The total generating capacity of the facility is 150 megawatts.

The residential portion (

\frac{7}{15}

) and commercial portion (

36\% = \frac{9}{25}

) combine to equal

\frac{62}{75}

of the total capacity. The municipal portion makes up the remaining

\frac{13}{75}

of the total, which is given as

26

MW. Setting

\frac{13}{75} T = 26

and solving yields

T = 150

MW.

Step-by-Step Solution

Convert the commercial percentage into a simplified fraction.

36\% = \frac{36}{100} = \frac{9}{25}

Converting all proportions to fractions allows for direct operations.

Find the combined fraction of capacity allocated to residential and commercial sectors.

\frac{7}{15} + \frac{9}{25} = \frac{35}{75} + \frac{27}{75} = \frac{62}{75}

Finding a common denominator (

75

) allows adding the two fractional parts.

Find the fraction corresponding to the remaining municipal portion.

1 - \frac{62}{75} = \frac{13}{75}

Subtracting the allocated fraction from

1

yields the unallocated fractional portion.

Calculate the total capacity by setting up a linear equation.

\frac{13}{75} \times \text{Total} = 26 \implies \text{Total} = 26 \times \frac{75}{13} = 150

Multiplying the known remaining value by the reciprocal of its fraction gives the whole total.

Key Concept

Combining fractions and percentages to find an unknown total amount.

Estimated Time:1m 30s

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