At the beginning of a dry season, a municipal water reservoir contains an initial volume o… — GRE General Test

At the beginning of a dry season, a municipal water reservoir contains an initial volume of $V$ gallons of water. During the first month, $\frac{1}{4}$ of the initial volume is released for agricultural irrigation. During the second month, $\frac{1}{5}$ of the remaining volume evaporates, after which $30,000$ gallons of water are pumped into the reservoir from an underground aquifer. If the final volume of water in the reservoir is equal to $\frac{3}{4}$ of the initial volume $V$ , what was the initial volume $V$ , in gallons?

A
75,000
B
150,000
200,000Cevap
D
250,000
E
600,000

Cevap

200,000 gallons

Subtracting the first month's release of

\frac{1}{4}V

leaves

\frac{3}{4}V

. Evaporating

\frac{1}{5}

of this remaining amount leaves

\frac{4}{5} \times \frac{3}{4}V = \frac{3}{5}V

. Adding

30,000

gallons gives a final volume of

\frac{3}{5}V + 30,000

. Setting this equal to

\frac{3}{4}V

results in

\frac{3}{4}V - \frac{3}{5}V = \frac{3}{20}V = 30,000

, which solves to

V = 200,000

Adım Adım Çözüm

Determine the volume remaining after the first month's irrigation release.

The volume remaining is

V - \frac{1}{4}V = \frac{3}{4}V

Releasing

\frac{1}{4}

of the initial volume leaves

1 - \frac{1}{4} = \frac{3}{4}

of the initial volume.

Calculate the volume remaining after the second month's evaporation.

The volume remaining after evaporation is

\left(1 - \frac{1}{5}\right) \times \frac{3}{4}V = \frac{4}{5} \times \frac{3}{4}V = \frac{3}{5}V

Evaporating

\frac{1}{5}

of the remaining water leaves

\frac{4}{5}

of that remaining amount.

Formulate the linear equation including the pumped volume and solve for

V

\frac{3}{5}V + 30,000 = \frac{3}{4}V \implies 30,000 = \frac{3}{4}V - \frac{3}{5}V = \frac{15 - 12}{20}V = \frac{3}{20}V \implies V = \frac{30,000 \times 20}{3} = 200,000$.

Equating the expression for the final water volume to

\frac{3}{4}V

yields a single linear equation in

V

Anahtar Kavram

Sequential fractional reduction of remaining amounts and solving linear equations involving rational numbers.

Soru

Cevap

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