In a school auditorium, \frac{3}{8} of the total seats are in the balcony, and the remaini… — ACT

In a school auditorium, $\frac{3}{8}$ of the total seats are in the balcony, and the remaining seats are on the main floor. For the spring play, $80\%$ of the balcony seats were occupied, and $60\%$ of the main floor seats were occupied. If there were exactly 117 empty seats in the auditorium, what was the total number of seats in the auditorium?

Cevap: 360 seats

Cevap

The total number of seats in the auditorium is 360.

The correct answer of 360 is found by expressing the empty seats from each section as a fraction of the total seats. Since the balcony represents

\frac{3}{8}

of the total seats and is

20\%

empty, the empty balcony seats equal

\frac{3}{40}

of the total seats. The main floor represents

\frac{5}{8}

of the total seats and is

40\%

empty, so the empty main floor seats equal

\frac{10}{40}

of the total seats. Summing these yields

\frac{13}{40}

of the total seats, which equals 117. Solving

\frac{13}{40}S = 117

gives a total of 360 seats.

Adım Adım Çözüm

Define the variable for the total seats and find the fraction of seats in each section.

Let the total number of seats be

S

. The balcony has

\frac{3}{8}S

seats, and the main floor has

1 - \frac{3}{8} = \frac{5}{8}S

seats.

Establishing fractional partitions of the total capacity is necessary to set up the equation.

Calculate the fraction of total seats that are empty in the balcony.

Empty balcony seats =

20\%

\frac{3}{8}S = 0.20 \times \frac{3}{8}S = \frac{1}{5} \times \frac{3}{8}S = \frac{3}{40}S

Since 80% of the balcony seats are occupied, the remaining 20% are empty.

Calculate the fraction of total seats that are empty on the main floor.

Empty main floor seats =

40\%

\frac{5}{8}S = 0.40 \times \frac{5}{8}S = \frac{2}{5} \times \frac{5}{8}S = \frac{10}{40}S

Since 60% of the main floor seats are occupied, the remaining 40% are empty.

Sum the empty seat fractions and set the sum equal to the total number of empty seats.

Total empty seats =

\frac{3}{40}S + \frac{10}{40}S = \frac{13}{40}S

. The equation is

\frac{13}{40}S = 117

Combining the empty seats from both sections represents the total empty seat count.

Solve the equation for the total number of seats

S

S = 117 \times \frac{40}{13} = 9 \times 40 = 360

Multiplying by the reciprocal of the empty seats fraction isolates the variable for the total seats.

Anahtar Kavram

Solving multi-step word problems involving fractions, decimal conversions, and percentages.

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