At a community theater performance, the ratio of adult tickets sold to student tickets sol… — ACT

At a community theater performance, the ratio of adult tickets sold to student tickets sold was $5 : 3$ . After $12$ additional student tickets were sold and no additional adult tickets were sold, the ratio of adult tickets to student tickets became $4 : 3$ . How many total tickets (adult and student combined) were sold originally?

A
48
B
80
C
96
128Answer
E
140

Answer

128 total tickets were sold originally.

Let the original number of adult tickets be

5x

and student tickets be

3x

. After 12 student tickets are added, the new ratio is

\\frac{5x}{3x + 12} = \\frac{4}{3}

. Cross-multiplying gives

15x = 12x + 48

, which simplifies to

3x = 48

and

x = 16

. The original total number of tickets is

5x + 3x = 8x = 8(16) = 128

Step-by-Step Solution

Define variables for the original quantities based on the initial ratio.

Let the original number of adult tickets be

5x

and the original number of student tickets be

3x

, where

x

is a common multiplier. The original total is

5x + 3x = 8x

Representing a ratio

5 : 3

with a multiplier allows algebraic manipulation when quantities change.

Set up an equation representing the new ratio after adding 12 student tickets.

\\frac{5x}{3x + 12} = \\frac{4}{3}

The number of adult tickets remains

5x

, while student tickets increase by 12 to

3x + 12

Solve for the multiplier

x

using cross-multiplication.

3(5x) = 4(3x + 12) \\implies 15x = 12x + 48 \\implies 3x = 48 \\implies x = 16

Equating cross-products isolates

x

to find the value of one ratio unit.

Calculate the original total number of tickets.

\\text{Original Total} = 8x = 8(16) = 128

Multiplying the common multiplier by 8 yields the combined original total.

Key Concept

Solving multi-step proportion problems involving changing part-to-part ratios.

Estimated Time:1m 15s

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