Word Problems: Tickets: Example 1

Help on this type of problem.

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Ticket problems are similar to the coin problem.
We are interested in the number of tickets in each price group, the value or price of each type (orchestra, balcony, etc.) and the total value of a collection of tickets. Consider the following problem. Orchestra seats to a show are $80 and balcony seats are $60. If a theater club spends $3760 on the purchase of 56 seats, how many orchestra seats were purchased?

What is the unknown in this problem?

The number of orchestra seats purchased is the unknown.

Define a variable for this problem.

Let x = the number of orchestra seats.

Express the number of balcony seats purchased in terms of x.

Since 56 seats were purchased in total,
the number of balcony seats is 56 – x.

Express the dollars spent on orchestra seats.

Since each seat costs $80, x seats must cost x times as much, or 80x.

Express the dollars spent on balcony seats.

Each seat costs $60. The number of seats is (56 – x). Multiply these together to get the cost of the seats: 60 * (56 – x).

How are these two amounts of money related to the $3760?

When added together, they are equal to the $3760.

Express this in terms of x.

80x + 60*(56 – x) = 3760

We need to solve for x. First distribute the factor of 60.

80x + 3360 – 60x = 3760

Collect like terms on the left side.

20x + 3360 = 3760

Subtract 3360 from both sides.

20x = 3760 – 3360, or
20x = 400

Solve for x.

Summarize these results.

20 orchestra seats and
56 – 20 = 36 balcony seats were purchased.

Check in the original words.

20 seats @ $80 = $1600
36 seats @ $60 = $2160
Total: 56 seats @ $3760. It checks!

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