Solving Ticket and Stamp Word Problems

Problems involving tickets or stamps are very much like coin problems. Each type of ticket and stamp has a value, just like each type of coin does. So to solve these problems, we will follow the same steps we used to solve coin problems.

Example

At a school concert, the total value of tickets sold was $1,506. Student tickets sold for $6 each and adult tickets sold for $9 each. The number of adult tickets sold was five less than three times the number of student tickets sold. How many student tickets and how many adult tickets were sold?

Solution

Step 1. Read the problem.

Determine the types of tickets involved. There are student tickets and adult tickets.
Create a table to organize the information.

$This table has three rows and four columns with an extra cell at the bottom of the fourth column. The top row is a header row that reads from left to right Type, Number, Value (\$), and Total Value (\$). The second row reads Student, blank, 6, and blank. The third row reads Adult, blank, 9, and blank. The extra cell reads 1506.$

Step 2. Identify what we are looking for.

We are looking for the number of student and adult tickets.

Step 3. Name. Represent the number of each type of ticket using variables.

We know the number of adult tickets sold was five less than three times the number of student tickets sold.

Let s be the number of student tickets.
Then $3s-5$ is the number of adult tickets

Multiply the number times the value to get the total value of each type of ticket.

$This table has three rows and four columns with an extra cell at the bottom of the fourth column. The top row is a header row that reads from left to right Type, Number, Value (\$), and Total Value (\$). The second row reads Student, s, 6, and 6s. The third row reads Adult, 3s minus 5, 9, and 9 times the quantity (3s minus 5). The extra cell reads 1506.$

Step 4. Translate. Write the equation by adding the total values of each type of ticket.

$6s+9\left(3s-5\right)=1506$

Step 5. Solve the equation.

$\begin{array}{ccc}\hfill 6s+27s-45& =\hfill & 1506\hfill \\ \hfill 33s-45& =\hfill & 1506\hfill \\ \hfill 33s& =\hfill & 1551\hfill \\ \hfill s& =\hfill & 47\phantom{\rule{0.2em}{0ex}}\text{student tickets}\hfill \end{array}$

$\begin{array}{c}3s-5\hfill \\ 3\left(47\right)-5\hfill \end{array}$

$\begin{array}{c}136\phantom{\rule{0.2em}{0ex}}\text{adult tickets}\hfill \end{array}$

Step 6. Check the answer.

There were 47 student tickets at $6 each and 136 adult tickets at $9 each. Is the total value $1,506? We find the total value of each type of ticket by multiplying the number of tickets times its value then add to get the total value of all the tickets sold.

$\begin{array}{ccc}\hfill 47·6& =\hfill & \phantom{\rule{0.8em}{0ex}}282\hfill \\ \hfill 136·9& =\hfill & \underset{\text{_____}}{1,224}\hfill \\ & & 1,506✓\hfill \end{array}$

Step 7. Answer the question. They sold 47 student tickets and 136 adult tickets.

We have learned how to find the total number of tickets when the number of one type of ticket is based on the number of the other type. Next, we’ll look at an example where we know the total number of tickets and have to figure out how the two types of tickets relate.

Suppose Bianca sold a total of 100 tickets. Each ticket was either an adult ticket or a child ticket. If she sold 20 child tickets, how many adult tickets did she sell?

Did you say ‘80’? How did you figure that out? Did you subtract 20 from 100?

If she sold 45 child tickets, how many adult tickets did she sell?

Did you say ‘55’? How did you find it? By subtracting 45 from 100?

What if she sold 75 child tickets? How many adult tickets did she sell?

The number of adult tickets must be $100-75.$ She sold 25 adult tickets.

Now, suppose Bianca sold x child tickets. Then how many adult tickets did she sell? To find out, we would follow the same logic we used above. In each case, we subtracted the number of child tickets from 100 to get the number of adult tickets. We now do the same with x.

We have summarized this below.

This table has five rows and two columns. The top row is a header row that reads from left to right Child tickets and Adult tickets. The second row reads 20 and 80. The third row reads 45 and 55. The fourth row reads 75 and 25. The fifth row reads x and 100 plus x.

We can apply these techniques to other examples

Example

Galen sold 810 tickets for his church’s carnival for a total of $2,820. Children’s tickets cost $3 each and adult tickets cost $5 each. How many children’s tickets and how many adult tickets did he sell?

Solution

Step 1. Read the problem.

Determine the types of tickets involved. There are children tickets and adult tickets.
Create a table to organize the information.

$This table has three rows and four columns with an extra cell at the bottom of the fourth column. The top row is a header row that reads from left to right Type, Number, Value (\$), and Total Value (\$). The second row reads Children, blank, 3, and blank. The third row reads Adult, blank, 5, and blank. The extra cell reads 2820.$

Step 2. Identify what we are looking for.

We are looking for the number of children and adult tickets.

Step 3. Name. Represent the number of each type of ticket using variables.

We know the total number of tickets sold was 810. This means the number of children’s tickets plus the number of adult tickets must add up to 810.
Let c be the number of children tickets.
Then $810-c$ is the number of adult tickets.
Multiply the number times the value to get the total value of each type of ticket.

$This table has three rows and four columns with an extra cell at the bottom of the fourth column. The top row is a header row that reads from left to right Type, Number, Value (\$), and Total Value (\$). The second row reads Children, c, 3, and 3c. The third row reads Adult, 810 minus c, 5, and 5 times the quantity (810 minus c). The extra cell reads 2820.$

Step 4. Translate.

Write the equation by adding the total values of each type of ticket.

Step 5. Solve the equation.

$\begin{array}{ccc}\hfill 3c+5\left(810-c\right)& =\hfill & 2,820\hfill \\ \hfill 3c+4,050-5c& =\hfill & 2,820\hfill \\ \hfill -2c& =\hfill & -1,230\hfill \\ \hfill c& =\hfill & 615\phantom{\rule{0.2em}{0ex}}\text{children tickets}\hfill \end{array}$

How many adults?

$810-c$

$810-615$

$195\phantom{\rule{0.2em}{0ex}}\text{adult tickets}$

Step 6. Check the answer. There were 615 children’s tickets at $3 each and 195 adult tickets at $5 each. Is the total value $2,820?

$\begin{array}{ccc}\hfill 615·3& =\hfill & 1845\hfill \\ \hfill 195·5& =\hfill & \phantom{\rule{0.3em}{0ex}}\underset{\text{____}}{975}\hfill \\ & & 2,820✓\hfill \end{array}$

Step 7. Answer the question. Galen sold 615 children’s tickets and 195 adult tickets.

Now, we’ll do one where we fill in the table all at once.

Example

Monica paid $8.36 for stamps. The number of 41-cent stamps was four more than twice the number of two-cent stamps. How many 41-cent stamps and how many two-cent stamps did Monica buy?

Solution

The types of stamps are 41-cent stamps and two-cent stamps. Their names also give the value!

“The number of 41-cent stamps was four more than twice the number of two-cent stamps.”

$\begin{array}{c}\text{Let}\phantom{\rule{0.2em}{0ex}}x=\text{number of 2-cent stamps.}\hfill \\ 2x+4=\text{number of 41-cent stamps}\hfill \end{array}$

$This table has three rows and four columns with an extra cell at the bottom of the fourth column. The top row is a header row that reads from left to right Type, Number, Value (\$), and Total Value (\$). The second row reads 41 cent stamps, 2x plus 4, 0.41, and 0.41 times the quantity (2x plus 4). The third row reads 2 cent stamps, x, 0.02, and 0.02x. The extra cell reads 8.36.$

\(\begin{array}{cccc}\text{Write the equation from the total values.}\hfill & & & \hfill 0.41\left(2x+4\right)+0.02x=8.36\hfill \\ \text{Solve the equation.}\hfill & & & \hfill \begin{array}{ccc}\hfill 0.82x+1.64+0.02x& =\hfill & 8.36\hfill \\ \hfill 0.84x+1.64& =\hfill & 8.36\hfill \\ \hfill 0.84x& =\hfill & 6.72\hfill \\ \hfill x& =\hfill & 8\hfill \end{array}\hfill \\ \begin{array}{c}\text{Monica bought eight two-cent stamps.}\hfill \\ \text{Find the number of 41-cent stamps she bought}\hfill \\ \text{by evaluating}\hfill \end{array}\hfill & & & \hfill \begin{array}{}\\ \\ \\ \hfill 2x+4\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}x=8.\hfill \\ \hfill 2x+4\hfill \\ \hfill 2\left(8\right)+4\hfill \\ \hfill 20\hfill \end{array}\hfill \\ \text{Check.}\hfill & & & \\ \hfill \begin{array}{ccc}\hfill 8\left(0.02\right)+20\left(0.41\right)& \stackrel{?}{=}\hfill & 8.36\hfill \\ \hfill 0.16+8.20& \stackrel{?}{=}\hfill & 8.36\hfill \\ \hfill 8.36& =\hfill & 8.36✓\hfill \end{array}\hfill & & & \\ & & & \begin{array}{c}\text{Monica bought eight two-cent stamps and 20}\hfill \\ \text{41-cent stamps.}\hfill \end{array}\hfill \end{array}\)

Solving Ticket and Stamp Word Problems