3.3 Solve mixture applications  (Page 4/10)

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.

We can apply these techniques to other examples

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.

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.

Step 4. Translate.

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

Step 5. Solve the equation.

$\begin{array}{ccc}\hfill 3c+5(810-c)& =\hfill & \mathrm{2,820}\hfill \\ \hfill 3c+\mathrm{4,050}-5c& =\hfill & \mathrm{2,820}\hfill \\ \hfill -2c& =\hfill & \mathrm{-1,230}\hfill \\ \hfill c& =\hfill & 615\text{children tickets}\hfill \end{array}$

How many adults?

$810-c$

$810-615$

$195\text{adult tickets}$

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?

$\begin{array}{ccc}\hfill 615·3& =\hfill & 1845\hfill \\ \hfill 195·5& =\hfill & \underset{\text{\_\_\_\_}}{975}\hfill \\ & & \mathrm{2,820}✓\hfill \end{array}$

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

Got questions? Get instant answers now!

Got questions? Get instant answers now!