Practice Solving Mixture Word Problems
Solve 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 3.29
At a school concert, the total value of tickets sold was $1,506. Student tickets sold for $6 each and adult tickets sold for $9 each. The number of adult tickets sold was five less than three times the number of student tickets sold. How many student tickets and how many adult tickets were sold?
Solution
Step 1. Read the problem.
- Determine the types of tickets involved. There are student tickets and adult tickets.
- Create a table to organize the information.
|
Type |
Number • Value($) = Total Value($) |
||
|
Student |
|
\(6\) |
|
|
Adult |
|
\(9\) |
|
|
|
\(1506\) |
||
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.
|
Type |
Number • Value($) = Total Value($) |
||
|
Student |
\(s\) |
\(6\) |
\(6s\) |
|
Adult |
\(3s-5\) |
\(9\) |
\(9(3s - 5) \) |
|
|
\(1506\) |
||
Step 4. Translate. Write the equation by adding the total values of each type of ticket.
\(6s+9(3 s-5)=1506\)
Step 5. Solve the equation.
| \(\begin{aligned} 6 s+27 s-45 &=1506 \\ 33 s-45 &=1506 \\ 33 s &=1551 \\ s &=47 \text { student tickets } \end{aligned}\) |
| \(3 s-5\) \(3(47)-5\) |
| 136 adult tickets |
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.
\(\begin{aligned} 47 \cdot 6 &=282 \\ 136 \cdot 9 &=\frac{1,224}{1,506 \, \text{✓}} \end{aligned}\)
Step 7. Answer the question. They sold 47 student tickets and 136 adult tickets.
Try It 3.57
The
first day of a water polo tournament the total value of tickets sold
was $17,610. One-day passes sold for $20 and tournament passes sold for
$30. The number of tournament passes sold was 37 more than the number of
day passes sold. How many day passes and how many tournament passes
were sold?
Try It 3.58
At the movie theater, the total value
of tickets sold was $2,612.50. Adult tickets sold for $10 each and
senior/child tickets sold for $7.50 each. The number of senior/child
tickets sold was 25 less than twice the number of adult tickets sold.
How many senior/child tickets and how many adult tickets were sold?
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.
| Child tickets |
Adult tickets |
|---|---|
| 20 |
80 |
| 45 |
55 |
| 75 |
25 |
| \(x\) |
\(100-x\) |
We can apply these techniques to other examples
Example 3.30
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 $5 each. How many
children's tickets and how many adult tickets did he sell?
Solution
Step 1. Read the problem.
- Determine the types of tickets involved. There are children tickets and adult tickets.
- Create a table to organize the information.
|
Type |
Number • Value($) = Total Value($) |
||
|
Children |
\(3\) |
||
|
Adult |
\(5\) |
||
|
|
\(2820\) |
||
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.
|
Type |
Number • Value($) = Total Value($) |
||
|
Children |
\(c\) |
\(3\) |
\(3c\) |
|
Adult |
\(810-c\) |
\(5\) |
\(5(810-c)\) |
|
|
\(2820\) |
||
Step 4. Translate.
Write the equation by adding the total values of each type of ticket.
Step 5. Solve the equation.
\(\begin{aligned} 3 c+5(810-c) &=2,820 \\ 3 c+4,050-5 c &=2,820 \\-2 c &=-1,230 \\ c &=615 \text { children tickets } \end{aligned}\)
How many adults?
\(\begin{array}{l} 810-c \\810-615 \\\text{195 adult tickets} \end{array}\)
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?
\(\begin{aligned} 615 \cdot 3 &=1845 \\ 195 \cdot 5 &=\frac{975}{2,820 \text{✓}} \end{aligned}\)
Step 7. Answer the question. Galen sold 615 children’s tickets and 195 adult tickets.
Try It 3.59
During
her shift at the museum ticket booth, Leah sold 115 tickets for a total
of $1,163. Adult tickets cost $12 and student tickets cost $5. How many
adult tickets and how many student tickets did Leah sell?
Try It 3.60
A
whale-watching ship had 40 paying passengers on board. The total
collected from tickets was $1,196. Full-fare passengers paid $32 each
and reduced-fare passengers paid $26 each. How many full-fare passengers
and how many reduced-fare passengers were on the ship?
Now, we'll do one where we fill in the table all at once.
Example 3.31
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".
Let \(x=\) number of 2 -cent stamps.
\(2 x+4=\) number of 41-cent stamps
|
Type |
Number• Value($) = Total Value($) |
||
|
41 cent stamps |
\(2x + 4\) |
\(0.41\) |
\(0.41(2x+ 4)\) |
|
2 cent stamps |
\(x\) |
\(0.02\) |
\(0.02x\) |
|
\(8.36\) |
|||
| Write the equation from the total values. | \(0.41(2 x+4)+0.02 x=8.36\) |
| Solve the equation. | \(\begin{aligned} 0.82 x+1.64+0.02 x &=8.36 \\ 0.84 x+1.64 &=8.36 \\ 0.84 x &=6.72 \\ x &=8 \end{aligned}\) |
| Monica bought eight two-cent stamps. | \(2 x+4\) for \(x=8\) |
| Find the number of 41 -cent stamps she bought be evaluating. | \( \begin{array}{c} 2 x+4 \\2(8)+4 \\ 20\end{array} \) |
| Check. |
\(\begin{aligned} 8(0.02)+20(0.41) & \stackrel{?}{=} 8.36 \\ 0.16+8.20 & \stackrel{?}{=} 8.36 \\ 8.36 &=8.36 \text{✓} \end{aligned}\) |
| Monica bought eight two-cent stamps and 20 41-cent stamps. |
Table 3.9
Try It 3.61
Eric
paid $13.36 for stamps. The number of 41-cent stamps was eight more
than twice the number of two-cent stamps. How many 41-cent stamps and
how many two-cent stamps did Eric buy?
Try It 3.62
Kailee paid $12.66 for stamps. The number of 41-cent stamps was four less than three times the number of 20-cent stamps. How many 41-cent stamps and how many 20-cent stamps did Kailee buy?