9.2 Solve Money Applications

Learning objectives

Learning objectives

• Solve coin word problems

• Solve ticket and stamp word problems

Solve Coin Word Problems

Imagine taking a handful of coins from your pocket or purse and placing them on your desk. How would you determine the value of that pile of coins?

If you can form a step-by-step plan for finding the total value of the coins, it will help you as you begin solving coin word problems.

One way to bring some order to the mess of coins would be to separate the coins into stacks according to their value. Quarters would go with quarters, dimes with dimes, nickels with nickels, and so on. To get the total value of all the coins, you would add the total value of each pile.

How would you determine the value of each pile? Think about the dime pile—how much is it worth? If you count the number of dimes, you'll know how many you have—the number of dimes.

But this does not tell you the value of all the dimes. Say you counted $17$ dimes, how much are they worth? Each dime is worth $\text{\textdollar 0.10}$—that is the value of one dime. To find the total value of the pile of $17$ dimes, multiply $17$ by $\text{\textdollar 0.10}$ to get $\text{\textdollar 1.70}.$ This is the total value of all $17$ dimes.

You could continue this process for each type of coin, and then you would know the total value of each type of coin. To get the total value of all the coins, add the total value of each type of coin.

Let's look at a specific case. Suppose there are $14$ quarters, $17$ dimes, $21$ nickels, and $39$ pennies. We'll make a table to organize the information – the type of coin, the number of each, and the value.

Table

9.1

The total value of all the coins is $\text{\textdollar 6.64}.$ Notice how Table 9.1 helped us organize all the information. Let's see how this method is used to solve a coin word problem.

Example 9.11

Solution

1. Express both coin counts. Let $d$ be the number of dimes. Since Adalberto has nine more nickels than dimes, the number of nickels is $d+9$.
2. Write the total-value equation. The dimes contribute $0.10d$ dollars and the nickels contribute $0.05(d+9)$ dollars, so add them to obtain $0.10d+0.05(d+9)=2.25$.
3. Solve the equation. Distributing gives $0.10d+0.05d+0.45=2.25$, hence $0.15d=1.80$ and $d=12$. The number of nickels is $d+9=21$.
4. Check both conditions. Twelve dimes are worth $12(0.10)=1.20$ dollars, and twenty-one nickels are worth $21(0.05)=1.05$ dollars. Their total is $2.25$ dollars, and the nickel count exceeds the dime count by $9$.
5. State the coin counts. Adalberto has $12$ dimes and $21$ nickels.

Takeaway: Writing the number of nickels as $d+9$ gives $0.10d+0.05(d+9)=2.25$, so Adalberto has $12$ dimes and $21$ nickels.

Example 9.12

Solution

1. Express both coin counts. Let $q$ be the number of quarters. Maria has twice as many pennies, so the number of pennies is $2q$.
2. Write the total-value equation. The quarter value is $0.25q$ dollars and the penny value is $0.01(2q)$ dollars, giving $0.25q+0.01(2q)=2.43$.
3. Combine and solve. The equation becomes $0.25q+0.02q=2.43$, so $0.27q=2.43$ and $q=9$. The number of pennies is $2q=18$.
4. Check the money and the ratio. Nine quarters are worth $9(0.25)=2.25$ dollars and eighteen pennies are worth $18(0.01)=0.18$ dollars. The total is $2.43$ dollars, and $18=2(9)$.
5. State the coin counts. Maria has $9$ quarters and $18$ pennies.

Takeaway: Writing the number of pennies as $2q$ gives $0.25q+0.01(2q)=2.43$, so Maria has $9$ quarters and $18$ pennies.

Example 9.13

Solution

1. Express the related coin counts. Let $p$ be the number of pennies. The number of nickels is ten times the penny count plus two, so it is $10p+2$.
2. Write the total-value equation. The pennies contribute $0.01p$ dollars and the nickels contribute $0.05(10p+2)$ dollars. Thus $0.01p+0.05(10p+2)=2.14$.
3. Combine and solve. Distributing gives $0.01p+0.50p+0.10=2.14$, hence $0.51p=2.04$ and $p=4$. The number of nickels is $10(4)+2=42$.
4. Check the money and relationship. Four pennies are worth $0.04$ dollars and forty-two nickels are worth $2.10$ dollars, totaling $2.14$ dollars. Also, $42=10(4)+2$, so the stated relationship holds.
5. State the coin counts. Danny has $4$ pennies and $42$ nickels.

Takeaway: Writing the number of nickels as $10p+2$ gives $0.01p+0.05(10p+2)=2.14$, so Danny has $4$ pennies and $42$ nickels.

Solve Ticket and Stamp Word Problems

The strategies we used for coin problems can be easily applied to some other kinds of problems too. Problems involving tickets or stamps are very similar to coin problems, for example. Like coins, tickets and stamps have different values; so we can organize the information in tables much like we did for coin problems.

Example 9.14

Solution

1. Express the ticket counts. Let $s$ be the number of student tickets. The number of adult tickets is five less than three times $s$, so it is $3s-5$.
2. Write the total-value equation. Student tickets contribute $6s$ dollars and adult tickets contribute $9(3s-5)$ dollars, so $6s+9(3s-5)=1506$.
3. Solve for the student count. Distributing and combining gives $6s+27s-45=1506$, hence $33s=1551$ and $s=47$. The adult count is $3(47)-5=136$.
4. Check the ticket values. The student tickets are worth $47(6)=282$ dollars and the adult tickets are worth $136(9)=1224$ dollars. Their total is $1506$ dollars, and $136=3(47)-5$.
5. State the ticket counts. They sold $47$ student tickets and $136$ adult tickets.

Takeaway: Writing the number of adult tickets as $3s-5$ gives $6s+9(3s-5)=1506$, so $47$ student tickets and $136$ adult tickets were sold.

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

Example 9.15

Solution

1. Express both stamp counts. Let $x$ be the number of $8$-cent stamps. The number of $49$-cent stamps is four more than twice $x$, so it is $2x+4$.
2. Write the total-cost equation. The $8$-cent stamps cost $0.08x$ dollars and the $49$-cent stamps cost $0.49(2x+4)$ dollars, giving $0.49(2x+4)+0.08x=10.44$.
3. Solve for the smaller count. Expanding gives $0.98x+1.96+0.08x=10.44$, so $1.06x=8.48$ and $x=8$. The other count is $2(8)+4=20$.
4. Check the total cost. Eight $8$-cent stamps cost $0.64$ dollars and twenty $49$-cent stamps cost $9.80$ dollars. The sum is $10.44$ dollars, and $20=2(8)+4$.
5. State the stamp counts. Monica bought $8$ eight-cent stamps and $20$ forty-nine-cent stamps.

Takeaway: Writing the number of 49-cent stamps as $2x+4$ gives $0.49(2x+4)+0.08x=10.44$, so Monica bought $20$ expensive stamps and $8$ eight-cent stamps.

Section 9.2 Exercises

Section 9.2 Exercises

Solve Coin Word Problems

In the following exercises, solve the coin word problems.

Solve Ticket and Stamp Word Problems

In the following exercises, solve the ticket and stamp word problems.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

1. Michaela has $\text{\textdollar 2.05}$ in dimes and nickels in her change purse. She has seven more dimes than nickels. How many coins of each type does she have?

2. Liliana has $\text{\textdollar 2.10}$ in nickels and quarters in her backpack. She has $12$ more nickels than quarters. How many coins of each type does she have?

1. Sumanta has $\text{\textdollar 4.20}$ in nickels and dimes in her desk drawer. She has twice as many nickels as dimes. How many coins of each type does she have?

2. Alison has three times as many dimes as quarters in her purse. She has $\text{\textdollar 9.35}$ altogether. How many coins of each type does she have?

1. Jesse has $\text{\textdollar 6.55}$ worth of quarters and nickels in his pocket. The number of nickels is five more than two times the number of quarters. How many nickels and how many quarters does Jesse have?

2. Elaine has $\text{\textdollar 7.00}$ in dimes and nickels in her coin jar. The number of dimes that Elaine has is seven less than three times the number of nickels. How many of each coin does Elaine have?

1. The first day of a water polo tournament, the total value of tickets sold was $\text{\textdollar 17,610}.$ One-day passes sold for $\text{\textdollar 20}$ and tournament passes sold for $\text{\textdollar 30}.$ 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?

2. At the movie theater, the total value of tickets sold was $\text{\textdollar 2,612.50}.$ Adult tickets sold for $\text{\textdollar 10}$ each and senior/child tickets sold for $\text{\textdollar 7.50}$ 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?

1. Eric paid $\text{\textdollar 16.64}$ for stamps so he could mail thank you notes for his wedding gifts. The number of $\mathrm{49-cent}$ stamps was eight more than twice the number of $\mathrm{8-cent}$ stamps. How many $\mathrm{49-cent}$ stamps and how many $\mathrm{8-cent}$ stamps did Eric buy?

2. Kailee paid $\text{\textdollar 14.84}$ for stamps. The number of $\mathrm{49-cent}$ stamps was four less than three times the number of $\mathrm{21-cent}$ stamps. How many $\mathrm{49-cent}$ stamps and how many $\mathrm{21-cent}$ stamps did Kailee buy?

1. Jaime has $\text{\textdollar 2.60}$ in dimes and nickels. The number of dimes is $14$ more than the number of nickels. How many of each coin does he have?

2. Ngo has a collection of dimes and quarters with a total value of $\text{\textdollar 3.50}.$ The number of dimes is $7$ more than the number of quarters. How many of each coin does he have?

1. Carolyn has $\text{\textdollar 2.55}$ in her purse in nickels and dimes. The number of nickels is $9$ less than three times the number of dimes. Find the number of each type of coin.

2. Chi has $\text{\textdollar 11.30}$ in dimes and quarters. The number of dimes is $3$ more than three times the number of quarters. How many dimes and nickels does Chi have?

1. A cash box of $\text{\textdollar 1}$ and $\text{\textdollar 5}$ bills is worth $\text{\textdollar 45}.$ The number of $\text{\textdollar 1}$ bills is $3$ more than the number of $\text{\textdollar 5}$ bills. How many of each bill does it contain?

2. In a cash drawer there is $\text{\textdollar 125}$ in $\text{\textdollar 5}$ and $\text{\textdollar 10}$ bills. The number of $\text{\textdollar 10}$ bills is twice the number of $\text{\textdollar 5}$ bills. How many of each are in the drawer?

1. Mukul has $\text{\textdollar 3.75}$ in quarters, dimes and nickels in his pocket. He has five more dimes than quarters and nine more nickels than quarters. How many of each coin are in his pocket?

2. The play took in $\text{\textdollar 550}$ one night. The number of $8 adult tickets was$10$less than twice the number of$\text{\textdollar 5}$ child tickets. How many of each ticket were sold?

1. The movie theater took in $\text{\textdollar 1,220}$ one Monday night. The number of $\text{\textdollar 7}$ child tickets was ten more than twice the number of $\text{\textdollar 9}$ adult tickets. How many of each were sold?

2. Julie went to the post office and bought both $\text{\textdollar 0.49}$ stamps and $\text{\textdollar 0.34}$ postcards for her office's bills She spent $\text{\textdollar 62.60}.$ The number of stamps was $20$ more than twice the number of postcards. How many of each did she buy?

1. Maria spent $\text{\textdollar 16.80}$ at the post office. She bought three times as many $\text{\textdollar 0.49}$ stamps as $\text{\textdollar 0.21}$ stamps. How many of each did she buy?

2. Hilda has $\text{\textdollar 210}$ worth of $\text{\textdollar 10}$ and $\text{\textdollar 12}$ stock shares. The numbers of $\text{\textdollar 10}$ shares is $5$ more than twice the number of $\text{\textdollar 12}$ shares. How many of each does she have?

1. Parent Volunteer As the treasurer of her daughter's Girl Scout troop, Laney collected money for some girls and adults to go to a $\mathrm{3-day}$ camp. Each girl paid $\text{\textdollar 75}$ and each adult paid $\text{\textdollar 30}.$ The total amount of money collected for camp was $\text{\textdollar 765}.$ If the number of girls is three times the number of adults, how many girls and how many adults paid for camp?

2. Suppose you have $6$ quarters, $9$ dimes, and $4$ pennies. Explain how you find the total value of all the coins.

1. In the table used to solve coin problems, one column is labeled “number” and another column is labeled ‘“value.” What is the difference between the number and the value?