3.3 Solve Mixture Applications

Learning objectives

Learning objectives

• Solve coin word problems

• Solve ticket and stamp word problems

• Solve mixture word problems

• Use the mixture model to solve investment problems using simple interest

Solve Coin Word Problems

In mixture problems, we will have two or more items with different values to combine together. The mixture model is used by grocers and bartenders to make sure they set fair prices for the products they sell. Many other professionals, like chemists, investment bankers, and landscapers also use the mixture model.

We will start by looking at an application everyone is familiar with—money!

Imagine that we take a handful of coins from a pocket or purse and place them on a desk. How would we determine the value of that pile of coins? If we can form a step-by-step plan for finding the total value of the coins, it will help us as we begin solving coin word problems.

So what would we do? To get some order to the mess of coins, we could separate the coins into piles 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, we would add the total value of each pile.

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

But this does not tell us the value of all the dimes. Say we counted 17 dimes, how much are they worth? Each dime is worth $0.10—that is the *value* of one dime. To find the total value of the pile of 17 dimes, multiply 17 by$0.10 to get $1.70. This is the total value of all 17 dimes. This method leads to the following model.

The number of dimes times the value of each dime equals the total value of the dimes.

We could continue this process for each type of coin, and then we 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.

The total value of all the coins is $6.64.

Notice how the chart helps organize all the information! Let’s see how we use this method to solve a coin word problem.

Example 3.26

Solution

Step 1. Read the problem. Make sure all the words and ideas are understood.

• Determine the types of coins involved. Think about the strategy we used to find the value of the handful of coins. The first thing we need is to notice what types of coins are involved. Adalberto has dimes and nickels.

• Create a table to organize the information. See chart below. Label the columns “type,” “number,” “value,” “total value.” List the types of coins. Write in the value of each type of coin. Write in the total value of all the coins. We can work this problem all in cents or in dollars. Here we will do it in dollars and put in the dollar sign ($) in the table as a reminder. The value of a dime is$0.10 and the value of a nickel is $0.05. The total value of all the coins is$2.25. The table below shows this information.

Step 2. Identify what we are looking for.

• We are asked to find the number of dimes and nickels Adalberto has.

Step 3. Name what we are looking for. Choose a variable to represent that quantity.

• Use variable expressions to represent the number of each type of coin and write them in the table.

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

Next we counted the number of each type of coin. In this problem we cannot count each type of coin—that is what you are looking for—but we have a clue. There are nine more nickels than dimes. The number of nickels is nine more than the number of dimes.

Fill in the “number” column in the table to help get everything organized.

Now we have all the information we need from the problem!

We multiply the number times the value to get the total value of each type of coin. While we do not know the actual number, we do have an expression to represent it.

And so now multiply $number\cdot{}value=total\,value.$ See how this is done in the table below.

Notice that we made the heading of the table show the model.

Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence. Translate the English sentence into an algebraic equation.

Write the equation by adding the total values of all the types of coins.

Step 5. Solve the equation using good algebra techniques.

Step 6. Check the answer in the problem and make sure it makes sense.

Does this check?

Table

3.7

Step 7. Answer the question with a complete sentence.

• Adalberto has twelve dimes and twenty-one nickels.

If this were a homework exercise, our work might look like the following.

Example 3.27

Solution

Step 1. Read the problem.

Determine the types of coins involved.

We know that Maria has quarters and pennies.

Create a table to organize the information.

• Label the columns “type,” “number,” “value,” “total value.”

• List the types of coins.

• Write in the value of each type of coin.

• Write in the total value of all the coins.

Step 2. Identify what you are looking for.

• We are looking for the number of quarters and pennies.

Step 3. Name. Represent the number of quarters and pennies using variables.

• We know Maria has twice as many pennies as quarters. The number of pennies is defined in terms of quarters.

• Let q represent the number of quarters.

• Then the number of pennies is 2*q*.

Multiply the ‘number’ and the ‘value’ to get the ‘total value’ of each type of coin.

Step 4. Translate. Write the equation by adding the ‘total value’ of all the types of coins.

Table

3.8

In the next example, we’ll show only the completed table—remember the steps we take to fill in the table.

Example 3.28

Solution

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

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.

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.

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

Step 5. Solve the equation.

• $\begin{array}{rll} 6s+27s-45 & = & 1506 \\ 33s-45 & = & 1506 \\ 33s & = & 1551 \\ s & = & 47\,\mathrm{student tickets} \\ \end{array}$

• $\begin{array}{cccr} & & & 3s-5 \\ & & & 3(47)-5 \\ \end{array}$

• 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.

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.

We can apply these techniques to other examples

Example 3.30

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.

How many adults?

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?

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 3.31

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.”

Table

3.9

Solve Mixture Word Problems

Now we’ll solve some more general applications of the mixture model. Grocers and bartenders use the mixture model to set a fair price for a product made from mixing two or more ingredients. Financial planners use the mixture model when they invest money in a variety of accounts and want to find the overall interest rate. Landscape designers use the mixture model when they have an assortment of plants and a fixed budget, and event coordinators do the same when choosing appetizers and entrees for a banquet.

Our first mixture word problem will be making trail mix from raisins and nuts.

Example 3.32

Solution

As before, we fill in a chart to organize our information.

The 10 pounds of trail mix will come from mixing raisins and nuts.

We enter the price per pound for each item.

We multiply the number times the value to get the total value.

Notice that the last line in the table gives the information for the total amount of the mixture.

We know the value of the raisins plus the value of the nuts will be the value of the trail mix.

We can also use the mixture model to solve investment problems using simple interest. We have used the simple interest formula, $I=Prt,$ where $t$ represented the number of years. When we just need to find the interest for one year, $t=1,$ so then $I=Pr.$

Example 3.33

Solution

We will fill in a chart to organize our information. We will use the simple interest formula to find the interest earned in the different accounts.

The interest on the mixed investment will come from adding the interest from the account earning 3% and the interest from the account earning 5% to get the total interest on the $20,000.

The amount invested is the principal for each account.

We enter the interest rate for each account.

We multiply the amount invested times the rate to get the interest.

Notice that the total amount invested, 20,000, is the sum of the amount invested at 3% and the amount invested at 5%. And the total interest, $0.045(20,000),$ is the sum of the interest earned in the 3% account and the interest earned in the 5% account.

As with the other mixture applications, the last column in the table gives us the equation to solve.

Section 3.3 Exercises

Section 3.3 Exercises

Solve Coin Word Problems

In the following exercises, solve each coin word problem.

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

Solve Mixture Word Problems

In the following exercises, solve each mixture word problem.

Everyday Math

Writing Exercises

ⓐ 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 $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 $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?

3. Sumanta has $4.20 in nickels and dimes in her piggy bank. She has twice as many nickels as dimes. How many coins of each type does she have?

1. Alison has three times as many dimes as quarters in her purse. She has $9.35 altogether. How many coins of each type does she have?

2. Jesse has $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?

3. Elane has $7.00 total in dimes and nickels in her coin jar. The number of dimes that Elane has is seven less than three times the number of nickels. How many of each coin does Elane have?

1. 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?

2. 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?

3. 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?

1. 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?

2. 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?

3. 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?

1. Orlando is mixing nuts and cereal squares to make a party mix. Nuts sell for $7 a pound and cereal squares sell for$4 a pound. Orlando wants to make 30 pounds of party mix at a cost of $6.50 a pound, how many pounds of nuts and how many pounds of cereal squares should he use?

2. Becca wants to mix fruit juice and soda to make a punch. She can buy fruit juice for $3 a gallon and soda for$4 a gallon. If she wants to make 28 gallons of punch at a cost of $3.25 a gallon, how many gallons of fruit juice and how many gallons of soda should she buy?

3. Remy has $14,000 to invest in two mutual funds. One fund pays interest at 4% per year and the other fund pays interest at 7% per year. How much should she invest in each fund if she wants to earn 6.1% interest on the total amount?

1. Marco has $8,000 to save for his daughter’s college education. He wants to divide it between one account that pays 3.2% interest per year and another account that pays 8% interest per year. How much should he invest in each account if he wants the interest on the total investment to be 6.5%?

2. Jaime has $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?

3. Ngo has a collection of dimes and quarters with a total value of $3.50. The number of dimes is seven more than the number of quarters. How many of each coin does he have?

1. A cash box of $1 and$5 bills is worth $45. The number of$1 bills is three more than the number of $5 bills. How many of each bill does it contain?

2. Rachelle has $6.30 in nickels and quarters in her coin purse. The number of nickels is twice the number of quarters. How many coins of each type does she have?

3. Harrison has $9.30 in his coin collection, all in pennies and dimes. The number of dimes is three times the number of pennies. How many coins of each type does he have?

1. In a cash drawer there is $125 in$5 and $10 bills. The number of$10 bills is twice the number of $5 bills. How many of each are in the drawer?

2. Carolyn has $2.55 in her purse in nickels and dimes. The number of nickels is nine less than three times the number of dimes. Find the number of each type of coin.

3. Chi has $11.30 in dimes and quarters. The number of dimes is three more than three times the number of quarters. How many of each are there?

1. Mukul has $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 school play sold $550 in tickets one night. The number of$8 adult tickets was 10 less than twice the number of $5 child tickets. How many of each ticket were sold?

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

1. The ice rink sold 95 tickets for the afternoon skating session, for a total of $828. General admission tickets cost$10 each and youth tickets cost $8 each. How many general admission tickets and how many youth tickets were sold?

2. The box office sold 360 tickets to a concert at the college. The total receipts were $4170. General admission tickets cost$15 and student tickets cost $10. How many of each kind of ticket was sold?

3. Julie went to the post office and bought both $0.41 stamps and$0.26 postcards. She spent $51.40. The number of stamps was 20 more than twice the number of postcards. How many of each did she buy?

1. Maria spent $12.50 at the post office. She bought three times as many$0.41 stamps as $0.02 stamps. How many of each did she buy?

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

3. Lauren in making 15 liters of mimosas for a brunch banquet. Orange juice costs her $1.50 per liter and champagne costs her$12 per liter. How many liters of orange juice and how many liters of champagne should she use for the mimosas to cost Lauren $5 per liter?

1. Kaapo is mixing Kona beans and Maui beans to make 25 pounds of coffee blend. Kona beans cost Kaapo $15 per pound and Maui beans cost$24 per pound. How many pounds of each coffee bean should Kaapo use for his blend to cost him $17.70 per pound?

2. Carmen wants to tile the floor of his house. He will need 1000 square feet of tile. He will do most of the floor with a tile that costs $1.50 per square foot, but also wants to use an accent tile that costs$9.00 per square foot. How many square feet of each tile should he plan to use if he wants the overall cost to be $3 per square foot?

3. Vartan was paid $25,000 for a cell phone app that he wrote and wants to invest it to save for his son’s education. He wants to put some of the money into a bond that pays 4% annual interest and the rest into stocks that pay 9% annual interest. If he wants to earn 7.4% annual interest on the total amount, how much money should he invest in each account?

1. Stephanie inherited $40,000. She wants to put some of the money in a certificate of deposit that pays 2.1% interest per year and the rest in a mutual fund account that pays 6.5% per year. How much should she invest in each account if she wants to earn 5.4% interest per year on the total amount?

2. Dominic pays 7% interest on his $15,000 college loan and 12$11,000 car loan. What average interest rate does he pay on the total $26,000 he owes? (Round your answer to the nearest tenth of a percent.)

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

1. Suppose you have six quarters, nine dimes, and four pennies. Explain how you find the total value of all the coins.

2. 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?”