# 3.3 Solve mixture applications

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By the end of this section, you will be able to:
• 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

Before you get started, take this readiness quiz.

1. Multiply: 14(0.25).
If you missed this problem, review [link] .
2. Solve: $0.25x+0.10\left(x+4\right)=2.5.$
If you missed this problem, review [link] .
3. The number of dimes is three more than the number of quarters. Let q represent the number of quarters. Write an expression for the number of dimes.
If you missed this problem, review [link] .

## 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.

Doing the Manipulative Mathematics activity Coin Lab will help you develop a better understanding of mixture word problems.

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. ## Total value of coins For the same type of coin, the total value of a number of coins is found by using the model $number·value=total\phantom{\rule{0.2em}{0ex}}value$ where number is the number of coins value is the value of each coin total value is the total value of all the coins The number of dimes times the value of each dime equals the total value of the dimes. $\begin{array}{ccc}\hfill number·value& =\hfill & total\phantom{\rule{0.2em}{0ex}}value\hfill \\ \hfill 17·0.10& =\hfill & \text{}1.70\hfill \end{array}$ 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.

Adalberto has $2.25 in dimes and nickels in his pocket. He has nine more nickels than dimes. How many of each type of coin does he have? ## Solution Step 1. Read the problem. Make sure all the words and ideas are understood. • Determine the types of coins involved. Think about the strategy we used to find the value of the handful of coins. The first thing we need is to notice what types of coins are involved. Adalberto has dimes and nickels. • Create a table to organize the information. See chart below. • Label the columns “type,” “number,” “value,” “total value.” • List the types of coins. • Write in the value of each type of coin. • Write in the total value of all the coins. We can work this problem all in cents or in dollars. Here we will do it in dollars and put in the dollar sign ($) 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. $\begin{array}{ccc}\hfill \text{Let}\phantom{\rule{0.2em}{0ex}}d& =\hfill & \text{number of dimes.}\hfill \\ \hfill d+9& =\hfill & \text{number of nickels}\hfill \end{array}$ 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 $number·value=total\phantom{\rule{0.2em}{0ex}}value.$ 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.  Now solve this equation. Distribute. Combine like terms. Subtract 0.45 from each side. Divide. So there are 12 dimes. The number of nickels is $d+9$ . 21 $\phantom{\rule{1.2em}{0ex}}$ Step 6. Check the answer in the problem and make sure it makes sense. Does this check? $\begin{array}{cccc}\text{12 dimes}\hfill & & & 12\left(0.10\right)=1.20\hfill \\ \text{21 nickels}\hfill & & & 21\left(0.05\right)=\underset{\text{____}}{1.05}\hfill \\ & & & \phantom{\rule{4.5em}{0ex}}2.25✓\hfill \end{array}$ 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. #### Questions & Answers Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height? genevieve Reply 6.25 Ciid 6.25 Big Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio. Katherine Reply complexed. could you please help me figure out? Ciid (sin=opp/adj) (tan= opp/adj) cos=hyp/adj tyler (sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up tyler (sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up tyler (sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up tyler SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent. tyler H=57 and O=285 figure out what the adjacent? tyler Amara currently sells televisions for company A at a salary of$17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of$29,000 plus a $20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal? Marisol Reply what is the quantity and price of the televisions for both options? karl Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+ Ciid Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 Ciid I'm mathematics teacher from highly recognized university. Mzo Reply here a question professor How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths as many sailors as soldiers? can you write out the college you went to with the name of the school you teach at and let me know the answer I've got it to be honest with you tyler is anyone else having issues with the links not doing anything? Helpful Reply Yes Val chapter 1 foundations 1.2 exercises variables and algebraic symbols theresa Reply June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold? Enter the answers in decimal form. Samer Reply Joseph would like to make 12 pounds of a coffee blend at a cost of$6.25 per pound. He blends Ground Chicory at $4.40 a pound with Jamaican Blue Mountain at$8.84 per pound. How much of each type of coffee should he use?
Samer
4x6.25= $25 coffee blend 4×4.40=$17.60 ground chicory 4x8.84= 35.36 blue mountain. In total they will spend for 12 pounds $77.96 they will spend in total tyler DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for three-quarters of an hour and Fabian rode his bike for half an hour to get to the park. Fabian’s speed was six miles per hour faster than DaMarcus’ speed. Find the speed of both soccer players. Sage Reply i need help how to do this is confusing Alvina Reply what kind of math is it? Danteii help me to understand Alvina Reply huh, what is the algebra problem Daniel How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths many sailors as soldiers? tyler What is the domain and range of heaviside Christopher Reply What is the domain and range of Heaviside and signum Christopher 25-35 Fazal The hypotenuse of a right triangle is 10cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle. Edi Reply Tickets for a show are$70 for adults and $50 for children. For one evening performance, a total of 300 tickets were sold and the receipts totaled$17,200. How many adult tickets and how many child tickets were sold?
A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used?
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold?