# 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 A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run. Elizabeth Reply ggfcc Mike Washing his dad’s car alone, eight year old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself? Gagan Reply 1,75hrs Mike I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3? John Oops I mean 1.75 John I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself. John Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball? Marie Reply Brandon has a cup of quarters and dimes with a total value of$3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have?
Tickets to a Broadway show cost $35 for adults and$15 for children. The total receipts for 1650 tickets at one performance were $47,150. How many adult and how many child tickets were sold? dana Reply 825 Carol Arnold invested$64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year? How do I do this Tanesia Reply how to square Fiona Reply easiest way to find the square root of a large number? Jackie the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0 Sabee Reply What is observation adeyemi Reply I'm confused by the question. Can you describe or explain the math question it pertains to? Melissa there is no math to it because all you use is your vision or gaze to the sorrounding areas Cesarp Teegan likes to play golf. He has budgeted$60 next month for the driving range. It costs him $10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month? Sunnyshay Reply 5 times max Anton Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her? Sophia Reply 35 min Debra Carmen wants to tile the floor of his house. He will need 1,000 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?
what you wanna get
Cesar
800 sq. ft @ $1.50 & 200 sq. ft @$9.00
Marco
Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be? 74.25 × .16 then get the total and that will be your tip David$74.25 x 0.16 = $11.88 total bill:$74.25 + $11.88 =$86.13
ericka
yes and tip 16% will be $11.88 David what is the shorter way to do it Cesar Reply Priam has dimes and pennies in a cup holder in his car. The total value of the coins is$4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?