# Math Models

## Solve Mixture Applications


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

### 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.
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?
1. 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?
1. 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?
1. The number of adult tickets must be 
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
Now, we’ll do one where we fill in the table all at once.

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

### Key Concepts

1. Total Value of Coins For the same type of coin, the total value of a number of coins is found by using the model.
 where number is the number of coins and value is the value of each coin; total value is the total value of all the coins
2. Problem-Solving Strategy—Coin Word Problems



### Practice Makes Perfect

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


### 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?