# 2.4 Solve Mixture and Uniform Motion Applications

Topics covered in this section are:

## 2.4.1 Solve Coin Word Problems

Using algebra to find the number of nickels and pennies in a piggy bank may seem silly. You may wonder why we just don’t open the bank and count them. But this type of problem introduces us to some techniques that will be useful as we move forward in our study of mathematics.

If we have a pile of dimes, how would we determine its value? 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 $23$ 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$23$dimes, multiply$23$by$\$0.10$ to get $\$2.30$. The number of dimes times the value of each dime equals the total value of the dimes. $number \cdot value = total \ value23 \cdot \$0.10 = \$2.30$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$number \cdot value = total \ value$• number is the number of coins • value is the value of each coin • total value is the total value of all the coins If we had several types of coins, 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. #### Example 1 Jesse has$\$3.02$ worth of pennies and nickels in his piggy bank. The number of nickels is three more than eight times the number of pennies. How many nickels and how many pennies does Jesse have?

Solution

The steps for solving a coin word problem are summarized below.

### HOW TO: Solve coin word problems.

1. Read the problem. Make sure all the words and ideas are understood.
• Determine the types of coins involved.
• Create a table to organize the information.
• Label the columns “type,” “number,” “value,” and “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.
2. Identify what you are looking for.
3. Name what you 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.
4. Translate into an equation.
• It may be helpful to restate the problem in one sentence with all the important information. Then, translate the sentence into an equation.
• Write the equation by adding the total values of all the types of coins.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

## 2.4.2 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.

Solution

## 2.4.3 Solve Mixture Word Problems

Now we’ll solve some more general applications of the mixture model. In mixture problems, we are often mixing two quantities, such as raisins and nuts, to create a mixture, such as trail mix. In our tables we will have a row for each item to be mixed as well as one for the final mixture.