2.4 Solve Mixture and Uniform Motion Applications

Topics covered in this section are:

• Solve coin word problems
• Solve ticket and stamp word problems
• Solve mixture word problems
• Solve uniform motion applications

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.

$number \cdot value = total \ value$ $23 \cdot \$0.10 = \$2.30$

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?

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.

Example 2

Danny paid $\$15.75$ for stamps. The number of $49$-cent stamps was five less than three times the number of $35$-cent stamps. How many $49$-cent stamps and how many $35$-cent stamps did Danny buy?

In most of our examples so far, we have been told that one quantity is four more than twice the other, or something similar. In our next example, we have to relate the quantities in a different way.

Suppose Aniket sold a total of $100$ tickets. Each ticket was either an adult ticket or a child ticket. If he sold $20$ child tickets, how many adult tickets did he sell?

  Did you say “$80$”? How did you figure that out? Did you subtract $20$ from $100$?

If he sold $45$ child tickets, how many adult tickets did he sell?

  Did you say “$55$”? How did you find it? By subtracting $45$ from $100$?

Now, suppose Aniket sold x child tickets. Then how many adult tickets did he 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 in the table.

We will apply this technique in the next example.

Example 3

A whale-watching ship had $40$ paying passengers on board. The total revenue 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?

2.4.3 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.

Example 4

Henning is mixing raisins and nuts to make $25$ pounds of trail mix. Raisins cost $\$4.50$ a pound and nuts cost $\$8$ a pound. If Henning wants his cost for the trail mix to be $\$6.60$ a pound, how many pounds of raisins and how many pounds of nuts should he use?

2.4.4 Solve Uniform Motion Applications

When you are driving down the interstate using your cruise control, the speed of your car stays the same—it is uniform. We call a problem in which the speed of an object is constant a uniform motion application. We will use the distance, rate, and time formula, $D=rt$, to compare two scenarios, such as two vehicles travelling at different rates or in opposite directions.

Our problem solving strategies will still apply here, but we will add to the first step. The first step will include drawing a diagram that shows what is happening in the example. Drawing the diagram helps us understand what is happening so that we will write an appropriate equation. Then we will make a table to organize the information, like we did for the coin, ticket, and stamp applications.

The steps are listed here for easy reference:

HOW TO: Solve a uniform motion application.

1. Read the problem. Make sure all the words and ideas are understood.
   • Draw a diagram to illustrate what is happening.
   • Create a table to organize the information.
     • Label the columns rate, time, distance.
     • List the two scenarios.
     • Write in the information you know.
2. Identify what you are looking for.
3. Name what you are looking for. Choose a variable to represent that quantity.
   • Complete the chart.
   • Use variable expressions to represent that quantity in each row.
   • Multiply the rate times the time to get the distance.
4. Translate into an equation.
   • Restate the problem in one sentence with all the important information.
   • Then, translate the sentence into an equation.
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.

Example 5

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne two hours to ride to the beach while it takes Dennis $1.5$ hours for the ride. Find the speed of both bikers.

In Example 5, we had two bikers traveling the same distance. In the next example, two people drive toward each other until they meet.

Example 6

Carina is driving from her home in Anaheim to Berkeley on the same day her brother is driving from Berkeley to Anaheim, so they decide to meet for lunch along the way in Buttonwillow. The distance from Anaheim to Berkeley is $395$ miles. It takes Carina three hours to get to Buttonwillow, while her brother drives four hours to get there. Carina’s average speed is $15$ miles per hour faster than her brother’s average speed. Find Carina’s and her brother’s average speeds.

As you read the next example, think about the relationship of the distances traveled. Which of the previous two examples is more similar to this situation?

Example 7

Two truck drivers leave a rest area on the interstate at the same time. One truck travels east and the other one travels west. The truck traveling west travels at $70$ mph and the truck traveling east has an average speed of $60$ mph. How long will they travel before they are $325$ miles apart?

It is important to make sure that the units match when we use the distance rate and time formula. For instance, if the rate is in miles per hour, then the time must be in hours.

Example 8

When Naoko walks to school, it takes her $30$ minutes. If she rides her bike, it takes her $15$ minutes. Her speed is three miles per hour faster when she rides her bike than when she walks. What is her speed walking and her speed riding her bike?

In the distance, rate and time formula, time represents the actual amount of elapsed time (in hours, minutes, etc.). If a problem gives us starting and ending times as clock times, we must find the elapsed time in order to use the formula.

Example 9

Cruz is training to compete in a triathlon. He left his house at 6:00 and ran until 7:30. Then he rode his bike until 9:45. He covered a total distance of $51$ miles. His speed when biking was $1.6$ times his speed when running. Find Cruz’s biking and running speeds.