2.4 Solve Mixture and Uniform Motion Applications

Learning objectives

Learning objectives

• Solve coin word problems

• Solve ticket and stamp word problems

• Solve mixture word problems

• Solve uniform motion applications

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.

This method leads to the following model.

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 2.39

Solution

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

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 2.40

Solution

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 2.41

Solution

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 2.42

Solution

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:

Example 2.43

Solution

Step 1. Read the problem. Make sure all the words and ideas are understood.

• Draw a diagram to illustrate what it happening. Shown below is a sketch of what is happening in the example.

• Create a table to organize the information. Label the columns “Rate,” “Time,” and “Distance.” List the two scenarios. Write in the information you know.

Step 2. Identify what you are looking for.

You are asked to find the speed of both bikers.

Notice that the distance formula uses the word “rate,” but it is more common to use “speed”

when we talk about vehicles in everyday English.

Step 3. Name what we are looking for. Choose a variable to represent that quantity.

• Complete the chart

• Use variable expressions to represent that quantity in each row.We are looking for the speed of the bikers. Let’s let r represent Wayne’s speed. Since Dennis’ speed is 7 mph faster, we represent that as $r+7$$\begin{array}{rll} r+7 & = & \mathrm{Dennis' speed} \\ r & = & \mathrm{Wayne's speed} \\ \end{array}$Fill in the speeds into the chart.

• Multiply the rate times the time to get the distance.

Step 4. Translate into an equation.

• Restate the problem in one sentence with all the important information.

• Then, translate the sentence into an equation.The equation to model this situation will come from the relation between the distances. Look at the diagram we drew above. How is the distance travelled by Dennis related to the distance travelled by Wayne?Since both bikers leave from Riverside and travel to the beach, they travel the same distance. So we write:

Step 5. Solve the equation using algebra techniques.

Step 6. Check the answer in the problem and make sure it makes sense.

$\begin{array}{ll} \mathrm{Dennis} & 28\,\mathrm{mph}(1.5\,\mathrm{hours})=42\,\mathrm{miles} \\ \mathrm{Wayne} & 21\,\mathrm{mph}(2\,\mathrm{hours})=42\,\mathrm{miles}\mathrm{✓} \\ \end{array}$

Step 7. Answer the question with a complete sentence.

Wayne rode at 21 mph and Dennis rode at 28 mph.

In Example 2.43, we had two bikers traveling the same distance. In the next example, two people drive toward each other until they meet.

Example 2.44

Solution

Step 1. Read the problem. Make sure all the words and ideas are understood.

• Draw a diagram to illustrate what it happening. Below shows a sketch of what is happening in the example.

• Create a table to organize the information. Label the columns rate, time, distance. List the two scenarios. Write in the information you know.

Step 2. Identify what we are looking for.

We are asked to find the average speeds of Carina and her brother.

Step 3. Name what we are looking for. Choose a variable to represent that quantity.

• Complete the chart.

• Use variable expressions to represent that quantity in each row.We are looking for their average speeds. Let’s let r represent the average speed of Carina's brother. Since Carina’s speed is 15 mph faster, we represent that as $r+15.$Fill in the speeds into the chart.

• Multiply the rate times the time to get the distance.

Step 4. Translate into an equation.

• Restate the problem in one sentence with all the important information.

• Then, translate the sentence into an equation.Again, we need to identify a relationship between the distances in order to write an equation. Look at the diagram we created above and notice the relationship between the distance Carina traveled and the distance her brother traveled.The distance Carina traveled plus the distance her brother travel must add up to 410 miles. So we write:

Step 5. Solve the equation using algebra techniques.

Step 6. Check the answer in the problem and make sure it makes sense.$\begin{array}{rlll} \mathrm{Carina drove} & 65\,\mathrm{mph}(3\,\mathrm{hours}) & = & \,195\,\mathrm{miles} \\ \mathrm{Her brother drove} & 50\,\mathrm{mph}(4\,\mathrm{hours}) & = & \underset{\mathrm{---------------}}{200\,\mathrm{miles}} \\ & & & \,395\,\mathrm{miles}\mathrm{✓} \\ \end{array}$

Step 7. Answer the question with a complete sentence.

Carina drove 65 mph and her brother 50 mph.

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 2.45

Solution

Step 1. Read the problem. Make all the words and ideas are understood.

• Draw a diagram to illustrate what it happening.

• Create a table to organize the information. Label the columns rate, time, distance. List the two scenarios. Write in the information you know.

Step 2. Identify what we are looking for.

We are asked to find the amount of time the trucks will travel until they are 325 miles apart.

Step 3. Name what we are looking for. Choose a variable to represent that quantity.

• Complete the chart.

• Use variable expressions to represent that quantity in each row.We are looking for the time travelled. Both trucks will travel the same amount of time.Let’s call the time t. Since their speeds are different, they will travel different distances.

• Multiply the rate times the time to get the distance.

Step 4. Translate into an equation.

• Restate the problem in one sentence with all the important information.

• Then, translate the sentence into an equation.We need to find a relation between the distances in order to write an equation. Looking at the diagram, what is the relationship between the distances each of the trucks will travel?The distance travelled by the truck going west plus the distance travelled by the truck going east must add up to 325 miles. So we write:

Step 5. Solve the equation using algebra techniques.

$\,\begin{array}{lccrll} \mathrm{Now solve this equation} & & & 70t+60t & = & 325 \\ & & & 130t & = & 325 \\ & & & t & = & 2.5 \\ \end{array}$

So it will take the trucks $2.5$ hours to be 325 miles apart.

Step 6. Check the answer in the problem and make sure it makes sense.

$\,\begin{array}{rlll} \mathrm{Truck going West} & 70\,\mathrm{mph}(2.5\,\mathrm{hours}) & = & \,175\,\mathrm{miles} \\ \mathrm{Truck going East} & 60\,\mathrm{mph}(2.5\mathrm{hours}) & = & \underset{\mathrm{---------------}}{150\,\mathrm{miles}} \\ & & & \,325\,\mathrm{miles}\mathrm{✓} \\ \end{array}$

Step 7. Answer the question with a complete sentence.It will take the trucks 2.5 hours to be 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 2.46

Solution

First, we draw a diagram that represents the situation to help us see what is happening.

We are asked to find her speed walking and riding her bike. Let’s call her walking speed r. Since her biking speed is three miles per hour faster, we will call that speed $r+3.$ We write the speeds in the chart.

The speed is in miles per hour, so we need to express the times in hours, too, in order for the units to be the same. Remember, 1 hour is 60 minutes. So:

We write the times in the chart.

Next, we multiply rate times time to fill in the distance column.

The equation will come from the fact that the distance from Naoko’s home to her school is the same whether she is walking or riding her bike.So we say:

Yes, either way Naoko travels 1.5 miles to school.

Naoko’s walking speed is 3 mph and her speed riding her bike is 6 mph.

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 2.47

Solution

A diagram will help us model this trip.

Next, we create a table to organize the information. We know the total distance is 51 miles. We are looking for the rate of speed for each part of the trip. The rate while biking is 1.6 times the rate of running. If we let r = the rate running, then the rate biking is 1.6*r*.

The times here are given as clock times. Cruz started from home at 6:00 a.m. and started biking at 7:30 a.m. So he spent 1.5 hours running. Then he biked from 7:30 a.m until 9:45 a.m. So he spent 2.25 hours biking.

Now, we multiply the rates by the times.

By looking at the diagram, we can see that the sum of the distance running and the distance biking is 255 miles.

Section 2.4 Exercises

Section 2.4 Exercises

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.

Solve Uniform Motion Applications

In the following exercises, solve.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

1. Jesse has $6.55 worth of quarters and nickels in his pocket. The number of nickels is five more than two times the number of quarters. How many nickels and how many quarters does Jesse have?

2. Elane has $7.00 total in dimes and nickels in her coin jar. The number of dimes that Elane has is seven less than three times the number of nickels. How many of each coin does Elane have?

3. Eric paid $19.88 for stamps. The number of 49-cent stamps was eight more than twice the number of 35-cent stamps. How many 49-cent stamps and how many 35-cent stamps did Eric buy?

1. Kailee paid $14.74 for stamps. The number of 49-cent stamps was four less than three times the number of 20-cent stamps. How many 49-cent stamps and how many 20-cent stamps did Kailee buy?

2. During her shift at the museum ticket booth, Leah sold 115 tickets for a total of $1,163. Adult tickets cost$12 and student tickets cost $5. How many adult tickets and how many student tickets did Leah sell?

3. Galen sold 810 tickets for his church’s carnival for a total revenue of $2,820. Children’s tickets cost$3 each and adult tickets cost $5 each. How many children’s tickets and how many adult tickets did he sell?

1. Orlando is mixing nuts and cereal squares to make a party mix. Nuts sell for $7 a pound and cereal squares sell for$4 a pound. Orlando wants to make 30 pounds of party mix at a cost of $6.50 a pound, how many pounds of nuts and how many pounds of cereal squares should he use?

2. Becca wants to mix fruit juice and soda to make a punch. She can buy fruit juice for $3 a gallon and soda for$4 a gallon. If she wants to make 28 gallons of punch at a cost of $3.25 a gallon, how many gallons of fruit juice and how many gallons of soda should she buy?

3. An express train and a local train leave Pittsburgh to travel to Washington, D.C. The express train can make the trip in four hours and the local train takes five hours for the trip. The speed of the express train is 12 miles per hour faster than the speed of the local train. Find the speed of both trains.

1. Jeromy can drive from his house in Cleveland to his college in Chicago in 4.5 hours. It takes his mother six hours to make the same drive. Jeromy drives 20 miles per hour faster than his mother. Find Jeromy’s speed and his mother’s speed.

2. Christopher and his parents live 115 miles apart. They met at a restaurant between their homes to celebrate his mother’s birthday. Christopher drove one and a half hours while his parents drove one hour to get to the restaurant. Christopher’s average speed was ten miles per hour faster than his parents’ average speed. What were the average speeds of Christopher and of his parents as they drove to the restaurant?

3. Ashley goes to college in Minneapolis, 234 miles from her home in Sioux Falls. She wants her parents to bring her more winter clothes, so they decide to meet at a restaurant on the road between Minneapolis and Sioux Falls. Ashley and her parents both drove two hours to the restaurant. Ashley’s average speed was seven miles per hour faster than her parents’ average speed. Find Ashley’s and her parents’ average speed.

1. Pierre and Monique leave their home in Portland at the same time. Pierre drives north on the turnpike at a speed of 75 miles per hour while Monique drives south at a speed of 68 miles per hour. How long will it take them to be 429 miles apart?

2. Thanh and Nhat leave their office in Sacramento at the same time. Thanh drives north on I-5 at a speed of 72 miles per hour. Nhat drives south on I-5 at a speed of 76 miles per hour. How long will it take them to be 330 miles apart?

3. Suzy takes 50 minutes to hike uphill from the parking lot to the lookout tower. It takes her 30 minutes to hike back down to the parking lot. Her speed going downhill is 1.2 miles per hour faster than her speed going uphill. Find Suzy’s uphill and downhill speeds.

1. Michaela has $2.05 in dimes and nickels in her change purse. She has seven more dimes than nickels. How many coins of each type does she have?

2. In a cash drawer there is $125 in$5 and $10 bills. The number of$10 bills is twice the number of $5 bills. How many of each type of bill is in the drawer?

3. Chi has $11.30 in dimes and quarters. The number of dimes is three more than three times the number of quarters. How many of each are there?

1. Mukul has $3.75 in quarters, dimes and nickels in his pocket. He has five more dimes than quarters and nine more nickels than quarters. How many of each coin are in his pocket?

2. The first day of a water polo tournament the total value of tickets sold was $17,610. One-day passes sold for$20 and tournament passes sold for $30. The number of tournament passes sold was 37 more than the number of day passes sold. How many day passes and how many tournament passes were sold?

3. Julie went to the post office and bought both $0.41 stamps and$0.26 postcards. She spent $51.40. The number of stamps was 20 more than twice the number of postcards. How many of each did she buy?

1. Hilda has $210 worth of$10 and $12 stock shares. The number of$10 shares is five more than twice the number of $12 shares. How many of each type of share does she have?

2. The ice rink sold 95 tickets for the afternoon skating session, for a total of $828. General admission tickets cost$10 each and youth tickets cost $8 each. How many general admission tickets and how many youth tickets were sold?

3. The box office sold 360 tickets to a concert at the college. The total receipts were $4,170. General admission tickets cost$15 and student tickets cost $10. How many of each kind of ticket was sold?

1. Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. Macadamia nuts cost $9 per pound and almonds cost$5.25 per pound. How many pounds of macadamia nuts and how many pounds of almonds should Macario use for the mixture to cost $6.50 per pound to make?

2. Riley is planning to plant a lawn in his yard. He will need nine pounds of grass seed. He wants to mix Bermuda seed that costs $4.80 per pound with Fescue seed that costs$3.50 per pound. How much of each seed should he buy so that the overall cost will be $4.02 per pound?

3. Vern sold his 1964 Ford Mustang for $55,000 and wants to invest the money to earn him 5.8% interest per year. He will put some of the money into Fund A that earns 3% per year and the rest in Fund B that earns 10% per year. How much should he invest into each fund if he wants to earn 5.8% interest per year on the total amount?

1. Liam borrowed a total of $35,000 to pay for college. He pays his parents 3$8,000 he borrowed from them and pays the bank 6.8% on the rest. What average interest rate does he pay on the total $35,000? (Round your answer to the nearest tenth of a percent.)

2. Kathy and Cheryl are walking in a fundraiser. Kathy completes the course in 4.8 hours and Cheryl completes the course in eight hours. Kathy walks two miles per hour faster than Cheryl. Find Kathy’s speed and Cheryl’s speed.

3. A commercial jet and a private airplane fly from Denver to Phoenix. It takes the commercial jet 1.6 hours for the flight, and it takes the private airplane 2.6 hours. The speed of the commercial jet is 210 miles per hour faster than the speed of the private airplane. Find the speed of both airplanes to the nearest 10 mph.

1. Charlie and Violet met for lunch at a restaurant between Memphis and New Orleans. Charlie had left Memphis and drove 4.8 hours towards New Orleans. Violet had left New Orleans and drove two hours towards Memphis, at a speed 10 miles per hour faster than Charlie’s speed. The distance between Memphis and New Orleans is 394 miles. Find the speed of the two drivers.

2. Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden two hours, they meet on the path. Ethan’s speed is six miles per hour faster than Leo’s speed. Find the speed of the two bikers.

3. 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.

1. Llewyn takes 45 minutes to drive his boat upstream from the dock to his favorite fishing spot. It takes him 30 minutes to drive the boat back downstream to the dock. The boat’s speed going downstream is four miles per hour faster than its speed going upstream. Find the boat’s upstream and downstream speeds.

2. Hamilton loves to travel to Las Vegas, 255 miles from his home in Orange County. On his last trip, he left his house at 2:00 p.m. The first part of his trip was on congested city freeways. At 4:00 pm, the traffic cleared and he was able to drive through the desert at a speed 1.75 times as fast as when he drove in the congested area. He arrived in Las Vegas at 6:30 p.m. How fast was he driving during each part of his trip?

3. Phuong left home on his bicycle at 10:00. He rode on the flat street until 11:15, then rode uphill until 11:45. He rode a total of 31 miles. His speed riding uphill was 0.6 times his speed on the flat street. Find his speed biking uphill and on the flat street.

1. Matt and Chris leave their uncle’s house in Phoenix at the same time. Matt drives west on I-60 at a speed of 76 miles per hour. Chris drives east on I-60 at a speed of 82 miles per hour. How many hours will it take them to be 632 miles apart?

2. Two boats leave the same dock in Cairo at the same time. One heads north on the Mississippi River while the other heads south. The northbound boat travels four miles per hour. The southbound boat goes eight miles per hour. How long will it take them to be 54 miles apart?

3. Julian rides his bike uphill for 45 minutes, then turns around and rides back downhill. It takes him 15 minutes to get back to where he started. His uphill speed is 3.2 miles per hour slower than his downhill speed. Find Julian’s uphill and downhill speed.

1. It takes Darline 20 minutes to drive to work in light traffic. To come home, when there is heavy traffic, it takes her 36 minutes. Her speed in light traffic is 24 miles per hour faster than her speed in heavy traffic. Find her speed in light traffic and in heavy traffic.

2. Aaron left at 9:15 to drive to his mountain cabin 108 miles away. He drove on the freeway until 10:45 and then drove on a mountain road. He arrived at 11:05. His speed on the freeway was three times his speed on the mountain road. Find Aaron’s speed on the freeway and on the mountain road.

3. Lizette is training for a marathon. At 7:00 she left her house and ran until 8:15 then she walked until 11:15. She covered a total distance of 19 miles. Her running speed was five miles per hour faster than her walking speed. Find her running and walking speeds.

1. Sarah wants to arrive at her friend’s wedding at 3:00. The distance from Sarah’s house to the wedding is 95 miles. Based on usual traffic patterns, Sarah predicts she can drive the first 15 miles at 60 miles per hour, the next 10 miles at 30 miles per hour, and the remainder of the drive at 70 miles per hour.

ⓐ How long will it take Sarah to drive the first 15 miles?

ⓑ How long will it take Sarah to drive the next 10 miles?

ⓒ How long will it take Sarah to drive the rest of the trip?

ⓓ What time should Sarah leave her house?

2. Do you find it helpful to use a table when solving coin problems? Why or why not?

3. When solving a uniform motion problem, how does drawing a diagram of the situation help you?