6.5 Applications of rates
Introduction
Janelle and her family are traveling around Europe. They're spending a few days in Spain and plan to travel to France, Switzerland, and Italy later. When they take a train from Madrid to Barcelona, Janelle asks her brother Andrew how long the journey will take. Andrew tells her that they will reach Barcelona in three hours.
Janelle is surprised to see that their train reaches Barcelona in three hours, just as Andrew said. She wonders if there's a relationship between the distance traveled and the time taken.
In this lesson, you'll relate the distance traveled to the time taken using a rate. You'll also study how rates apply to other everyday situations.
Watch the video on the next screen how a rate can be used to find the speed of a train.
Transcript
There are many types of rates. One of the most common rates is speed.
An object speed is the distance it travels over one unit of time. We can use this equation to find an object’s speed.
Let's look at an example, Janelle is traveling by train from Madrid to Barcelona. She knows that the distance between these two cities is three hundred and ninety miles because she also knows that the train covers this distance in three hours. Janelle can find the train speed using the speed formula.
She divides the distance traveled. By the time it takes to get from Madrid to Barcelona. Let's look at the map to see if this calculation makes sense.
The distance between Madrid and Barcelona is three hundred and ninety miles divided by three hours of travel. Each hour the train moves a distance of one hundred thirty miles, so the rate of travel is one hundred thirty miles per hour.
Next stop. Barcelona.
Question
Select the correct answer.
Rosemarie drove from Philadelphia, Pennsylvania, to Pittsburgh, Pennsylvania. The distance between Philadelphia and Pittsburgh is 305 miles. Rosemarie reached Pittsburgh 5 hours after she left Philadelphia. What is the speed at which she drove her car?
A. 81 miles/hour
B. 61 miles/hour
C. 51 miles/hour
D. 71 miles/hour
Distance
You can find the distance between two places if you know the speed and the travel time by writing the formula Speed = Distance / Time so that distance is on the left side: Distance = Speed x Time
Here's an example of how we can use the formula. Theodore is riding a bicycle at a speed of 15 miles/hour. We'll find the distance he covers in 5 hours:
distance = speed × time
distance = 15 miles/hour × 5 hours = 75 miles.
You can use another version of this formula to find the time taken to travel a given distance if you know the speed:
Time = Distance / Speed
Questions
Answer the question below. Type your response in the space provided.
Althea traveled 280 miles at a speed of 70 miles/hour. How much time did she take to cover this distance?
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Unit Price
Unit prices are another type of ratio. We come across unit prices in everyday situations such as buying groceries. The unit price is the amount of money paid for a single item. A unit price is equal to the ratio of the total cost to the number of units sold:
unit price = total cost / total number of units sold
You can also use this formula to find total cost when you already know the unit cost. To do so, rewrite the equation above in this form:
total cost = unit price x total number of units sold
For example, let's say that oranges cost $2 per pound, and you want to find the cost of 4 pounds of oranges. You can plug the known values into this equation:
total cost = 4 pounds x $2 per pound
So, 4 pounds of oranges will cost $8.
Let's look at another example involving unit price. Annie bought 6 pounds of berries for $12 and wants to buy another 10 pounds. She wants to know how much money she needs to pay for 10 pounds.
Annie first needs to find the unit price, or the price of 1 pound:
cost of 6 pounds of berries = $12
unit price = total cost / total number of units
unit price of berries = $12 / 6 = $2 per pound
Now that we know the cost per pound, we can calculate the cost of 10 pounds:
total cost of 10 pounds = unit price × number of pounds
total cost = $2 per pound × 10 pounds = $20.
Lesson Activity
Applications of Rates
The Lesson Activity will help you meet these educational goals:
- Content Knowledge—You will solve unit rate problems, including those that involve unit pricing and constant speed.
- Mathematical Practices—You will make sense of problems and solve them.
- STEM—You will use mathematical processes and analysis in scientific investigation to solve real-world design problems.
- 21st Century Skills—You will use critical-thinking and problem-solving skills.
Directions
Read the instructions for this self-checked activity. Type in your response to each question, and check your answers. At the end of the activity, write a brief evaluation of your work.
Activity
Janelle and her family are traveling around Europe by train. Their trip will begin in Madrid, Spain, and will end in Rome, Italy.
The first part of their train journey, from Madrid, Spain, to Barcelona, Spain, took 3 hours. Use this information to solve the problems below.
Part A
If the train went 390 miles in 3 hours, what was the speed of the train in miles per hour?
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Part B
The distance from Barcelona, Spain, to Paris, France, is 650 miles. If the train from Barcelona to Paris travels 130 miles per hour, how long does it take to get from Barcelona to Paris?
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Part C
If the trip from Paris, France, to Geneva, Switzerland, takes 5 hours, and the train travels at 130 miles per hour, how many miles from Paris is Geneva?
Part D
The train from Geneva, Switzerland, to Milan, Italy, travels at 100 miles per hour, and the two cities are 200 miles apart. How long does the journey from Geneva to Milan take?
Part E
If the train from Milan, Italy, to Rome, Italy, takes 6 hours at a speed of 130 miles per hour, how many miles from Milan is Rome?
Part F
Janelle’s parents got a special offer of $75 per person for each train trip. If there are 4 people in Janelle’s family, what is the total cost of each train trip?
Part G
What is the cost of 5 train trips for all 4 family members?
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Question 11/15
Gaston is buying fruit at a local grocery store. This table shows the cost per pound of each type of fruit. Match each quantity of fruit with its total cost. Drag the items on the left to the correct location on the right.
12/15
Other Ways to Use Rates
In the Lesson Activity, you used the speed to find the distance traveled and the time taken. You can apply the concept of rates to other situations too.
Let's look at an example of a rate calculation. If someone mows 8 lawns in 2 days, the rate at which the person works is
number of lawn mowed per day = (number of lawns mowed / number of days = 8/2 = 4
The person mows 4 lawns per day.
Let's look at another example in which we know the unit rate. If someone who can mow 3 lawns per day mows 18 lawns, you can find the time needed to mow 18 lawns:
number of days need to mow 18 lawns = number of lawn mowed / number of lawns mowed per day
number of days needed to mow 18 lawns = 18/3 = 6 days
13/15
Another way you can use rates is to find a worker's wages for a specific number of units of work: total wages = wages per unit × number of units of work. Units of work can be days, hours, or items such as lawns mowed, cars washed, and windows cleaned.
Here's an example of how to apply wages per unit of work. Tony washes cars at the local garage. He earned $400 for washing 10 cars last week. This week, his wage for washing each car has been increased by $3. Let's calculate how much he will earn if he washes 10 cars this week.
First we calculate Anthony's wages per car last week:
wages per car = total wages / number of cars washed = $400 / 10 = $40
Because Anthony's wage has gone up by $3 per car, he will earn $43 per car this week. If he again washes 10 cars:
total wages = wages per car x number of cars washed = $43 per car x 10 cars = $430
Anthony will earn $430 if he washes 10 cars this week.
