8. Ratio Tables and Graphs
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Chapter 1
8.
Ratio Tables and Graphs
Ratio tables are useful tools for understanding the relationships between different quantities. This lesson focuses on how to use ratio tables to represent equivalent ratios and visualize their scaling on a coordinate plane. By graphing these tables, students can better understand proportional relationships and how they change across different values. This skill is crucial in various fields, such as science, economics, and everyday problem-solving, where comparisons between quantities are needed. The lesson provides practical guidance on interpreting and creating graphs that show the connections between different ratios, helping to visualize data in a way that makes it easier to understand and analyze.
| Student Learning Objectives: |
|---|
|
| | 11 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Ratio Tables and Graphs
Slide of 11
A ratio shows the relationship between two quantities. In some cases, more than one ratio can represent the same relationship. This lesson will introduce ratio tables and explore how they can be used to represent relationships.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Explore
Comparing the Simplest Forms of a Pair of Ratios
Take a look at these ratios and see how they can be reduced to their simplest form.
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Discussion
Equivalent Ratios
A ratio is a comparison of two quantities that describes how much of one thing there is compared to another. Ratios are commonly represented using colon notation or as fractions. They are read as the ratio of a to b,
where b is a non-zero number.
ccc Colon Notation &&Fraction a:b && a/b
Ratios that express the same relationship between quantities are called equivalent ratios. For instance, consider the ratios of pages read per minute by Tearrik and by Zain. Tearrik's Ratio& &Zain's Ratio 27/15& &45/25 These ratios can be simplified by finding the greatest common factor of their numerator and denominator. That factor can then be used to rewrite each ratio.
| Fraction Form | Greatest Common Factor | Rewrite | Simplify | |
|---|---|---|---|---|
| Tearrik | 27/15 | GCF(27,15)= 3 | 9* 3/5* 3 | 9/5 |
| Zain | 45/25 | GCF(45,25)= 5 | 9* 5/5* 5 | 9/5 |
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Discussion
Scaling Ratios
Scaling is a way of creating equivalent ratios by multiplying or dividing the numerator and denominator of a ratio by the same number.
| 3/6 | |
|---|---|
| Scaling Back | Scaling Forward |
| 3÷3/6÷3 | 3*3/6*3 |
| 1/2 | 9/18 |
Sometimes we need to scale back and forward to find an equivalent ratio for a particular situation. Consider the following scenario.
Chocolate bars are on sale at 14 for $6. Let's scale the ratio up and down to find the cost for 21 chocolate bars.
| 14/6 | |
|---|---|
| Scaling Back | Scaling Forward |
| 14÷2/6÷2 | 7*3/3*3 |
| 7/3 | 21/9 |
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Pop Quiz
Are Two Ratios Equivalent?
Determine whether the given pair of ratios is equivalent.
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Example
Mark's Favorite Shot
One of Mark's favorite prints is of the Walter Pyramid. The picture has a height-to-width ratio of 3:4.
He wants to print it out and hang it on his bedroom wall, but he wants to print it in two different sizes equivalent to the original one.
a
For the first size option, Mark would like to enlarge the original image until it is 4 times its original size. What will be the ratio of the height to the width of the new image? Write the answer in fraction form.
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b
Mark needs to increase the size of his image by a factor of 7 for the second print. What ratio represents the height-to-width ratio of this image? Answer in fraction form.
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Hint
a
Write the original image's ratio in fraction form. Scale the original ratio by multiplying the numerator and the denominator by 4.
b
Multiply the original ratio by 7 to get the ratio for the second size.
Solution
a
Let's start by writing the size ratio of the original image as a fraction.
3in.:4in. ⇔ 3in./4in. Next, we will scale this ratio to find the ratio for the first size that Mark wants. He wants to enlarge the original image by a factor of 4, so we multiply the numerator and the denominator of the original ratio by 4. 3in.* 4/4in.* 4=12in./16in. The second print will be 12 inches high and 16 inches wide.
b
Let's follow a similar process as in Part A to find the size ratio for the second print. Mark wants to increase the size of his image by a factor of 7, so we multiply the numerator and denominator of the original ratio by 7.
3in.* 7/4in.* 7=21in./28in. The second print will be 21 inches high and 28 inches wide.
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Discussion
Ratio Table
Tables of equivalent ratios are called ratio tables. Consider the ratio table containing the distance a person runs and the time it takes him to cover that distance.
| Distance Run (Miles) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Time (Minutes) | 10 | 20 | 30 | 40 |
The ratios 110, 220, 330, and 440 are equivalent since each of them simplifies to a ratio of 110. A ratio table can be created by multiplying or dividing each quantity in a ratio by the same number, or by repeatedly adding or subtracting the ratio.
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Example
Exploring the Aspect Ratio of Mark's New Phone Camera
Mark's phone has a unique camera.
The aspect ratio, which represents the relationship between height and width, of the main camera is 2:3. Mark created a ratio table to explore the equivalent ratios that his photos can have.
| Height | 2 | A | 6 | 18 |
|---|---|---|---|---|
| Width | 3 | 6 | 9 | B |
However, the table is missing some values.
a
What are the values of A and B that complete the ratio table Mark created?
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b
Which of the following graphs represents the ratio table for the aspect ratio of Mark's new phone camera?
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Hint
a
A ratio table can be created by multiplying or dividing both quantities in a ratio by the same number. It can also be created by repeatedly adding or subtracting the original ratio.
b
Graph the ratio table created in Part A on a coordinate plane. The x-axis will represent the heights and the y-axis will represent the widths.
Solution
a
To complete a ratio table, multiply or divide both quantities by the same number. We can also add or subtract equal groups of the original ratio. Let's look at the ratio table.
| Height | 2 | A | 6 | 18 |
|---|---|---|---|---|
| Width | 3 | 6 | 9 | B |
Let's use addition and the initial ratio of 2: 3 to find the ratio for the second column. We added 3 to the width of the first ratio to get 6. If we add the original numerator 2 to the height of the first ratio, we will find A, the height matching the width of 6.
Now let's use multiplication and the 6:9 ratio to find the width B for a height of 18. Since 6* 3=18, we can multiply 9 by 3 to get the value of B.
Therefore, A= 4 and B= 27.
b
Let's graph the ratio table we made in Part A on a coordinate plane. The x-axis will represent the heights and the y-axis the widths of the camera aspect ratio. Consider the ratio table from Part A.
| Height, x | 2 | 4 | 6 | 18 |
|---|---|---|---|---|
| Width, y | 3 | 6 | 9 | 27 |
Because the table contains only positive values, we only need the first quadrant. Starting at the origin, we move along the x-axis according to the number of units specified by the height. Then we move vertically the number of units specified by the width to graph each ratio.
This graph corresponds to option A.
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Example
Mark and Maya's Five-Day Photo Challenge
The following table shows the number of photos Mark and Maya took over a 5 day photo challenge.
| Maya | Mark | ||
|---|---|---|---|
| Days | Photos | Days | Photos |
| 1 | 5 | 1 | 7 |
| 2 | 10 | 2 | 14 |
| 3 | 15 | 3 | 21 |
| 4 | 20 | 4 | 28 |
| 5 | 25 | 5 | 35 |
a
Choose the graph that accurately displays the information recorded in the tables.
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b
Use both the table and the graph to determine which of the statements below are true. Select all that apply.
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Hint
a
Graph both data sets on a coordinate plane. The x-coordinate will be the day and the y-coordinate will be the number of photos taken.
b
Draw a line through each data. Which line is steeper?
Solution
a
Let's graph both data sets on a coordinate plane. The x-axis will indicate the days, while the y-axis will show the number of photos taken. We will start by writing the ordered pair (x,y) for each column of the table recorded by Maya.
| Maya | ||
|---|---|---|
| Days | Photos | (x,y) |
| 1 | 5 | (1,5) |
| 2 | 10 | (2,10) |
| 3 | 15 | (3,15) |
| 4 | 20 | (4,20) |
| 5 | 25 | (5,25) |
To graph the data, let's start at the origin and move along the x-axis the number of units specified by the day. Then, we will move up the number of units specified by the photos taken to graph each ordered pair.
Before we add Mark's data to the graph, let's write each ordered pair of his data set.
| Mark | ||
|---|---|---|
| Days | Photos | (x,y) |
| 1 | 7 | (1,7) |
| 2 | 14 | (2,14) |
| 3 | 21 | (3,21) |
| 4 | 28 | (4,28) |
| 5 | 35 | (5,35) |
Let's now add the ordered pairs to the graph.
The graph matches option B.
b
We need to compare the ratios of photos taken to days passed for Mark and Maya. Let's consider the first row of both data sets to see their unit ratios.
| Maya | Mark | |
|---|---|---|
| Day, x | 1 | 1 |
| Photos, y | 5 | 7 |
Maya's photo-to-day ratio is 5:1 and Mark's is 7:1. Now let's look at the graph we drew in Part A.
Notice that both sets of points seem to create straight lines, but the line for Mark's data is steeper than the line for Maya's data. Let's now decide which of the given statements are true.
| Statement | True? |
|---|---|
| Maya's ratio of photos to days is 5:1. | ✓ |
| Mark's ratio of photos to days is 7:1. | ✓ |
| The line for Mark's data is steeper than the line for Maya's data. | ✓ |
| Maya's ratio of days to photos is 5:1. | * |
| Mark's ratio of days to photos is 7:1. | * |
| The line for Maya's data is steeper than the line for Mark's data. | * |
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Example
How Much Money Will Mark Earn?
Mark is thinking about selling his nature photos to a website. The graph below shows how much he would earn per photo sold.
The graph does not tell how much Mark would earn if he sold 6 photos. If the ratio of photos sold to money earned remains constant, how much money would Mark earn from selling 6 photos?
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Hint
Use the relationship between a pair of consecutive points on the graph to find the ratio of money earned to photos sold. Use that ratio to find the point for selling 6 photos on the graph.
Solution
Mark wants to know how much money he would earn for selling 6 photos to the website. We know that the ratio of money earned per photo sold remains constant. We can find it using the given graph since the points appear to lie on a straight line.
The point (4,6) is 2 units to the right and 3 units up from point (2,3).
This means that the ratio of the money earned to photos sold is 3: 2 — Mark will earn $3 for every 2 photos sold. Therefore, if he sells 6 = 2 * 3 photos, he will earn $3 * 3 = $9.
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Closure
Graphing Ratio Tables on a Coordinate Plane
This lesson covered some uses of equivalent ratios and how ratio tables can be used to find them. Equivalent ratios are particularly useful when scaling ratios to maintain proportions. Take a look at a ratio table that shows how many photos Maya's printer can print per minute.
| Photos | 4 | 8 | 12 |
|---|---|---|---|
| Time (minutes) | 5 | 10 | 15 |
Now consider the ratio table that shows the same information for Mark's printer.
| Photos | 20 | 24 | 28 |
|---|---|---|---|
| Time (minutes) | 25 | 30 | 35 |
The two ratio tables may seem unrelated at first glance, but notice that we can simplify the first ratio in Mark's table. 20÷ 5/25÷ 5=4/5 The ratios 20:25 and 4:5 are the equivalent. This suggests that all the ratios in both tables are also equivalent. Consider what would happen if these two ratio tables were plotted on a coordinate plane.
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Ratio Tables and Graphs
Exercise 2.1
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We want to determine how many sweaters Kevin can knit in 15 days. We know that he can knit 6 sweaters in 10 days, so let's use this information to create a ratio table that relates the number of sweaters knit to the amount of time it takes to knit them.
| Sweaters | 6 | ||
|---|---|---|---|
| Time | 10 |
We can scale the ratio of 6 sweaters in 10 days, 6:10, back by dividing both quantities by 2. This will tell us how many sweaters Kevin can knit in 5 days.
| Sweaters | 6 | 6÷2=3 | |
|---|---|---|---|
| Time | 10 | 10÷2=5 |
We found an equivalent ratio of 3 sweaters knit in 5 days. We can use this ratio to find how many sweaters Kevin can knit in 15 days. Since 5* 3=15, we can scale the 3:5 ratio forward by 3 to find the missing information.
| Sweaters | 6 | 6÷2=3 | 3* 3=9 |
|---|---|---|---|
| Time | 10 | 10÷2=5 | 5* 3=15 |
We found that Kevin can knit 9 sweaters in 15 days.
We want to determine how many days it takes Kevin to knit 12 sweaters, given that he can knit 6 sweaters in 10 days. Let's use this information to write the ratio of time spent knitting to the number of sweaters knit.
10days/6sweaters
We know that 6* 2=12. This means we can multiply both quantities in this ratio by 2 to get an equivalent ratio with a denominator of 12 sweaters.
10days* 2/6sweaters* 2=20days/12sweaters
We got 20 days in the numerator. This means that it took Kevin 20 days to knit 12 sweaters.
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We can create a ratio table to determine the number of snacks the vending machine would dispense in 7 minutes. We are given that the vending machine dispenses 70 snacks in 5 minutes. Let's add this information to the ratio table.
| Snacks | 70 | ||
|---|---|---|---|
| Time | 5 |
Note that there is no integer we can multiply 5 by to get 7. Let's first find the unit rate of snacks to time by scaling the ratio 70:5 back. We can do this by dividing both quantities in the ratio by 5 and recording the result in the second column of the ratio table. Remember, a unit ratio must have a denominator of 1.
| Snacks | 70 | 70÷5=14 | |
|---|---|---|---|
| Time | 5 | 5÷5=1 |
Next, we can scale the unit ratio forward to 7 minutes by multiplying it by 7. This will give us the number of snacks the vending machine would dispense in 7 minutes.
| Number of Snacks | 70 | 14 | 14*7=98 |
|---|---|---|---|
| Time | 5 | 1 | 1*7=7 |
Therefore, the vending machine would dispense 98 snacks in 7 minutes.
Let's start by recalling that the vending machine dispenses 70 snacks in 5 minutes. We can represent this information as a ratio of time to snacks dispensed.
5minutes/70snacks
We can use this ratio to create a ratio table that will help us to determine how long it would take for the vending machine to dispense 420 snacks. Let's multiply the numerator and the denominator of the ratio by 1, 2, 3, and so on, up to 7, to get the corresponding time and snacks for each equivalent ratio.
| Time | 5 | 10 | 15 | 20 | 25 | 30 | 35 |
|---|---|---|---|---|---|---|---|
| Snacks | 70 | 140 | 210 | 280 | 350 | 420 | 490 |
This means that it would take the machine 30 minutes to dispense 420 snacks. In this case, we were able to find the solution because 420 is a multiple of 70. Another number of snacks might require numbers beyond integers.
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Dylan and Emily are putting together a charity fundraiser at their school. They have both been selling tickets for the event. Dylan has sold 3 out of every 5 tickets.
At the end of the day, Dylan realized he had sold 5 more tickets than Emily. How many tickets have they sold altogether?
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Dylan and Emily sold tickets for a charity fundraiser event. We know that Dylan sold 3 out of every 5 tickets sold. Let's represent this information as the ratio of tickets sold by Dylan to the total number of tickets sold. 3:5 We want to determine the total number of tickets sold if Dylan sold five more tickets than Emily. First, let's create a ratio table with ratios that are equivalent to 3:5.
| Tickets Sold by Dylan | 3 | 6 | 9 | 12 | 15 | 18 |
|---|---|---|---|---|---|---|
| Total Tickets Sold | 5 | 10 | 15 | 20 | 25 | 30 |
We can find the number of tickets sold by Emily by calculating the difference between the total number of tickets sold and the number of tickets sold by Dylan.
| Total Tickets Sold | Tickets Sold by Dylan | Difference | Tickets Sold by Emily |
|---|---|---|---|
| 5 | 3 | 5-3 | 2 |
| 10 | 6 | 10-6 | 4 |
| 15 | 9 | 15-9 | 6 |
| 20 | 12 | 20-12 | 8 |
| 25 | 15 | 25-15 | 10 |
| 30 | 18 | 30-18 | 12 |
We now have the total number of tickets sold by both Dylan and Emily. Let's subtract the number of tickets sold by Emily from the number of tickets sold by Dylan to find the difference between the total tickets sold by Dylan and Emily.
| Tickets Sold by Dylan | Tickets Sold by Emily | Difference | Extra Tickets Sold by Dylan |
|---|---|---|---|
| 3 | 2 | 3-2 | 1 |
| 6 | 4 | 6-4 | 2 |
| 9 | 6 | 9-6 | 3 |
| 12 | 8 | 12-8 | 4 |
| 15 | 10 | 15-10 | 5 |
| 18 | 12 | 18-12 | 6 |
Now, we were told that that Dylan sold 5 tickets more than Emily. The only option in the table that meets this description is if 25 total tickets were sold.
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Let's write the rate of the distance each vehicle travels to the amount of time it spent traveling. We know that the delivery truck travels 220 miles in 4 hours and that the delivery van travels 180 miles in 3 hours. cc Delivery Truck & & Delivery Van 220mi:4h & & 180mi:3h Next, we can determine the unit rates for each vehicle. Remember, finding the unit rate means that we need to get 1 on the right-hand side of each rate. Let's divide the delivery truck's rate by 4 to find its unit rate.
This means that the delivery truck travels 55 miles in 1 hour. Now let's calculate the unit rate for the delivery van by dividing both quantities by 3.
The delivery van travels 60 miles in 1 hour. Since the delivery van has a higher unit rate, it is faster than the delivery truck.
Consider the unit rates found in Part A for each vehicle.
cc
Delivery Truck & & Delivery Van
55mi:1h & & 60mi:1h
We can create two equivalent rates by scaling up these rates by 5. That will show us the distance traveled by both vehicles after 5 hours.
cc
Delivery Truck& &Delivery Van
55mi* 5:1h* 5& &60mi* 5:1h* 5
⇓& &⇓
275mi:5h& &300mi:5h
The delivery truck will travel 275 miles after 5 hours, while the delivery van will travel 300 miles. Let's subtract the distance the truck can travel from the distance the van can travel to find out how much further the delivery van will go.
300-275=25miles
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Ratio Tables and Graphs
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