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
Exercises
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Start by writing the given ratio in fraction form. 5:9 ⇔ 5/9 We multiply both the numerator and the denominator of a ratio by the same number to get an equivalent ratio. In this case, we need to scale the given ratio forward by 5 to find a ratio equivalent to 5:9. Let's use the fraction form of the ratio to find the equivalent ratio. 5* 5/9* 5=25/45 Therefore, scaling the given ratio 5:9 forward by 5 results in the equivalent ratio 25:45.
We will follow a similar process as in Part A to scale the given ratio back. Let's write it in fraction form first.
77:49 ⇔ 77/49
Since we want to scale the given ratio back by 7, we must divide both the numerator and denominator by 7 instead of multiplying. Let's do it!
77÷ 7/49÷ 7=11/7
Scaling the given ratio 77:49 back by 7 results in the equivalent ratio 11:7.
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Solution
Consider the following ratio table.
| x | 2 | A | 4 |
|---|---|---|---|
| y | B | 6 | 8 |
What are the values of A and B that complete the ratio table?
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We want to find the values of A and B in the ratio table.
| x | 2 | A | 4 |
|---|---|---|---|
| y | B | 6 | 8 |
Let's use the last column of the table to write the ratio of y to x. 8:4 ⇔ 8/4 Next, we can determine the unit ratio by dividing both quantities in this ratio by 4. 8÷ 4/4÷ 4=2/1 Let's use this unit ratio to find the missing values in the table. Remember, all ratios in a ratio table are equivalent. 2/1 ⇔ B/2 The denominator of B2 is twice the denominator of 21. This means that the numerator B should be twice the numerator 2. B=2* 2 ⇔ B=4 We can follow a similar process to find the value of A. 2/1 ⇔ 6/A In this case, the numerator of 6A is three times greater than the numerator of 21, the denominator A should be three times greater than the denominator 2. A=1* 3 ⇔ A=3 We found that A=3 and B=4. Let's complete the given ratio table.
| x | 2 | 3 | 4 |
|---|---|---|---|
| y | 4 | 6 | 8 |
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| Ramsha | Tadeo | ||
|---|---|---|---|
| Time (minutes) | Words Typed | Time (minutes) | Words Typed |
| 1 | 25 | 1 | 40 |
| 2 | 50 | 2 | 80 |
| 3 | 75 | 3 | 120 |
| 4 | 100 | 4 | 160 |
| 5 | 125 | 5 | 200 |
| 6 | 150 | 6 | 240 |
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Let's plot the data on a coordinate plane to determine which graph accurately displays the information from the ratio tables. In this case, the x-axis will represent the time and the y-axis will represent the number of words typed. Let's consider Ramsha's table.
| Ramsha | |
|---|---|
| Time (minutes) | Words Typed |
| 1 | 25 |
| 2 | 50 |
| 3 | 75 |
| 4 | 100 |
| 5 | 125 |
| 6 | 150 |
We will move along the x-axis away from the origin the number of units specified by the time. We will then move vertically the number of units specified by the number of words typed to plot each ordered pair.
Now let's consider Tadeo's table.
| Tadeo | |
|---|---|
| Time (minutes) | Words Typed |
| 1 | 40 |
| 2 | 80 |
| 3 | 120 |
| 4 | 160 |
| 5 | 200 |
| 6 | 240 |
We will follow a similar process to graph Tadeo's information.
This graph corresponds to option D.
We can determine which statements about the data are true by comparing the unit ratios in the first row of the ratio table.
| Ramsha | Tadeo | ||
|---|---|---|---|
| Time (minutes) | Words Typed | Time (minutes) | Words Typed |
| 1 | 25 | 1 | 40 |
Ramsha typed 25 words per minute, while Tadeo typed 40 words per minute. Ramsha& &Tadeo 25words:1min & &40words:1min We can also consider the graph we drew in Part A.
This graph shows that both data sets follow a straight line. However, Tadeo's line is steeper than Ramsha's. With these findings, we can identify which statements are true about their data.
| Statement | True? |
|---|---|
| Ramsha's ratio of words typed to time is 25:1. | ✓ |
| Tadeo's ratio of words typed to time is 40:1. | ✓ |
| The line for Tadeo's data is steeper than the line for Ramsha's. | ✓ |
| Ramsha's ratio of words typed to time is 40:1. | * |
| Tadeo's ratio of words typed to time is 25:1. | * |
| The line for Ramsha's data is steeper than the line for Tadeo's. | * |
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Magdalena is working on a new logo for her mother's business. She made a logo that is 2 inches wide and 1 inch tall. The logo is intended for a banner and needs to be eight times bigger while maintaining the same aspect ratio. What are the dimensions of the enlarged logo?
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Magdalena made a logo that is 2 inches wide by 1 inch tall. Let's write the ratio of the width to height for Magdalena's logo. 2in.:1in. ⇔ 2in./1in. The enlarged logo must have the exact same proportions as Magdalena's logo, but it has to be eight times bigger. We can multiply the numerator and the denominator of Magdalena's logo ratio by 8 to get an equivalent ratio for the enlarged logo. 2 in.* 8/1 in. * 8=16in./8in. This means the enlarged logo's dimensions should be 16 inches in width and 8 inches in height.
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A package of 6 cupcakes costs $5. How much does a package of 12 cupcakes cost if the price per cupcake is the same for every package size?
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We will find the cost of a package of 12 cupcakes, given that a package of 6 cupcakes costs $5. We can use this information to write the ratio for the number of cupcakes to their cost. 6cupcakes/$5 Since the price per cupcake is the same for every package, we can use this ratio to find the cost of 12 cupcakes. Let's write an equivalent ratio with a numerator of 12 cupcakes. 6cupcakes/$5=12cupcakes/ We are missing the denominator of this new ratio. Recall that to find an equivalent ratio, we multiply both quantities in the original ratio by the same number. Since 6* 2=12, we can multiply both quantities in the first ratio by 2 to find this equivalent ratio. 6cupcakes* 2/$5* 2=12cupcakes/$10 This means that a package of 12 cupcakes costs $10.
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Ratio Tables and Graphs
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