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Pre-Algebra View details
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.
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Ratio Tables and Graphs
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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.

Explore

Comparing the Simplest Forms of a Pair of Ratios

Take a look at these ratios and see how they can be simplified down to their simplest form.
Simplify Two Random Ratios
What conclusions can be made about the ratio pairs?
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 Fraction && Colon Notation 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
These ratios are equivalent because both simplify to 95. Equivalent ratios can be created by multiplying or dividing the numerator and denominator of a ratio by the same number.
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. Sometimes it is necessary to scale back and forward to find an equivalent ratio for a particular situation. Consider the following scenario.

Chocolate bars

Chocolate bars are on sale at 14 for $6. This ratio can be scaled up and down to find the cost of 21 chocolate bars.

14/6
Scaling Back Scaling Forward
14÷2/6÷2 7*3/3*3
7/3 21/9
As shown, the cost of 21 chocolate bars is $9.
Pop Quiz

Determining if Two Ratios Are Equivalent

Determine whether the given pair of ratios is equivalent.

An applet that provides two ratios and asks whether they are equivalent.
Example

Mark's Favorite Shot

Mark loves taking pictures with his camera. One of his favorite shots is of the Walter Pyramid. The picture has a height-to-width ratio of 3:4.

WalterPyramid.jpg

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 four times. What will be the ratio of the height to the width of the new image? Write the answer in fraction form.
b Mark needs to increase the size of his image by a factor of seven for the second print. What ratio represents the height-to-width ratio of this image? Answer in fraction form.

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 Start by writing the size ratio of the original image as a fraction.

3in.:4in. ⇔ 3in./4in. Next, scale this ratio to find the ratio for the first size that Mark wants. Remember, he wants to enlarge the original image four times. This means he should multiply the numerator and the denominator of the original ratio by 4 to get the ratio for the new image. 3in.* 4/4in.* 4=12in./16in. The size of the first print will be 12 inches in height and 16 inches in width.

b Follow a similar process as in Part A to find the size ratio for the second print. This time, Mark wants to increase the size of his image by a factor of seven. Multiply the numerator and denominator of the original ratio by 7 to get the size ratio for the second image.

3in.* 7/4in.* 7=21in./28in. The second print will be 21 inches in height and 28 inches in width. Mark printed out his favorite photo of the Walter Pyramid in two sizes and hung them on his bedroom wall. Now his bedroom looks fantastic with the art!

Discussion

Ratio Table

Tables of equivalent ratios are called ratio tables. Consider the ratio table containing the distance Zain 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.

Four ways to produce ratio tables.
Example

Exploring the Aspect Ratio of Mark's New Phone Camera

Mark received a new phone from his parents that has an excellent camera for his photography hobby.

A brand new cellphone

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?
b Which of the following graphs represents the ratio table for the aspect ratio of Mark's new phone camera?
Four coordinate planes.

Hint
a A ratio table can be created by multiplying or dividing each quantity in a ratio by the same number or by adding or subtracting in increments of the 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, the widths.

Solution
a There are two methods to find ratios for a ratio table. The first method is to multiply or divide both quantities in a ratio by the same number. The second method is to add and subtract quantities in increments of the original ratio. With this in mind, consider the given ratio table.
Height 2 A 6 18
Width 3 6 9 B

Use the initial ratio of 2:3 to determine the ratio for the second column. Note that adding 3 to the width of the first ratio gives 6. This means that 2 can be added to the height of the first ratio to find the height that corresponds to a width of 6, which is the value of A.

Addition applied to the first column to find the value of A in the second column.

Use the ratio of 6:9 to determine the corresponding width for a height of 18. Since 6*3=18, multiply 9 by 3 to get the value of B.

The ratio in the third column multiplied by 3 to find the value of B

Therefore, A=4 and B=27.

b Graph the ratio table found in Part A on a coordinate plane. The x-axis will represent the heights and the y-axis the widths. 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, only the first quadrant is required. Begin at the origin and move along the x-axis according to the number of units specified by the height. Then, move vertically the number of units specified by the width to graph each ratio.

Graph of the ratio table.

This graph corresponds to option A.

Example

Mark and Maya's Five-Day Photo Challenge

Mark and Maya met up to compare the results of a five-day photo challenge. They each recorded the total number photos taken on each day in a table so that they could make a collage after finishing the challenge. The following table shows the data recorded by Maya and Mark.

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.
Four coordinate planes
b Using both the table and the graph to determine which of the statements below are true. Select all that apply.

Hint
a Graph both tables on a coordinate plane. Begin at the origin and move along the x-axis by the number of units specified by the day. Next, move vertically the number of units specified by the photos taken to graph each data.
b Use the first column of each table to compare both ratios. Draw a line through each data. Which line is steeper?

Solution
a The tables will be graphed one by one on a coordinate plane. The x-axis will indicate the days, while the y-axis will show the number of photos taken. Begin 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, begin at the origin and move along the x-axis the number of units specified by the day. Then, move the number of units specified by the photos taken vertically to graph each ordered pair.

Maya's data on a coordinate plane.

Before adding Mark's data to the graph, write each column of the table as an ordered pair.

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)

Each ordered pair can now be added to the graph.

Maya's and Mark's data on a coordinate plane.

The graph corresponds to the one given in option B.

b In this part, the ratios of photos taken to days passed for Mark and Maya will be compared. Consider the first column of both tables 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, look at the graph drawn in Part A.

A line drawn through each data.

Notice that both sets of points appear to fit in straight lines. However, the line for Mark is steeper than the line for Maya. The statements that are true about the ratios of photos taken to days passed have been identified.

Statement True?
The ratio of photos to days for Maya is 5:1. ✓
The ratio of photos to days for Mark is 7:1. ✓
The line for Mark's data is steeper than the line for Maya's. ✓
The ratio of days to photos for Maya is 5:1. *
The ratio of days to photos for Mark is 7:1. *
The line for Maya's data is steeper than the line for Mark's. *
Example

How Much Money Will Mark Earn?

Maya suggested Mark post his outstanding artwork on a website for people to buy. As a beginner seller, Mark received a graph that showed how much he would earn per photo sold.

Graph that shows the amount earned per photo sold.
However, the graph does not provide information on Mark's earnings if he sells six photos. Assuming the ratio of photos sold to earnings remains constant, how much money would Mark earn from selling six photos?

Hint

Use the relationship between a pair of consecutive points on the graph to determine the ratio of money earned to photos sold. Apply that ratio to find the point for selling six photos on the graph.

Solution

Mark wants to know how much money he would earn for selling six photos. It is given that the ratio of money earned per photo sold remains constant. Consider the graph carefully. The points appear to fall in a straight line.

A straight line through the points representing the amount earned and photos sold.

The point (4,6) is 2 units to the right and 3 units up from point (2,3).

Showing the x- and y-units to get from point (2,3) to (4,6) on the graph.

This means that the ratio of the money earned to photos sold is 3: 2 — Mark will earn $3 for every 2 photos sold. The amount of money he will earn by selling six photos can be determined using this ratio. Move 2 units to the right and 3 units up from (4,2) to locate this point.

Moving 2 units to the right and 3 units up from point (4,6) to locate the missing point (6,9) on the graph.

Therefore, Mark would earn $9 for selling six photos.

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. However, notice that the initial ratio in Mark's table can be simplified. 20÷ 5/25÷ 5=4/5 This means that the ratios 20:25 and 4:5 are the equivalent, which implies that all the ratios in both tables are also equivalent. A question that arises here is what would happen if these two ratio tables were plotted on a coordinate plane.

Two ratio tables plotted on a coordinate plane, with the x-axis representing time in minutes and the y-axis representing photos printed per minute. A line connects points since ratios are equivalent.
Notice that both data sets fit on the same line. This indicates that they are collinear. In conclusion, if two ratio tables with equivalent initial ratios are graphed, they will lie on the same line, meaning they are collinear.
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Ratio Tables and Graphs
Exercise 3.1
A survey of students was conducted. Out of the total students surveyed, 9 in 10 have a pet. The difference between those who have pets and those who do not is 112. How many students were surveyed?

We are told that 9 out of 10 students surveyed own a pet. This means that only 1 out of 10 students surveyed does not own a pet. We can use this information to write the ratio of students who own a pet to those who do not. 9:1 For every 10 students, the difference of students who own a pet and who do not is 9−1=8. We know that the total difference for all students surveyed is 112. We can scale the ratio and find equivalent ratios to create a ratio table to see if any give us the desired difference. Consider that 9*10=90, which is closer to our target difference of 112. Let's start scaling this ratio by 10 and upwards.

Students Who Own a Pet Students Who Do Not Own a Pet
9 1
90 10
99 11
108 12
117 13
126 14
135 15

Let's find the difference between the students who own a pet and the students who do not own a pet for each row of the table to see if any of them gives a difference of 112.

Students Who Own a Pet Students Who Do Not Own a Pet Difference
9 1 9-1=8
90 10 90-10=80
99 11 99-11=88
108 12 108-12=96
117 13 117-13=104
126 14 126-14=112
135 15 135-15=120

We found that the ratio of 126:14 has a difference of 112. We can add these quantities to find the total number of students surveyed. Let's do it! Total Number of Students Surveyed 126+14=140students

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Ratio Tables and Graphs

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