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5. Ratios & Proportional Relationships
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Grade 6

5.04 Proportional relationships

Lesson
Practice

Proportional relationships

We say that two quantities have a proportional relationship if the values are always represented by the same ratio. When two quantities are proportional, we can use a ratio table to show equivalent ratios and find unknown values.

If a cookie recipe calls for 2 cups of sugar for every 4 cups of flour, we could write this as the ratio {4:2}.

Putting this in a ratio table, we have:

A table showing the relationship between cups of sugar (x) and cups of flour (y). The top row lists the values of x as 1, 2, 3, and 5. The bottom row lists the corresponding values of y as 2, 4, 6, and 10, indicating that y is twice the value of x. Red arrows point from x to y with the multiplication factor ×2.

This relationship is proportional because the ratio, \dfrac{y}{x}, of flour to sugar is constant:

\dfrac{2}{1} = \dfrac{4}{2} = \dfrac{6}{3} = \dfrac{10}{5} = 2

Let's consider another scenario. The cost to rent a scooter and time rented are shown in the table below:

A table showing the relationship between time in minutes and cost in dollars. The top row lists time values as 3, 6, 9, and 15 minutes. The bottom row lists the corresponding costs as 5 dollars, 7 dollars, 9 dollars, and 13 dollars.

This relationship is not proportional because the ratios between cost and time are not constant:

\dfrac{5}{3} \neq \dfrac{7}{6} \neq \dfrac{9}{9} \neq \dfrac{13}{15}

We see what each relationship looks like in a graph by turning each column from the table into an ordered pair \left(x, y \right).

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\text{Sugar (cups)}
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\text{Flour (cups)}
Graph shows the proportional relationship between cups of flour and cups of sugar.
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\text{Time (min)}
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\text{Cost} (\$)
Graph shows the non-proportional relationship between cost of a scooter and time.

In both graphs, we connected the points with a line because the values in between the points make sense for the context. Fore example, we could make a recipe with 3.5 cups of sugar or (depending on the renting rules) we could probably rent a scooter for 7.25 minutes.

If the contexts were changed so the values in between the points did not make sense, we would not connect them. For example, if we were comparing the ratio of \text{flour} : \text{eggs} in a recipe we would probably not use a fraction of an egg. Or if the scooter rental only allowed us to rent for specific amounts of time, then it would not make sense to calculate cost times outside of that.

We call the relationship continuous if it would make sense to include the values between points and discrete if only the points make sense in the context.

Two different graphs can be represented by the data depending on the order we choose. The above graph shows the relationship between cups of flour and cups of sugar where y represents the cups of flour and x represents the cups of sugar. This graph represents the ratio of y to x as 4:2.

A graph plotting cups of sugar against cups of flour. The x axis represents flour (cups) and the y axis represents sugar (cups). Points at (1,1), (2,2), (3,3), (4,4), and (5,5) form a straight line indicating a 1:1 ratio.

We can also create a graph for the ratio of y to x as 2:4 where y represents the cups of sugar and x represents the cups of flour. Notice the similarities and differences between the two graphs. They both pass through (0,0) but the ratio 4:2 is much steeper than the ratio 2:4.

It is important to state which quantity is represented by x and which quantitiy is represented by y.

A graph is proportional if the graph is linear, meaning it looks like a straight line, and it passes through the origin, (0,0).

The graph of a proportional relationship always includes (0,0) because we can create the equivalent ratio 0:0 by multiplying both parts of the ratio by 0. For the sugar and flour example, this would mean that a recipe that calls for 0 cups of flour would need 0 cups of sugar.

Examples

Example 1

If the ratio of y to x is represented by 3:1, plot the ratio as a single point on the coordinate plane.

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Worked Solution
Create a strategy

A ratio of the form y:x, means that the x represents the horizontal position and y the vertical position of the point.

Apply the idea

The horizontal position is 1, and the vertical position is 3.

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Example 2

Consider the following graph:

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\text{green}
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a

What ratio of y:x has been plotted?

Worked Solution
Create a strategy

Use the graph to find the ratio, y : x. Often, we choose the point where x=1 if y is a whole number as well. This will give us the most simplified ratio.

Apply the idea

The point (1, 2) lies on the line. At this point x=1 and y=2. So the ratio is 2 : 1.

Reflect and check

We could have chosen any point on the line in our graph. They all represent equivalent ratios. For example, if we chose the point (5, 10), we know:

\dfrac{10}{5} = \dfrac{2}{1}

So, the ratio 10:5 is an equivalent ratio to 2:1.

b

Which of the following could be represented by this graph and ratio?

A
For every 1 green candy, there are 2 red candies.
B
For every 2 green candies, there is 1 red candy.
Worked Solution
Create a strategy

The x-axis is labeled "green", and the y-axis is labeled "red".

Apply the idea

So the ratio y:x=2:1 is read as 2 \text{ reds}: 1\text{ green}.

So the answer is option A.

Example 3

Determine whether each of the following shows a proportional relationship:

a
x01234
y0714628
Worked Solution
Create a strategy

If the table represents a proportional relationship, it will be made up entirely of equivalent ratios. We can check to see if each ratio y:x is equivalent by first writing the ratios in fraction form.

Apply the idea

We can determine whether each ratio is equivalent by checking to see if we can simplify the fraction to be equal to the unit rate.

\displaystyle \dfrac{7}{1}\displaystyle =\displaystyle \dfrac{7}{1}
\displaystyle \dfrac{14 \div 2}{2 \div 2}\displaystyle =\displaystyle \dfrac{7}{1}
\displaystyle \dfrac{28 \div 4}{4 \div 4}\displaystyle =\displaystyle \dfrac{7}{1}
\displaystyle \dfrac{6 \div 3}{3 \div 3}\displaystyle =\displaystyle \dfrac{2}{1}

As we can see, all but one of the ratios are equal. \dfrac{6}{3} simplifies to \dfrac{2}{1}, which is not equal to the rest of the ratios which simplify to \dfrac{7}{1}.

Reflect and check

We can also determine whether the relationship is proportional using the relationship x:y.

\dfrac{1}{7} = \dfrac{2}{14} = \dfrac{4}{28} \neq \dfrac{3}{6}

In a proportional relationship all these ratios must be equal. As we can see, all but one of the ratios are equal. \dfrac{3}{6} simplifies to \dfrac{1}{2}, which is not equal to the rest of the ratios which simplify to \dfrac{1}{7}.

If the relationship x:y is not proportional then the relationship y:x is also not proportional.

b
x036912
y01234
Worked Solution
Create a strategy

Similar to part (a), if the table represents a proportional relationship, the ratio of y:x will be constant for every coordiante pair.

Apply the idea

Fractions are equivalent if we can multiply the numerator and denominator by the same number. Let's determine if the ratio of each coordinate pair is equivalent to the unit rate:

\displaystyle \dfrac{1}{3}\displaystyle =\displaystyle \dfrac{1}{3}
\displaystyle \dfrac{1}{3} \cdot \dfrac{2}{2}\displaystyle =\displaystyle \dfrac{2}{6}
\displaystyle \dfrac{1}{3} \cdot \dfrac{3}{3}\displaystyle =\displaystyle \dfrac{3}{9}
\displaystyle \dfrac{1}{3} \cdot \dfrac{4}{4}\displaystyle =\displaystyle \dfrac{4}{12}

All of the ratios are equal because we can create each fraction by multiplying \dfrac{1}{3} by a constant. Since the ratio of y:x is constant, this table represents a proportional relationship.

c
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Worked Solution
Create a strategy

In a proportional relationship, the coordinates will fall on a straight line that goes through the origin.

Apply the idea

The graph shows a straight line that goes through the origin. Therefore, this graph represents a proportional relationship.

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Worked Solution
Create a strategy

In a proportional relationship, the coordinates will fall on a straight line which goes through the origin.

Apply the idea

The graph shows a straight line that does not go through the origin. Therefore, this graph does not represent a proportional relationship.

e

Mark has \$500 in savings and the amount in his savings account doubles every year.

Worked Solution
Create a strategy

In a proportional relationship, a situation with increase or decrease at the same rate. It will also make sense to have 0 of both quantities.

Apply the idea

At the start (time 0), Mark had saved \$500. If the amount of money doubles every year, this is not a constant rate. We can create a table of values to confirm this:

Years passed01234
Total savings5001000200040008000

The first year, the account grows by \$500. The next year, the account grows by \$1000. Since the rate of increase is not constant and there are \$500 in his account at the beginning, this situation is not proportional.

Reflect and check

We could have determined this relationship was not proportional from the start. The point (0,0) could not fit this relationship, because Mark has \$500 at the start which means this relationship begins at (0, 500).

f

It just started raining, and it rains half an inch every hour all day.

Worked Solution
Create a strategy

In a proportional relationship, a situation with increase or decrease at the same rate. It will also fit the pattern to have 0 of both quantities at some point.

Apply the idea

At time 0, there were 0 inches of rain. Then, the rain fell the same amount each hour.

Time passed01234
Total rain0\dfrac{1}{2}11\dfrac{1}{2}2

Since the rate of increase is constant and there are 0 inches of rain at the beginning, this situation is proportional.

Example 4

The ratio table represents the proportional relationship between number of pens purchased and cost.

a

Complete the table:

\text{Pens}1020304050
\text{Cost (dollars)}11.6029.0058.00
Worked Solution
Create a strategy

In a ratio table, every column must represent an equivalent ratio. We can multiply or divide the number of pens by any value as long as we do the same to the cost in that column.

Apply the idea

Since 20 pens cost \$11.60, 10 pens (half of 20) cost \$11.60 \div 2 = \$5.80.

30 pens cost 3 times more than 10 pens.

40 pens cost 4 times more than 10 pens.

58=29 \cdot 2, so the number of pens is 100.

\text{Pens}1020304050100
\text{Cost (dollars)}5.8011.6017.4023.2029.0058.00
Reflect and check

We could have also found the unit rate by dividing the cost by the number of pens.

\dfrac{\$11.60}{20} = \dfrac{\$0.58}{1} We could then multiply \$0.58 by any number of pens to find the total cost.

b

Calculate the cost of buying 90 pens.

Worked Solution
Create a strategy

We know the cost of of 10 pens is \$5.80. Consider how many groups of 10 pens we have.

Apply the idea
\displaystyle \text{Cost}\displaystyle =\displaystyle 9 \cdot 5.80Find the cost of 9 lots of 10 pens
\displaystyle =\displaystyle \$52.20Evaluate
c

How much would you expect to pay for 5 pens?

Worked Solution
Create a strategy

5 pens is half the amount of 10 pens so they will cost half as much.

Apply the idea
\displaystyle \text{Cost}\displaystyle =\displaystyle 5.80 \div 2Halve the cost of 10 pens
\displaystyle =\displaystyle \$2.90Evaluate
Reflect and check

We could also use the unit rate we found in part (a). The unit rate was \$0.58. For 5 pens:

\$0.58 \cdot 5 = \$2.90

This is the same result as our other strategy.

Example 5

Zoe eats 6 sour candies every minute.

a

Does this situation represent a proportional relationship?

Worked Solution
Create a strategy

In a proportional relationship, a situation will increase or decrease at the same rate. It will also fit the pattern to have 0 of both quantities at some point.

Apply the idea

If 0 minutes has passed, Zoe will have eaten 0 candies. The ratio of candy eaten to time is always 6:1 so this situation represents a proportional relationship.

b

Create a table and a graph that represent this situation.

Worked Solution
Create a strategy

Let y be the number of candies Zoe eats and x be the time in minutes. We can use the ratio of y:x, which is 6:1, to create both a table and a graph.

Apply the idea

The ratio of candies to time is always 6:1 in this table:

x\text{ (Time)}1234510
y\text{ (Candies)}61218243060

To graph, we will assume that Zoe is continuously eating the candy. We will use our ratio to graph and connect our points with a line:

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Reflect and check
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In this problem, we assumed Zoe was eating the candy continuously, so we drew the line on the graph to connect the points. If we instead knew that she was eating each piece instantaneously at each minute we would not connect the points because the values in between would have no meaning since she does not eat additional candy between each minute.

c

Compare the characteristics that are easier or harder to see from the written context, the table, and the graph.

Worked Solution
Create a strategy

Consider the different parts of a proportional relationship that are important. Use the representation from part (a) and (b) to answer the question.

Apply the idea

The context makes it easy to see the connection between the ratio and real life. We can easily see the unit rate of 6 candies per minute. It is harder to see that the situation includes the point (0,0).

The table highlights that every ratio of y:x is equivalent and we can see how many candies were eaten for each of the different minutes. However, the table won't have all of the possible values.

The graph makes it easy to see that the relationship goes through the origin. It is easy to see that the relationship increases at a constant rate because it is linear, but it takes a little more work to see what the rate actually is.

Idea summary

We can use ratio tables to determine unknown values by multiplying or dividing. All of the values in the table will be equivalent ratios.

The graph of a proportional relationship (if extended far enough) is a straight line that passess through the origin \left(0,0 \right).

Outcomes

6.PFA.2

The student will identify and represent proportional relationships between two quantities, including those in context (unit rates are limited to positive values).

6.PFA.2b

Determine a missing value in a ratio table that represents a proportional relationship between two quantities using a unit rate.

6.PFA.2c

Determine whether a proportional relationship exists between two quantities, when given a table of values, context, or graph.

6.PFA.2d

When given a contextual situation representing a proportional relationship, find the unit rate and create a table of values or a graph.

6.PFA.2e

Make connections between and among multiple representations of the same proportional relationship using verbal descriptions, ratio tables, and graphs.

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