Proportional Relationships and Rate of Change

Lesson: Unit Rates and Complex Fractions

When two quantities are compared using division, we call the comparison a rate. Rates are used in real life all the time. For example, miles per hour, dollars per pound, and words per minute are all rates.

A unit rate is a rate with a denominator of 1. It tells how much of one quantity there is for 1 unit of another quantity. For example, if 180 miles are traveled in 3 hours, then the unit rate is 60 miles per 1 hour, or 60 miles per hour.

In 8th grade, unit rates often involve fractions and decimals. Sometimes the rate itself looks like a fraction divided by another fraction. This is called a complex fraction.

For example, the expression \(\frac{3}{4} \div \frac{1}{2}\) is a complex fraction because one fraction is being divided by another fraction. Complex fractions are useful because they help us find unit rates when amounts are less than 1 or are written as fractions.

Why unit rates matter

• They help compare situations fairly.
• They show how much happens for 1 unit.
• They connect arithmetic to algebra and proportional relationships.
• They help us decide which deal, speed, or method is better.

How to find a unit rate

To find a unit rate, divide the first quantity by the second quantity.

General form:

$$\text{unit rate} = \frac{\text{amount}}{\text{number of units}}$$

If the denominator is not 1, divide to make it 1.

For example, if 12 notebooks cost $3, then the cost per notebook is:

$$\frac{3}{12} = 0.25$$

So the unit rate is $0.25 per notebook.

Unit rates with fractions

Sometimes one or both quantities are fractions. Then we still divide, but we may need to divide by a fraction.

Remember this rule:

To divide by a fraction, multiply by its reciprocal.

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

The reciprocal of a fraction is found by flipping it. For example:

• The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
• The reciprocal of \(\frac{5}{4}\) is \(\frac{4}{5}\).

Understanding complex fractions as unit rates

A complex fraction often appears when we write a rate with fractional amounts. For example, suppose a machine uses \(\frac{3}{4}\) liter of fuel in \(\frac{1}{2}\) hour. The rate in liters per hour is:

$$\frac{\frac{3}{4}}{\frac{1}{2}}$$

This means:

$$\frac{3}{4} \div \frac{1}{2}$$

Now multiply by the reciprocal:

$$\frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2} = 1.5$$

So the unit rate is 1.5 liters per hour.

Steps for solving unit rates with complex fractions

1. Write the rate as a division problem.
2. If needed, rewrite mixed numbers as improper fractions.
3. Divide by multiplying by the reciprocal.
4. Simplify the answer.
5. Add units so the answer makes sense.

Worked Example 1: Simple fraction unit rate

A runner jogs \(\frac{3}{5}\) mile in \(\frac{1}{10}\) hour. Find the unit rate in miles per hour.

Step 1: Write the rate as division.

$$\frac{3}{5} \div \frac{1}{10}$$

Step 2: Multiply by the reciprocal.

$$\frac{3}{5} \times \frac{10}{1} = \frac{30}{5} = 6$$

Answer: The unit rate is 6 miles per hour.

This means the runner travels 6 miles in 1 hour if the speed stays constant.

Worked Example 2: Decimal unit rate

Three pounds of apples cost $4.50. What is the cost per pound?

We divide cost by number of pounds:

$$\frac{4.50}{3} = 1.50$$

Answer: The unit rate is $1.50 per pound.

This example uses decimals, but the idea is the same. A unit rate always tells the amount for 1 unit.

Worked Example 3: Complex fraction with mixed numbers

A recipe uses \(1\frac{1}{2}\) cups of flour to make \(\frac{3}{4}\) batch of muffins. How many cups of flour are needed for 1 full batch?

Step 1: Rewrite the mixed number.

$$1\frac{1}{2} = \frac{3}{2}$$

Step 2: Write the unit rate.

$$\frac{\frac{3}{2}}{\frac{3}{4}} = \frac{3}{2} \div \frac{3}{4}$$

Step 3: Multiply by the reciprocal.

$$\frac{3}{2} \times \frac{4}{3} = \frac{12}{6} = 2$$

Answer: The unit rate is 2 cups of flour per batch.

This tells us how much flour is needed for exactly 1 batch.

Worked Example 4: Mixed units

A car travels 126 miles using 4.5 gallons of gas. Find the unit rate in miles per gallon.

Divide miles by gallons:

$$\frac{126}{4.5} = 28$$

Answer: The car gets 28 miles per gallon.

This is a unit rate with mixed units because we are comparing miles and gallons.

How to interpret a unit rate

Finding the number is only part of the job. You must also understand what it means.

• 6 miles per hour means 6 miles are traveled in 1 hour.
• $1.50 per pound means each pound costs $1.50.
• 2 cups per batch means each full batch needs 2 cups.
• 28 miles per gallon means the car travels 28 miles on 1 gallon.

Common mistakes to avoid

• Reversing the division: Be careful about which quantity goes on top. For miles per hour, divide miles by hours, not hours by miles.
• Forgetting the reciprocal: When dividing by a fraction, flip the second fraction and multiply.
• Leaving out units: Always label the answer, such as dollars per item or miles per hour.
• Not changing mixed numbers: Rewrite mixed numbers as improper fractions before dividing.

Quick check: Which unit rate should you use?

If the question asks:

• cost per item, divide total cost by number of items
• miles per hour, divide miles by hours
• words per minute, divide words by minutes
• ounces per serving, divide ounces by servings

Connection to proportional relationships

In a proportional relationship, the unit rate is the constant amount that connects the two quantities. If a relationship is proportional, the unit rate stays the same.

For example, if oranges cost $2 per pound, then:

• 1 pound costs $2
• 2 pounds cost $4
• 3 pounds cost $6

The cost changes, but the unit rate stays constant at $2 per pound. This constant unit rate is an important idea in understanding linear relationships later.

Summary

A rate compares two quantities using division, and a unit rate tells the amount for 1 unit. When fractions are involved, unit rates may look like complex fractions, which are solved by dividing fractions.

To solve a complex fraction, rewrite it as division and multiply by the reciprocal of the second fraction. Then simplify and include the correct units.

If you keep track of what the units mean and divide in the correct order, unit rates become a powerful way to compare situations and solve real-world problems.

Constant of Proportionality

In many real-life situations, one quantity changes at a constant rate compared to another quantity. When this happens, the relationship is called a proportional relationship.

The constant of proportionality is the number that tells how much one quantity changes for each 1 unit of the other quantity. It is the same multiplier every time.

For example, if each notebook costs $3, then the total cost is always 3 times the number of notebooks. The number 3 is the constant of proportionality.

In a proportional relationship, the equation has the form:

$$y = kx$$

Here:

• \(x\) is one quantity

• \(y\) is the other quantity

• \(k\) is the constant of proportionality

You can find the constant of proportionality by dividing:

$$k = \frac{y}{x}$$

This works only when the relationship is proportional and the ratio stays the same.

How to recognize a proportional relationship

• The ratio \(\frac{y}{x}\) is the same for all pairs of values.

• The equation can be written as \(y = kx\).

• Its graph is a straight line that goes through the origin, which is the point \((0,0)\).

• In words, one quantity is a constant multiple of the other.

Finding the constant of proportionality from a table

Look at each pair of values and divide \(y\) by \(x\). If the result is always the same, that result is the constant of proportionality.

Worked Example 1: Table

A table shows the number of bags of apples and the total cost.

$$ \begin{array}{c|c} \text{Bags} & \text{Cost} \\ \hline 1 & 4 \\ 2 & 8 \\ 3 & 12 \\ 5 & 20 \end{array} $$

Let \(x\) = number of bags and \(y\) = cost.

Now divide cost by number of bags:

$$\frac{4}{1} = 4, \quad \frac{8}{2} = 4, \quad \frac{12}{3} = 4, \quad \frac{20}{5} = 4$$

The ratio is always 4, so the relationship is proportional.

The constant of proportionality is:

$$k = 4$$

This means each bag costs $4. The equation is:

$$y = 4x$$

Finding the constant of proportionality from an equation

If an equation is already written in the form \(y = kx\), then the constant of proportionality is just the number in front of \(x\).

Worked Example 2: Equation

Find the constant of proportionality in:

$$y = 7x$$

The equation matches the form \(y = kx\).

So:

$$k = 7$$

This means \(y\) is always 7 times \(x\).

For example:

• If \(x = 1\), then \(y = 7\)

• If \(x = 3\), then \(y = 21\)

Finding the constant of proportionality from a graph

On a graph of a proportional relationship, the line passes through the origin. To find the constant of proportionality, choose a point on the line and divide \(y\) by \(x\).

Worked Example 3: Graph

A graph shows a line passing through \((0,0)\) and \((4,10)\).

Use the point \((4,10)\):

$$k = \frac{y}{x} = \frac{10}{4} = \frac{5}{2} = 2.5$$

The constant of proportionality is:

$$k = 2.5$$

So the equation is:

$$y = 2.5x$$

This means for every 1 unit increase in \(x\), \(y\) increases by 2.5.

Finding the constant of proportionality from words

Sometimes a problem describes the relationship in a sentence. Look for a phrase like:

• "for each"

• "per"

• "times as much"

These often tell you the constant of proportionality.

Worked Example 4: Verbal Description

A machine fills 6 bottles every minute. Let \(x\) be the number of minutes and \(y\) be the number of bottles filled.

The machine fills 6 bottles for each 1 minute, so the constant of proportionality is 6.

$$k = 6$$

The equation is:

$$y = 6x$$

If the machine runs for 4 minutes, then:

$$y = 6(4) = 24$$

So it fills 24 bottles.

How to tell when a relationship is not proportional

Not every table, graph, or equation shows a proportional relationship.

A relationship is not proportional if:

• The ratio \(\frac{y}{x}\) is not constant

• The graph does not go through \((0,0)\)

• The equation is not in the form \(y = kx\)

For example, the equation

$$y = 3x + 2$$

is not proportional because of the +2. It does not match the form \(y = kx\).

Common mistakes to avoid

• Mixing up the order of division: If the relationship is written as \(y = kx\), find \(k\) using \(\frac{y}{x}\), not \(\frac{x}{y}\).

• Assuming every straight line is proportional: A graph must be a straight line and pass through the origin.

• Using a table without checking all rows: Make sure the ratio is the same for every pair.

• Forgetting what \(k\) means: It tells how much \(y\) there is for each 1 unit of \(x\).

Quick check

1. In the equation \(y = 9x\), what is the constant of proportionality?

2. If a table has pairs \((2,6)\), \((4,12)\), and \((5,15)\), what is \(k\)?

3. A runner goes 8 miles per hour. If \(x\) is hours and \(y\) is miles, write the equation.

4. Is \(y = 5x + 1\) proportional? Why or why not?

Answers

1. \(k = 9\)

2. \(\frac{6}{2} = 3\), \(\frac{12}{4} = 3\), \(\frac{15}{5} = 3\), so \(k = 3\)

3. \(y = 8x\)

4. No. It is not proportional because it is not in the form \(y = kx\).

Summary

The constant of proportionality is the constant multiplier in a proportional relationship. It tells how much one quantity changes for each 1 unit of another quantity.

You can find it from a table, graph, equation, or verbal description. In every case, look for the same idea: a relationship that can be written as

$$y = kx$$

and where

$$k = \frac{y}{x}$$

If the ratio stays constant, the relationship is proportional. If it does not, then it is not proportional.

Graphing Proportional Relationships

In 8th grade, you learn that some quantities change together in a very special way. When one quantity is always a constant multiple of another, the relationship is called proportional.

For example, if apples cost \(\$2\) each, then the total cost is always \(2\) times the number of apples. If you buy 1 apple, the cost is \(\$2\). If you buy 3 apples, the cost is \(\$6\). If you buy 10 apples, the cost is \(\$20\). The ratio stays the same, so this is a proportional relationship.

When we graph a proportional relationship, the graph has an important pattern: it forms a straight line that passes through the origin.

The origin is the point \((0,0)\). It means that when one quantity is 0, the other quantity is also 0. This makes sense in proportional situations. If you buy 0 apples, the cost is \(\$0\).

1. What makes a relationship proportional?

A relationship is proportional if it can be written in the form

$$y = kx$$

In this equation:

• \(x\) is one quantity,
• \(y\) is the other quantity,
• \(k\) is the constant of proportionality.

The constant of proportionality tells how much \(y\) changes for each 1 unit of \(x\). It is also the unit rate.

For example, in

$$y = 3x$$

the constant of proportionality is \(3\). This means for every 1 unit increase in \(x\), \(y\) increases by 3.

2. Key features of the graph

A graph of a proportional relationship has these features:

• It is a straight line.
• It passes through the origin, \((0,0)\).
• The slope, or steepness, is the constant of proportionality.

If the constant of proportionality is larger, the line is steeper. If it is smaller, the line is less steep.

3. How to graph a proportional relationship

There are two common ways to graph a proportional relationship.

Method A: Start with an equation

1. Write the equation in the form \(y = kx\).
2. Plot the origin, \((0,0)\).
3. Choose some \(x\)-values.
4. Use the equation to find the matching \(y\)-values.
5. Plot the points.
6. Draw a straight line through the points.

Method B: Start with a table

1. Check whether the ratios \(\frac{y}{x}\) are the same for all nonzero \(x\)-values.
2. If the ratio is constant, the relationship is proportional.
3. Plot the ordered pairs.
4. Make sure the graph forms a straight line through \((0,0)\).

4. Why does the graph go through the origin?

In a proportional relationship, if \(x = 0\), then

$$y = k(0) = 0$$

So the point \((0,0)\) must be on the graph. This is one of the easiest ways to recognize a proportional graph.

If a straight line does not go through the origin, then it is not proportional.

Worked Example 1: Graphing from an equation

Graph the relationship

$$y = 2x$$

Step 1: Identify the constant of proportionality.

Here, \(k = 2\).

Step 2: Make a table of values.

• If \(x=0\), then \(y=2(0)=0\)
• If \(x=1\), then \(y=2(1)=2\)
• If \(x=2\), then \(y=2(2)=4\)
• If \(x=3\), then \(y=2(3)=6\)

So the points are:

$$ (0,0), (1,2), (2,4), (3,6) $$

Step 3: Plot the points and draw a line.

The points form a straight line through the origin. This confirms the relationship is proportional.

What does the graph mean?

For every 1 unit increase in \(x\), the value of \(y\) increases by 2.

Worked Example 2: Graphing from a table

A car travels at a constant speed. The table shows time and distance.

• 1 hour \(\) 50 miles
• 2 hours \(\) 100 miles
• 3 hours \(\) 150 miles
• 4 hours \(\) 200 miles

Let \(x\) be time in hours and \(y\) be distance in miles.

Step 1: Check the ratios.

$$\frac{50}{1} = 50, \quad \frac{100}{2} = 50, \quad \frac{150}{3} = 50, \quad \frac{200}{4} = 50$$

The ratio \(\frac{y}{x}\) is always 50, so this is a proportional relationship.

Step 2: Write the equation.

$$y = 50x$$

Step 3: Graph the points.

$$ (1,50), (2,100), (3,150), (4,200) $$

Also include the origin:

$$ (0,0) $$

Step 4: Draw the line.

The graph is a straight line through the origin.

Interpretation: The car travels 50 miles each hour. The constant of proportionality is 50.

Worked Example 3: Deciding whether a graph is proportional

Suppose a line is straight, but it passes through \((0,3)\) instead of \((0,0)\).

Is it proportional?

No. Even though the graph is a straight line, it does not pass through the origin. That means it cannot represent a proportional relationship.

A proportional graph must match the form

$$y = kx$$

But a line through \((0,3)\) would look more like

$$y = kx + 3$$

This is linear, but not proportional.

Worked Example 4: Finding the constant of proportionality from a graph

A graph passes through the points \((0,0)\) and \((4,12)\). Is the relationship proportional, and what is the constant of proportionality?

Step 1: Check for the origin.

Since the graph passes through \((0,0)\), it could be proportional.

Step 2: Find the constant of proportionality.

Use

$$k = \frac{y}{x}$$

Using the point \((4,12)\):

$$k = \frac{12}{4} = 3$$

So the equation is

$$y = 3x$$

Conclusion: Yes, the relationship is proportional, and the constant of proportionality is 3.

5. Connecting graphs, tables, and equations

For proportional relationships, these three forms all show the same idea:

• Table: The ratio \(\frac{y}{x}\) stays constant.
• Equation: The rule is \(y = kx\).
• Graph: The points lie on a straight line through the origin.

If you can move between these forms, you understand the relationship well.

6. Common mistakes to avoid

• Forgetting the origin: A proportional graph must go through \((0,0)\).
• Only checking if it is a straight line: Not every straight line is proportional.
• Mixing up \(x\) and \(y\): Be careful which quantity goes on each axis.
• Not checking the ratio: In a proportional relationship, \(\frac{y}{x}\) must stay the same.

7. Quick checklist for graphing proportional relationships

• Can the relationship be written as \(y = kx\)?
• Is the ratio \(\frac{y}{x}\) constant?
• Does the graph make a straight line?
• Does the line pass through \((0,0)\)?

If the answer to all of these is yes, then the relationship is proportional.

Summary

A proportional relationship can be written as \(y = kx\), where \(k\) is the constant of proportionality. On a graph, proportional relationships form straight lines that pass through the origin.

To graph one, plot \((0,0)\), use the equation or table to find more points, and draw a straight line. Always remember: straight line + origin = proportional relationship.

Unit Rate as Slope

In 8th grade math, you learn that a proportional relationship can be shown in a table, an equation, a graph, or a real-world situation.

When a relationship is proportional, one quantity is always a constant multiple of the other. This means the ratio stays the same.

That constant ratio is called the unit rate. On a graph, this same idea is called the slope.

This lesson will help you see how unit rate, slope, and the steepness of a graph are connected.

1. What is a unit rate?

A rate compares two different kinds of quantities, like miles and hours, dollars and pounds, or pages and minutes.

A unit rate tells how much of one quantity there is for 1 unit of the other quantity.

For example, if a car travels 120 miles in 3 hours, the unit rate is:

$$\frac{120}{3} = 40$$

So the car travels 40 miles per hour.

This means that for every 1 hour, the car goes 40 miles.

2. What is slope?

Slope tells how much the output changes when the input changes.

On a graph, slope measures how steep a line is.

Slope is often written as:

$$\text{slope} = \frac{\text{rise}}{\text{run}}$$

Rise means the vertical change, and run means the horizontal change.

It can also be written using coordinates:

$$m = \frac{y_2-y_1}{x_2-x_1}$$

Here, the slope compares the change in \(y\) to the change in \(x\).

3. How are unit rate and slope connected?

In a proportional relationship, the graph is a straight line that goes through the origin, which is \((0,0)\).

The equation of a proportional relationship is:

$$y = kx$$

The number \(k\) is the constant of proportionality.

That same number \(k\) is also the unit rate and the slope.

So in a proportional relationship:

• constant of proportionality = unit rate = slope

This is a very important idea. It means that if you know one of these, you know all three.

4. Why does slope represent unit rate?

Think about a graph where \(x\) is hours and \(y\) is miles traveled.

If the slope is 40, that means:

$$\frac{\text{change in miles}}{\text{change in hours}} = 40$$

So for every 1 hour, the miles increase by 40.

That is exactly what a unit rate means: 40 miles for 1 hour.

So the slope tells the unit rate when the relationship is proportional.

5. Steepness and slope

The slope also tells how steep the line looks on the graph.

• A greater positive slope means a steeper line going upward from left to right.
• A smaller positive slope means a less steep line going upward.
• If the slope is 0, the line is flat.

For proportional relationships in this lesson, we usually focus on positive slopes.

Compare these ideas:

• A line with slope 5 rises 5 units for every 1 unit it runs.
• A line with slope 2 rises 2 units for every 1 unit it runs.

The line with slope 5 is steeper because it goes up more quickly.

6. Finding unit rate from a table

If a relationship is proportional, you can find the unit rate by dividing \(y\) by \(x\).

If the value of \(\frac{y}{x}\) is always the same, then the relationship is proportional, and that value is the slope.

Worked Example 1: Table to unit rate and slope

A bicyclist rides the distances shown below.

• 1 hour \(\rightarrow\) 12 miles
• 2 hours \(\rightarrow\) 24 miles
• 3 hours \(\rightarrow\) 36 miles

Find the unit rate and slope.

Step 1: Divide distance by time.

$$\frac{12}{1}=12, \quad \frac{24}{2}=12, \quad \frac{36}{3}=12$$

Step 2: Since the ratio is always 12, the relationship is proportional.

Step 3: The unit rate is 12 miles per hour.

Step 4: The slope is also 12.

If you graph the points, the line goes through the origin and rises 12 for every 1 it runs.

7. Finding slope from a graph

When points are on a graph, you can count rise and run between two points.

If the graph represents a proportional relationship, the slope you find is the unit rate.

Worked Example 2: Graph idea using points

Suppose a line goes through \((0,0)\) and \((4,20)\).

Find the slope and explain the unit rate.

Step 1: Use rise over run.

$$\text{slope} = \frac{20-0}{4-0} = \frac{20}{4} = 5$$

Step 2: The slope is 5.

Step 3: Because the line goes through the origin, this is a proportional relationship.

Step 4: The unit rate is 5.

This means the quantity increases by 5 units of \(y\) for every 1 unit of \(x\).

8. Finding slope from an equation

In a proportional relationship, the equation has the form:

$$y = kx$$

The number multiplying \(x\) is the slope and the unit rate.

Worked Example 3: Equation to unit rate and slope

A relationship is given by:

$$y = 3.5x$$

Find the slope and explain what it means.

Step 1: Compare the equation to \(y = kx\).

Here, \(k = 3.5\).

Step 2: The slope is 3.5.

Step 3: The unit rate is also 3.5.

This means that for every increase of 1 in \(x\), \(y\) increases by 3.5.

For example, if \(x\) represents pounds and \(y\) represents cost, then the cost is $3.50 per pound.

9. Using two points to find slope

Even if you are not given the origin, you can still find slope using any two points on the line.

If the relationship is proportional, the line will still pass through the origin, even if it is not shown in the problem.

Worked Example 4: Two points on a proportional line

A line includes the points \((2,14)\) and \((5,35)\).

Find the slope and unit rate.

Step 1: Use the slope formula.

$$m = \frac{35-14}{5-2} = \frac{21}{3} = 7$$

Step 2: The slope is 7.

Step 3: Check the ratio \(\frac{y}{x}\).

$$\frac{14}{2}=7 \quad \text{and} \quad \frac{35}{5}=7$$

Since the ratio is constant, the relationship is proportional.

Step 4: The unit rate is 7.

This means \(y\) increases by 7 for every 1 increase in \(x\).

10. How to tell whether unit rate equals slope

The unit rate equals the slope when the relationship is proportional.

You can check for a proportional relationship in these ways:

• The graph is a straight line through \((0,0)\).
• The equation is in the form \(y = kx\).
• The ratio \(\frac{y}{x}\) is constant in a table.

If these are true, then the slope is the unit rate.

11. Common mistakes to avoid

• Mixing up rise and run: Slope is \(\frac{\text{rise}}{\text{run}}\), not the other way around.
• Using subtraction in the wrong order: In \(\frac{y_2-y_1}{x_2-x_1}\), keep the order the same in the top and bottom.
• Assuming every line shows a proportional relationship: The line must go through the origin.
• Forgetting the meaning: Slope is not just a number. It tells how much \(y\) changes for each 1 unit of \(x\).

12. Quick check for understanding

Ask yourself these questions when solving a problem:

1. Is the relationship proportional?
2. What does 1 unit of \(x\) represent?
3. How much does \(y\) change for that 1 unit?
4. Does the graph’s steepness match the slope value?

13. Summary

In a proportional relationship, the unit rate and the slope are the same number.

This number tells how much \(y\) changes for every 1 unit increase in \(x\).

On a graph, slope also shows the steepness of the line. A greater slope means a steeper line.

You can find this value from a table, a graph, two points, or an equation like \(y = kx\).

Remember: in proportional relationships, constant of proportionality = unit rate = slope.

Comparing Proportional Relationships

In 8th Grade maths, you will often see situations where two quantities change together. For example, the number of notebooks you buy and the total cost, or the number of hours worked and the amount of money earned.

When one quantity is always a constant multiple of the other, the relationship is called a proportional relationship.

A big skill is being able to compare two proportional relationships, even when they are shown in different ways, such as:

• an equation,
• a table,
• a graph, or
• a word description.

This lesson will help you figure out which relationship has the greater rate, how to read the unit rate from different forms, and how to compare them correctly.

1. What is a proportional relationship?

A proportional relationship can be written in the form

$$y = kx$$

Here, k is the constant of proportionality. It tells how much y changes for each 1 unit of x.

This constant is also called the unit rate or the rate of change in proportional situations.

For example, if

$$y = 4x$$

then the constant of proportionality is 4. This means for every 1 increase in x, y increases by 4.

2. Key idea when comparing proportional relationships

To compare two proportional relationships, find the unit rate for each one.

Then ask:

• Which one has the greater constant of proportionality?
• Which one grows faster?
• Which gives more output for the same input?

If one relationship has a larger unit rate, then it has the steeper graph and the larger value of k.

3. How to recognize a proportional relationship in different forms

From an equation:

If the equation is written as

$$y = kx$$

then the constant of proportionality is the number multiplying x.

Example: In \(y = 7x\), the unit rate is 7.

From a table:

Divide \(y\) by \(x\) for any pair of values.

If the ratio \(\frac{y}{x}\) is always the same, then the relationship is proportional, and that common ratio is the unit rate.

From a graph:

A proportional relationship is shown by a straight line that passes through the origin, which is \((0,0)\).

To find the unit rate from the graph, choose a point on the line and compute

$$\text{unit rate} = \frac{y}{x}$$

This is also the slope of the line in a proportional relationship.

From a word description:

Look for a constant rate such as:

• "$3 per notebook"
• "travels 60 miles each hour"
• "earns $12 for each hour worked"

That constant rate is the unit rate.

4. Comparing relationships shown in different ways

Sometimes one relationship is given as an equation and the other as a graph. Or one might be in a table and the other in words. The method is always the same:

1. Find the unit rate for the first relationship.
2. Find the unit rate for the second relationship.
3. Compare the two rates.

The relationship with the greater unit rate has:

• the larger constant of proportionality,
• the steeper line on a graph,
• and the greater output for the same input.

Worked Example 1: Equation vs. Equation

Compare these two proportional relationships:

• Relationship A: \(y = 5x\)
• Relationship B: \(y = 3x\)

Step 1: Find the unit rate for each.

• For A, the constant of proportionality is 5.
• For B, the constant of proportionality is 3.

Step 2: Compare.

Since \(5 > 3\), Relationship A has the greater rate of change.

Conclusion: For the same value of \(x\), Relationship A gives a larger value of \(y\).

Worked Example 2: Table vs. Equation

Compare the relationship shown in the table with the equation \(y = 6x\).

x	y
1	4
2	8
3	12

Step 1: Find the unit rate from the table.

Compute \(\frac{y}{x}\):

• \(\frac{4}{1} = 4\)
• \(\frac{8}{2} = 4\)
• \(\frac{12}{3} = 4\)

The table shows a proportional relationship with unit rate 4.

Step 2: Find the unit rate from the equation.

In \(y = 6x\), the unit rate is 6.

Step 3: Compare.

Since \(6 > 4\), the equation \(y = 6x\) represents the greater proportional relationship.

Conclusion: The equation grows faster than the relationship in the table.

Worked Example 3: Graph vs. Equation

Suppose Relationship A is shown by a graph that passes through the points \((0,0)\) and \((2,10)\). Relationship B is given by

$$y = 4x$$

Which relationship has the greater unit rate?

Step 1: Find the unit rate from the graph.

Use the point \((2,10)\):

$$\frac{y}{x} = \frac{10}{2} = 5$$

So Relationship A has unit rate 5.

Step 2: Find the unit rate from the equation.

Relationship B has unit rate 4.

Step 3: Compare.

Since \(5 > 4\), the graph represents the greater proportional relationship.

Conclusion: The line on the graph is steeper than the graph of \(y = 4x\).

Worked Example 4: Word description vs. Graph

Relationship A: A taxi charges $2 per mile.

Relationship B is shown on a graph passing through \((0,0)\) and \((3,9)\).

Which relationship has the greater rate?

Step 1: Find the unit rate for Relationship A.

"$2 per mile" means the unit rate is 2.

Step 2: Find the unit rate for Relationship B.

Use the point \((3,9)\):

$$\frac{9}{3} = 3$$

So Relationship B has unit rate 3.

Step 3: Compare.

Since \(3 > 2\), Relationship B has the greater rate.

Conclusion: Relationship B increases more for each 1 unit of input.

5. Important graph facts

• A proportional relationship must be a straight line.
• It must pass through the origin, \((0,0)\).
• A steeper line means a greater unit rate, if the lines are increasing.
• The ratio \(\frac{y}{x}\) stays constant for every point on the line, except when \(x=0\).

6. Common mistakes to avoid

• Do not just compare points without finding the unit rate. Two relationships may use different \(x\)-values, so compare the rate, not just the \(y\)-values.
• Do not assume every line is proportional. If a graph does not pass through \((0,0)\), it is not a proportional relationship.
• Do not mix up \(x\) and \(y\). For proportional relationships, use \(\frac{y}{x}\) to find the constant of proportionality.
• Do not forget the meaning of the rate. A unit rate describes how much output changes for each 1 unit of input.

7. Quick strategy you can always use

When you need to compare two proportional relationships, ask yourself:

1. Is each relationship proportional?
2. What is the unit rate of the first one?
3. What is the unit rate of the second one?
4. Which unit rate is larger?

If the unit rates are equal, then the two relationships are the same proportional relationship, even if they are shown in different forms.

8. Final summary

Comparing proportional relationships means comparing their constant of proportionality, also called the unit rate.

No matter how the relationship is shownby equation, table, graph, or wordsyou can find the rate and compare it.

Remember: in a proportional relationship, the equation looks like \(y = kx\), the graph is a straight line through the origin, and the ratio \(\frac{y}{x}\) stays constant.

Once you know the unit rates, you can decide which relationship increases faster or whether they are equal.

Percent Applications Review

Percent means “out of 100”. When we say 25%, we mean 25 out of 100, or \(\frac{25}{100}\), which is the same as \(0.25\).

In real life, percents are used in many situations, such as sales, taxes, tips, markups, markdowns, discounts, and simple interest. To solve these problems, it helps to remember that percent problems are really proportional reasoning problems.

In this lesson, you will review how to:

• find a percent of a number,
• find a new price after a markup or markdown,
• add sales tax,
• find simple interest.

1. Changing a percent into a decimal

To use a percent in calculations, change it to a decimal by dividing by 100, or moving the decimal point two places left.

• \(15\% = 0.15\)
• \(8\% = 0.08\)
• \(125\% = 1.25\)

This is important because to find a percent of an amount, you multiply by the decimal form of the percent.

2. Finding a percent of a number

To find \(p\%\) of a number, use:

$$\text{percent amount} = (\text{decimal form of percent})(\text{whole amount})$$

For example, to find 20% of 50:

$$0.20 \times 50 = 10$$

So, 20% of 50 is 10.

This idea is the foundation for all percent applications. In markup, markdown, tax, and interest, you first find the percent amount, then combine it with the original amount.

3. Markup

A markup is an amount added to the original price. Stores may mark up items to make a profit.

To solve a markup problem:

1. Find the markup amount.
2. Add it to the original price.

Formula:

$$\text{markup} = (\text{markup rate})(\text{original price})$$ $$\text{new price} = \text{original price} + \text{markup}$$

4. Markdown

A markdown is an amount subtracted from the original price. This usually happens during a sale or discount.

To solve a markdown problem:

1. Find the markdown amount.
2. Subtract it from the original price.

Formula:

$$\text{markdown} = (\text{markdown rate})(\text{original price})$$ $$\text{sale price} = \text{original price} - \text{markdown}$$

5. Sales tax

Sales tax is an extra amount added to the price of an item when you buy it.

To solve a sales tax problem:

1. Find the tax amount.
2. Add it to the original price.

Formula:

$$\text{tax} = (\text{tax rate})(\text{price})$$ $$\text{total cost} = \text{price} + \text{tax}$$

6. Simple interest

Simple interest is money earned or paid based only on the original amount of money.

The original amount is called the principal.

The simple interest formula is:

$$I = Prt$$

where:

• \(I\) = interest
• \(P\) = principal
• \(r\) = annual interest rate as a decimal
• \(t\) = time in years

After finding the interest, you can find the total amount:

$$A = P + I$$

7. One-step shortcut using percent multipliers

Sometimes you can find the final amount in one step.

• For a 15% markup or tax, multiply by \(1.15\).
• For a 20% markdown, multiply by \(0.80\).

Why does this work?

If an item increases by 15%, then the new amount is:

$$100\% + 15\% = 115\% = 1.15$$

If an item decreases by 20%, then the new amount is:

$$100\% - 20\% = 80\% = 0.80$$

This shortcut is helpful, but make sure you understand the two-step method first.

Worked Example 1: Finding a markdown

A jacket costs \(\$60\). It is on sale for 25% off. What is the sale price?

Step 1: Find the markdown amount.

$$0.25 \times 60 = 15$$

The discount is \(\$15\).

Step 2: Subtract the markdown from the original price.

$$60 - 15 = 45$$

Answer: The sale price is \(\$45\).

You could also use the shortcut:

$$60 \times 0.75 = 45$$

Since 25% off means you pay 75% of the original price.

Worked Example 2: Adding sales tax

A video game costs \(\$48\). The sales tax rate is 7.5%. What is the total cost?

Step 1: Change the tax rate to a decimal.

$$7.5\% = 0.075$$

Step 2: Find the tax amount.

$$0.075 \times 48 = 3.60$$

The tax is \(\$3.60\).

Step 3: Add the tax to the original price.

$$48 + 3.60 = 51.60$$

Answer: The total cost is \(\$51.60\).

You could also use the multiplier \(1.075\):

$$48 \times 1.075 = 51.60$$

Worked Example 3: Markup

A store buys a backpack for \(\$32\) and marks it up 40%. What is the new price?

Step 1: Find the markup.

$$0.40 \times 32 = 12.80$$

The markup is \(\$12.80\).

Step 2: Add the markup to the original price.

$$32 + 12.80 = 44.80$$

Answer: The new price is \(\$44.80\).

Using the multiplier method:

$$32 \times 1.40 = 44.80$$

Worked Example 4: Simple interest

You deposit \(\$500\) into a savings account that earns 6% simple interest each year for 3 years. How much interest will you earn, and what will be the total amount?

Use the simple interest formula:

$$I = Prt$$

Substitute the values:

$$I = 500(0.06)(3)$$ $$I = 90$$

The interest earned is \(\$90\).

Now find the total amount:

$$A = P + I = 500 + 90 = 590$$

Answer: You will earn \(\$90\) in interest, and the total amount will be \(\$590\).

How to decide whether to add or subtract

A common mistake is using the percent correctly but then doing the wrong operation. Use these clues:

• Markup means add.
• Tax means add.
• Markdown, discount, or sale means subtract.
• Simple interest earned is usually added to the principal to get the total.

Common mistakes to avoid

• Forgetting to change the percent to a decimal.
• Subtracting when you should add, or adding when you should subtract.
• Finding only the tax, markup, or discount amount, but not the final total.
• In simple interest problems, forgetting that \(t\) is in years.

Helpful problem-solving steps

1. Read the problem carefully.
2. Identify the original amount.
3. Identify the percent rate.
4. Decide whether the percent amount should be added or subtracted.
5. Calculate carefully and label your answer.

Quick Review

• Percent means out of 100.
• To find a percent of a number, multiply by the decimal form.
• Markup and tax increase a price.
• Markdown decreases a price.
• Simple interest uses the formula \(I = Prt\).

Summary

Percent applications are all about finding a part of an amount and then using it in context. In shopping problems, you may add a markup or tax, or subtract a markdown. In money problems, simple interest tells how much extra money is earned or owed based on the original amount.

If you remember to change the percent to a decimal, multiply correctly, and then decide whether to add or subtract, you can solve many real-world percent problems with confidence.