Proportions and ratios

A proportion is an equality between two ratios. Another way to think about proportional relationships is through equivalent fractions.

By multiplying these numbers together, we find that:

$\begin{array}{rl}3\times 12=& 4\times 9\\ 36=& 36\end{array}$

Both numbers are the same, so these fractions are indeed proportional.

If you would like, you can complete the next series of practice exercises using TVO Mathify. You can also use your notebook or the following fillable and printable document.

Task 1

Use cross-multiplication to check if the following are proportional or not. If they are not proportional, make an edit to the fractions to make them proportional.

The strategy of cross-multiplying can help us to find a missing unit in a proportion.

Example:

If three apples cost $4, how much would six apples cost?

When we use cross-multiplication, we will use a letter to represent the missing value. In this case, our missing value is the cost of six apples, so:

Let x represent the cost of six apples.

Then we can write our proportion and cross multiply:

$\begin{array}{rl}\frac{\text{\$}4}{3\text{ apples}}=& \frac{x}{6\text{ apples}}\\ 4\times 6=& 3x\\ 24=& 3x\\ x=& 8\end{array}$

So, it would cost $8 for 6 apples.

Why do I need this strategy?

Task 2

For each question, write a proportion statement using a letter to represent the missing value. Then, cross-multiply and solve for the missing value.

• If two shirts cost $29.95, how much would five shirts cost?
• You get six pieces of gum in a package that costs $0.87. If you could buy a package of the same gum with three pieces in it, how much would it cost?
• You are working on a recipe which calls for 3.25 cups of flour to make 4 servings. How much flour would you need for 10 servings?

Apples and math

One of the beautiful things about mathematics is its ability to show the same problem (and solutions) in different ways. We are going to take our apple example from earlier. We can use skills in arithmetic, patterning, and graphing.

For example, let’s take our apples example here. It costs $4 for 3 apples. We wrote out a proportion statement like so:

$\frac{\text{\$}4}{3\text{ apples}}=\frac{\text{\$}8}{6\text{ apples}}$

There are certainly more statements we could write. For example:

$\frac{\mathrm{\$4}}{3\text{ apples}}=\frac{\text{\$}12}{9\text{ apples}}$

Question: How do we create more proportion statements? How many proportions statements could you make?

What if we wanted to create a formula for the cost of any number of apples? What would that be like?

• If possible, take time now to discuss this with a partner. Try to identify a pattern and use a variable to construct a formula for the cost of the apples.

If you would like, you can complete the next series of activities using TVO Mathify. You can also use your notebook or the following fillable and printable documents.

The cost of the apples depends on how many apples you want to buy. In this situation, 3 apples costs $4 or ($\frac{\mathrm{\$4}}{3}$ ). If we wanted to buy 12 apples, we need to calculate the cost. Let's use the variable a to represent the “cost of apples". The proportion statement would appear as such: $\frac{\mathrm{\$4}}{3}=\frac{\mathrm{a}}{12}$.

In this case, the cost of apples would be:

$\begin{array}{rl}a& =\frac{4}{3}\times 12\\ & =\frac{4\times 12}{3}\\ & =\frac{48}{3}\\ & =16\end{array}$

So, 12 apples would cost $16.

If we cross-multiply, then:

$\begin{array}{rl}\frac{\mathrm{\$4}}{3}& =\frac{a}{12}\\ 3a& =\mathrm{\$4}\times 12\\ 3a& =\mathrm{\$48}\\ a& =\frac{\mathrm{\$48}}{3}\\ a& =\mathrm{\$16}\end{array}$

If cost of apples is a and number of apples is n, we can write this new equation as:

$a=\frac{\mathrm{\$4}}{3}\times n$

Let's determine the cost of various numbers of apples.

Finally, we can take this to its furthest point by plotting our new equation for the cost of apples onto the Cartesian plane. Our points (x,y) will be (number of apples, cost).

Try to do this on your own first by describing what the graph will be like on the Cartesian plane. You may want to use a graphing software for this.

Question: What do you notice about the points on the graph?

Being able to take real life situations and proportions and construct mathematical objects such as equations and graphs is an essential tool in mathematics. This allows us to understand patterns in real life situations that we might not have been able to understand before.

For example: now we know that the pattern between the cost of the apples and the number of apples purchased is a line when graphed. In fact, all proportions can be made into line graphs!

Practice

For each situation, complete the following:

• construct an equation for the general pattern
• describe/create a line graph showing the pattern is linear

Reflect

Consider the questions you completed and their associated linear graphs.

• Are all the graphs realistic for any number?
• What about negative numbers on the graph? Do they make sense for any of these situations?
• Consider the third practice question involving Danny’s running time. How might this graph be represented in a realistic situation? How would the shape of the graph change over time?

Real world problems

Task 1

Use cross-multiplication to solve for the variable in each problem:

Task 2

Create two proportion-related world problems of your own. Keep the following things in mind:

• Try to keep your problems closely related to a real-life situation.
• Provide a full solution to your problems.
• You should be able to create an equation for each of your problems. Construct one and describe/create a graph showing the equation.
• When finished, exchange problems with a partner (if possible) and solve each other’s problems.

Reflection

As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.

Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.

When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.

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