Exploring fractions

Consider the following fractions:

$\frac{4}{5}\text{ and }\frac{24}{30}$

How are these two fractions related?

These two fractions are said to be equivalent, which means they are equal.  

$\frac{4}{5}=\frac{24}{30}$

Since these two fractions are in fact equal, we say that the equality statement is a proportion.

To use in real life, we often need both the numerator and denominator to have a unit attached to it.

Example: At the grocery store, there was a sale on chocolate bars: $4 for 5 bars. How much would it cost to purchase 30 chocolate bars?

$\frac{\mathrm{\$}4}{5\mathrm{ }\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{ }\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{s}}\mathrm{=}\frac{\mathrm{?}}{30\mathrm{ }\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{ }\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{s}}$

This fraction is exactly like our equivalent fractions and so we know that it would cost $24 for 30 chocolate bars.

The fraction on the left:

$\frac{\text{\$}4}{5\text{ chocolate bars}}$

Is called a rate because it is a ratio (think: fraction of two numbers) where each number has a different unit.

Here are some examples of ratios, rates, and proportions.

• When reading the examples, think about why they are different from each other between the three columns.
• Construct 2 more examples in each column.

If you would like, you can complete this activity using TVO Mathify. You can also use your notebook or the fillable document.

Practice

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

For each situation listed, determine whether it is an example of a proportion. For each case, explain why it is or is not a proportion.

Math and problem-solving

From our earlier example, we were able to use the rates provided in the problem to help create a proportion and use it to solve the problem. This is the general strategy that you can use for any problem involving things that are proportional.

Example:

Henry’s Grocer sells apples at the price of $3 per 100 grams. How much would 1 kilogram of apples cost?

Set up the problem by writing the proportion statement:

$\frac{\text{\$}3}{100g}=\frac{\text{?}}{1kg}$

Use your knowledge of equivalent fractions to calculate the cost of the 1 kg of apples using the proportion statement given.

Task 1

Solve the following problems by writing a proportion based on the information provided.

Sometimes, you may need to compare different rates that are not proportional.

Example:

Henry’s Grocer sells apples at the price of $3 per 100 grams. Flo’s Food Market sells apples at the price of $28 per 1 kilogram. Which store has the better deal on apples?

You need to compare two different rates:

$\frac{\text{\$}3}{100g}\text{ and }\frac{\text{\$}28}{1kg}$

Finding the unit rate

In this work we use the word unit for a value of 1. The unit rate is a rate where the denominator has a value of 1. In other words, it is the rate for 1 unit of the item. In our example, we can find the unit rate of Henry’s Grocer apples:

$\frac{\text{\$}3}{100g}=\frac{\text{\$}0.03}{1g}$

Why is this a valid proportion statement? In other words, explain why these two fractions are equivalent.

Calculate the unit rate for Flo’s Food Market apples and compare the unit rate. Which store has the better deal?

Task 2

If you would like, you can complete this activity using TVO Mathify. You can also use your notebook or the following fillable and printable document.

Solve the following problems by calculating the unit rate and comparing.

Cooking inquiry

Ratios, rates, and proportions are essential to sharing recipes. A typical recipe might be something like:

Crispy Roasted Potatoes

Serves 4

• 4 pounds of Yukon Gold potatoes
• $\frac{1}{2}$ teaspoon baking soda
• 5 tablespoons of olive oil
• 3 cloves garlic
• 2 teaspoons salt
• 1 small handful of fresh rosemary

All of the items on the list are ratios:

If you would like, you can complete this activity using TVO Mathify. You can also use your notebook or the following fillable and printable document.

Find a recipe that you love. You can:

• ask someone in your learning space about a meal that they have made that you love
• speak to family members about meals that you have had with them
• research meals that you like and search for a recipe

When you have your recipe:

• construct a list of all of the ratios of items you need
• try to be as specific as possible
• include an estimated serving amount (how many people will be eating the amount you provided)

Using your recipe:

• calculate the ratios needed for a different serving size other than the one you provided. For example, for the crispy roasted potato recipe we could find ratios for 8, 10, or 14 people

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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