Representing Proportional Relationships with Equivalent Ratios

Hello. In the last lesson, you set up a retrieve-check-correct cycle: attempt from memory first, keep answers hidden until the attempt is complete, then correct specific gaps. You will use that same approach here rather than simply rereading ratio examples.

This lesson focuses on one precise skill: taking a relationship that stays constant and representing it as two equivalent ratios. This is the foundation for proportions, which you will solve in the next lesson.

---

A ratio has meaning as well as numbers

A ratio compares two quantities by division. You can write the same ratio in three forms:

$$\frac{a}{b}$$ $$a:b$$

or “$a$ to $b$.”

The order matters because each position has a meaning. If a reader completes 1 book in 2 days, the ratio of books to days is:

$$\frac{1\text{ book}}{2\text{ days}}$$

Writing $\frac{2\text{ days}}{1\text{ book}}$ is not wrong, but it is a different comparison: days to books. On a quiz, always begin by naming the quantities in the requested order.

A proportional relationship is one where the comparison stays the same as the quantities grow or shrink together. For the reader:

$$\frac{1\text{ book}}{2\text{ days}}=\frac{5\text{ books}}{10\text{ days}}$$

Both ratios mean one book for every two days. The second situation is larger, but the pace has not changed.

When two ratios have the same value and represent the same ordered quantities, they are equivalent ratios. Writing them with an equals sign creates a proportion.

---

Keep the relationship constant

The central rule is simple:

To create an equivalent ratio, multiply or divide both parts of the ratio by the same number.

For any ratio $\frac{a}{b}$, multiplying both quantities by the same scale factor $k$ preserves its value:

$$\frac{a}{b}=\frac{ka}{kb}$$

For example, suppose 3 cups of flour are used with 1 cup of butter:

$$\frac{3\text{ cups flour}}{1\text{ cup butter}}$$

To describe three batches, multiply both quantities by 3:

$$\frac{3\text{ cups flour}}{1\text{ cup butter}}=\frac{9\text{ cups flour}}{3\text{ cups butter}}$$

The amounts changed, but the recipe did not. It still has 3 cups of flour for every 1 cup of butter.

The whiteboard example represents a constant pace of 1 level per 2 hours:

$$\frac{1\text{ level}}{2\text{ hours}}=\frac{4\text{ levels}}{8\text{ hours}}$$

The factor from the first ratio to the second is 4:

$$1\times 4=4$$ $$2\times 4=8$$

Because the same factor was applied to both quantities, the ratios are equivalent.

A useful way to state the meaning in words is:

“For every 1 level completed, 2 hours are played” is the same relationship as “For every 4 levels completed, 8 hours are played.”

---

Build a correct proportion step by step

When a word problem gives a proportional situation, use this five-part routine.

1. Name the quantities and their order.
   For example: posters to minutes.

2. Write the starting ratio with units.
   Suppose a printer makes 4 posters in 6 minutes:

   $$\frac{4\text{ posters}}{6\text{ minutes}}$$

3. Choose one scale factor.
   To describe three times as much work, use 3.

4. Scale both quantities by that factor.

   $$4\times 3=12$$ $$6\times 3=18$$

5. Write the two equivalent ratios as a proportion.

   $$\frac{4\text{ posters}}{6\text{ minutes}}=\frac{12\text{ posters}}{18\text{ minutes}}$$

Notice that the ratio stays in the same order on both sides: posters first, minutes second.

There is also a reverse way to find an equivalent ratio: divide both parts by the same common factor. For instance:

$$12:8=6:4=3:2$$

The first division uses 2, and the second uses 4. Each ratio describes the same comparison.

---

How to tell whether two ratios are equivalent

Two ratios are equivalent only if they pass both checks.

Check	What to look for
Same ordered quantities	If the first ratio is apples to bags, the second must also be apples to bags.
Same scale factor	The number used to change the first part must also change the second part.

Consider these two statements:

$$\frac{2\text{ red tiles}}{3\text{ blue tiles}}=\frac{8\text{ red tiles}}{12\text{ blue tiles}}$$

This is valid because both quantities were multiplied by 4.

Now consider:

$$\frac{2\text{ red tiles}}{3\text{ blue tiles}}\ne \frac{8\text{ red tiles}}{9\text{ blue tiles}}$$

The first number was multiplied by 4, while the second number was multiplied by 3. Since the factors differ, the color comparison changed.

A second quick test is to simplify both ratios. For the valid example:

$$\frac{8}{12}=\frac{2}{3}$$

Since both ratios reduce to $\frac{2}{3}$, they are equivalent.

Be careful with a common error: adding the same number to both values does not usually preserve a ratio. Starting with $2:3$, adding 2 gives $4:5$. But $2:3$ and $4:5$ do not have the same value. Equivalent ratios come from multiplying or dividing both terms by one common factor.

---

A short closed-note check

Use the study method from the previous lesson for the final few minutes. Close this lesson and any notes. On a blank page, write:

• the definition of an equivalent ratio in your own words;
• one proportion using quantities and units;
• the scale factor that connects the two ratios;
• one sentence explaining why both ratios describe the same relationship.

Do not reopen the lesson until all four parts are complete. Then check your work using this checklist:

• Did both ratios compare the same quantities in the same order?
• Did you multiply or divide both values by the same number?
• Did you include units, or clearly state what each number represents?
• Does your written sentence describe one unchanged “for every” relationship?

If one part is unclear, mark it P for partial or X for incorrect, correct it, and try writing a new example from memory.

---

Equivalent ratios express the same comparison with different-sized numbers. To represent a proportional relationship as two equivalent ratios, preserve the order and units, then multiply or divide both quantities by the same factor.

In the next lesson, you will use this idea to solve a proportion with one unknown value and verify that your solution makes the two ratios equivalent.