Understanding Ratios and Equivalent Ratios

A ratio is a mathematical way to compare two quantities. It tells us how much of one thing there is compared to another. For example, if a recipe calls for 2 cups of flour and 1 cup of sugar, the ratio of flour to sugar is 2 to 1.

Writing Ratios in Different Forms

You can write a ratio in three different ways. Let's say a class has 8 boys and 12 girls. The ratio of boys to girls can be written as:

1. Using a colon: $8:12$
2. Using the word "to": $8$ to $12$
3. As a fraction: $\frac{8}{12}$

Order matters in ratios! The ratio of girls to boys would be $12:8$, which is different from $8:12$.

Equivalent Ratios

Equivalent ratios are ratios that express the same relationship between two quantities, even if they use different numbers. They work exactly like equivalent fractions.

To find an equivalent ratio, you can multiply or divide both parts of the ratio by the same non-zero number.

Example: Find three equivalent ratios to $2:5$.

• Multiply by 2: $(2 \times 2) : (5 \times 2) = 4:10$
• Multiply by 3: $(2 \times 3) : (5 \times 3) = 6:15$
• Multiply by 10: $(2 \times 10) : (5 \times 10) = 20:50$

So, $2:5$, $4:10$, $6:15$, and $20:50$ are all equivalent ratios.

Simplifying Ratios

Just like fractions, ratios are often easiest to understand when written in their simplest form. To simplify a ratio, divide both sides by their Greatest Common Factor (GCF).

Example: Simplify the ratio $15:25$.

1. Find the GCF of 15 and 25, which is 5.
2. Divide both numbers by 5.
3. $15 \div 5 = 3$
4. $25 \div 5 = 5$

The simplified ratio is $3:5$.

Using Ratio Tables

A ratio table is a handy tool to organize equivalent ratios. It helps you see patterns and find missing values easily.

Boys	Girls
2	3
4	6
6	9
8	12

In this table, every row represents the same ratio ($2:3$). You can see that multiplying the first row by 2 gives the second row, multiplying by 3 gives the third row, and so on.