Introduction to Ratios & Notation - Sixth Grade Mathematics

Ratios are fundamental mathematical tools used to compare two quantities multiplicatively, forming the cornerstone of proportional reasoning, scientific formulas, and financial analysis.

What is a Ratio?

A ratio is a mathematical comparison of two numbers or quantities by division. Instead of asking how much larger one quantity is than another by subtraction (an additive comparison), a ratio asks how many times greater one quantity is than another, or what proportion of a total each quantity represents (a multiplicative comparison).

Additive Comparison:       "There are 4 more apples than oranges." (8 - 4 = 4)
Multiplicative (Ratio):    "There are twice as many apples as oranges." (8 / 4 = 2)

The Three Standard Ratio Notations

A ratio comparing quantity a to quantity b can be written in three distinct, mathematically equivalent notations.

+-------------------------------------------------------------------------+
|                        THREE NOTATIONS FOR RATIOS                       |
+-------------------+-----------------------+-----------------------------+
| Notation Format   | Written Expression    | Spoken Aloud                |
+-------------------+-----------------------+-----------------------------+
| Word Form         | a to b                | "a to b"                    |
|                   | Example: 3 to 5       | "Three to five"             |
+-------------------+-----------------------+-----------------------------+
| Colon Notation    | a : b                 | "a to b"                    |
|                   | Example: 3 : 5        | "Three to five"             |
+-------------------+-----------------------+-----------------------------+
| Fraction Notation | a / b                 | "a to b"  (or "a over b")   |
|                   | Example: 3 / 5        | "Three to five"             |
+-------------------+-----------------------+-----------------------------+

The Order of Terms is Paramount

The order in which a ratio is expressed must strictly correspond to the language of the prompt. The ratio of dogs to cats is NOT the same as the ratio of cats to dogs. If a shelter has 12 dogs and 8 cats: The ratio of dogs to cats is 12 to 8 (or 12:8, which simplifies to 3:2). The ratio of cats to dogs is 8 to 12 (or 8:12, which simplifies to 2:3). Reversing the terms reverses the meaning of the relationship.

Simplifying Ratios to Simplest Form

Like fractions, ratios should be simplified to simplest form (lowest terms) by dividing all terms by their Greatest Common Factor (GCF).

Simplifying the ratio 18 : 24
Step 1: Find GCF(18, 24). The greatest factor of both 18 and 24 is 6.
Step 2: Divide both terms by 6:
        18 / 6 = 3
        24 / 6 = 4
Step 3: Write the simplified ratio: 3 : 4 (or 3 to 4, or 3/4).

Ratios Involving Measurement Units

When forming a ratio between two measurements of the same dimension (such as length or time), convert both measurements into the same unit before writing and simplifying the ratio. Example: Write the ratio of 8 inches to 2 feet in simplest form. Step 1: Convert 2 feet into inches: 2 feet * 12 inches/foot = 24 inches. Step 2: Write the ratio in common units: 8 inches to 24 inches. Step 3: Simplify: 8/24 = 1/3 (or 1:3). Notice that once units match, the units cancel out: a pure ratio comparing like measurements has no unit label.

Exercises

Exercise 1. In a classroom, there are 15 boys and 12 girls. Write the ratio of girls to boys in all three mathematical notations, and reduce to simplest form.

Exercise 2. A recipe calls for 4 cups of flour and 6 tablespoons of sugar. Can you immediately write the ratio as 4 to 6? Explain why or why not, and provide the correct converted ratio in tablespoons (1 cup = 16 tablespoons).

Exercise 3. Express the ratio of 45 minutes to 2 hours in simplest form using colon notation.

Solutions

Solution to Exercise 1. The problem asks for the ratio of girls to boys. The count of girls is 12, and the count of boys is 15. The unsimplified ratio is 12 to 15. The GCF of 12 and 15 is 3. Divide both terms by 3: 12 / 3 = 4, and 15 / 3 = 5. In simplest form: Word form: 4 to 5; Colon notation: 4 : 5; Fraction notation: 4/5.

Solution to Exercise 2. No, you cannot write 4 to 6 directly because cups and tablespoons are different units of measurement. You must convert to identical units first. Since 1 cup = 16 tablespoons, 4 cups = 4 * 16 = 64 tablespoons. Now write the ratio in tablespoons: 64 tablespoons to 6 tablespoons. Simplify by dividing by GCF(64, 6) = 2: 64 / 2 = 32, and 6 / 2 = 3. The correct converted ratio in simplest form is 32 to 3 (or 32:3).

Solution to Exercise 3. Both measurements describe time. Convert 2 hours to minutes: 2 * 60 = 120 minutes. Write the ratio of 45 minutes to 120 minutes: 45:120. Find the GCF of 45 and 120, which is 15. Divide both terms by 15: 45 / 15 = 3, and 120 / 15 = 8. In colon notation, the ratio in simplest form is 3 : 8.