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\( \frac{\color{#818cf8}3}{4} \)

Writing & Simplifying
Ratios

Learn to write, simplify, and interpret ratios as part-to-part and part-to-whole relationships using the greatest common factor.

Pre-Algebra

What Is a Ratio?

A ratio is a comparison of two quantities by division. It tells you how much of one thing there is relative to another thing.

3 blue 3 green

Three Ways to Write a Ratio

Every ratio can be expressed in three equivalent notations. All three mean the same comparison.

Colon Form
\( 3 : 5 \)
Read as "3 to 5"
Fraction Form
\( \dfrac{3}{5} \)
Looks like a fraction
Word Form
"3 to 5"
Written in words

Why Do Ratios Matter?

Ratios appear everywhere in real life: recipes, maps, sports stats, and science. Understanding ratios helps you compare quantities and scale things proportionally.

Recipes
\( 2 : 3 \) cups flour to sugar
Maps
\( 1 : 100{,}000 \) scale
Sports
\( 12 : 8 \) wins to losses

Simplify with the GCF

To simplify a ratio, divide both quantities by their Greatest Common Factor (GCF). A simplified ratio uses the smallest possible whole numbers.

\( 12 : 8 \)
 Original
\( \div 4 \)
\( 3 : 2 \)
 Simplified
\( \text{GCF}(12, 8) = 4 \)

Finding the GCF

The Greatest Common Factor is the largest number that divides evenly into both quantities. Here are two methods:

List Factors
Write all factors of each number, then pick the largest one they share.
Divide Repeatedly
Keep dividing both by common factors (2, 3, 5...) until nothing divides evenly.

Part-to-Part vs Part-to-Whole

Ratios can compare a part to another part, or a part to the whole group. They answer different questions.

4 blue 6 amber Part-to-Part:
\( 4 : 6 = 2 : 3 \)
 Part-to-Whole:
\( 4 : 10 = 2 : 5 \)
Part-to-Part
Compares one group to another group: blue to amber \( = 2 : 3 \)
Part-to-Whole
Compares one group to the total: blue to all \( = 2 : 5 \)

How to Simplify Any Ratio

1
Identify the two quantities being compared and note which comes first (order matters).
2
Write the ratio using colon, fraction, or word form: \( a : b \).
3
Find the GCF of the two numbers.
4
Divide both parts by the GCF: \( \dfrac{a}{\text{GCF}} : \dfrac{b}{\text{GCF}} \).
5
Check: confirm the GCF of the simplified ratio is \( 1 \).

Example 1: Simplify \( 15 : 25 \)

Start
\( 15 : 25 \)
 GCF
\( \text{GCF} = 5 \)
 Result
\( 3 : 5 \)
 Work:
Factors of \(15\): \(1, 3, 5, 15\)
Factors of \(25\): \(1, 5, 25\)
\( \text{GCF} = 5 \)
so \( 15 \div 5 = 3 \), \( 25 \div 5 = 5 \)

Example 2: Part-to-Whole Ratio

A bag has 8 red marbles and 12 blue marbles. Write the ratio of red marbles to total marbles in simplest form.

8 red 12 blue

Example 3: Three-Part Ratio

A smoothie recipe calls for 6 strawberries, 4 bananas, and 2 cups of yogurt. Write the simplified ratio.

1
Write the ratio in order: \( 6 : 4 : 2 \)
2
Find the GCF of all three: \(\text{GCF}(6, 4, 2) = 2\)
3
Divide each part by 2: \( \dfrac{6}{2} : \dfrac{4}{2} : \dfrac{2}{2} = 3 : 2 : 1 \)

Equivalent Ratios

Just like equivalent fractions, you can multiply or divide both parts of a ratio by the same number to create equivalent ratios.

\( 2 : 3 \)
 Simplest Form
\( \times 2 \)
\( 4 : 6 \)
\( \times 3 \)
\( 6 : 9 \)

Ratio vs Fraction vs Rate

Ratio Fraction Rate
Compares Two quantities Part to whole Two different units
Example \( 3 : 5 \) \( \frac{3}{8} \) \( \frac{60 \text{ miles}}{1 \text{ hr}} \)
Units Same or different Same Always different
Can exceed 1? Yes (\( 5 : 2 \)) Yes (\( \frac{5}{2} \)) Yes

Common Mistakes

✕ Wrong Order
Writing "dogs to cats" as \( \text{cats} : \text{dogs} \). Always match the order stated in the problem.
✕ Not Simplifying
Leaving a ratio as \( 6 : 9 \) instead of simplifying to \( 2 : 3 \) by dividing by the GCF.
✕ Mixing Up Types
Confusing part-to-part with part-to-whole. Read the question carefully for "to total" clues.

Quick Reference Summary

Concept Key Idea Example
Write a ratio Compare two quantities in order \( 5 : 8 \)
Simplify Divide both parts by GCF \( 10 : 15 = 2 : 3 \)
Part-to-Part One group vs another group Boys : Girls = \( 3 : 4 \)
Part-to-Whole One group vs total Boys : Class = \( 3 : 7 \)
Equivalent Multiply/divide both by same # \( 2 : 3 = 6 : 9 \)

Question 1 of 5

Simplify the ratio \( 24 : 36 \).

Question 2 of 5

A class has 10 boys and 15 girls. What is the part-to-whole ratio of boys to all students, in simplest form?

Question 3 of 5

Which ratio is equivalent to \( 3 : 7 \)?

Question 4 of 5

A recipe uses 4 cups of flour and 6 cups of milk. What is the GCF used to simplify the ratio \( 4 : 6 \)?

Question 5 of 5

A parking lot has 18 cars, 6 trucks, and 12 motorcycles. What is the simplified ratio of cars to trucks to motorcycles?