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Understand
Ratio Concept & Language

Learn how to use ratios to compare two quantities, write them in multiple forms, and describe relationships using precise ratio language.

6th Grade Math

What Is a Ratio?

A ratio is a comparison of two quantities that tells you how much of one thing there is relative to another thing. Ratios describe a relationship between two numbers, not just a single amount.

Seeing Ratios Visually

Stars to Circles : \(3\) stars \(2\) circles Ratio = 3 : 2

For every \(3\) stars, there are \(2\) circles. The ratio of stars to circles is \(3 : 2\).

Why Do Ratios Matter?

Ratios appear everywhere in daily life. They help us compare quantities, scale recipes, read maps, and understand data. Mastering ratio language is the first step toward proportional reasoning.

★ Recipes
"For every \(2\) cups of flour, use \(1\) cup of sugar."
★ Maps
"Every \(1\) inch on the map represents \(10\) miles."
★ Sports
"The team's win-to-loss ratio is \(5 : 3\)."

Three Ways to Write a Ratio

Every ratio can be written in three equivalent forms. All three mean exactly the same thing:

Colon Form
\(a : b\)
Example: \(3 : 2\)
Word Form
\(a\) to \(b\)
Example: \(3\) to \(2\)
Fraction Form
\(\frac{a}{b}\)
Example: \(\frac{3}{2}\)

Using Ratio Language

Ratio language uses specific phrases to describe the relationship between two quantities. Here are the key phrases to recognize and use:

1
"The ratio of A to B is..." Directly states the comparison. Example: "The ratio of boys to girls is \(4 : 5\)."
2
"For every A, there are B..." Shows a repeating pattern. Example: "For every \(2\) wings, there is \(1\) beak."
3
"For each A, there are B..." Same as "for every." Example: "For each vote A gets, C gets \(3\) votes."

Part-to-Part vs Part-to-Whole

3 blue 2 green Part-to-Part: 3 : 2 Part-to-Whole: 3 : 5
Part-to-Part
Compares one part of a group to another part. Blue to green: \(3 : 2\)
Part-to-Whole
Compares one part to the total. Blue to all: \(3 : 5\)

How to Write a Ratio

1
Identify the two quantities being compared. Ask: "What am I comparing?"
2
Determine the order. Which quantity comes first? The order in the sentence matches the order in the ratio.
3
Count or find each quantity. Use the given information to get the numbers.
4
Write the ratio in any form: \(a : b\), \(a\) to \(b\), or \(\frac{a}{b}\).
5
Simplify if possible. Divide both numbers by their greatest common factor (GCF).

Example 1: Bird House

A bird house at the zoo has birds with a total of \(10\) wings and \(5\) beaks. Write the ratio of wings to beaks.

Wings 10 Beaks 5 :

Example 2: Classroom Colors

A classroom has \(12\) blue markers, \(8\) red markers, and \(4\) green markers. Find the ratio of red markers to the total number of markers.

12 blue 8 red 4 green Total: 12 + 8 + 4 = 24

Example 3: Equivalent Ratios

A recipe calls for \(3\) cups of flour for every \(2\) cups of milk. If you use \(9\) cups of flour, how much milk do you need?

Original Ratio 3 : 2 → ×3 Scaled Ratio 9 : 6 Multiply both parts by 3 3 × 3 = 9 flour 2 × 3 = 6 milk

Simplifying Ratios

To simplify a ratio, divide both quantities by their greatest common factor (GCF). A simplified ratio uses the smallest whole numbers possible.

1
Find the GCF of the two numbers. For \(12\) and \(8\), the GCF is \(4\).
2
Divide both numbers by the GCF: \(12 \div 4 = 3\) and \(8 \div 4 = 2\).
3
Write the simplified ratio: \(12 : 8\) becomes \(3 : 2\).

Ratio vs Fraction vs Rate

Ratio Fraction Rate
What it compares Any two quantities Part to whole Two different units
Example \(3 : 2\) \(\frac{3}{5}\) \(60\) miles per hour
Written forms \(a : b\), \(a\) to \(b\), \(\frac{a}{b}\) \(\frac{a}{b}\) only \(a\) per \(b\)
Units Same or different Same unit Always different

Order Matters in Ratios

Cats to Dogs 4 : 2 Dogs to Cats 2 : 4 ≠

Common Mistakes

✕ Reversing the Order
Writing "cats to dogs" as \(2 : 4\) instead of \(4 : 2\). The first quantity named goes first.
✕ Adding Instead of Comparing
Saying "\(4\) cats and \(2\) dogs equals \(6\)" instead of writing the ratio \(4 : 2\). A ratio compares, it does not combine.
✕ Forgetting to Simplify
Leaving a ratio as \(10 : 5\) when it simplifies to \(2 : 1\). Always check if both numbers share a common factor.

Ratio Cheat Sheet

Concept Key Idea Example
Ratio definition Compares two quantities \(3 : 2\)
Three forms Colon, word, fraction \(3 : 2\), \(3\) to \(2\), \(\frac{3}{2}\)
Part-to-Part Part vs another part Blue to red: \(5 : 3\)
Part-to-Whole Part vs total Blue to all: \(5 : 8\)
Simplifying Divide by GCF \(10 : 4 \to 5 : 2\)
Order First named = first number Cats to dogs ≠ dogs to cats

Question 1 of 5

A bag has \(6\) red marbles and \(9\) blue marbles. What is the ratio of red marbles to blue marbles in simplest form?

Question 2 of 5

A zoo has \(8\) penguins and \(12\) flamingos. Which sentence correctly uses ratio language?

Question 3 of 5

In a class of \(30\) students, \(18\) are girls. What is the part-to-whole ratio of girls to all students?

Question 4 of 5

A recipe uses flour and sugar in the ratio \(5 : 2\). If you use \(15\) cups of flour, how many cups of sugar do you need?

Question 5 of 5

The ratio of dogs to cats at a shelter is \(4 : 7\). Which of the following is NOT a correct way to express this ratio?