Understand the Concept of a Ratio

In this lesson, you will:

• See why ratios are multiplicative — not additive — comparisons
• Write ratios three ways and classify them as part-to-part or part-to-whole

What You Will Learn Today

By the end of this lesson, you will:

1. Distinguish additive from multiplicative comparisons
2. Use ratio language — "for every ___ there are ___"
3. Write a ratio three ways: a to b, a:b, a/b
4. Recognize that order matters — a:b ≠ b:a
5. Classify ratios as part-to-part or part-to-whole
6. Write a complete ratio statement for a real-world context

Two People, Same Tray — Are Both Right?

A tray holds 6 apples and 2 oranges.

• Person A says: "There are 4 more apples than oranges."
• Person B says: "There are 3 apples for every orange."

Are they both correct? Are they saying the same thing?

Additive vs. Multiplicative: Two Comparisons

The ratio is the multiplicative comparison — the "for every" answer.

A Ratio Is a Multiplicative Comparison

• A ratio compares two quantities multiplicatively
• It asks: how many of one quantity for every one of the other?
• 6 apples, 2 oranges → 3 apples for every 1 orange

Not a ratio: "4 more apples than oranges" ← this is an additive comparison

Ratio Language — Say It This Way

The ratio of wings to beaks in a bird house was 2:1.

• "For every 2 wings, there is 1 beak."
• "The ratio of wings to beaks is 2 to 1."
• "2 wings for every 1 beak."

Three canonical phrasings — all equivalent.

Quick Check — Write It in Ratio Language

A shelter has 4 cats and 6 dogs.

Write the cats-to-dogs comparison in ratio language.

Use the template: "For every ___ there are ___."

Ratio Language Requires Units and Labels

Cats to dogs: "For every 4 cats, there are 6 dogs."

Flour to sugar: "For every 3 cups of flour, there are 2 cups of sugar."

Always name what you're comparing — bare numbers are incomplete.

Ratio Language Is Clear but Long

We've been writing:
"For every 3 cups of flour, there are 2 cups of sugar."

That's clear — but long. How would you write it shorter?

What notation would you choose?

Three Ways to Write a Ratio

• Word form: a to b → "3 to 2"
• Colon form: a:b → "3:2" ← most common
• Fraction form: a/b → "3/2"

a/b in a part-to-part ratio is notation only

Order Matters — a:b ≠ b:a

• Wings to beaks: 2:1 — twice as many wings as beaks
• Beaks to wings: 1:2 — half as many beaks as wings

Same tray. Two different ratios. Both true.

The Votes Example — Approximately Right

"For every vote A received, C received nearly three votes."

• A:C ≈ 1:3 → 1 to 3, 1:3, 1/3
• C:A ≈ 3:1 ← reversed order, same situation

"Nearly 3" is not imprecision — approximation is the point.

Your Turn — All Three Notations

The tray has 6 apples and 2 oranges.

1. Write the apples-to-oranges ratio in all three notations
2. Write the oranges-to-apples ratio in colon form

Try both before advancing.

Check — Notations and Order

Apples to oranges:

• 6 to 2    |    6:2    |    6/2

Oranges to apples:

• 2:6

Same tray — two different ratios. Order anchors the meaning.

Wings and Beaks — What Is the Whole?

Wings:beaks = 2:1 — both parts of the same animal.

"What fraction of all animal parts are wings?"

• Wings: 2, Beaks: 1 → Total: 2 + 1 = 3
• Wings to total parts = 2:3

2:3 is a different ratio — it compares one part to the total.

Part-to-Part: Two Parts of the Same Group

A part-to-part ratio compares two parts of the same group.

• Wings to beaks: 2:1
• Apples to oranges: 6:2
• Boys to girls: 12:18

No whole needed — compare the parts to each other.

Part-to-Whole: One Part to the Total

A part-to-whole ratio compares one part to the entire group.

• Wings to total: 2:3
• Boys to class: 12:30
• Cats to all pets: 4:10

Compute the whole first — add all the parts.

Boys/Girls — Both Ratios Side by Side

Part-to-part: boys:girls = 2:3. Part-to-whole: boys:class = 2:5.

Reading a/b as a Fraction of the Whole

Boys:class = 12:30 = 2:5
→ Boys are 2/5 of the class ✓

Boys:girls = 12:18 = 2:3
→ Boys are 2/3 of the class ✗ (girls are not the whole class)

a/b reads as a fraction-of-whole ONLY when b is the actual whole.

Quick Check — Part-to-Whole and the Fraction

Wings:beaks = 2:1 (part-to-part)

1. Write the part-to-whole ratio for wings (wings:all parts)
2. Is writing "wings are 2/1 of beaks" a valid fraction-of-whole reading? Explain.

Worked Example — Two Ratios, Same Situation

Recipe: 3 cups flour, 2 cups sugar

• Flour:sugar = 3:2 → Part-to-part
• Flour:total dry = 3:5 → Part-to-whole; flour is 3/5 of the mix ✓

Shelter: 4 cats, 6 dogs

• Cats:dogs = 4:6 = 2:3 → Part-to-part
• Cats:all pets = 4:10 = 2:5 → Part-to-whole

The Key Insight — Same Situation, Different Ratios

The same context can produce multiple legitimate ratios.

• Which ratio you write depends on what question is being asked
• Always name: what are the two quantities? what is the whole (if part-to-whole)?

A ratio is not a number in isolation — it is a statement about a relationship.

Complete Write-Up — Wings and Beaks

"The ratio of wings to beaks was 2:1 — for every 2 wings there was 1 beak."

1. Identify: wings, beaks
2. Order: wings first
3. Notations: 2 to 1, 2:1, 2/1
4. Language: "For every 2 wings, there is 1 beak"
5. Classify: Part-to-part

Complete Write-Up — Votes A and C

"For every vote A received, C received nearly three votes."

1. Identify: votes for A, votes for C
2. Order: A first
3. Notations: about 1 to 3, ≈1:3, ≈1/3
4. Language: "For every 1 vote for A, C received about 3"
5. Classify: Part-to-part (total votes unknown)

Your Turn — Write the Cats-to-Dogs Ratio

A park has 4 cats and 6 dogs.

Write the complete cats-to-dogs ratio:

1. Identify the quantities (with units)
2. Write all three notations
3. State in ratio language
4. Classify — and explain why

Try all four steps before advancing.

Find the Error — Boys and Girls

A classmate's work:

"A class has 12 boys and 18 girls. The ratio of boys to girls is 12:18 = 2/3. So boys are 2/3 of the class."

Find the error. Write the correct statement.

Full Write-Up Practice — Red and Blue Marbles

A bag holds 5 red and 7 blue marbles.

Write two complete ratio statements:

1. The red-to-blue ratio (part-to-part)
2. The red-to-total ratio (part-to-whole)

For each: all three notations, ratio language, classification, and — for #2 — the fraction of the whole.

No hints. Full write-up for both.

Three Mistakes to Watch For

Subtracted instead of ratio → "4 more" is additive; "3 for every 1" is the ratio.

Reversed the order → Wings:beaks 2:1 ≠ beaks:wings 1:2.

Part-to-part read as fraction-of-whole → Boys:girls = 2:3 ≠ boys are 2/3 of class. Boys:class = 2:5.

Five Things to Remember About Ratios

✓ Ratios are multiplicative — "for every ___, there are ___"
✓ Three notations: a to b, a:b, a/b
✓ Order matters — a:b ≠ b:a
✓ Classify every ratio: part-to-part or part-to-whole
✓ Only part-to-whole ratios read as a fraction of the whole

Label your ratios

Coming Up Next — Unit Rates (6.RP.A.2)

You can now write any ratio three ways and classify it.

"For 2 cups sugar there are 3 cups flour. How much flour per ONE cup of sugar?"

That per-one-unit answer is a unit rate — what the fraction form gives you when you evaluate it as a number.