Introduction to Ratios

🍎 Introduction to Ratios

Part 1 of 5 — Concept Introduction

A ratio compares two quantities by division. It tells you how much of one thing there is compared to another thing.

Imagine a fruit bowl with 3 apples and 5 oranges. A ratio lets us describe that relationship with numbers:

"There are 3 apples for every 5 oranges."

Ratios are everywhere — mixing paint, following recipes, comparing team scores, even sharing snacks fairly. Once you can read and write a ratio, you can compare almost anything!

Three Ways to Write a Ratio

The ratio of apples to oranges (3 to 5) can be written in three equal ways:

Form	How it looks	How you say it
Using "to"	3 to 5	"three to five"
Using a colon	$3:5$	"three to five"
As a fraction	$\frac{3}{5}$	"three to five"

All three mean exactly the same thing. The order matters: apples come first, so the apple number is written first in all three forms.

⚠️ Order Matters!

A ratio is like a sentence — the order of the words changes the meaning.

• The ratio of apples to oranges is $3:5$.
• The ratio of oranges to apples is $5:3$.

These are not the same! Always match the order of your numbers to the order named in the problem.

Part-to-Part vs Part-to-Whole

Ratios can compare two parts, or a part to the whole group.

• Part-to-Part compares one part to another part.
  • Example: 3 boys to 5 girls $\to 3:5$
• Part-to-Whole compares one part to the total.
  • Example: 3 boys out of 8 total students $\to 3:8$

To find the whole, add the parts together: $3 + 5 = 8$ students in all.

Concept Check 🎯

A box has 4 red marbles and 7 blue marbles.

📝 Worked Examples

Part 2 of 5 — Worked Examples

Just like fractions, ratios should be written in simplest form. To simplify, divide both numbers by the same number (their greatest common factor).

Example 1: Simplify $6:8$

• Both $6$ and $8$ can be divided by $2$.
• $6 \div 2 = 3$ and $8 \div 2 = 4$.

$6:8 = 3:4$

So $6:8$ in simplest form is $3:4$. ✅

Example 2: Write and Simplify a Ratio

A classroom has 10 boys and 15 girls. Write the ratio of boys to girls in simplest form.

Step 1 — Write it: boys first, so $10:15$.

Step 2 — Find a common factor: Both $10$ and $15$ divide by $5$.

Step 3 — Divide both parts: $10 \div 5 = 2$ and $15 \div 5 = 3$.

$10:15 = 2:3$

The ratio of boys to girls is $2:3$. 🎉

Your Turn 🧮

A bag has 8 green counters and 12 yellow counters. Simplify the ratio of green to yellow step by step.

1. Write the ratio of green to yellow before simplifying, using a colon. (Type it like $8:12$.)

2. Both numbers divide by the same factor. What is the largest number that divides both $8$ and $12$? (Type a single number.)

3. Write the ratio in simplest form, using a colon. (Type it like $2:3$.)

🧭 Guided Practice

Part 3 of 5 — Guided Practice

Read each problem carefully and watch the order of the numbers. Simplify when you can.

Part-to-Part or Part-to-Whole? 🔍

A garden has 4 roses and 6 tulips, for 10 flowers in all. Choose the correct ratio for each description.

🌍 Application & Word Problems

Part 4 of 5 — Real-World Problems

Ratios that mean the same comparison are called equivalent ratios. You make them by multiplying or dividing both parts by the same number.

$2:3 = 4:6 = 6:9 = 8:12$

Each pair was found by multiplying both numbers: $2 \times 2 = 4$ and $3 \times 2 = 6$, and so on.

Lemonade Recipe 🍋

A lemonade recipe uses 2 cups of lemon juice for every 3 cups of water, a ratio of $2:3$.

If you want to make a bigger batch using 6 cups of water, how much lemon juice do you need?

• Water went from $3$ to $6$ — that's $\times 2$.
• So lemon juice must also be $\times 2$: $2 \times 2 = 4$ cups.

The bigger batch uses $4:6$, which is equivalent to $2:3$. The taste stays the same! 🎉

Recipe Time 🧮

A trail-mix recipe uses 3 scoops of nuts for every 4 scoops of raisins, a ratio of $3:4$. You want a bigger batch with 12 scoops of raisins.

1. The raisins went from $4$ scoops to $12$ scoops. What number did you multiply by? (Type a single number.)

2. Multiply the nuts by that same number. How many scoops of nuts do you need? (Type a single number.)

3. Write the new equivalent ratio of nuts to raisins, using a colon. (Type it like $9:12$.)

Apply It 🎯

A paint mix uses 1 part blue to 4 parts white, a ratio of $1:4$.

🏆 Review & Challenge

Part 5 of 5 — Putting It All Together

You've learned a lot about ratios! Here is a quick summary of the big ideas.

Idea	What it means	Example
Ratio	Compares two quantities	3 apples to 5 oranges
Three forms	"to", colon, fraction	3 to 5, $3:5$, $\frac{3}{5}$
Order matters	First named comes first	apples:oranges $= 3:5$
Simplify	Divide both parts by a common factor	$6:8 = 3:4$
Part-to-part	One part vs another part	boys:girls $= 3:5$
Part-to-whole	One part vs the total	boys:total $= 3:8$
Equivalent	Multiply/divide both parts equally	$2:3 = 4:6$

Remember: A ratio compares amounts, the order must match the words, and the simplest form keeps the numbers as small as whole numbers can be. Now try the final challenge! 💪

Final Challenge 🎯

A class has 12 girls and 8 boys. Mix together everything you've learned.