Understanding Ratios

⚖️ Understanding Ratios

Part 1 of 5 — What Is a Ratio?

Topics in This Part

🔑 Key Concept: A ratio compares two amounts. If a fruit bowl has $3$ apples and $2$ oranges, the ratio of apples to oranges is $3$ to $2$ — it tells you how the amounts compare, not how many there are in total.

What a Ratio Compares

A ratio is a comparison of two quantities by how much of one there is for each amount of the other.

Imagine a box of 5 blue marbles and 3 red marbles:

• The ratio of blue to red is $5$ to $3$.
• The ratio of red to blue is $3$ to $5$.
• The ratio of blue to the total is $5$ to $8$ (because $5 + 3 = 8$).

💡 Notice: A ratio doesn't have to compare a part to a part. It can also compare a part to the whole, like blue marbles to all the marbles.

Real Examples of Ratios

The Three Ways to Write a Ratio

The same ratio can be written three different ways. All three mean exactly the same thing:

So "$3$ to $4$", "$3 : 4$", and $\dfrac{3}{4}$ are three names for the same ratio.

⚠️ Watch out: A ratio written as a fraction like $\dfrac{3}{4}$ does not always mean "three-fourths of the whole." Here it means 3 cats for every 4 dogs — a comparison, not a piece of one pizza.

Concept Check 🎯

Why the Order Matters

In a ratio, the order you write the numbers tells you what is being compared to what.

If a recipe needs 2 cups of water for every 1 cup of rice:

• Water to rice is $2 : 1$ ✅
• Rice to water is $1 : 2$ ✅

These are not the same! $2 : 1$ means more water than rice, but $1 : 2$ means less water than rice.

🔑 Key Idea: Always read a ratio left-to-right in the same order the words are given. "Cats to dogs" puts the cat number first.

Match the Words to the Ratio 🔽

A classroom has 8 boys and 12 girls.

Read the Picture 🧮

A pet store window shows 5 puppies and 4 kittens.

1) What is the first number in the ratio of puppies to kittens? 2) What is the second number in that ratio? 3) How many animals are in the window in total?

⚖️ Understanding Ratios

Part 2 of 5 — Equivalent Ratios & Simplest Form

🔑 The Idea: Just like $\dfrac{1}{2} = \dfrac{2}{4}$, two ratios can look different but mean the same comparison. We make them by multiplying or dividing both numbers by the same amount.

Equivalent Ratios

Two ratios are equivalent if you can get from one to the other by multiplying (or dividing) both numbers by the same number.

Start with $2 : 3$. Multiply both parts by $2$, then by $3$, then by $4$:

So $2 : 3 = 4 : 6 = 6 : 9 = 8 : 12$. They are all equivalent ratios.

⚠️ Both numbers, same multiplier! Multiplying only one part breaks the comparison. $2 : 3$ is not equal to $4 : 3$.

Concept Check 🎯

Find the Missing Number 🧮

Each pair of ratios is equivalent. Find the missing value.

1) $2 : 5 = 8 : \,?$ 2) $3 : 4 = \,? : 12$ 3) $10 : 6 = 5 : \,?$

Simplest Form (Reducing a Ratio)

A ratio is in simplest form when the two numbers share no common factor except $1$ — exactly like reducing a fraction.

To simplify, divide both parts by their greatest common factor (GCF).

Example: Simplify $12 : 18$

The GCF of $12$ and $18$ is $6$:

$12 : 18 \;\xrightarrow{\;\div 6\;}\; 2 : 3$

So $12 : 18$ in simplest form is $2 : 3$.

Example: Simplify $9 : 15$

The GCF of $9$ and $15$ is $3$:

$9 : 15 \;\xrightarrow{\;\div 3\;}\; 3 : 5$

💡 Tip: If you can't spot the GCF at once, just keep dividing by any common factor ($2$, $3$, $5$…) until nothing divides both numbers.

Simplify Each Ratio 🧮

Write each ratio in simplest form. Enter the first number, then the second number.

1) $10 : 15 = \,? : \,?$ 2) $8 : 12 = \,? : \,?$

⚖️ Understanding Ratios

Part 3 of 5 — Ratio Tables & Unit Rates

🔑 Why this matters: A ratio table lets you scale a ratio up or down to answer real questions like "If 3 tickets cost $12, how much do 7 tickets cost?" The secret is the unit rate — the amount for just one.

Ratio Tables

A ratio table is a row of equivalent ratios. Each column keeps the same comparison.

If $1$ bag holds $4$ apples, then:

Every column is the ratio $1 : 4$ scaled up. To fill a table, multiply both rows by the same number to move right, or divide to move left.

💡 You can jump straight to any column: $7$ bags $\rightarrow 7 \times 4 = 28$ apples.

Fill In the Ratio Table 🧮

A muffin recipe uses $3$ eggs for every $12$ muffins. Use the ratio $3 : 12$ to complete the table.

Unit Rate: The Amount for "One"

A unit rate is a ratio that compares an amount to exactly one of the other thing. The second number becomes $1$.

To find a unit rate, divide so the second quantity equals $1$.

Example: 12 dollars for 3 tickets

$\frac{12}{3 \text{ tickets}} = \frac{12 \div 3}{3 \div 3} = \frac{4}{1 \text{ ticket}}$ which is $4 per ticket.

The unit rate is $4 per ticket.

Example: 150 miles in 3 hours

$\frac{150 \text{ mi}}{3 \text{ hr}} = \frac{150 \div 3}{3 \div 3} = \frac{50 \text{ mi}}{1 \text{ hr}} = 50 \text{ mph}$

🔑 Once you know the unit rate, you can find any amount — just multiply. $7$ tickets $= 7 \times 4 = 28$, giving $28.

Concept Check 🎯

Find the Unit Rate 🧮

Find each unit rate by dividing.

1) $20$ dollars for $5$ shirts $= \,?$ dollars per shirt 2) $45$ words typed in $9$ minutes $= \,?$ words per minute 3) A $7$-ticket bundle costs $4 each. Total cost $= \,?$ dollars

⚖️ Understanding Ratios

Part 4 of 5 — Part-to-Part, Part-to-Whole & Word Problems

🔑 Big Idea: A part-to-part ratio (like boys to girls) can be turned into a part-to-whole ratio (boys to everyone) by adding the parts to get the total. This unlocks real word problems.

Part-to-Part vs. Part-to-Whole

Suppose a bag of fruit snacks has red to green in the ratio $3 : 2$.

• Part-to-part: red to green is $3 : 2$.
• The total number of "parts" is $3 + 2 = 5$.
• Part-to-whole: red to all is $3 : 5$, and green to all is $2 : 5$.

💡 The total is the sum of the parts. Add the two numbers of a part-to-part ratio to find how many equal "shares" the whole is divided into.

Part or Whole? 🔽

A team's wins to losses are in the ratio $5 : 3$.

Solving Ratio Word Problems

Example: Sharing in a ratio

Maria and Jon share $20$ stickers in the ratio $3 : 2$. How many does each get?

Step 1 — Find the total parts: $3 + 2 = 5$ parts.

Step 2 — Find one part: $20 \div 5 = 4$ stickers per part.

Step 3 — Multiply each share: $\text{Maria} = 3 \times 4 = 12, \qquad \text{Jon} = 2 \times 4 = 8$

✅ Check: $12 + 8 = 20$ ✓ and $12 : 8$ simplifies to $3 : 2$ ✓

Example: Scaling a recipe

A recipe uses flour to sugar in the ratio $4 : 1$. If you use $12$ cups of flour, how much sugar?

From $4$ to $12$ is $\times 3$, so sugar $= 1 \times 3 = 3$ cups.

Concept Check 🎯

Solve the Word Problems 🧮

1) A class of $24$ students has a boys-to-girls ratio of $1 : 2$. How many girls are there? 2) A smoothie mixes juice to yogurt in the ratio $3 : 2$. With $9$ cups of juice, how many cups of yogurt? 3) $$28$ is shared in the ratio $4 : 3$. How many dollars is the larger share?

⚖️ Understanding Ratios

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) write ratios three ways, (2) build equivalent ratios and simplify them, (3) use ratio tables and unit rates, and (4) solve part-to-whole word problems. Let's put it all together.

Quick Reference

⚠️ Two reminders: keep the order consistent, and when scaling, change both numbers by the same amount.

Mixed Practice 🎯

One More Round 🧮

1) $4 : 7 = 12 : \,?$ (find the missing number) 2) Simplify $30 : 45$. Enter the first number, then the second. 3) $$36$ is shared in the ratio $5 : 1$. The larger share is $$\,?$

Exit Quiz ✅

Answer all three to finish the lesson.