Unit Rates

🚗 Unit Rates

Part 1 of 5 — Concept Introduction

Have you ever wondered which box of cereal is the better deal, or how fast a car is really going? The secret tool behind these everyday questions is the unit rate.

What Is a Rate?

A rate is a special kind of ratio that compares two different units. The two quantities have different labels (like miles and hours, or dollars and books).

Examples of rates:

• $60$ miles in $2$ hours
• $15 for $3$ books
• $150$ words in $5$ minutes

Notice that each one connects two different kinds of measurements. That is what makes it a rate instead of a plain ratio like $3$ cats to $4$ dogs (same unit: animals).

What Is a Unit Rate? ⭐

A unit rate is a rate where the second quantity is exactly 1. The word unit means "one," so a unit rate tells you "how much for just one."

The little word "per" is your clue. It means "for each one." Miles per hour means miles for each one hour.

How to Find a Unit Rate

To turn any rate into a unit rate, divide the first quantity by the second quantity:

$\text{Unit Rate} = \frac{\text{First Quantity}}{\text{Second Quantity}}$

Worked Example: $12 for $4$ pounds of apples.

$\frac{12 \text{ dollars}}{4 \text{ pounds}} = 3 \text{ dollars per pound}$

So the apples cost $3 per pound. Dividing by the second number "shrinks" it down to $1$, which is exactly what a unit rate needs. 🍎

Concept Check 🎯

Make sure you can spot a unit rate from a regular rate.

📝 Worked Examples

Part 2 of 5 — Worked Examples

Let's slow down and find unit rates one careful step at a time.

Example 1: Speed → $240$ miles in $4$ hours

Step 1 — Set up the rate as a fraction (first quantity on top):

$\frac{240 \text{ miles}}{4 \text{ hours}}$

Step 2 — Divide the top by the bottom to make the bottom equal $1$:

$240 \div 4 = 60$

Step 3 — Write the unit rate with its label:

$60 \text{ miles per } 1 \text{ hour} = 60 \text{ mph}$

The car travels $60$ miles per hour. 🚗

Example 2: Price → $18 for $6$ smoothies

Step 1 — Set up the rate:

$\frac{18 \text{ dollars}}{6 \text{ smoothies}}$

Step 2 — Divide the first quantity by the second:

$18 \div 6 = 3$

Step 3 — Write the unit rate:

$\$3 \text{ per } 1 \text{ smoothie}$

Each smoothie costs $3. Now if a friend wants $10$ smoothies, you can quickly predict the cost: $10 \times 3 = 30$, i.e. $30. That is the real power of a unit rate! 🥤

Your Turn 🧮

A printer makes $90$ pages in $3$ minutes. Find the unit rate step by step.

1. What number do you divide by to make the time equal $1$ minute? (Type a single number.)

2. Now compute $90 \div 3$. What is the unit rate in pages per minute? (Type a single number.)

3. Using that unit rate, how many pages would the printer make in $1$ minute? (Type a single number.)

🧭 Guided Practice

Part 3 of 5 — Guided Practice

Find each unit rate by dividing the first quantity by the second. Watch your labels!

Fill In the Unit Rates 🔍

A car uses $24$ gallons of gas to travel $600$ miles. Complete the two facts about its unit rates.

🌍 Application & Word Problems

Part 4 of 5 — Real-World Problems

Unit rates are everywhere in real life. The most powerful use is comparing prices to find the better deal — this is called comparison shopping.

Which Is the Better Buy? 🛒

A small bag of pretzels is $4 for $8$ ounces. A large bag is $9 for $18$ ounces. Which is cheaper per ounce?

Find the unit price (dollars per ounce) for each:

• Small bag: $\dfrac{4}{8} = 0.50$, i.e. $0.50 per ounce
• Large bag: $\dfrac{9}{18} = 0.50$, i.e. $0.50 per ounce

Surprise — they cost the same per ounce! Without unit rates, the large bag looks more expensive because $9 > $4. Unit rates reveal the truth. 🥨

Unit rates also help you make predictions and plan budgets: once you know the cost for $1$ item, you can multiply to find the cost for any number of items.

Better Buy Practice 🧮

Juice comes in two sizes. Brand A is $6 for $3$ liters. Brand B is $10 for $5$ liters.

1. What is Brand A's unit price in dollars per liter ($6 \div 3$)? (Type a single number.)

2. What is Brand B's unit price in dollars per liter ($10 \div 5$)? (Type a single number.)

3. Which brand is the better buy? Type A, B, or same.

Prediction Problem 📋

🏆 Review & Challenge

Part 5 of 5 — Review & Challenge

You made it! Here is everything about unit rates in one place.

Quick Summary Table

Remember:

• A unit rate always has a denominator of $1$.
• To find a unit rate, divide the first quantity by the second.
• Unit rates help you compare prices, calculate speed, make predictions, and plan budgets.
• The lower unit price is the better buy.

Now try these mixed challenge problems! 🎯

Challenge Round 🎯

These mix every idea together. Take your time and divide carefully!