🎯⭐ INTERACTIVE LESSON

Rates and Unit Rates

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Rates and Unit Rates - Complete Interactive Lesson

Part 1: From Ratios to Rates

🏎️ Rates and Unit Rates

Part 1 of 5 — From Ratios to Rates

Topics in This Part

Section
What Makes a Ratio a Rate
The Two Numbers and Their Units
Reading a Rate Out Loud

🔑 Key Concept: A rate is a ratio that compares two quantities with different units — like miles and hours, or dollars and pounds. Almost every "per" you've ever heard ("miles per hour", "dollars per pound") is a rate.

Ratio vs. Rate

A ratio compares two amounts. When those two amounts have different units, we give the ratio a special name: a rate.

Comparison	Units	Ratio or Rate?
3 cats to 2 dogs	animals to animals (same)	ratio
4 cups flour to 1 cup sugar	cups to cups (same)	ratio
120 miles to 2 hours	miles to hours (different)	rate
$6 for 3 pounds	dollars to pounds (different)	rate

💡 Tip: If the two things being compared are measured in different units, you've got a rate. The little word "per" is your signal: miles per hour, dollars per pound, students per teacher.

Concept Check 🎯

Writing a Rate

A rate is usually written as a fraction, with each quantity labeled by its unit:

$\text{rate} = \frac{\text{first quantity}}{\text{second quantity}}$

Example: A car travels $120$ miles in $2$ hours. As a rate:

Build the Rate 🔽

A printer prints $48$ pages in $6$ minutes. Choose the correct piece for each blank.

One More Check 🎯

Recap

A ratio compares two amounts.
A rate is a special ratio comparing different units (the word per is the clue).
Write a rate as a fraction with the units kept on: $\dfrac{\text{quantity 1}}{\text{quantity 2}}$ .

Right now our rates still have both numbers ( $120$ miles per $2$ hours). In Part 2 we'll simplify every rate down to a single, super-useful number called the .

Part 2: The Unit Rate

🏎️ Rates and Unit Rates

Part 2 of 5 — The Unit Rate

🔑 The Idea: A unit rate tells you "how much for exactly ONE" — one hour, one pound, one minute. You find it by making the bottom number equal to $1$ .

What Is a Unit Rate?

A unit rate is a rate with a denominator of $1$ . The word unit means "one."

To get there, divide the top number by the bottom number:

$\text{unit rate} = \frac{\text{first quantity}}{\text{second quantity}} \;=\; \text{(first} \div \text{second) per } 1$

Part 3: Comparing with Unit Rates (Better Buy)

🏎️ Rates and Unit Rates

Part 3 of 5 — Comparing with Unit Rates (Better Buy)

🔑 Why it works: Two deals are hard to compare when they have different sizes. Boil each down to its unit price ("per one") and the comparison becomes obvious — just pick the smaller price per item.

Finding the Better Buy

When you shop, the lower unit price is the better deal (you pay less for each one).

Worked Example

Store A: $12$ granola bars for $\$ 6$
Store B: $8$ granola bars for $\$ 4.80$

Find each unit price (dollars per bar) by dividing cost by quantity:

$Store A: \frac{$ 6}{}$

Part 4: Using Unit Rates to Solve Problems

🏎️ Rates and Unit Rates

Part 4 of 5 — Using Unit Rates to Solve Problems

🔑 Big Payoff: Once you know the unit rate, you can scale up or down to any amount — multiply the unit rate to find a total, or divide a total to find an amount of time, distance, or money.

Scaling Up: Multiply by the Unit Rate

If you know "how much for one," multiply to find "how much for many."

$\text{total} = \text{unit rate} \times \text{number of units}$

Worked Example

Apples cost $\$ 2 $p e r p o u n d . H o w m$ 7$ pounds?

Part 5: Mixed Practice & Mastery Check

🏎️ Rates and Unit Rates

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) tell a rate from a ratio, (2) find a unit rate, (3) compare deals and speeds, and (4) use a unit rate to scale up or down. Let's put it all together.

Quick Reference

Goal	Key move
Spot a rate	different units → it's a rate (look for per)
Find a unit rate	divide top $\div$ bottom → "per $1$ "
Better buy	compute unit price; pick the lower one
Faster speed	compute unit rate; pick the higher one
Find a total	unit rate number of units

$\frac{120 \text{ miles}}{2 \text{ hours}}$

We read this as " $120$ miles per $2$ hours." The word per means "for every" or "divided by."

🔑 Key Idea: Keep the labels (units) attached to the numbers. The labels tell you what the rate means and stop you from mixing things up.

second quantityfirst quantity=

Worked Example: $120$ miles in $2$ hours

Divide miles by hours:

$\frac{120 \text{ miles}}{2 \text{ hours}} = \frac{120 \div 2}{2 \div 2} = \frac{60 \text{ miles}}{1 \text{ hour}}$

The unit rate is $60$ miles per hour (often written $60$ mph). That means in one hour the car goes $60$ miles.

💡 Notice we divided both the top and the bottom by $2$ so the bottom became $1$ . The value of the rate doesn't change — we just renamed it "per one hour."

Concept Check 🎯

Worked Example: $\$ 6 $for$ 3$ pounds of apples

Divide dollars by pounds:

$\frac{\$6}{3 \text{ lb}} = \frac{6 \div 3}{3 \div 3} = \frac{\$2}{1 \text{ lb}}$

The unit rate is $2 per pound. This is also called the unit price — the cost of exactly one pound.

Worked Example: $45$ words in $5$ minutes

$\frac{45 \text{ words}}{5 \text{ min}} = \frac{45 \div 5}{5 \div 5} = \frac{9 \text{ words}}{1 \text{ min}}$

The unit rate is $9$ words per minute.

🔑 Two special names to know:

A unit rate for price is called a unit price (dollars per item).

A unit rate for speed is just called speed (miles per hour).

Find the Unit Rate 🧮

Divide the top by the bottom. Enter just the number for each unit rate.

1) $\dfrac{84 \text{ miles}}{4 \text{ hours}} = \,?$ miles per hour 2) $\dfrac{\$ 20}{5 \text{ shirts}} = $,? $per shirt **3)**$ \dfrac{144 \text{ beats}}{2 \text{ minutes}} = ,?$ beats per minute

Name That Unit Rate 🔽

Match each unit rate to its common name and value.

Recap

To turn any rate into a unit rate:

Write it as $\dfrac{\text{first}}{\text{second}}$ .
Divide the top by the bottom.
The answer is "per 1" of the second quantity.

In Part 3 we'll see why unit rates are so handy: they let you instantly compare two deals and find the better buy.

Store A: \frac{$6}{12} = $0.50 per bar

$\text{Store B: } \frac{\$4.80}{8} = \$0.60 \text{ per bar}$

Since $\$ 0.50 < $0.60$, Store A is the better buy — each bar costs less.

💡 Golden rule for shopping: lower dollars-per-item = better deal. (For speed or production, you usually want the higher unit rate instead — more done per unit of time.)

Concept Check 🎯

Comparing Speeds (Higher Is Faster)

Unit rates also compare speed. Here the bigger unit rate is faster.

Worked Example

Runner 1: $300$ meters in $50$ seconds
Runner 2: $400$ meters in $80$ seconds

$\text{Runner 1: } \frac{300}{50} = 6 \text{ m/s}$

$\text{Runner 2: } \frac{400}{80} = 5 \text{ m/s}$

Runner 1 covers $6$ meters each second vs. $5$ — so Runner 1 is faster.

⚠️ Watch the direction! For price, smaller is better. For speed or work done, larger is better. Always ask: "Do I want this number high or low?"

High or Low? 🔽

Decide whether you want the unit rate to be bigger or smaller to "win."

Compare the Deals 🧮

Find each unit price, then answer.

1) $6$ pens for $\$ 3 $. Unit price$ = $,? $per pen. **2)**$ 10 $pens for$ $4 $. Unit price$ = $,?$ per pen (decimal). 3) Which pack is the better buy — type 1 or 2?

p er p o u n d . Ho w m u c h f or

$\$2 \text{ per lb} \times 7 \text{ lb} = \$14$

A car drives $60$ miles per hour. How far in $4$ hours?

$60 \text{ mph} \times 4 \text{ h} = 240 \text{ miles}$

💡 The unit rate is like a recipe per one — repeat it as many times as you need.

Concept Check 🎯

Scaling Down: Divide by the Unit Rate

If you know the unit rate and the total, divide to find the missing amount.

$\text{number of units} = \frac{\text{total}}{\text{unit rate}}$

Worked Example

A car goes $60$ miles per hour. How long to drive $180$ miles?

$\frac{180 \text{ miles}}{60 \text{ mph}} = 3 \text{ hours}$

Worked Example

Stickers cost $\$ 0.25 $each. How many can you buy with$ $2.00$?

$\frac{\$2.00}{\$0.25 \text{ per sticker}} = 8 \text{ stickers}$

⚠️ Check your units! When you multiply a unit rate by units, the bottom unit cancels and you get a total. When you divide a total by a unit rate, you get back a number of units.

Multiply or Divide? 🔽

For each problem, decide the operation and the answer. Use the unit rate given.

Put It to Work 🧮

1) A faucet fills $4$ liters per minute. How many liters in $9$ minutes? $\,?$ L 2) Oranges cost $\$ 3 $per bag. How many bags for$ $18 $?$ ,? $bags **3)** A train travels$ 80 $miles per hour. How long to go$ 400 $miles?$ ,?$ hours

⚠️ The classic trap: lower is better for price, but higher is better for speed/work. Always ask whether you want the number big or small.

Mixed Practice 🎯

One More Set 🧮

1) $\$ 1.50 $for$ 5 $apples. Unit price$ = $,? $per apple *(decimal)*. **2)** At$ 7 $pages per minute, pages read in$ 6 $minutes$ = ,? $pages. **3)** Pack X:$ 4 $for$ $1.00 $($ $0.25 $each). Pack Y:$ 10 $for$ $2.00 $($ $0.20$ each). Which is the better buy — type X or Y?

Exit Quiz ✅

Answer all three to finish the lesson.

Rates and Unit Rates

Rates and Unit Rates - Complete Interactive Lesson

Part 1: From Ratios to Rates

🏎️ Rates and Unit Rates

Topics in This Part

Ratio vs. Rate

Writing a Rate

Recap

Part 2: The Unit Rate

🏎️ Rates and Unit Rates

What Is a Unit Rate?

Part 3: Comparing with Unit Rates (Better Buy)

🏎️ Rates and Unit Rates

Finding the Better Buy

Worked Example

Part 4: Using Unit Rates to Solve Problems

🏎️ Rates and Unit Rates

Scaling Up: Multiply by the Unit Rate

Worked Example

Part 5: Mixed Practice & Mastery Check

🏎️ Rates and Unit Rates

Quick Reference

Worked Example: 120120120 miles in 222 hours

Worked Example: \6forforfor3$ pounds of apples

Worked Example: 454545 words in 555 minutes

Recap

Comparing Speeds (Higher Is Faster)

Worked Example

Worked Example

Scaling Down: Divide by the Unit Rate

Worked Example

Worked Example

Worked Example: $120$ miles in $2$ hours

Worked Example: $\$ 6 $for$ 3$ pounds of apples

Worked Example: $45$ words in $5$ minutes