Rates and Unit Rates

🏎️ Rates and Unit Rates

Part 1 of 5 — From Ratios to Rates

Topics in This Part

🔑 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.

💡 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:

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

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.

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 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:

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

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:

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

so $6 for $3$ lb gives $2 per $1$ 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) $\dfrac{20}{5} = \,?$   ($20 for $5$ shirts, in $ 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:

1. Write it as $\dfrac{\text{first}}{\text{second}}$.
2. Divide the top by the bottom.
3. 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.

🏎️ 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:

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

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

That is $0.50 per bar vs. $0.60 per bar. Since $0.50 is less than $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 $= \,?$   (in $ per pen). 2) $10$ pens for $4. Unit price $= \,?$   (in $ per pen, decimal). 3) Which pack is the better buy — type 1 or 2?

🏎️ 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 per pound. How much for $7$ pounds?

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

so the total is $14.

Worked Example

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

🏎️ 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

⚠️ 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 $= \,?$   (in $ 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.