Unit Rates

⚡ Unit Rates

Part 1 of 5 — From Ratios to Rates to Unit Rates

Topics in This Part

🔑 Key Concept: A unit rate tells you how much of one quantity goes with exactly one of another — like miles per 1 hour or dollars per 1 pound. Once you can find a unit rate, comparing deals, speeds, and recipes becomes easy.

Ratios, Rates, and Units

A ratio compares two quantities. A rate is a special ratio that compares two quantities with different units.

A rate is usually written as a fraction, keeping the units attached:

$\frac{120 \text{ miles}}{2 \text{ hours}} \qquad \frac{6 \text{ dollars}}{3 \text{ pounds}}$

💡 The word "per" is your signal. "Miles per hour," "dollars per pound," and "students per teacher" are all rates.

Concept Check 🎯

What Makes It a Unit Rate

A unit rate is a rate whose second quantity is exactly 1.

$\frac{120 \text{ miles}}{2 \text{ hours}} \;\longrightarrow\; \frac{60 \text{ miles}}{1 \text{ hour}} = 60 \text{ miles per hour}$

To get there, we divided both the top and bottom by 2 so the bottom became $1$:

$\frac{120 \div 2}{2 \div 2} = \frac{60}{1}$

🔑 Key Idea: A unit rate always has a denominator of 1, so we usually just say the single number and the units — like "$60$ miles per hour."

Read the Unit Rate 🔽

For each rate, pick the matching unit rate (denominator $= 1$).

Your First Unit Rates 🧮

Find each unit rate (the "per 1" amount). Enter just the number.

1) $\dfrac{18 \text{ cookies}}{3 \text{ kids}}$ → ? cookies per kid 2) $\dfrac{24 \text{ dollars}}{6 \text{ tickets}}$ → ? dollars per ticket 3) $\dfrac{250 \text{ miles}}{5 \text{ hours}}$ → ? miles per hour

⚡ Unit Rates

Part 2 of 5 — Computing Unit Rates by Division

🔑 The One Rule: To find a unit rate, divide the first quantity by the second quantity. The unit you want "per 1" of always goes on the bottom (the divisor).

The Division Method

A unit rate is just a division problem. The trick is putting the right quantity on the bottom.

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

Whatever follows the word "per" is the denominator (the thing you want one of).

Worked Example: dollars per pound

$15 buys $4$ pounds of grapes. Find the price per pound. "Per pound" means pounds go on the bottom:

$\frac{15 \text{ dollars}}{4 \text{ lb}} = 15 \div 4 = 3.75 \text{ per pound}$

That is $3.75 per pound.

Worked Example: miles per hour

A car travels $180$ miles in $3$ hours:

$\frac{180 \text{ mi}}{3 \text{ hr}} = 180 \div 3 = 60 \text{ miles per hour}$

💡 Two different unit rates live inside every rate. From "$15 for $4$ lb" you can also find pounds per dollar: $4 \div 15 \approx 0.27$ lb per dollar. Pick whichever the question asks for.

Which Goes on the Bottom? 🔽

The word after "per" is the denominator. Choose the correct division for each.

Compute the Unit Rate 🧮

Divide to find each unit rate. Decimals are fine; round to the hundredth if it doesn't come out evenly.

1) $312$ miles on $12$ gallons → ? miles per gallon 2) $5.40 for $6$ apples → ? dollars per apple 3) $144$ pages read in $18$ minutes → ? pages per minute

Unit Rates With Fractions

Sometimes a quantity is a fraction. The rule is the same: divide the top by the bottom. To divide by a fraction, multiply by its reciprocal.

Worked Example

A snail crawls $\dfrac{3}{4}$ mile in $\dfrac{1}{2}$ hour. Find its speed in miles per hour. "Per hour" means hours on the bottom:

$\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \cdot \frac{2}{1} = \frac{6}{4} = \frac{3}{2} = 1.5 \text{ mph}$

So the snail travels $1.5$ miles per hour.

⚠️ Watch out: dividing by a number less than 1 makes the answer bigger. Going $\frac{3}{4}$ mile in only half an hour means more than $\frac{3}{4}$ mile in a full hour — that's why $1.5 > 0.75$.

Concept Check 🎯

⚡ Unit Rates

Part 3 of 5 — Unit Price & Finding the Best Buy

🔑 Real-Life Power Tool: A unit price is the cost of one unit (one ounce, one pound, one item). To find the better deal, compare unit prices — the lower unit price is the better buy.

Unit Price = Cost ÷ Amount

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

Worked Example

A $20$-ounce bottle of juice costs $3.20. Find the unit price (per ounce):

$\frac{3.20 \text{ dollars}}{20 \text{ oz}} = 3.20 \div 20 = 0.16 \text{ per ounce}$

That comes to $0.16 per ounce, or $16$ cents per ounce.

💡 Stores often print the unit price on the shelf tag (like "$0.16 /oz") precisely so shoppers can compare. You're learning the math behind that tag.

Find the Unit Price 🧮

Divide cost by amount. Enter the dollar amount (e.g. type $0.25$ for $25$ cents).

1) $12$ oz of cereal for $4.80 → ? dollars per ounce 2) $8$ rolls of paper towels for $10.00 → ? dollars per roll 3) $5$ pounds of rice for $6.45 → ? dollars per pound

Comparing to Find the Best Buy

When two packages are different sizes, you can't just compare total prices — you compare unit prices.

Worked Example: Which is the better buy?

Option B costs $0.21 per ounce vs. $0.24 per ounce for A. Since $0.21 < 0.24$, Option B is the better buy even though its total price is higher.

⚠️ The bigger package is usually cheaper per unit — but not always! Always compute the unit price. The lower unit price wins.

Concept Check 🎯

Best Buy Walkthrough 🔽

A $32$-oz sports drink costs $2.56, and a $20$-oz bottle costs $1.80. Work it out.

⚡ Unit Rates

Part 4 of 5 — Using Unit Rates to Solve Problems

🔑 The Payoff: Once you know a unit rate, you can scale up to find any amount by multiplying, or scale down by dividing. Speed, cost, and "how long will it take" problems all become one-step calculations.

Multiply by the Unit Rate to Scale Up

If you know the cost (or distance, or pay) for one unit, multiply to find the cost for many.

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

Worked Example: total cost

Bananas cost $0.60 per pound. How much do $7$ pounds cost?

$0.60 \times 7 = 4.20$

So $7$ pounds cost $4.20.

Worked Example: distance from speed

A train moves at $50$ miles per hour. How far in $4$ hours?

$50 \text{ mph} \times 4 \text{ hr} = 200 \text{ miles}$

💡 Units cancel like numbers. "$\text{dollars per lb}$" times "$\text{lb}$" leaves dollars. Tracking units this way tells you if you set the problem up correctly.

Scale Up With a Unit Rate 🧮

Multiply the unit rate by the amount. Enter just the number.

1) Apples are $1.25 per pound. Cost of $6$ pounds? → ? dollars 2) A typist does $40$ words per minute. Words in $9$ minutes? → ? words 3) A car gets $32$ miles per gallon. Miles on $11$ gallons? → ? miles

Divide by the Unit Rate to Find "How Many" or "How Long"

If you know the total and the unit rate, divide to find the number of units.

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

Worked Example: how many pounds?

Cherries are $4 per pound. How many pounds can you buy with $10?

$10 \div 4 = 2.5 \text{ pounds}$

Worked Example: how long?

A cyclist rides at $12$ miles per hour. How long to ride $30$ miles?

$30 \div 12 = 2.5 \text{ hours}$

⚠️ Multiply or divide? If you're given the unit rate and want a total, multiply. If you're given a total and want the number of units, divide.

Multiply or Divide? 🔽

Decide the operation, then give the answer.

Concept Check 🎯

⚡ Unit Rates

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) tell a rate from a ratio, (2) compute a unit rate by dividing, (3) compare unit prices to find the best buy, and (4) multiply or divide by a unit rate to solve real problems. Let's put it all together.

Quick Reference

⚠️ Remember: a unit rate always has a denominator of 1, and dividing by a fraction less than $1$ makes the answer larger.

Mixed Practice 🧮

Use everything from this lesson. Enter just the number.

1) $204$ miles in $4$ hours → ? miles per hour 2) $\frac{3}{4}$ cup flour makes $\frac{1}{4}$ batch. Cups per FULL batch? → ? 3) Pears are $1.40 per pound. Cost of $3$ pounds? → ? dollars

One More Best Buy 🔽

Bulk almonds: $3$ lb for $13.50. Small bag: $1$ lb for $4.80. Find the better buy.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.