🎯⭐ INTERACTIVE LESSON

Unit Rates with Fractions

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Unit Rates with Fractions - Complete Interactive Lesson

Part 1: Ratios, Rates, and "Per One"

🏃 Unit Rates with Fractions

Part 1 of 5 — Ratios, Rates, and "Per One"

Topics in This Part

Section
Ratios vs. Rates
What Makes a Rate a Unit Rate
Finding a Unit Rate by Dividing

🔑 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. To find it, you divide. This whole lesson is about doing that division when the numbers are fractions.

Ratios, Rates, and Unit Rates

A ratio compares two quantities, like $3$ apples to $2$ oranges, written $3 : 2$ or $\dfrac{3}{2}$ .

A rate is a ratio that compares two quantities with different units, like $120$ miles in $2$ hours.

A unit rate is a rate written with a denominator of $1$ :

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

Phrase	What it is	Example
" $3$ to $2$ "	ratio (same idea)	$3:2$
" $120$ miles in $2$ hours"

🔑 The word "per" means "for every one." Miles per hour = miles for every one hour. Dollars per pound = dollars for every one pound.

Name That Comparison 🔽

Decide whether each phrase is a plain ratio, a rate, or a unit rate.

Finding a Unit Rate

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

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

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

Concept Check 🎯

Find the Unit Rate 🧮

Divide the first quantity by the second. Enter just the number.

1) $\$ 20 $for$ 5 $notebooks$ ;\Rightarrow; $dollars per notebook$ = ,?2404;\Rightarrow;= ,?363;\Rightarrow;= ,?$

Where We're Headed

So far every division came out to a whole number. But real problems sound like this:

"A snail crawls $\frac{1}{2}$ meter in $\frac{1}{4}$ hour. How fast is that ?"

Part 2: Complex Fractions: Fraction ÷ Fraction

🏃 Unit Rates with Fractions

Part 2 of 5 — Complex Fractions: Fraction ÷ Fraction

🔑 The Idea: A unit rate with fractions looks like $\dfrac{1/2}{1/4}$ — a fraction on top of a fraction. That's a complex fraction, and it just means " $\frac{1}{2}$ divided by ." The trick:

Part 3: Computing Unit Rates with Fractions

🏃 Unit Rates with Fractions

Part 3 of 5 — Computing Unit Rates with Fractions

🔑 Putting it together: A unit rate is "first quantity $\div$ second quantity." When those quantities are fractions, you divide fractions — exactly the skill from Part 2. The answer comes out per one of the bottom unit.

The Method

To find a unit rate with fractions:

Write the rate as a (complex) fraction: $\dfrac{\text{first quantity}}{\text{second quantity}}$ .

Part 4: Real-World Applications

🏃 Unit Rates with Fractions

Part 4 of 5 — Real-World Applications

🔑 Why this matters: Unit rates let you compare and predict. Which jar of peanut butter is the better deal? Who is reading faster? How long will the trip take? All of these become easy once you find the rate "per one."

Comparing with Unit Rates (Better Buy)

To find the better deal, compute the price per one unit for each option, then compare.

Example: which is cheaper per pound?

Option A: $\$ 3 $for$ \dfrac{3}{4}$ pound
Option B: $\$ 5 $for$ \dfrac{5}{4}$ pound

Part 5: Mixed Practice & Mastery Check

🏃 Unit Rates with Fractions

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) recognize unit rates, (2) divide fractions with Keep·Change·Flip, (3) compute a unit rate from a complex fraction, and (4) use rates to compare and predict. Let's put it all together.

Quick Reference

Goal	Key move
Find a unit rate	first quantity $\div$ second quantity
Divide by a fraction	Keep·Change·Flip (multiply by reciprocal)
" $A$ per $B$ "	on , on the

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

Example: $150$ words in $5$ minutes

$\frac{150 \text{ words}}{5 \text{ min}} = \frac{30 \text{ words}}{1 \text{ min}} = 30 \text{ words per minute}$

💡 The unit rate is just the answer to a division problem. Once the numbers become fractions (next part), the idea stays exactly the same — divide the top by the bottom.

That's still a unit rate — we still divide $\frac{1}{2}$ by $\frac{1}{4}$ — but now we're dividing a fraction by a fraction. Part 2 gives you the one tool that makes that easy: the complex fraction.

dividing by a fraction is the same as multiplying by its reciprocal.

Keep · Change · Flip

To divide by a fraction, Keep the first fraction, Change $\div$ to $\times$ , and Flip the second fraction (use its reciprocal):

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

The reciprocal of a fraction just swaps top and bottom:

Fraction	Reciprocal
$\dfrac{1}{4}$	$\dfrac{4}{1} = 4$

⚠️ Flip only the second fraction — the one you're dividing by. The first fraction stays exactly as it is.

Reciprocals 🔽

Choose the reciprocal of each number.

Worked Examples: Dividing Fractions

Example: $\dfrac{1}{2} \div \dfrac{1}{4}$

Keep $\frac{1}{2}$ , change to $\times$ , flip $\frac{1}{4}$ to $\frac{4}{1}$ :

$\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2$

Example: $\dfrac{2}{3} \div \dfrac{4}{9}$

$\frac{2}{3} \div \frac{4}{9} = \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2}$

Example: $\dfrac{3}{4} \div 6$

Write $6$ as $\frac{6}{1}$ , flip to $\frac{1}{6}$ :

$\frac{3}{4} \div 6 = \frac{3}{4} \times \frac{1}{6} = \frac{3}{24} = \frac{1}{8}$

💡 Always simplify the answer. $\frac{18}{12}$ and $\frac{3}{2}$ are equal, but is the finished form.

Concept Check 🎯

Divide the Fractions 🧮

Use Keep–Change–Flip and simplify. Enter your answer as a fraction (like 3/2) or a whole number.

1) $\dfrac{1}{2} \div \dfrac{1}{3} = \,?$ 2) $\dfrac{3}{4} \div \dfrac{1}{2} = \,?$ 3) $\dfrac{5}{6} \div \dfrac{5}{12} = \,?$

Divide — Keep·Change·Flip.

Simplify, and attach the units: "(answer) per one of the bottom unit."

Worked Example: the snail

A snail crawls $\dfrac{1}{2}$ meter in $\dfrac{1}{4}$ hour. How many meters per hour?

$\frac{1/2 \text{ m}}{1/4 \text{ hr}} = \frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2$

The snail crawls $2$ meters per hour.

✅ Does it make sense? It moved $\frac{1}{2}$ m in just a quarter hour. In a whole hour (four quarters) it would go $4 \times \frac{1}{2} = 2$ m. ✓

Worked Example: paint

A painter uses $\dfrac{3}{4}$ gallon of paint to cover $\dfrac{1}{2}$ wall. How many gallons per whole wall?

We want gallons per wall, so gallons go on top and walls on the bottom:

$\frac{3/4 \text{ gal}}{1/2 \text{ wall}} = \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2}$

That's $\dfrac{3}{2} = 1\dfrac{1}{2}$ gallons per wall.

⚠️ Watch what's on top! "Gallons per wall" puts gallons on top. "Walls per gallon" would flip it. The unit you say last ("per ___") goes on the bottom.

Set It Up 🔽

You walk $\dfrac{3}{4}$ mile in $\dfrac{1}{3}$ hour and want your speed in miles per hour.

Concept Check 🎯

Compute the Unit Rate 🧮

Find each unit rate. Enter a fraction (like 8/3) or a whole number.

1) $\dfrac{1}{3}$ liter in $\dfrac{1}{6}$ hour $\;\Rightarrow\;$ liters per hour $= \,?$ 2) $\dfrac{2}{5}$ mile in $\dfrac{1}{10}$ hour $\;\Rightarrow\;$ miles per hour $= \,?$ 3) $\dfrac{3}{8}$ kg for $\dfrac{3}{4}$ box $\;\Rightarrow\;$ kg per box $= \,?$

\text{A: } \frac{\$3}{3/4} = 3 \div \frac{3}{4} = 3 \times \frac{4}{3} = \frac{12}{3} = \$4 \text{ per lb}

A: \frac{$3}{3/4} = 3 \div \frac{3}{4} = 3 \times \frac{4}{3} = \frac{12}{3} = $4 per lb

$\text{B: } \frac{\$5}{5/4} = 5 \div \frac{5}{4} = 5 \times \frac{4}{5} = \frac{20}{5} = \$4 \text{ per lb}$

💡 They cost the same per pound — $\$ 4$ each! A bigger price tag doesn't mean a worse deal until you check the unit price.

Speed, Work, and Predicting

A unit rate also lets you scale up to answer "how much in a full hour / full day / full anything."

Example: jogging speed

Jordan jogs $\dfrac{2}{3}$ mile in $\dfrac{1}{6}$ hour. Find the speed, then the distance in a full hour.

$\text{speed} = \frac{2}{3} \div \frac{1}{6} = \frac{2}{3} \times \frac{6}{1} = \frac{12}{3} = 4 \text{ mph}$

So in one full hour Jordan jogs $4$ miles. (And in $2$ hours, $8$ miles.)

✅ Sanity check: $\frac{1}{6}$ hour is $10$ minutes. Going $\frac{2}{3}$ mile every minutes is miles in minutes. ✓

Pick the Better Rate 🔽

A factory robot paints $\dfrac{3}{5}$ of a car in $\dfrac{1}{4}$ hour.

Apply It 🧮

1) A pipe fills $\dfrac{1}{4}$ of a tank in $\dfrac{1}{8}$ hour. Tanks per hour $= \,?$ (whole number) 2) Sugar costs $\$ 2 $for$ \dfrac{2}{3} $pound. Dollars **per pound**$ = ,? $*(whole number)* **3)** A runner goes$ \dfrac{1}{2} $mile in$ \dfrac{1}{12} $hour. Speed in miles per hour$ = ,?$ (whole number)

⚠️ Two reminders: flip only the divisor (the second fraction), and the unit you name after "per" always belongs on the bottom.

Mixed Practice 🎯

One More Set 🧮

Enter a fraction (like 5/2) or a whole number.

1) $\dfrac{4}{5} \div \dfrac{2}{5} = \,?$ 2) $\dfrac{5}{6}$ liter in $\dfrac{1}{3}$ hour $\;\Rightarrow\;$ liters per hour $= \,?$ 3) $\$ 4 $for$ \dfrac{2}{3} $pound$ ;\Rightarrow; $dollars per pound$ = ,?$

Exit Quiz ✅

Answer all three to finish the lesson.

6 \times \frac{2}{3} = 4

Unit Rates with Fractions

Unit Rates with Fractions - Complete Interactive Lesson

Part 1: Ratios, Rates, and "Per One"

🏃 Unit Rates with Fractions

Topics in This Part

Ratios, Rates, and Unit Rates

Finding a Unit Rate

Example: \6forforf3$ pounds of apples

Where We're Headed

Part 2: Complex Fractions: Fraction ÷ Fraction

🏃 Unit Rates with Fractions

Part 3: Computing Unit Rates with Fractions

🏃 Unit Rates with Fractions

The Method

Part 4: Real-World Applications

🏃 Unit Rates with Fractions

Comparing with Unit Rates (Better Buy)

Example: which is cheaper per pound?

Part 5: Mixed Practice & Mastery Check

🏃 Unit Rates with Fractions

Quick Reference

Example: 150150150 words in 555 minutes

Keep · Change · Flip

Worked Examples: Dividing Fractions

Example: 12÷14\dfrac{1}{2} \div \dfrac{1}{4}21​÷41​

Example: 23÷49\dfrac{2}{3} \div \dfrac{4}{9}32​÷94​

Example: 34÷6\dfrac{3}{4} \div 643​÷6

Worked Example: the snail

Worked Example: paint

Speed, Work, and Predicting

Example: jogging speed

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

Example: $150$ words in $5$ minutes

Example: $\dfrac{1}{2} \div \dfrac{1}{4}$

Example: $\dfrac{2}{3} \div \dfrac{4}{9}$

Example: $\dfrac{3}{4} \div 6$