Unit Rates with Fractions

🏃 Unit Rates with Fractions

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

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. 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 $1$:

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

🔑 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

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

i.e. $2 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.

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 $= \,?$ 2) $240$ words in $4$ minutes $\;\Rightarrow\;$ words per minute $= \,?$ 3) $36$ students on $3$ buses $\;\Rightarrow\;$ students per bus $= \,?$

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 per hour?"

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.

🏃 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 $\frac{1}{4}$." The trick: 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:

⚠️ 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 $\frac{3}{2}$ 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) $\dfrac{5}{6} \div \dfrac{5}{12} = \,?$

🏃 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:

1. Write the rate as a (complex) fraction: $\dfrac{\text{first quantity}}{\text{second quantity}}$.
2. Divide — Keep·Change·Flip.
3. 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 $= \,?$

🏃 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

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

so $4 per lb.

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

so $4 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.

Better Buy 🎯

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 $10$ minutes is $6 \times \frac{2}{3} = 4$ miles in $60$ 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)

🏃 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

⚠️ 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.