Unit Rates with Ratios of Fractions

Section 1

Where Do Unit Rates Come From?

People have been comparing amounts since the very beginning of civilization. Whenever someone asked, "How much do I get for this?" they were thinking about a rate. The idea is as old as trading grain, measuring land, and building pyramids. Let's look at some key moments.

~3000 BCE

Ancient Egypt

Egyptian scribes kept careful records of how many loaves of bread could be made from a set amount of grain. They were already computing rates — like "loaves per bushel" — to make sure rations were fair.

~300 BCE

Ancient Greece

Euclid wrote about ratios in his famous book Elements. He showed how to compare two quantities and decide if they were in the same proportion, which is exactly what we do when we find unit rates.

~600–800 CE

Indian Mathematicians

Mathematicians like Brahmagupta developed clearer rules for working with fractions. Their work made it possible to compute rates even when the numbers weren't whole — the exact skill you'll learn today.

1600s–1700s

The Scientific Revolution

Scientists like Galileo needed to describe speeds such as "miles per hour." As experiments grew more precise, rates involving fractions became essential for physics, astronomy, and engineering.

Today

Everyday Life

You see unit rates everywhere: price per ounce at the grocery store, miles per gallon on a car, and calories per serving on a nutrition label. When those measurements involve fractions, you need the skill covered in this lesson.

Throughout history, the question has stayed the same: "How much per one?" When the quantities are whole numbers, the answer is a simple division. But what happens when you're dividing fractions by fractions? That's the problem this lesson solves.

Section 2

Core Definitions

Before we dive in, let's make sure four key ideas are crystal clear. These are the building blocks for everything that follows.

Ratio

A ratio compares two quantities. For example, "3 cups of flour to 2 cups of sugar" is the ratio 3 : 2. Ratios can also be written as fractions.

Rate

A rate is a special ratio that compares two quantities with different units — like miles and hours, or dollars and pounds.

Unit Rate

A unit rate tells you how much of the first quantity goes with exactly one of the second quantity. "60 miles per 1 hour" is a unit rate.

Complex Fraction

A complex fraction has a fraction in the numerator, the denominator, or both. Example: ½ over ¾. You simplify it by dividing the top fraction by the bottom fraction.

Here's the main idea of this lesson: when a rate is written as a fraction divided by another fraction, you can still find the unit rate. You just need to know how to divide fractions. That's it!

✦ KEY TAKEAWAY

Think of a unit rate like a "per one" answer. If you share ½ of a pizza equally among ¼ of an hour, the unit rate tells you how much pizza you'd eat in a whole hour at that pace. Finding it means dividing the top fraction by the bottom fraction.

Section 3

Seeing It Visually

Let's look at a picture to understand what a unit rate with fractions really means. Imagine you walk ¾ of a mile in ½ of an hour. How fast are you going per one hour?

Number line diagram showing ¾ mile walked in ½ hour and how it extends to 1½ miles in 1 hour

In the diagram above, the top number line shows time and the bottom shows distance. The solid shading shows what was given: you walked ¾ mile in ½ hour. The dashed sections show what happens if you keep that same pace for the other half-hour. You'd walk another ¾ mile, reaching 1½ miles in 1 full hour. That's your unit rate!

Mathematically, we got there by dividing: ¾ ÷ ½ = ¾ × 2 = ³⁄₂ = 1½. The next section shows you the step-by-step method for any problem like this.

Section 4

The Step-by-Step Method

Finding a unit rate when fractions are involved comes down to one big idea: divide the first quantity by the second quantity. Since both quantities might be fractions, you need to remember how to divide fractions. Here's the formula.

Unit Rate Formula

Unit Rate = first quantity / second quantity = first quantity ÷ second quantity

"First quantity" is what you're measuring. "Second quantity" is "per what."

And here's the rule for dividing fractions that makes it all work:

Dividing Fractions

(a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d) / (b × c)

To divide by a fraction, multiply by its reciprocal (flip the second fraction).

Let's break the full process into three clear steps:

Step 1 — Set up the division

Write the rate as a fraction: the first quantity on top, the second quantity on the bottom. This might look like a "fraction over a fraction" — that's totally okay!

Step 2 — Keep, Change, Flip

Keep the first fraction the same. Change the division sign to a multiplication sign. Flip (find the reciprocal of) the second fraction. Then multiply straight across: numerator × numerator and denominator × denominator.

Step 3 — Simplify

Reduce the answer to lowest terms. If the answer is an improper fraction, you can also write it as a mixed number. Don't forget to attach the correct units — like "miles per hour" or "dollars per pound."

✦ KEY TAKEAWAY

Dividing fractions is like a recipe swap: you "flip" the second ingredient and then multiply. It's the same move every time — Keep, Change, Flip. Once you master that, unit rates with fractions are no harder than unit rates with whole numbers.

Section 5

Types of Unit Rate Problems

Unit rate problems with fractions show up in lots of different situations. The table below organizes the most common types you'll see. Notice that the setup is always the same: divide the "what" by the "per what."

Scenario	Rate Expression	Unit Rate
A painter uses ²⁄₃ gallon of paint for ¼ of a wall	²⁄₃ gal ÷ ¼ wall	2⅔ gal per wall
A recipe calls for ¾ cup of sugar for ½ batch	¾ cup ÷ ½ batch	1½ cups per batch
A runner covers ⅚ mile in ⅓ hour	⅚ mi ÷ ⅓ hr	2½ mi per hr
A store charges $⅘ for ⅔ lb of trail mix	⅘ $ ÷ ⅔ lb	$1⅕ per lb
A faucet leaks ⅛ gallon in ¾ hour	⅛ gal ÷ ¾ hr	⅙ gal per hr

Now let's see a visual flowchart that shows the decision process from start to finish.

Flowchart showing three-step process: identify quantities, set up division (Keep-Change-Flip), simplify and label units

Section 6

Worked Example

Let's walk through a full problem together, nice and slow.

ℹ Problem

Mia uses ²⁄₃ of a cup of lemon juice for every ³⁄₈ of a pitcher of lemonade. How many cups of lemon juice does she use per whole pitcher?

Worked Example: Mia's Lemonade

Step 1 — Identify the quantities and set up the divisionThe two quantities are ²⁄₃ cup of lemon juice and ³⁄₈ pitcher. We want "cups per pitcher," so cups go on top and pitchers go on the bottom.

Unit Rate = ²⁄₃ ÷ ³⁄₈

Step 2 — Keep, Change, FlipKeep ²⁄₃. Change ÷ to ×. Flip ³⁄₈ to get ⁸⁄₃.

²⁄₃ × ⁸⁄₃

Step 3 — Multiply straight acrossNumerators: 2 × 8 = 16. Denominators: 3 × 3 = 9.

(2 × 8) / (3 × 3) = 16/9

Step 4 — Simplify and add units¹⁶⁄₉ is already in lowest terms (16 and 9 share no common factor besides 1). As a mixed number: 16 ÷ 9 = 1 remainder 7, so ¹⁶⁄₉ = 1⁷⁄₉.

1⁷⁄₉ cups of lemon juice per pitcher

That means for every full pitcher of lemonade, Mia needs a little more than 1¾ cups of lemon juice. Makes sense — if ²⁄₃ cup only fills ³⁄₈ of a pitcher, a whole pitcher needs a lot more juice!

Section 7

Whole-Number Rates vs. Fraction Rates

You already know how to find unit rates with whole numbers — like "150 miles in 3 hours → 50 miles per hour." Fraction unit rates use the exact same logic. The table below compares the two side by side so you can see how similar they really are.

Feature	Whole-Number Rate	Fraction Rate
Example	150 mi ÷ 3 hr	¾ mi ÷ ½ hr
Operation	Regular division	Division of fractions (Keep, Change, Flip)
Result	50 mi/hr	1½ mi/hr
Extra step?	None — just divide	Flip the second fraction, then multiply
When to simplify	If the answer isn't a whole number	Always check — reduce and convert to mixed number if needed
Common mistakes	Dividing in the wrong order	Forgetting to flip, or flipping the wrong fraction

✦ KEY TAKEAWAY

Finding a unit rate with fractions is like following the same recipe but adding one extra ingredient. You still divide the "what" by the "per what." The only new step is that you flip the second fraction before you multiply. If you can divide fractions, you can find any unit rate!

Section 8

What Comes Next?

Now that you can compute unit rates with fractions, you're building a bridge to more advanced math. Here's a quick peek at where this skill leads.

This Lesson	Coming Soon
Find a unit rate from a single ratio of fractions	Use unit rates to check if two ratios form a proportion
Work with simple fraction ÷ fraction	Solve proportional equations where the unknown is in one of the fractions
Attach units (miles per hour, dollars per pound)	Graph proportional relationships on a coordinate plane and interpret the constant of proportionality
Compute one rate at a time	Compare multiple unit rates to decide the better deal or the faster speed

Every proportional relationship you'll study in 7th grade and beyond relies on the idea of a unit rate. The skills you practiced today — dividing fractions and interpreting the result with units — will show up again and again in algebra, science, and real life. You're setting a strong foundation!

Section 9

Practice Problems

Try these five problems on your own. Start from the top and work your way down — they get a little harder each time. Click "Show Answer" when you're ready to check your work.

PROBLEM 1 — CONCEPTUAL

In your own words, what does a unit rate tell you? If someone says the unit rate is "12 pages per hour," what does the "per hour" part mean?

PROBLEM 2 — BASIC CALCULATION

A snail crawls ¹⁄₄ of a meter in ¹⁄₂ of an hour. What is the snail's speed in meters per hour?

PROBLEM 3 — INTERMEDIATE

A baker uses ⁵⁄₆ of a cup of flour for ²⁄₃ of a batch of cookies. How many cups of flour does the baker need for one whole batch?

PROBLEM 4 — APPLIED / WORD PROBLEM

At a farmer's market, ¾ of a pound of organic blueberries costs $⁹⁄₂ (which is $4.50). Meanwhile, ⅖ of a pound of regular blueberries costs $⁶⁄₅ (which is $1.20). Find the unit price (dollars per pound) for each type. Which is cheaper per pound?

PROBLEM 5 — CHALLENGE

Jayden runs ⁷⁄₈ of a mile in ³⁄₁₀ of an hour. Keisha runs ²⁄₃ of a mile in ¹⁄₅ of an hour. Who has a faster speed (higher unit rate in miles per hour)? By how much?

Summary

Lesson Summary

A unit rate tells you how much of one quantity corresponds to exactly one of another. When the given amounts are fractions, you find the unit rate by setting up a complex fraction — the first quantity over the second — and then dividing. The key technique is Keep, Change, Flip: keep the first fraction, change division to multiplication, and flip the second fraction (use its reciprocal). After multiplying straight across, simplify your answer and attach the correct units.

This skill lets you compare prices, speeds, recipes, and other real-world rates even when the measurements aren't nice whole numbers. Whether it's dollars per pound, miles per hour, or cups per batch, the process is always the same: divide the "what" by the "per what," simplify, and interpret your answer in context.