Unit Rates with Fractions

Lesson 1 of 3: Extending What You Know

In this lesson:

• Connect fractional unit rates to what you already know from Grade 6
• Set up and compute unit rates when the given quantities are fractions

Learning Objectives

By the end of this three-lesson series, you will be able to:

1. Identify the two quantities and their units in a ratio of fractions
2. Set up the complex fraction that represents a unit rate
3. Compute a unit rate by dividing a fraction by a fraction and simplifying
4. Interpret the unit rate in context with correct units
5. Apply unit rate reasoning to lengths, areas, and other quantities

Quick Review: Unit Rates You Already Know

A car travels 240 miles in 3 hours. What is the speed?

• Unit rate = value per one unit of the other quantity
• "Per one hour" means we divide until the denominator becomes 1
• The setup: distance ÷ time → miles per hour

The Same Question — New Numbers

A runner covers mile in hour. What is the speed?

• Same question: How far per one hour?
• Same setup: Speed = distance ÷ time
• New computation: Dividing a fraction by a fraction

A fraction in the numerator and a fraction in the denominator — this is a complex fraction.

Whole-Number vs. Fractional: Same Structure

The reasoning is identical — only the arithmetic differs.

Worked Example: Runner's Speed

mile in hour → speed in miles per hour

Step 1: Set up the complex fraction

Step 2: Multiply by the reciprocal

Step 3: Simplify with units

Quick Check: Setting Up the Direction

A price tag says: pound costs .

Which complex fraction gives the price per pound?

Decide before the next slide — read the answer unit first.

Worked Example: Price per Pound

lb for → price per pound

Answer unit: dollars per pound → dollars on top

Worked Example: Recipe Rate

A recipe uses cup oil for cup vinegar.

Cups of oil per cup of vinegar:

Both units are "cups" — they cancel, leaving a pure ratio:

Your Turn

Compute the unit rate. Show all three steps for each.

1. A cyclist rides mile in hour. Speed in mph?

2. Paint costs for liter. Cost per liter?

Step 1: Write the answer unit → Step 2: Set up complex fraction → Step 3: Multiply by reciprocal and simplify with units.

Answers

Problem 1: Cyclist speed

Problem 2: Paint cost

Same question. Same setup. Fraction ÷ fraction.

Key Takeaways

✓ Unit rate = "per one unit" — same definition as Grade 6
✓ Same setup: answer unit in numerator, reference unit in denominator
✓ Same computation: divide first quantity by second — even when both are fractions

Watch out: set up before you compute

• "Per pound" → dollars in numerator; "per hour" → miles in numerator
• Read the answer unit first, then build the complex fraction

Watch out: multiply by the reciprocal

• — flip the second fraction
• is multiplication, not division — a different operation entirely

Coming Up: Lesson 2

Complex Fractions — Writing Them, Simplifying Them

In Lesson 2, we will:

• Name the form and understand what it means formally
• Simplify complex fractions efficiently using canceling before multiplying
• Work through the standard's canonical example: mph

Before Lesson 2: review dividing fractions — keep-change-flip: