Complex Fractions

Lesson 2 of 3: Writing Them, Simplifying Them

In this lesson:

• Define and recognize the complex fraction form
• Simplify complex fractions using canceling before multiplying
• Verify the CCSS canonical walking-speed example

What You Will Learn in Lesson Two

By the end of this lesson, you should be able to:

1. Identify quantities and units in a ratio of fractions
2. Set up a complex fraction representing a unit rate
3. Compute a unit rate by dividing fraction by fraction and simplifying
4. Interpret the unit rate in context with correct units
5. Use unit rate reasoning in varied contexts

Connecting Complex Fractions to Lesson One

From Lesson 1: runner's speed =

This has a fraction in the numerator and a fraction in the denominator.

That is called a complex fraction:

• The horizontal bar means division
• Two notations — one meaning

Three Equivalent Forms of Division

The CCSS Example: Walking Speed

A person walks mile in each hour.

Verify: hour × 4 = 1 hour, so walk miles. ✓

The Common Error: Multiplying Instead of Dividing

The two-step rule — write both steps every time:

1. Rewrite division as multiplication:
2. Flip the second fraction: — write this step explicitly

Canceling Before Multiplying Saves Steps

Before: Multiply out first, then simplify

Better: Cancel common factors first, then multiply

Look across the × sign: numerator of the first and denominator of the second — any common factor?

Applying Diagonal Canceling to Fractions

Also check the other diagonal: denominator of first and numerator of second.

Example: → cancel 2 and 4 →

Worked Example: Three-Step Full Process

Simplify :

Step 1: Rewrite as division

Step 2: Multiply by reciprocal

Step 3: Cancel, then multiply

Worked Example: Fractions with Different Denominators

Simplify :

Step 1:

Step 2:

Step 3: Cancel 2 and 4 (GCF = 2):

Practice: Simplify Four Complex Fractions

Show all steps. Include units where context is given.

1.    2.    3. (hint: write 3 as first)

2. A pipe discharges gallon in minute. Gallons per minute?

Set up, apply reciprocal, cancel if possible, simplify, write units.

Answers to the Four Practice Problems

4. gal/min

Three Key Takeaways from Lesson Two

✓ Complex fraction: numerator or denominator is itself a fraction
✓ Fraction bar = division;
✓ Simplify: rewrite as ×, flip divisor, cancel, then multiply

Unsimplified ≠ final answer — denominator must equal 1

Flip the second fraction (the divisor), not the first

Preview: Context Drives the Unit Rate

Choosing Direction — Lesson 3

In Lesson 3, we will:

• Compute unit rates in both directions from the same ratio
• Work with like-unit ratios — units cancel to give a scale factor
• Apply to area contexts and mixed-unit problems