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

Learning Objectives

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. Apply unit rate reasoning to varied contexts

Connecting to Lesson 1

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

The CCSS Example: Walking Speed

The standard gives this example directly:

A person walks mile in each hour.

Verify with reasoning: hour × 4 = 1 full 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 to its reciprocal:

Never combine these two steps into one.

Canceling Before Multiplying

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?

Canceling in Action

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

Example: → cancel 2 and 4 →

Worked Example: Full Process

Simplify :

Step 1: Rewrite as division

Step 2: Multiply by reciprocal

Step 3: Cancel, then multiply

Worked Example: Different Denominators

Simplify :

Step 1:

Step 2:

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

Your Turn: Simplify These

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

4. gal/min

Key Takeaways

✓ Complex fraction = fraction whose numerator or denominator (or both) are fractions
✓ — the fraction bar means division, always
✓ Simplify: rewrite as ×, flip the divisor, cancel across the × sign, then multiply

Watch out: a complex fraction is not the final answer

• still has denominator — not per one unit yet
• The unit rate is complete only when the denominator simplifies to 1

Watch out: flip the divisor (second fraction), not the dividend (first)

• — flip the one you are dividing by

Coming Up: Lesson 3

Choosing Direction — Context Drives the Unit Rate

In Lesson 3, we will:

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

Key question to think about: given one ratio, how many different unit rates can you compute?