Neft Teacher · Learn It

Unit 6 · Standard 6.NOS.1

Divide Mixed Numbers

Lesson 6-10 · Learn It — how the math works

Why we're learning this

Agent Chen is mapping a suspect's route. The total distance is 3 1/2 miles, and each segment between landmarks is 1/4 mile. She needs to know how many segments are on the route to place surveillance cameras at every landmark.

✱ Math Workbench

Learn It

How do I divide with mixed numbers?

A mixed number, like 3 1/2, has a whole part and a fraction part. Before you can divide, you change each mixed number into an improper fraction (top heavier than bottom). Then ask how many of the divisor fit — with common denominators or by multiplying by the reciprocal — and check by multiplying the quotient by the divisor.

Picture itSee what the math looks like before you read about it

Agent Chen maps a 3 1/2-mile suspect route and splits it into equal 1/4-mile segments so she can place a surveillance marker at each segment.
Where you see this in real lifeAgent Chen maps a 3 1/2-mile suspect route and splits it into equal 1/4-mile segments so she can place a surveillance marker at each segment.

Watch me solve itI do — read one step at a time

The problem looks like this
Fraction bars: 3 1/2 miles is 14 fourths, so 14 segments of 1/4 mile fit.14/4
  1. Agent Chen has a 3 1/2-mile route cut into 1/4-mile segments. I want 3 1/2 ÷ 1/4.3 1/2 ÷ 1/4
  2. First I 3 1/2. Multiply the whole by the bottom: 3 × 2 = 6. Add the top: 6 + 1 = 7. So 3 1/2 = 7/2.3 × 2 = 6
  3. Now I ask how many 1/4s fit in 7/2. In fourths, 7/2 = 14/4, and 14 fourths ÷ 1 fourth = 14 — the common-denominator method.7/2 = 14/4
  4. The shortcut gives the same answer:7/2 × 4/1 = 28/2 = 14
  5. Check by multiplying: 14 × 1/4 = 14/4 = 3 1/2. So 3 1/2 ÷ 1/4 = 14 segments.14 × 1/4 = 14/4 = 3 1/2

Watch out — mistakes students make here

  • 1 7/9 is 8/3 × 2/3 — the divisor was never flipped. 2 2/3 = 8/3, and 8 thirds hold 4 groups of 2 thirds.
  • 2/3 is just the divisor. Count how many 2/3s fit in 2 2/3: in thirds, 8/3 ÷ 2/3 = 8 ÷ 2 = 4.
  • 8/3 is 2 2/3 written as an improper fraction — that is only the first step. Now divide: 8/3 ÷ 2/3 = 8 ÷ 2 = 4.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Let's try 2 1/2 ÷ 1/2. First 2 1/2: 2 × 2 = 4, plus 1 = 5, so 2 1/2 = 5/2.2 × 2 = 4
  2. How many halves fit in 5/2?
    Check this step
    Five — the halves are already counted: 5/2 ÷ 1/2 = 5. The shortcut agrees:5/2 × 2/1 = 10/2 = 5
  3. Check by multiplying: 5 × 1/2 = 5/2 = 2 1/2. So 2 1/2 ÷ 1/2 = 5.5 × 1/2 = 5/2 = 2 1/2

On your ownYou do — solve this one by yourself

Before you start, remember:

  • Now fill in the table: each to an , divide by multiplying by the , then check each quotient by multiplying it by the divisor.

Solve: Complete the table.

ProblemConvert to ImproperThe reciprocal shortcutQuotient
3 1/2 ÷ 1/4
1 3/4 ÷ 1/2
2 1/3 ÷ 7/6
4 1/5 ÷ 3/5

Stuck? Open one hint at a time.

Hint 1

The columns walk the whole method: convert, the reciprocal shortcut, then simplify.

Hint 2

The KCF column shows each multiplication already set up — your job is to finish and reduce it.

Hint 3

Row one reaches 28/2. Divide 28 by 2 to finish, then work each row the same way.

Check my answer
✅ Answers:
  • 3 1/2 ÷ 1/4 → Convert to Improper: 7/2 ÷ 1/4
  • 3 1/2 ÷ 1/4 → The reciprocal shortcut: 7/2 × 4/1 = 28/2
  • 3 1/2 ÷ 1/4 → Quotient: 14
  • 1 3/4 ÷ 1/2 → Convert to Improper: 7/4 ÷ 1/2
  • 1 3/4 ÷ 1/2 → The reciprocal shortcut: 7/4 × 2/1 = 14/4
  • 1 3/4 ÷ 1/2 → Quotient: 3 1/2
  • 2 1/3 ÷ 7/6 → Convert to Improper: 7/3 ÷ 7/6
  • 2 1/3 ÷ 7/6 → The reciprocal shortcut: 7/3 × 6/7 = 42/21
  • 2 1/3 ÷ 7/6 → Quotient: 2
  • 4 1/5 ÷ 3/5 → Convert to Improper: 21/5 ÷ 3/5
  • 4 1/5 ÷ 3/5 → The reciprocal shortcut: 21/5 × 5/3 = 105/15
  • 4 1/5 ÷ 3/5 → Quotient: 7
Each row walks the same method, so you only finish and simplify the last column: 28/2 = 14, 14/4 = 3 1/2, 42/21 = 2, and 105/15 = 7. Notice the quotients are larger than the mixed numbers you started with, because dividing by a fraction less than 1 asks how many small pieces fit inside.

Turn & TalkSay it out loud with a partner

Agent Chen maps a 3 1/2-mile route divided into 1/4-mile segments. Why can't she just divide 3 1/2 by 1/4 while it is still a ?

Try starting with:

  • I cannot divide yet because 3 1/2 is a ___ and I need an ___.No puedo dividir todavía porque 3 1/2 es un ___ y necesito una ___.
  • First I must convert 3 1/2 into ___.Primero debo convertir 3 1/2 en ___.
Use these wordsmixed numberimproper fractionconvert