Neft Teacher · Learn It

Unit 6 · Standard 6.NOS.1

Division Expressions with Fractions and Mixed Numbers

Lesson 6-2 · Learn It — how the math works

Why we're learning this

Agent Rivera is dividing a 3/4-pound evidence bag into portions that each weigh 1/8 pound. She needs to know exactly how many portions she can make for testing. Can you solve the case?

✱ Math Workbench

Learn It

How do I divide one fraction by another fraction?

When you divide a fraction by a fraction, you keep the first fraction, change ÷ to ×, and flip the second fraction (the divisor) to its reciprocal. Then multiply and simplify.

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

Agent Reyes has 5/6 pound of fingerprint powder in the evidence locker. Each sample tube for the crime lab holds 1/12 pound.
Where you see this in real lifeAgent Reyes has 5/6 pound of fingerprint powder in the evidence locker. Each sample tube for the crime lab holds 1/12 pound.

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

The problem looks like this
Fraction bar cut into 8 equal parts with 6 shaded: 3/4 pound is 6 eighths, so 6 portions of 1/8 pound fit.6/8
  1. Agent Rivera has 3/4 pound and each portion is 1/8 pound. I want 3/4 ÷ 1/8.3/4 ÷ 1/8
  2. It asks how many 1/8-pound portions fit in 3/4 pound. I rename 3/4 in eighths:3/4 = 6/8
  3. 6 eighths ÷ 1 eighth = 6, the same way 6 apples ÷ 1 apple = 6. That is the common-denominator method.
  4. The shortcut gives the same answer: 3/4 × 8/1 = 24/4 = 6, because each pound holds 8 eighths. Check: 6 × 1/8 = 6/8 = 3/4. She can make 6 portions.3/4 × 8/1 = 24/4 = 6
  5. One more move to know: → : multiply the bottom by the whole number, add the top, keep the same bottom. 3 1/2 → (2 × 3) + 1 = 7 → 7/2.(2 × 3) + 1 = 7

Watch out — mistakes students make here

  • 5/72 is 5/6 × 1/12 — the divisor was never flipped. The question asks how many 1/12-foot pieces fit, so rename 5/6 in twelfths and count them.
  • Only 2 scoops of 1/6 cup makes 2/6 = 1/3 cup — not enough sugar. Use the reciprocal shortcut on 2/3 ÷ 1/6 and simplify what you get.
  • 3 scoops of 1/6 cup make 3/6 = 1/2 cup — not enough. 2/3 = 4/6, so you need 4 scoops.

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

Same steps, new numbers.

  1. Let's try 1/2 ÷ 1/4. How many 1/4s fit in 1/2?1/2 ÷ 1/4
    Check this step
    Rename 1/2 as 2/4.
  2. 2 fourths ÷ 1 fourth = 2. The shortcut agrees:1/2 × 4/1 = 4/2 = 2
  3. Check by multiplying: 2 × 1/4 = 2/4 = 1/2. So 1/2 ÷ 1/4 = 2.2 × 1/4 = 2/4 = 1/2

On your ownYou do — solve this one by yourself

Before you start, remember:

  • Try 2 1/2 ÷ 1/4. The first number is a , so it cannot be flipped or multiplied as it stands.
  • Rewrite it as an : 2 1/2 = 5/2, because two wholes are 4 halves plus 1 more half.
  • Now multiply by the of the 1/4: 5/2 × 4/1 = 20/2.
  • : 20/2 = 10. So 2 1/2 ÷ 1/4 = 10. Check by multiplying: 10 × 1/4 = 10/4 = 2 1/2 — ten quarter-pieces really do fill two and a half wholes.

Solve: A recipe needs 2/3 cup of sugar. You only have a 1/6-cup scoop. How many scoops do you need?

My answer

Stuck? Open one hint at a time.

Hint 1

The recipe total is 2/3 cup; one scoop holds 1/6 cup — you're counting scoops, not multiplying.

Hint 2

Rewrite 2/3 in sixths so both amounts use the same size pieces.

Hint 3

Multiply top and bottom of 2/3 by 2 to get sixths, then count how many 1/6-scoops that makes.

Check my answer
✅ Answer: 4 scoops
2/3 ÷ 1/6 = 2/3 × 6/1 = 12/3 = 4 scoops.

Turn & TalkSay it out loud with a partner

Agent Rivera splits a 3/4-pound evidence bag into 1/8-pound portions. Will she get more or fewer than 1 portion? Estimate before solving.

Try starting with:

  • She will get ___ than 1 portion because ___.Ella obtendrá ___ que 1 porción porque ___.
  • Since 3/4 is the dividend and 1/8 is the divisor, ___.Como 3/4 es el dividendo y 1/8 es el divisor, ___.
Use these wordsdividenddivisorquotient