Solve Mixed Number Division: 6⅔ ÷ 2½ Step-by-Step

Q: Why do I need to convert mixed numbers to improper fractions?

Mixed numbers have two parts (whole + fraction) that make division complicated. Converting to improper fractions gives you single fractions that are much easier to divide using the standard "multiply by reciprocal" rule.

Q: How do I remember the conversion formula?

Use this pattern: $ a\frac{b}{c} = \frac{a \times c + b}{c} $. For $ 6\frac{2}{3} $: multiply whole number by denominator (6×3=18), add numerator (+2=20), keep same denominator = $ \frac{20}{3} $!

Q: What's the reciprocal and why do I flip it?

The reciprocal flips numerator and denominator. For $ \frac{5}{2} $, the reciprocal is $ \frac{2}{5} $. Division by a fraction equals multiplication by its reciprocal - it's a fundamental rule that makes fraction division possible!

Q: How do I convert the final improper fraction back?

Divide the numerator by denominator. For $ \frac{8}{3} $: 8÷3 = 2 remainder 2, so the answer is $ 2\frac{2}{3} $. The quotient becomes the whole number, the remainder becomes the new numerator!

Q: Should I always simplify before converting back?

Yes! Always simplify improper fractions first. $ \frac{40}{15} $ simplifies to $ \frac{8}{3} $ (÷5), which is much easier to convert to $ 2\frac{2}{3} $ than working with the larger numbers.

Mixed Number Division with Improper Fractions

$6\frac{2}{3}:2\frac{1}{2}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Convert mixed fractions to fractions

00:18 Calculate the numerators

00:25 Write division as a fraction

00:34 Multiply by the reciprocal

00:44 Calculate the products

00:51 Now convert to mixed fraction

00:54 Break down 40 into 30 plus 10

00:58 Break down the fraction into whole number and remainder

01:04 Reduce what we can

01:08 Convert proper fraction to whole number, and add to mixed number

01:15 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$6\frac{2}{3}:2\frac{1}{2}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Convert each mixed number to an improper fraction.
Step 2: Divide the fractions by multiplying by the reciprocal of the divisor.
Step 3: Simplify the resulting improper fraction and convert it back to a mixed number.

Let's work through these steps:

Step 1: Convert the mixed numbers:
$6\frac{2}{3} = \frac{6 \times 3 + 2}{3} = \frac{20}{3}$
$2\frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}$

Step 2: Divide the fractions:
To divide $\frac{20}{3}$ by $\frac{5}{2}$ , we multiply by the reciprocal of $\frac{5}{2}$ :
$\frac{20}{3} \times \frac{2}{5} = \frac{20 \times 2}{3 \times 5} = \frac{40}{15}$

Step 3: Simplify and convert back:
Simplify $\frac{40}{15}$ :
The greatest common divisor of 40 and 15 is 5:
$\frac{40}{15} = \frac{8}{3}$
Convert $\frac{8}{3}$ to a mixed number:
$8\div3 = 2$ remainder $2$ , so $\frac{8}{3} = 2\frac{2}{3}$ .

Therefore, the solution to the problem is $2\frac{2}{3}$ .

Final Answer

$2\frac{2}{3}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert mixed numbers to improper fractions before dividing
Technique: Multiply by reciprocal: $\frac{20}{3} \times \frac{2}{5} = \frac{40}{15}$
Check: Verify by multiplying answer by divisor: $2\frac{2}{3} \times 2\frac{1}{2} = 6\frac{2}{3}$ ✓

Common Mistakes

Avoid these frequent errors

Dividing mixed numbers without converting to improper fractions
Don't try to divide $6\frac{2}{3} ÷ 2\frac{1}{2}$ by working with whole and fractional parts separately = confusing wrong answers! This creates multiple division steps that are hard to track. Always convert both mixed numbers to improper fractions first, then divide using the reciprocal method.

Practice Quiz

Test your knowledge with interactive questions

$1\frac{4}{5}\times1\frac{1}{3}=$

FAQ

Everything you need to know about this question

Why do I need to convert mixed numbers to improper fractions?

Mixed numbers have two parts (whole + fraction) that make division complicated. Converting to improper fractions gives you single fractions that are much easier to divide using the standard "multiply by reciprocal" rule.

How do I remember the conversion formula?

Use this pattern: $a\frac{b}{c} = \frac{a \times c + b}{c}$ . For $6\frac{2}{3}$ : multiply whole number by denominator (6×3=18), add numerator (+2=20), keep same denominator = $\frac{20}{3}$ !

What's the reciprocal and why do I flip it?

The reciprocal flips numerator and denominator. For $\frac{5}{2}$ , the reciprocal is $\frac{2}{5}$ . Division by a fraction equals multiplication by its reciprocal - it's a fundamental rule that makes fraction division possible!

How do I convert the final improper fraction back?

Divide the numerator by denominator. For $\frac{8}{3}$ : 8÷3 = 2 remainder 2, so the answer is $2\frac{2}{3}$ . The quotient becomes the whole number, the remainder becomes the new numerator!

Should I always simplify before converting back?

Yes! Always simplify improper fractions first. $\frac{40}{15}$ simplifies to $\frac{8}{3}$ (÷5), which is much easier to convert to $2\frac{2}{3}$ than working with the larger numbers.

Can I check my answer without doing the whole problem backwards?

Yes! Multiply your answer by the original divisor. If you get the original dividend, you're correct: $2\frac{2}{3} \times 2\frac{1}{2}$ should equal $6\frac{2}{3}$ ✓

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