How mixed number math works (in plain English)

A mixed number is a whole number plus a fraction, like 2 3/4. It literally means “two and three fourths.” Mixed numbers are handy in real life because they read naturally: 1 1/2 cups, 3 3/8 inches, 4 2/3 miles. But for calculations, math becomes much easier if you first convert mixed numbers into improper fractions.

To convert W N/D into an improper fraction:

• Multiply the whole number by the denominator: W × D.
• Add the numerator: (W × D) + N.
• Put that result over the same denominator: ((W × D)+N) / D.

Example: 2 3/4 becomes: ((2 × 4) + 3) / 4 = (8 + 3) / 4 = 11/4. Now you can do arithmetic using standard fraction rules.

Once both mixed numbers are improper fractions, the operations are:

• Add/Subtract: Make common denominators, then add/subtract numerators.
• Multiply: Multiply numerators together and denominators together.
• Divide: Multiply by the reciprocal (flip the second fraction).

For addition and subtraction, the “clean” method is: a/b ± c/d = (ad ± bc) / bd. Teachers may also want you to use the LCM first to keep numbers smaller, but the final answer is the same after simplification.

A fraction is simplified when the numerator and denominator share no common factor greater than 1. To simplify, divide both by the greatest common factor (GCF): (p/q) → (p ÷ g)/(q ÷ g). If you ever see an even numerator and denominator, try dividing by 2 first; if both are divisible by 3, divide by 3, etc.

To convert an improper fraction p/q into a mixed number:

• Divide: p ÷ q gives a whole number W and remainder r.
• Write: W r/q, and simplify r/q if needed.

Example: 17/6 becomes 2 5/6 because 17 ÷ 6 = 2 remainder 5.

Worked examples you can copy

Example 1 (Add): 2 3/4 + 1 1/2

• Convert: 2 3/4 = 11/4, and 1 1/2 = 3/2.
• Add: 11/4 + 3/2 = 11/4 + 6/4 = 17/4.
• Convert back: 17/4 = 4 1/4.

Example 2 (Subtract): 5 1/3 − 2 5/6

• Convert: 5 1/3 = 16/3, and 2 5/6 = 17/6.
• Common denominator: 16/3 = 32/6.
• Subtract: 32/6 − 17/6 = 15/6 = 5/2.
• Convert: 5/2 = 2 1/2.

Example 3 (Multiply): 1 2/5 × 3 1/2

• Convert: 1 2/5 = 7/5, and 3 1/2 = 7/2.
• Multiply: (7/5)×(7/2) = 49/10.
• Convert: 49/10 = 4 9/10.

Example 4 (Divide): 2 1/4 ÷ 1 1/2

• Convert: 2 1/4 = 9/4, and 1 1/2 = 3/2.
• Divide: 9/4 ÷ 3/2 = 9/4 × 2/3 = 18/12 = 3/2.
• Convert: 3/2 = 1 1/2.

If your teacher wants “show work,” the result box above gives an easy step list you can paste into a message.

Why converting first makes everything easier

Mixed numbers are great for humans and terrible for algebra. The reason is simple: operations like multiplication and division don’t have a clean “mixed number” rule. But fractions do. When you convert to an improper fraction, you’re converting “whole + part” into a single rational number that you can multiply, divide, or compare without ambiguity.

Another hidden advantage is that improper fractions keep your math exact. If you convert to decimals too early, you may introduce rounding errors (for example, 1/3 = 0.333…). By staying in fractions until the end, your simplified answer is correct and you can optionally show a decimal approximation afterward.

This is also why schools emphasize “common denominators” and “simplify.” It’s not just tradition. Common denominators let you combine pieces of the same size (fourths with fourths, sixths with sixths). Simplification then compresses your answer into the most readable form—because 15/6 and 5/2 mean the same amount, but 5/2 is the one you can recognize instantly (and it turns into 2 1/2 cleanly).

When you see a mixed number like 3 8/9, you can think: “That’s almost 4, but short by 1/9.” When you see 35/9, you can think: “That’s 3 remainder 8.” They’re the same, but each is useful in a different moment. This calculator shows both so you can pick the format that matches your assignment.

The most common convention is to put the negative sign in front of the whole mixed number: -1 2/3 means -(1 + 2/3). In improper fraction form that is -(5/3) = -5/3. This calculator uses that convention because it matches most textbooks and avoids confusing forms like “1 -2/3.”

If the numerator and denominator are both divisible by 2, 3, 5, 7, or 11, simplify right away. If you don’t see obvious factors, the safest approach is to compute the GCF using the Euclidean algorithm (that’s what the calculator does behind the scenes).

Frequently Asked Questions

• What is a mixed number?

  A mixed number combines a whole number and a proper fraction, like 4 1/2. It’s another way to write an improper fraction (for example, 9/2).

• How do I convert a mixed number to an improper fraction?

  Multiply the whole number by the denominator, add the numerator, and put it over the same denominator: ((W × D) + N) / D. Example: 3 2/5 → (3×5+2)/5 = 17/5.

• How do I simplify a fraction?

  Divide numerator and denominator by their greatest common factor (GCF). Example: 12/18 simplifies to 2/3 because the GCF is 6.

• Why do we need common denominators to add or subtract?

  You can only combine pieces that are the same size. “Fourth + half” is like “apples + oranges” until you rewrite them with a shared denominator: 1/4 + 1/2 = 1/4 + 2/4.

• How do you divide mixed numbers?

  Convert to improper fractions first, then multiply by the reciprocal (flip the second fraction). Example: 9/4 ÷ 3/2 = 9/4 × 2/3.

• Does this calculator round decimals?

  The fraction result is exact. The decimal is shown as an approximation (rounded to a reasonable number of digits). If your assignment requires specific rounding, use the fraction answer and round at the end.

• Can I enter an improper fraction directly?

  Yes—just set the whole number to 0 and enter the fraction part. For example, 11/4 can be entered as whole = 0, numerator = 11, denominator = 4.

How people actually share this

• Homework chats: copy the step list into group chats (it reads like a mini solution).
• Tutoring: use the Example button in sessions to generate quick practice problems.
• Real life: recipe scaling and woodworking measurements often use mixed numbers.
• Quick “check me” posts: screenshot the result box—clean formatting is intentional.

Want even more stickiness? Pair this with your Fraction Calculator and Simplify Fractions tool.