Adding and subtracting mixed numbers uses two clear stages: convert each mixed number to an improper fraction, then add or subtract using a common denominator exactly as you would with ordinary fractions. Converting the answer back to a mixed number gives the simplest final form.
What is a mixed number and why does converting help?
A mixed number like 2¾ combines a whole number (2) and a proper fraction (¾). Although you could sometimes add the whole-number and fraction parts separately, that shortcut breaks down when the fractions give a total greater than 1 or when you are subtracting. Converting to improper fractions first gives a single consistent method that works every time.
Converting a mixed number to an improper fraction:
Multiply the whole number by the denominator, then add the numerator. Use the same denominator.
Formula: whole number w and fraction a/b → (w × b + a) / b
| Mixed number | Calculation | Improper fraction |
|---|---|---|
| 2¾ | (2 × 4 + 3) / 4 | 11/4 |
| 3⅖ | (3 × 5 + 2) / 5 | 17/5 |
| 5⅓ | (5 × 3 + 1) / 3 | 16/3 |
How do you add mixed numbers?
- Convert each mixed number to an improper fraction.
- Find a common denominator (use the LCM of both denominators).
- Add the numerators; keep the common denominator.
- Simplify and convert back to a mixed number.
Worked example: 2¾ + 1⅗
- Convert: 2¾ = 11/4; 1⅗ = 8/5.
- LCM(4, 5) = 20. Equivalent fractions: 11/4 = 55/20; 8/5 = 32/20.
- Add: 55/20 + 32/20 = 87/20.
- Convert: 87 ÷ 20 = 4 remainder 7 → 4 7/20.
Check: 2¾ ≈ 2.75, 1⅗ = 1.6; sum ≈ 4.35. And 4 7/20 = 4.35 ✓
How do you subtract mixed numbers?
The method is identical — the only change is that you subtract the numerators in step 3.
Worked example: 5⅓ − 2⅝
- Convert: 5⅓ = 16/3; 2⅝ = 21/8.
- LCM(3, 8) = 24. Equivalent fractions: 16/3 = 128/24; 21/8 = 63/24.
- Subtract: 128/24 − 63/24 = 65/24.
- Convert: 65 ÷ 24 = 2 remainder 17 → 2 17/24.
Check: 5⅓ ≈ 5.333, 2⅝ = 2.625; difference ≈ 2.708. And 2 17/24 ≈ 2.708 ✓
What is the "separate whole and fraction" method — and when does it fail?
An alternative method adds the whole-number parts and the fraction parts separately:
2¾ + 1⅓: whole parts 2 + 1 = 3; fraction parts ¾ + ⅓ = 9/12 + 4/12 = 13/12 = 1 1/12; total = 3 + 1 1/12 = 4 1/12.
This works when the fractions add to less than 1. It fails on subtraction when the fraction being subtracted is larger:
4⅙ − 1¾: fraction parts ⅙ − ¾ = 2/12 − 9/12 = negative. Here you must "borrow" 1 from the whole-number part, which is error-prone. Converting to improper fractions first is safer and always works.
How do you use this in word problems?
Problem: A plank is 3⅔ m long. A carpenter cuts off 1⅖ m. How long is the piece remaining?
- Convert: 3⅔ = 11/3; 1⅖ = 7/5.
- LCM(3, 5) = 15: 11/3 = 55/15; 7/5 = 21/15.
- Subtract: 55/15 − 21/15 = 34/15.
- Convert: 34 ÷ 15 = 2 remainder 4 → 2 4/15 m.
What mistakes should you avoid?
Mistake 1 — Adding the whole numbers and fractions separately when the fractions produce a "carry". If ¾ + ⅔ = 17/12 = 1 5/12, you must remember to carry the 1 across to the whole-number total. Convert to improper fractions to avoid this trap.
Mistake 2 — Using the wrong LCM. Finding a common denominator that is not the lowest common multiple still gives a correct answer, but produces larger numbers. 4/12 + 3/12 is correct, but if you used 24 as your common denominator unnecessarily, the arithmetic is harder than needed.
Mistake 3 — Forgetting to convert back. The answer 87/20 is a valid fraction, but the expected form on a KS3 paper is the mixed number 4 7/20. Always convert unless the question specifically asks for an improper fraction.
Frequently asked questions
Is there a shortcut for adding mixed numbers with the same denominator?
Yes — if the denominators are already equal, just add the numerators and whole-number parts in one step. For 3⅖ + 1⅘: add fractions first: ⅖ + ⅘ = 6/5 = 1⅕; then add whole numbers: 3 + 1 + 1 = 5⅕. With a common denominator already present, no conversion is needed until the fraction total exceeds 1.
How do I convert an improper fraction back to a mixed number?
Divide the numerator by the denominator. The quotient is the whole-number part and the remainder is the new numerator, with the same denominator. For 87/20: 87 ÷ 20 = 4 remainder 7, so the mixed number is 4 7/20.
Do I need to simplify the final fraction?
Always check whether the fraction part simplifies. For example, 4 6/8 should be simplified to 4¾. The fully simplified mixed number is the expected form on exam papers and avoids losing presentation marks.
Why do we need the lowest common multiple rather than just any common multiple?
You can use any common multiple — the answer will still be correct. However, using the LCM keeps the numbers as small as possible throughout the calculation, reducing arithmetic errors. If you use a larger common multiple, you must simplify a larger fraction at the end.
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