Mixed Numbers
Convert between mixed numbers and improper fractions, compare them, and see why 2 1/2 is really 5 halves.
1 Two names for the same amount
A mixed number (212) and an improper fraction (52) can name the exact same amount.
Mixed → improper: multiply and add
212 = 2×2+12 = 52
Multiply the whole by the denominator, add the numerator, keep the denominator.
Improper → mixed: divide
73 = 213 (7 ÷ 3 = 2 R 1)
Divide the numerator by the denominator: the quotient is the whole, the remainder is the new numerator.
Comparing mixed numbers
Compare the wholes first. 314 beats 278 no matter the fractions. If the wholes tie, compare the fraction parts.
2 See it: 2 1/2 as halves
Two whole circles plus a half circle — count the halves.
2 wholes = 4 halves, plus 1 half = 5 halves: 2 1/2 = 5/2.
3 Worked examples
Convert both directions and compare.
Example 1 Mixed to improper: 2 1/2
- Thinking: multiply the whole by the denominator: 2 × 2 = 4.
- Add the numerator: 4 + 1 = 5. Keep the denominator 2.
- So 2 1/2 = 5/2.
2 1/2 = 5/2
Example 2 Improper to mixed: 7/3
- Thinking: divide: 7 ÷ 3 = 2 R 1.
- The quotient 2 is the whole; the remainder 1 is the new numerator.
- So 7/3 = 2 1/3.
7/3 = 2 1/3
Example 3 Compare: 3 1/4 vs 2 7/8
- Thinking: compare the wholes first: 3 > 2.
- The fractions do not matter — 3 wholes already win.
- So 3 1/4 > 2 7/8.
3 1/4 > 2 7/8
Example 4 Compare: 1 2/5 vs 7/5
- Thinking: convert: 1 2/5 = (1 × 5 + 2)/5 = 7/5.
- They are the same amount!
- So 1 2/5 = 7/5.
1 2/5 = 7/5
4 Common mistakes
Two traps catch almost everyone.
Mistake 1: multiplying instead of adding the numerator
Wrong
2 1/2 = 2×2×12 = 4/2 — “multiply everything.”Right
2 1/2 = 2×2+12 = 5/2: multiply the whole, then add the numerator.Fix: say it: multiply the whole, ADD the numerator, keep the bottom.
Mistake 2: forgetting the wholes when comparing
Wrong
2 7/8 > 3 1/4 — “7/8 is bigger than 1/4, so it wins.”Right
3 1/4 > 2 7/8: the whole numbers decide first — 3 beats 2.Fix: compare wholes first; only compare fractions when the wholes tie.
5 Key vocabulary
- Mixed number
- A whole number plus a fraction, like 2 1/2.
- Improper fraction
- A fraction with numerator ≥ denominator, like 5/2.
- Proper fraction
- A fraction smaller than 1, like 1/2.
- Convert
- Changing form without changing value.
6 Quick check
Try these, then reveal the answers.
1. Write 2 3/4 as an improper fraction.
Answer
11/4 ((2 × 4 + 3)/4).2. Write 17/5 as a mixed number.
Answer
3 2/5 (17 ÷ 5 = 3 R 2).3. Compare: 4 1/3 __ 4 5/6.
Answer
< (wholes tie; 1/3 < 5/6).Key points to remember
- Mixed → improper: (whole × denominator + numerator) over the denominator.
- Improper → mixed: divide; quotient = whole, remainder = new numerator.
- Converting never changes the value — only the form.
- Compare mixed numbers: wholes first, then fractions.
- Simplify the fraction part of every mixed number.