A mixed number is a sum with the plus sign left out. 2 3/4 means 2 + 3/4, and almost every difficulty people have with them comes from forgetting that one fact. It explains the sign. −2 3/4 is −(2 + 3/4), which is −11/4. It is not −2 + 3/4, and the difference is a whole one and a half. The minus co...
2½ × 3½ is not 6¼. Multiplying the whole parts and the fraction parts separately feels natural and throws away two of the four products the sum expands into — the answer is 8¾, and the missing 2½ is not a rounding difference. Across 100,000 random pairs the shortcut was wrong 94.6% of the time, and whenever both numbers had a whole part and a fraction part it was wrong every single time. A mixed number is a sum in disguise, so the only safe move is to write both as single fractions first. This page does that and shows every line of it.
FIRST
SECOND
2 3/4 + 1 5/6
4 7/12
55/12 · 4.58(3)
Worked as exact fractions. The decimal never ends — a block of 1 digit repeats forever — so the fraction is the exact answer and the decimal is not.
2 1/5 is what you get by adding the denominators too. The denominator names the size of the pieces, not a count of them, so it cannot be added. Halves plus quarters are not thirds. The answer is 4 7/12.
AS AN IMPROPER FRACTION
55/12
the form to calculate with
AS A DECIMAL
4.58(3)
never ends — the fraction is exact, this is not
COMMON DENOMINATOR
12
smaller than the product, 24
STEPS
4
shown on the right
WHAT IS HAPPENING
2 3/4 = 11/4 · 1 5/6 = 11/6
A mixed number is a sum, so multiply the whole part by the denominator and add the numerator. Nothing can be done with the parts while they are apart.
11/4 → 33/12 · 11/6 → 22/12
The least common denominator of 4 and 6 is 12, not 24 — they share a factor of 2, so the product is larger than it needs to be.
33/12 + 22/12 = 55/12
The denominator does not change. It says what size the pieces are, and adding pieces of the same size cannot change their size.
55/12 = 4 7/12
Divide 55 by 12: it goes 4 whole times with 7 left over, and the leftover keeps the same denominator.
Why the improper form comes first. While a number is split into a whole and a fraction, every operation has to deal with two things at once, and the rules for combining them are different in every case. Written as one fraction it is a single object again, and the four operations become the four rules you already know. Converting back at the end is presentation, not arithmetic.
Exact fractions throughout · answers always returned in lowest terms
Type each number as a whole part, a fraction, or both with a space between — 2 3/4, or 3/4, or 5, or 2.5. A minus sign in front covers the whole quantity, so -2 3/4 is minus eleven quarters.
Choose the operation. The answer appears as a mixed number, as an improper fraction and as a decimal, with the decimal marked when it repeats forever rather than quietly cut off.
Read the steps on the right. Each one names what is being done and why, from converting to single fractions through finding the common denominator to cancelling and converting back.
Look at the picture below the steps. For multiplication it is the rectangle that shows which two of the four products the usual shortcut throws away; for addition it shows both fractions re-cut into pieces of the same size.
Take 2 3/4 + 1 5/6. Convert both to single fractions first. 2 3/4 is (2 × 4 + 3)/4 = 11/4, and 1 5/6 is (1 × 6 + 5)/6 = 11/6. Nothing useful can be done while each number is still in two pieces. Now the denominators. 4 and 6 share a factor of 2, so the least common denominator is 12 rather than 24. Rewriting: 11/4 becomes 33/12, and 11/6 becomes 22/12. Add the numerators and leave the denominator alone: 33 + 22 = 55, so the answer is 55/12. The denominator says how big the pieces are, and adding pieces of one size cannot change their size. Convert back: 12 goes into 55 four times with 7 left over, so 55/12 is 4 7/12. Notice what would have happened working in columns. The whole parts give 3, and the fraction parts give 3/4 + 5/6 = 19/12, which is more than a whole. That extra whole has to be carried across to make 4 7/12. Leaving it as 3 19/12 is the same number but is not a finished answer. Now the one worth doing carefully: 2½ × 3½. As single fractions, 5/2 × 7/2 = 35/4 = 8¾. The tempting shortcut is 2 × 3 = 6 and ½ × ½ = ¼, giving 6¼. It is wrong by 2½, and the reason is visible the moment you draw it as a rectangle. A rectangle 2½ wide and 3½ tall splits into four pieces: 2 × 3, 2 × ½, ½ × 3 and ½ × ½. The shortcut counts the first and the last and ignores the two rectangles along the edges, which together come to 1 + 1½ = 2½. Multiplying sums means multiplying every part of one by every part of the other. Finally, division: 3½ ÷ 1¼. As single fractions, 7/2 ÷ 5/4. Dividing by a number is multiplying by its reciprocal, so this is 7/2 × 4/5 = 28/10. Top and bottom share a factor of 2, so it cancels to 14/5, which is 2 4/5. Checking the sense of it: 1¼ fits into 3½ twice with 1 left over, and that 1 is four fifths of another 1¼ — which is exactly what the fraction part of the answer says.
| Rule | What it says | Why |
|---|---|---|
| What a mixed number is | 2 3/4 means 2 + 3/4 | A sum written without the plus. Every other rule follows from this. |
| To an improper fraction | (whole × den + num) / den | 2 3/4 → (2×4+3)/4 = 11/4. Do this before any arithmetic. |
| Back to a mixed number | divide, keep the remainder | 11/4 → 4 goes into 11 twice with 3 left → 2 3/4. |
| A leading minus | −2 3/4 = −(2 + 3/4) | It covers the whole quantity: −11/4, not −5/4. Never −2 + 3/4. |
| Adding | common denominator first | Pieces must be the same size before they can be counted together. |
| The denominator to use | the least common multiple | The product always works but is often bigger than it needs to be. |
| Adding | never add the denominators | 3/4 + 5/6 is not 8/10. The denominator names the size, not a count. |
| Multiplying | straight across | No common denominator needed — that is an addition requirement only. |
| Multiplying | never part by part | 2½ × 3½ is 8¾, not 6¼. Two of the four products get dropped. |
| Dividing | flip and multiply | a ÷ b = a × 1/b, because the reciprocal is what undoes it. |
| Carrying | 3/4 + 5/6 = 1 7/12 | When the fraction parts pass a whole, the extra whole joins the wholes. |
| Simplifying | cancel the common factor | An unsimplified answer is not wrong, but it is not finished. |
| Improper is not rude | 11/4 is fine | Preferred in algebra. Mixed form is for measurement and everyday reading. |
| As a decimal | may never terminate | 2 1/3 is 2.333… exactly. The fraction is the exact form, not the decimal. |
Convert both to improper fractions, put them over a common denominator, add the numerators, then convert back. Working in columns — wholes with wholes, fractions with fractions — also works, but only if you remember to carry when the fraction parts add to a whole or more. Among random pairs where both numbers had a fraction part, that carry was needed 52.5% of the time, so it is not an occasional detail.
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Last updated: August 5, 2026 · Exact fractions throughout, never floating point · −2 3/4 means −(2 + 3/4).