How do you convert an improper fraction to a mixed number?
Short answer. Divide the numerator by the denominator. The whole-number answer is the whole part, and the remainder goes over the original denominator. For 17/5: 17 ÷ 5 = 3 remainder 2, so 17/5 = 3 2/5. Simplify the fraction part if you can. If the remainder is 0, the answer is a whole number.
An improper fraction is one where the top is at least as big as the bottom: 7/4, 17/5, 9/9. Nothing is wrong with it. It is a perfectly good number, and for arithmetic it is often the better form. But nobody asks for 7/4 cups of flour. People want 1 3/4. Converting is one division with a remainder.
The method: divide, keep the remainder
Ask how many times the denominator fits into the numerator. That count is the whole number. Whatever is left over stays a fraction, over the same denominator:
17/5 17 ÷ 5 = 3 remainder 2 (3 × 5 = 15, 17 − 15 = 2) whole number 3 remainder over 5 2/5 17/5 = 3 2/5 check: 3 × 5 + 2 = 17 ✓
The check at the bottom is worth the two seconds. Multiply the whole number by the denominator, add the remainder, and you should land back on the original numerator. If you do not, the division was off.
Why it works
Split the numerator into the biggest multiple of the denominator that fits, plus what is left. The multiple becomes whole ones:
17/5 = 15/5 + 2/5
= 3 + 2/5
= 3 2/5
17/5 = 3.4
3 2/5 = 3 + 0.4 = 3.4 same number
A mixed number is just a sum with the plus sign left out. 3 2/5 means 3 + 2/5, never 3 × 2/5. That is also why you can add mixed numbers by adding the whole parts and the fraction parts separately.
More worked examples
7/4 7 ÷ 4 = 1 r 3 = 1 3/4 22/7 22 ÷ 7 = 3 r 1 = 3 1/7 43/8 43 ÷ 8 = 5 r 3 = 5 3/8 100/7 100 ÷ 7 = 14 r 2 = 14 2/7 (14 × 7 = 98) 9/2 9 ÷ 2 = 4 r 1 = 4 1/2
The remainder is always smaller than the denominator. If yours is not, the whole number is too small: 17 ÷ 5 as "2 remainder 7" is technically true and means you can fit another 5 in.
When the remainder is zero
Then there is no fraction part. The improper fraction was a whole number in disguise:
12/4 = 12 ÷ 4 = 3 r 0 = 3 5/5 = 1 40/8 = 5
Yes, 5/5 counts as improper. The top is not smaller than the bottom, and that is the whole definition.
Simplify the fraction part
The remainder over the denominator is not always in lowest terms. Simplify it the usual way, or simplify the improper fraction before you divide. Either order works:
26/4 = 6 r 2 = 6 2/4 = 6 1/2 or first: 26/4 = 13/2 = 6 r 1 = 6 1/2 1234/56 = 22 r 2 (22 × 56 = 1232) = 22 2/56 = 22 1/28 or first: 1234/56 = 617/28 = 22 r 1 = 22 1/28
Negative improper fractions
Convert the digits and keep the sign in front of the whole thing. The minus applies to the whole mixed number, not only to the whole-number part:
−17/5 = −3 2/5 which means −(3 + 2/5) = −3.4 not −3 + 2/5 = −2.6
A programming aside, because it catches people: many languages round integer division down, toward negative infinity. Python's divmod(-17, 5) returns (−4, 3), and −4 + 3/5 is also −3.4. Correct number, wrong shape. No textbook writes −17/5 as "−4 and 3/5", so if you are writing code that prints mixed numbers, divide the absolute value and put the sign back afterwards.
On a calculator
A calculator does the division instantly. The remainder is where the care goes:
17 ÷ 5 = 3.4 whole number is 3 17 − 3 × 5 = 2 remainder 17/5 = 3 2/5
Get the remainder by subtracting, not by multiplying the decimal part back up. For 17/5 the decimal part is 0.4 and 0.4 × 5 = 2, fine. But in double-precision floating point, the format most programming languages and spreadsheets use, (17 ÷ 5 − 3) × 5 comes out as 1.9999999999999996, and (22 ÷ 7 − 3) × 7 as 0.9999999999999996. Subtraction of whole numbers has no such problem. 22 − 3 × 7 is 1, every time.
It is the same drift behind 0.1 + 0.2 = 0.30000000000000004. 22/7 is a repeating decimal (3.142857 142857 ...), so no finite decimal can hold it. As a fraction, 3 1/7 is exact and fits on one line. More on that in converting fractions to decimals.
Going back the other way
Multiply the whole number by the denominator, add the numerator, keep the denominator. It is the check from above, promoted to a method:
3 2/5 = (3 × 5 + 2)/5 = 17/5 1 3/4 = (1 × 4 + 3)/4 = 7/4
You need this direction whenever you multiply or divide mixed numbers, and it is the escape hatch when subtracting mixed numbers would otherwise need borrowing.
Where this actually comes up
Recipes scaled up: three batches of 3/4 cup is 9/4 cups, which is 2 1/4 cups. Wood and bolts: 33/8 inches is 4 1/8 inches, and nobody sells a 33/8 inch anything. Time: 150 minutes is 150/60 hours, which is 5/2, which is 2 1/2 hours.
Common mistakes, quickly
| Mistake | What happens | Fix |
|---|---|---|
| Remainder written as a decimal17 ÷ 5 = 3 r 2, written 3.2 | 3.2 is 16/5, not 17/5 | The remainder goes over 5: 3 2/5 = 3.4 |
| Remainder over the quotient17/5 as 3 2/3 | 3 2/3 is 11/3 | Keep the original denominator: 3 2/5 |
| Remainder too big17/5 as 2 7/5 | Still has an improper part | Divide again: 3 2/5 |
| Not simplifying26/4 as 6 2/4 | Correct value, not lowest terms | Reduce: 6 1/2 |
| Sign on the whole part only−17/5 as −3 + 2/5 | That is −2.6, not −3.4 | The minus covers both: −(3 + 2/5) |
Every number on this page was computed exactly and checked before publishing.
Or type 17 ÷ 5 and read 3 2/5
Prism's fraction mode computes exactly and shows answers as mixed numbers in lowest terms, so 1234/56 comes back as 22 1/28, not a long decimal. Fractions are free forever, no subscription. See the app.