How do you convert a mixed number to an improper fraction?
Short answer. Multiply the whole number by the denominator, add the numerator, and put the result over the same denominator. For 3 2/5: 3 × 5 = 15, 15 + 2 = 17, so 3 2/5 = 17/5. For a negative mixed number, convert the digits and keep the minus sign in front: −2 3/4 = −11/4.
A mixed number is a whole number and a fraction sitting side by side: 1 3/4, 3 2/5, 12 3/7. It is the form people read comfortably. An improper fraction, like 17/5, is the form arithmetic prefers. Going from the first to the second is one multiplication and one addition.
The method: multiply, add, keep the bottom
3 2/5 3 × 5 = 15 fifths in the 3 whole ones 15 + 2 = 17 plus the 2 fifths you had 3 2/5 = 17/5 check: 17 ÷ 5 = 3 remainder 2 = 3 2/5 ✓
The denominator never changes. You are counting fifths the whole time; you are only finding out how many there are in total.
Why it works
A mixed number is an addition with the plus sign left out. 3 2/5 means 3 + 2/5, never 3 × 2/5. Write the 3 as fifths and the addition becomes ordinary adding with a common denominator:
3 2/5 = 3 + 2/5
= 15/5 + 2/5
= 17/5
3 2/5 = 3.4
17/5 = 3.4 same number
More worked examples
1 3/4 ( 1 × 4 + 3)/4 = 7/4 2 1/3 ( 2 × 3 + 1)/3 = 7/3 4 5/8 ( 4 × 8 + 5)/8 = 37/8 6 1/2 ( 6 × 2 + 1)/2 = 13/2 12 3/7 (12 × 7 + 3)/7 = 87/7 10 9/10 (10 × 10 + 9)/10 = 109/10
A quick sanity check: the new numerator should be bigger than the old whole number times the denominator, and smaller than the next whole number times it. 12 3/7 sits between 12 and 13, so 87/7 must sit between 84/7 and 91/7. It does.
Whole numbers
A whole number is a mixed number with nothing in the fraction part. Put it over 1, or over whatever denominator the rest of the problem is using:
5 = 5/1 5 = 5 × 4 / 4 = 20/4 if you are working in quarters
Does the result need simplifying?
Only if the fraction part did. If the fraction part of the mixed number is already in lowest terms, the improper fraction is too, because adding a multiple of the denominator to the numerator does not change what the two share. So simplify first or last, whichever you notice:
3 2/4 = (3 × 4 + 2)/4 = 14/4 = 7/2 or first: 3 2/4 = 3 1/2 = (3 × 2 + 1)/2 = 7/2
Negative mixed numbers
The minus sign belongs to the whole mixed number. Convert the digits as if it were positive, then put the sign back:
−2 3/4 = −(2 + 3/4) = −(2 × 4 + 3)/4 = −11/4 = −2.75 not (−2 × 4 + 3)/4 = −5/4 = −1.25
The wrong version treats the minus as if it only touched the 2, and quietly adds the 3/4 back in the other direction.
Why bother: multiplying and dividing
Adding mixed numbers works fine in mixed form. Multiplying does not, and this is where the conversion earns its keep:
2 1/2 × 1 3/5 right: 5/2 × 8/5 = 40/10 = 4 wrong: (2 × 1) + (1/2 × 3/5) = 2 + 3/10 = 2 3/10
The piece-by-piece version forgets that 2 also has to be multiplied by 3/5, and 1/2 by 1. Converting first means there is nothing to forget. Division is the same story:
3 3/4 ÷ 1 1/4 = 15/4 ÷ 5/4 = 15/4 × 4/5 = 60/20 = 3
It is also the way out of borrowing when you subtract mixed numbers:
4 1/5 − 1 2/3 = 21/5 − 5/3 = 63/15 − 25/15 = 38/15 = 2 8/15
That last step, 38/15 back to 2 8/15, is the reverse trip. It has its own walkthrough: divide, and the remainder goes over the denominator.
On a calculator
The multiply-and-add is whole-number arithmetic, so do it that way: 12 × 7 + 3 = 87, then write 87/7. The tempting shortcut is to type the mixed number as a decimal and multiply back up, and that is where rounding gets in. Round 2 1/3 to 2.33 and 2.33 × 3 is 6.99, not 7. Even without rounding by hand, double-precision floating point, the format most programming languages and spreadsheets use, computes (4 + 1/7) × 7 as 29.000000000000004.
It is the same drift behind 0.1 + 0.2 = 0.30000000000000004. 1/7 has no finite decimal form, so it is already rounded before the multiplication starts. As integers, 4 × 7 + 1 = 29 exactly, every time.
Common mistakes, quickly
| Mistake | What happens | Fix |
|---|---|---|
| Multiplying by the numerator3 2/5 as (3 × 2 + 5)/5 | Gives 11/5, which is 2 1/5 | Multiply by the denominator: (3 × 5 + 2)/5 = 17/5 |
| Forgetting to add the numerator3 2/5 as 15/5 | 15/5 is just 3 | Add the 2: 17/5 |
| Changing the denominator3 2/5 as 17/15 | 17/15 is barely more than 1 | The bottom stays 5 |
| Reading it as multiplication3 2/5 as 3 × 2/5 | Gives 6/5 | A mixed number is a sum: 3 + 2/5 |
| Sign on the whole part only−2 3/4 as (−8 + 3)/4 | Gives −5/4, which is −1.25 | The minus covers both: −11/4 |
Every number on this page was computed exactly and checked before publishing.
Or type 2 1/2 × 1 3/5 and read 4
Prism's fraction mode takes mixed numbers as you write them and computes exactly on the numerator and denominator, so nothing gets rounded along the way. Fractions are free forever, no subscription. See the app.