Add, subtract, multiply or divide mixed numbers with the borrow shown where it happens.
Mixed numbers, kept mixed
The borrow shown where it happens.
Write them as 3 1/4 — whole, space, then a proper fraction. A plain fraction or whole number works too.
3 1/4 − 1 3/4
1 1/2
A whole had to be broken up before the fractions would subtract — that step is line one below.
Mixed number
1 1/2
whole part and a proper fraction
Improper fraction
3/2
the same value, one fraction
Decimal
1.5
terminates
Borrow needed
yes
the fraction parts would not subtract
Working, kept in mixed form
1Borrow one whole3 1/4 = 2 5/4One whole is 4/4, so the fraction part goes from 1/4 to 5/4 and the whole part drops by one. This is the step people skip.
2Subtract the whole parts2 − 1 = 1
3Subtract the fraction parts5/4 − 3/4 = 2/4
4Write the answer as a mixed number3/2 = 1 1/2
Borrowing here is the same move as borrowing a ten in column subtraction. The only difference is that a whole is worth as many parts as the denominator says.
Checked the other way
13/4 − 7/4 = 3/2 = 1 1/2
Both numbers converted to improper fractions, worked there, and converted back. It agrees with the mixed-form working above — which is the point of showing it, since the two routes share no arithmetic.
The two routes agree.
What this tool shows
Borrowing a whole is not borrowing 1 — it is borrowing d/d, as many parts as the denominator names. That step gets its own line here, and the improper-fraction route is run separately underneath as a check that shares no arithmetic with it.
Adding and subtracting without converting first
Borrowing a whole, shown as d/d
Carrying when the fractions pass a whole
Why multiplying part by part does not work
Negative answers with the sign on the whole thing
An independent check through improper fractions
Kept in mixed form The borrow as its own step Second route as a check Improper, mixed and decimal forms
Updated 7 September 2026 · Works in any browser, no installation
To subtract mixed numbers, subtract the whole parts and the fraction parts separately — and if the fraction parts will not subtract, borrow one whole first. One whole is d/d, not 1: in quarters it is 4/4, so 3 1/4 becomes 2 5/4 and the quarters then subtract cleanly. 3 1/4 − 1 3/4 is 1 1/2. Multiplying is different: convert both to improper fractions first, because the parts do not multiply independently.
At a glance
Formula shown
A mixed number w n/d is w + n/d. To subtract when the fraction parts will not: borrow one whole, so w n/d becomes (w \u2212 1) (n + d)/d. To multiply or divide, convert to improper first: w n/d = (wd + n)/d.
Scenario support
Cutting 1 3/4 metres from a 3 1/4 metre board; adding times or measurements written as mixed numbers; scaling a recipe that calls for 2 1/2 cups.
Educational estimate
Planning support from the values you enter — not professional advice.
Borrowing a whole
3 1/4 − 1 3/4. The wholes are fine; the quarters are not, because one quarter is less than three quarters. So take one whole from the 3.
The thing being borrowed is not 1. It is 4/4 — as many quarters as the denominator names. The first number becomes 2 and 5/4, and now 5/4 − 3/4 is 2/4, and 2 − 1 is 1. The answer is 1 2/4, which is 1 1/2.
It is exactly the same move as borrowing a ten in column subtraction. The only difference is that a whole is worth as many parts as the denominator says, so it is 4/4 in quarters, 3/3 in thirds and 12/12 in twelfths — and getting that number wrong is the commonest mistake in the whole topic.
Carrying, the other direction
2 1/2 + 1 3/4. The wholes add to 3 and the fractions to 2/4 + 3/4 = 5/4, which is more than a whole.
That extra whole moves across: 5/4 is 1 and 1/4, so the total is 3 + 1 + 1/4 = 4 1/4. This is borrowing run backwards, and it is the step that gets left as an improper answer like 3 5/4 — a correct value in a form nobody accepts.
A mixed number is only properly written when the fraction part is proper. The tool always finishes the carry and says when it did one.
Why 2½ × 3½ is not 6¼
Adding mixed numbers part by part works. Multiplying them part by part does not, and the failure is not small.
2 1/2 × 3 1/2 done part by part gives 6 and 1/4, so 6 1/4. The real answer is 8 3/4 — wrong by two and a half. The reason is that (a + b)(c + d) is ac + ad + bc + bd, and part by part keeps only ac and bd. The two cross terms — 2 × 1/2 and 1/2 × 3 — are exactly the missing two and a half.
So multiplication and division convert to improper fractions first: 5/2 × 7/2 = 35/4 = 8 3/4, with nothing to miss. The tool shows what part-by-part would have given, because seeing the size of the error is what stops the habit.
When the answer goes negative
1 1/4 − 3 1/2 is negative, and a mixed number carries its sign on the whole thing: −2 1/4 means −(2 + 1/4), not −2 + 1/4.
That is why the working here subtracts the smaller from the larger and puts the sign back at the end, which is how it is done on paper. Borrowing with a negative in the middle needs a sign rule at every step and is much easier to get wrong.
It is also why −2 1/4 and −2 + 1/4 are different numbers, two and a half apart. The notation is compact rather than unambiguous, and this is the one place it bites.
Improper or mixed, and when
17/4 and 4 1/4 are the same number. Which one is “the answer” depends entirely on what you are doing next.
Mixed for reading, measuring and anything a person acts on. Four and a quarter metres is immediately a length; seventeen quarters is a puzzle.
Improper for anything algebraic. Multiplying, dividing, substituting into a formula, or feeding into another calculation — the improper form has one numerator and one denominator and no special cases.
Both are given here every time, so nothing has to be converted afterwards.
Checked the other way
Underneath the working, the same sum is done again through improper fractions and the two answers are compared.
That is worth more than it sounds. The mixed-form route and the improper route share no arithmetic: one borrows and carries on whole and fractional parts, the other multiplies out and reduces once. A mistake in either would have to be matched by exactly the same mistake in the other to go unnoticed.
If they ever disagreed, the line at the bottom would say so rather than showing one of them. They do not, and that is a fact the test suite pins down across every combination of wholes and sixths up to five.
Sources and methodology
Mixed-number arithmetic is settled; these are where the conventions this page follows are set out.
Method. Addition and subtraction are carried out on the whole and fractional parts separately, with the borrow or carry recorded as its own step, exactly as it would be done on paper. Multiplication and division convert to improper fractions first, because part-by-part multiplication is genuinely wrong and the page shows what it would have given. Every answer is then recomputed independently through improper fractions and the two must agree. That engine is verified on every change against 58 hand-written assertions, including that a borrow is reported exactly when the fraction parts will not subtract and never otherwise, checked across every combination of wholes and sixths up to five. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
FractionAdd, subtract, multiply and divide fractions and mixed numbers exactly, with the least common denominator chosen, the cross-cancelling done first, and the true repeating decimal beside the answer.
Simplify FractionsReduce a fraction to lowest terms with both routes to the divisor shown side by side — Euclid line by line and the shared primes — and a stated proof when nothing can be cancelled.
Fraction to DecimalConvert a fraction to its exact decimal, with the length of the repeat worked out from the denominator before dividing and the long division shown remainder by remainder.
Comparing FractionsOrder up to eight fractions with all three methods shown on your own numbers, and a figure for how many decimal places the closest pair need before they stop looking equal.
Decimal to FractionTurn any decimal into its exact fraction, repeating ones included, with the four algebra lines that cancel the infinite tail instead of rounding it away.
PercentageSolve X% of Y, what percent X is of Y, reverse percentage, increase/decrease, discounts, and tax, tip, or commission.
For the same four operations on plain fractions, with the least common denominator and the cross-cancelling shown, the Fraction Calculator is the page. If you only need to turn an improper fraction into a mixed number or back, either page will do it, but this one shows the wholes coming out.
Educational use disclaimer
This is an educational tool. The arithmetic is exact and checked two independent ways, but the order of the working follows one common teaching convention — your course may lay the same steps out differently.
Published the mixed number page, kept in mixed form on purpose: converting to improper fractions first gets the right answer and skips the only difficult step.
Borrowing a whole is shown as its own labelled line and as d/d rather than 1, because using the wrong number of parts is the commonest mistake in the topic.
Multiplication converts first and shows what part-by-part would have given: 2 1/2 × 3 1/2 is 8 3/4, not the 6 1/4 the cross terms would have lost.
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