Mixed Number Calculator

Improper numerator = |whole| × denominator + numerator Then: a/b + c/d = (a×d + c×b) / (b×d) b/d × c/f = (b×c) / (d×f)

Whole number part

The integer in front of the fraction, in either mixed number: 2 in 2 1/4, or 1 in 1 1/2. It may be negative, and when it is, the minus belongs to the whole mixed number — -1 1/2 is minus one and a half, not minus a half. It is the part that carries the sign.

Numerator

The top of the fraction, in either mixed number: 1 in 2 1/4. It must be zero or more: a minus sign here would make the mixed number ambiguous, so the page refuses it rather than choosing a reading for you.

Denominator

The bottom of the fraction, in either mixed number: 4 in 2 1/4. It must be positive, for the same reason a fraction's denominator always is — the sign lives on the numerator, and there is only one place for it to live.

Operation

Add, subtract, multiply or divide. All four run through the same conversion, so the choice only decides which of the four combinations of numerator and denominator is formed at the end.

Mixed number

The primary reading: the result written as a whole number and a fraction. 15/4 is printed as 3 3/4. When the result is a whole number the fraction disappears entirely, and when it is less than one the whole number does — 3/4 has no 0 in front of it.

Improper fraction

The same result as a single fraction in lowest terms, 15/4. It is exact, and it is the form the mixed number has to be turned into before it can be added, subtracted, multiplied or divided by anything.

Before reducing

The result of the arithmetic before any cancelling: 30/8 in the default case. It is there so the working is visible — 30/8 and 15/4 are the same number, and the second line is what that number looks like once it is in lowest terms.

The first use is homework, and it is the reason the second line exists: mixed numbers have to be converted before they can be operated on, so a question asked in mixed form is answered by going through improper form, and both are printed. The second is checking work already done, where entering the two mixed numbers and comparing the third line against what you wrote shows whether the raw arithmetic was right before any cancelling. The third is subtracting across zero: 1 1/2 minus 4 is -2 1/2, and the sign convention matters most in exactly those cases. The fourth is a division that comes out whole — 2 1/2 divided by 1 1/4 is 2, and the fraction simply is not there in the answer, which is worth seeing once. The fifth is any recipe or measurement that arrives in mixed form and has to be doubled, halved or scaled: doubling 2 1/4 is a multiplication by 2 0/1, and the fourth field takes it.

1. 2 1/4 + 1 1/2

   1. To improper form: 2 1/4 is 2 × 4 + 1 = 9, so 9/4; and 1 1/2 is 1 × 2 + 1 = 3, so 3/2
   2. Add across: (9 × 2 + 3 × 4) ÷ (4 × 2) = (18 + 12) ÷ 8 = 30/8
   3. Reduce: 30/8 divides by 2, giving 15/4
   4. Back to mixed form: 15 ÷ 4 is 3 remainder 3, so 3 3/4

   The default case, and the one that shows why three readings are printed rather than one: 30/8, 15/4 and 3 3/4 are the same number at three different stages. The first step is the conversion the whole page is built on — the whole number multiplied by the denominator and added to the numerator — and the last step reverses it, which is why the answer comes back in the form the question was asked in. Reading the four steps through is also the check: if the reduced fraction and the mixed number disagree, one of them was printed wrong.

2. 3 1/2 − 1 3/4

   1. To improper form: 3 1/2 is 3 × 2 + 1 = 7, so 7/2; and 1 3/4 is 1 × 4 + 3 = 7, so 7/4
   2. Subtract across: (7 × 4 − 7 × 2) ÷ (2 × 4) = (28 − 14) ÷ 8 = 14/8
   3. Reduce: 14/8 divides by 2, giving 7/4
   4. Back to mixed form: 7 ÷ 4 is 1 remainder 3, so 1 3/4

   A subtraction, where the whole numbers are subtracted through the same conversion rather than separately: 3 minus 1 would give 2, and the answer is 1 3/4, because the fractions take back more than a whole. Doing the subtraction on the whole numbers first is the mistake this shape is designed to catch, and one look at the reduced line shows it — 7/4 is less than 2, so the answer cannot be 2 and a bit. Note the numerator comes out positive here; when it comes out negative the sign goes in front of the whole mixed number.

3. 1 2/3 × 2 1/4

   1. To improper form: 1 2/3 is 1 × 3 + 2 = 5, so 5/3; and 2 1/4 is 2 × 4 + 1 = 9, so 9/4
   2. Multiply across: (5 × 9) ÷ (3 × 4) = 45/12
   3. Reduce: 45/12 divides by 3, giving 15/4
   4. Back to mixed form: 15 ÷ 4 is 3 remainder 3, so 3 3/4

   Multiplication, which is the one operation where the mixed numbers do not have to be aligned in any way: 45/12 and its reduced form 15/4 are the same number, and both are arrived at by multiplying numerators and denominators straight across. Note that the same answer, 3 3/4, came out of the default addition case as well — different question, same result, which is a coincidence of these two examples and a reminder to read the fields rather than the answer. Note also that the third line, 45/12, is the largest number on the page: the raw product of two fractions is always bigger than its reduced form.

4. 2 1/2 ÷ 1 1/4

   1. To improper form: 2 1/2 is 5/2, and 1 1/4 is 5/4
   2. Divide by inverting the second: (5 × 4) ÷ (2 × 5) = 20/10
   3. Reduce: 20/10 is 2
   4. The mixed form of 2 is just 2 — there is no fraction left to write

   A division that comes out whole, which is the case that shows the mixed-number notation giving up: 2 is printed as 2, not as 2 0/1, because a numerator of zero has nothing to write above the line. The arithmetic itself is the same conversion as everywhere else, with the second fraction turned over — 20/10 reduces to 2, and the raw line is there so that the 20 and the 10 can be seen before they cancel. This is also the shape where a mixed number is genuinely the wrong tool: the answer was never a mixed number at all.

5. -1 1/2 + 2 1/4

   1. To improper form: -1 1/2 is −(1 × 2 + 1) = −3, so −3/2; and 2 1/4 is 9/4
   2. Add across: (−3 × 4 + 9 × 2) ÷ (2 × 4) = (−12 + 18) ÷ 8 = 6/8
   3. Reduce: 6/8 divides by 2, giving 3/4
   4. The result is less than one, so the mixed form is just the fraction

   The sign rule doing its work: the minus belongs to the whole mixed number, so -1 1/2 converts to −3/2 and not to −1/2. The first step is where that shows, and the multiplication by the denominator makes it concrete — the whole part contributes −1 × 2, not +1 × 2 with a stray minus somewhere else. The answer is between zero and one, so the mixed form collapses to a plain fraction and the first two lines agree exactly; the third line is the only one that still has the uncancelled 6/8 in it.

The numerator of either mixed number must be zero or more. A minus sign there would be ambiguous — 1 -1/2 could mean one minus a half or one plus a negative half, and -1 -1/2 has no reading at all — so the page refuses it and asks you to put the sign on the whole number, where it has one meaning. The whole-number part may be negative, which is how a negative mixed number is written. The denominators must be positive, since the sign lives on the numerator. Both mixed numbers are limited in size: the whole-number parts and the numerators are capped at 100,000, and the improper numerator a mixed number converts to is capped so that the two of them can be multiplied together without leaving the range of exactly representable integers — a product of two numbers past roughly ninety-four million would be quietly inexact, so the page refuses the entry instead. Division by zero is refused, where zero means the whole second mixed number being zero, which includes forms like 0 0/4 that do not look like a zero at first glance. The result is shown to no fixed number of decimal places, because there are no decimals: every reading on this page is exact, which is why fractions are worth the trouble in the first place. What the page will not do is decide whether a mixed number is the right way to write your answer — a mathematician would say it usually is not, and that objection is real, though it is not arithmetic.

How do I add mixed numbers?

Convert both to improper fractions, do the arithmetic, then convert back. 2 1/4 is 2 × 4 + 1 = 9, so 9/4; 1 1/2 is 3/2. Adding gives (9 × 2 + 3 × 4) ÷ (4 × 2) = 30/8, which reduces to 15/4, which is 3 3/4. The page runs all four steps and prints the last three of them.

Why is the answer printed three times?

Because they are three different things you might need. The mixed number is the answer in the form the question was asked. The improper fraction is the form the working has to pass through, and the form a fraction can be used in. The third line is the result before any cancelling, which is what you compare against if you are checking your own working by hand.

Can I enter a negative mixed number?

Yes, by putting the minus on the whole-number field: -1 1/2 is minus one and a half. The numerator field will not accept a minus sign, and that is deliberate — 1 -1/2 can be read as one minus a half or as one plus a negative half, and -1 -1/2 is worse. One sign, in the one place where it has a single meaning.

Does the page use a common denominator?

No, and it does not print one either. Addition and subtraction multiply the two denominators together and cross-multiply the numerators, which gives the same answer as finding the lowest common denominator — 30/8 rather than 15/4 in the default case, the same number. The common denominator is only worth the extra work when the panel is showing it, which is what the adding fractions page does.

Is 2 1/4 the right way to write two and a quarter?

It is the way it is usually written in a recipe or at school, and it is ambiguous. Mathematicians read two symbols side by side as a multiplication, so 2 1/4 says two times a quarter, which is a half. MathWorld's own entry on the subject records its author losing marks on a calculus exam for writing it that way. This page prints it because it is the notation you asked about, and keeps 9/4 on the line below.

What happens if the answer is a whole number?

The fraction disappears. Dividing 2 1/2 by 1 1/4 gives 20/10, which reduces to 2, and 2 is printed as 2 rather than as 2 0/1 — a numerator of zero has nothing to put above the line. The same rule runs the other way: an answer below 1 is printed as a plain fraction, with no zero in front of it.

Why will it not divide by 0 0/4?

Because 0 0/4 is zero. The divisor is the whole second mixed number, and a mixed number is zero when its integer part and its fraction are both zero — which is a shape that does not look like a zero until you convert it. Zero has no reciprocal, so the division has no answer, and the page says so rather than printing something.

• Mixed Fraction — the definition of a mixed fraction as an improper fraction written as a whole number beside a fraction, the ambiguity of that notation in algebra, and the author's own account of losing marks on a calculus exam for using it — Wolfram MathWorld (United States)
• Improper Fraction — a fraction whose numerator is at least its denominator, which is the form both mixed numbers are converted into before the arithmetic runs — Wolfram MathWorld (United States)
• Proper Fraction — a fraction whose numerator is smaller than its denominator, which is what is left over beside the whole number when a result is written as a mixed number — Wolfram MathWorld (United States)
• Reduced Fraction — the lowest terms of a fraction, which is the step between the raw result on the third line and the two readings above it — Wolfram MathWorld (United States)
• Greatest Common Divisor — the largest integer dividing two given integers, used to cancel the raw result down to lowest terms — Wolfram MathWorld (United States)