How to calculate mixed fractions, step by step
To calculate mixed fractions, take each mixed number and convert it to an improper fraction first. A mixed number is a whole number plus a fraction: 3 1/2 means 3 + 1/2. To convert it, multiply the whole part by the denominator and add the numerator, then keep the denominator: 3 × 2 + 1 is 7, over 2, so 3 1/2 is 7/2. Do the arithmetic with the improper fractions, then convert the answer back to a mixed number by dividing the numerator by the denominator. That is the whole method: convert in, work in improper fractions, convert back. Work it out in the mixed-fraction-calculator.
The worked example: converting 3 1/2 to 7/2
The worked example, step by step. Step 1 — the whole part: 3. Step 2 — the denominator: 2. Step 3 — multiply the whole part by the denominator: 3 × 2 is 6. Step 4 — add the numerator: 6 + 1 is 7. Step 5 — keep the denominator: 7 over 2 is 7/2. Step 6 — check it the other way: 7 ÷ 2 is 3 remainder 1, so 7/2 is 3 1/2, which is where you started. The check matters: converting back and landing on the number you started with is how you know the conversion is right. The same five steps work for any mixed number: multiply, add, keep the denominator, and check by dividing back.
The worked example: adding 1 2/3 + 2 1/4
The worked example, step by step. Step 1 — convert the first number: 1 2/3 is 1 × 3 + 2 = 5, over 3, so it is 5/3. Step 2 — convert the second number: 2 1/4 is 2 × 4 + 1 = 9, over 4, so it is 9/4. Step 3 — find a common denominator: 3 and 4 share 12. Step 4 — convert both fractions: 5/3 is 20/12 and 9/4 is 27/12. Step 5 — add the numerators: 20 + 27 is 47, over 12, so the sum is 47/12. Step 6 — convert back: 47 ÷ 12 is 3 remainder 11, so 47/12 is 3 11/12. Step 7 — check it: three wholes is 36/12, plus 11/12 is 47/12, which is the sum you found. The answer is 3 11/12, and every step is checkable by hand.
What a mixed number is
A mixed number is a whole number plus a fraction, written together: 3 1/2 is three wholes and one half. It is the way people say amounts in the kitchen, the workshop and the classroom, because three and a half cups is easier to hear than seven halves. The improper fraction is the same amount written for arithmetic: 7/2 is seven halves, and seven halves is three wholes and a half. The two forms describe the same amount, and the conversion between them is the whole skill: the mixed form is for reading, the improper form is for working. Whenever you add, subtract, multiply or divide mixed numbers, convert them to improper fractions first, do the arithmetic, and convert the answer back.
Subtracting mixed fractions
Subtracting works the same way as adding, with one difference: the order matters. Convert both mixed numbers to improper fractions, find a common denominator, subtract the numerators, and convert the answer back. If the second number is larger than the first, the answer is negative, and the sign lives on the whole part: a negative mixed number means minus the whole value, so the sign goes with the whole part, never with the numerator alone. The same check applies as for adding: convert the answer back to an improper fraction and compare it with the subtraction you did. If the two agree, the answer is right; if they disagree, the error is in the conversion, the common denominator, or the subtraction, and reworking the steps in order will find it.
Multiplying and dividing mixed fractions
Multiplying and dividing do not need a common denominator, which is the one mercy in the subject. Convert both mixed numbers to improper fractions, then multiply straight across: numerator times numerator, denominator times denominator, and reduce the answer to lowest terms. Dividing means flipping the second fraction upside down and multiplying: the second improper fraction is turned over, and the two are multiplied as above. Dividing by a fraction smaller than one gives an answer bigger than the number you started with, and that is correct, not a mistake: splitting a quantity into smaller pieces makes more pieces. Reduce every answer to lowest terms before you convert it back to a mixed number.
Common mistakes
The first is adding the whole parts and the fraction parts separately without a common denominator: the fraction parts only add directly when the denominators already match, and with unlike denominators the answer is wrong. The second is putting the minus sign on the fraction part instead of the whole part: the sign lives on the whole part, and a negative numerator is a different number, not a shorthand. The third is forgetting to convert back: an answer left as an improper fraction is not wrong, but it is unfinished, and the question asked for a mixed number. The fourth is dividing without flipping: dividing by a fraction means multiplying by its reciprocal, and forgetting the flip gives the reciprocal of the right answer. The fifth is not reducing: an unreduced answer is correct but not finished, and the calculator always reduces. Work it out in the mixed-fraction-calculator, and check each step against the worked examples above.
A quick check you can do in your head
You cannot do every conversion in your head, but you can sanity-check an answer. A mixed number is always bigger than its whole part and smaller than the next whole number: 3 1/2 sits between 3 and 4, and any answer outside that range for a conversion of 3 1/2 is wrong. An addition of two positive mixed numbers is always bigger than either one: 1 2/3 + 2 1/4 is 3 11/12, which is bigger than both 1 2/3 and 2 1/4, and a sum smaller than either addend is wrong. A product of two mixed numbers bigger than one is always bigger than either factor. If a calculator says 3 1/2 is 5/2, or that 1 2/3 + 2 1/4 is 3 1/2, the number is wrong — the answers are 7/2 and 3 11/12, and anything far from them is a conversion error. Run your numbers through the mixed-fraction-calculator and compare them with the worked examples above.