Improper fractions and mixed numbers: what they are and how to convert them

The recipe says 2 1/2 cups of flour, and the only clean measuring cup in the drawer holds 1/2 cup. Your child counts the scoops: 1, 2, 3, 4, 5. Five half cups is 5/2, and 5/2 is the same amount as 2 1/2, which is the whole idea of improper fractions and mixed numbers in one scoop count.

What are improper fractions and mixed numbers?

Improper fractions and mixed numbers are two ways to write an amount that is 1 or more. The improper fraction 7/4 counts seven quarters; the mixed number 1 3/4 says the same thing as one whole and three quarters left over.

A fraction is proper when its numerator is smaller than its denominator, as in 3/4, and improper when the numerator is equal or bigger. OpenStax Prealgebra defines it exactly that way: "The fraction a/b is a proper fraction if a<b and an improper fraction if a≥b." The same chapter says that "a mixed number consists of a whole number and a fraction."

Whole numbers fit the same picture. Any whole number can be written over 1, so 3 = 3/1, and an improper fraction whose top is a multiple of the bottom is a whole number in disguise: 8/4 = 2, because eight quarters fill two wholes.

Why schools teach both forms

Schools teach both forms because each one is easier for a different job. A mixed number shows the size of an amount at a glance (1 3/4 is between 1 and 2), and an improper fraction is easier to multiply and divide.

The Common Core standards start in grade 4. Standard 4.NF.B.3b asks children to decompose a fraction into a sum of fractions with the same denominator, with the example "2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8", and 4.NF.B.3c asks them to "add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction". In England, the national curriculum asks Year 5 pupils to "recognise mixed numbers and improper fractions and convert from one form to the other".

How to convert an improper fraction to a mixed number, step by step

Converting an improper fraction to a mixed number is a division with a remainder. The fraction bar already means "divided by", which is how Common Core standard 5.NF.B.3 puts it: "Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b)." Here is 17/5.

1. Divide the numerator by the denominator. 17 ÷ 5 = 3 remainder 2, because 3 × 5 = 15 and 17 − 15 = 2.
2. Write the quotient as the whole number. The quotient is 3, so the mixed number starts with 3.
3. Write the remainder as the new numerator. The remainder 2 becomes the top of the fraction part.
4. Keep the same denominator. The pieces are still fifths, so the bottom stays 5, and 17/5 = 3 2/5.
5. Simplify the fraction part. For 22/6, 22 ÷ 6 = 3 remainder 4, giving 3 4/6, and 4/6 divides by 2 to make 3 2/3.

The division in step 1 is the same one children meet in long division, where the remainder is the part that does not fill another group. Quick division facts make every conversion faster.

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The part people get wrong: changing the denominator

The denominator in a conversion never changes, and changing it is the most common error. A child who divides 17 by 5 sometimes writes 3 2/3, putting the quotient on the bottom, or 3 2/17, putting the old numerator there.

Ask what size the pieces are. Seventeen fifths are fifths before the conversion and fifths after it; regrouping five of them into each whole leaves every piece a fifth. Saying the units out loud ("seventeen fifths is three wholes and two fifths") catches the slip.

How to convert a mixed number to an improper fraction

Converting a mixed number to an improper fraction runs the division backwards: multiply the whole number by the denominator, add the numerator, and keep the denominator. For 3 2/5, the three wholes hold 3 × 5 = 15 fifths, plus 2 more fifths makes 17 fifths, so 3 2/5 = (3 × 5 + 2)/5 = 17/5.

The flour from the opening works the same way. For 2 1/2 cups, the two whole cups hold 2 × 2 = 4 half cups, plus 1 more half cup gives 5/2, which is five scoops of the 1/2 cup. OpenStax lists the same three moves: "Multiply the whole number by the denominator", "Add the numerator to the product found in Step 1", "Write the final sum over the original denominator."

Why converting first makes the math easier

Converting a mixed number to an improper fraction first makes multiplying safe, because the whole part and the fraction part travel together. Multiplying 2 1/2 × 2 part by part tempts children to drop the half; 5/2 × 2 = 10/2 = 5 leaves nothing to drop. The guide on how to multiply fractions shows the full method for mixed numbers, and the guide on how to subtract fractions shows when to borrow a whole from a mixed number.

Kitchen-table tips for improper fractions and mixed numbers

Kitchen objects make improper fractions and mixed numbers visible, because a child can count the pieces and watch the wholes form. Ten minutes with food and a pencil does more than a page of drills.

• Pizza slices: Cut two pizzas into quarters and lay out 7 slices. Ask how many quarters there are (7/4), then how many whole pizzas you can rebuild (1) and how many slices are left (3).
• Measuring cups: Fill a bowl with 2 1/2 cups of rice using only the 1/2 cup and count the scoops. Repeat with the 1/3 cup for 1 2/3 cups, which takes 5 scoops, so 1 2/3 = 5/3.
• A number line: Draw a line from 0 to 4 and mark every fifth. Count 17 fifths from 0 and see where you land: 2 marks past 3, so 17/5 = 3 2/5.

Say the units out loud

Saying the unit name keeps the denominator in place. "Nine halves is four wholes and one half" is easier to get right than "nine over two is four and one over two", and the spoken version shows why the bottom number cannot change. The guide to fractions explained for parents uses the same counting language for numerators and denominators.

How to check progress with improper fractions and mixed numbers

Five short questions show whether improper fractions and mixed numbers have clicked. Ask your child to say each step, because a right answer from a guess hides a gap.

• Basic conversion: Write 11/4 as a mixed number. The answer is 2 3/4, because 11 ÷ 4 = 2 remainder 3.
• Back again: Write 4 1/3 as an improper fraction. The answer is 13/3, because 4 × 3 + 1 = 13.
• Needs simplifying: Write 14/4 as a mixed number. The answer is 3 2/4, which simplifies to 3 1/2.
• Comparing: Ask which is bigger, 9/4 or 2 1/2. The fraction 9/4 is 2 1/4, so 2 1/2 is bigger.
• Placing it: Ask between which two whole numbers 23/5 lies. The fraction 23/5 = 4 3/5, so it lies between 4 and 5.

A two-way test

A two-way test converts the same amount in both directions on one page: change 19/6 to a mixed number (3 1/6), then change the answer back. If the round trip returns 19/6, the method is solid. Repeat weekly with new numbers, and move on after two clean weeks.

The math game for 4th graders page collects practice for the year when this skill arrives.

Common mistakes and edge cases

Four mistakes cause most wrong answers with improper fractions and mixed numbers.

• Changing the denominator: Writing 17/5 as 3 2/3 swaps fifths for thirds. Keep the original denominator.
• Adding instead of multiplying: Turning 3 2/5 into (3 + 2)/5 = 5/5 counts each whole as a single fifth. Multiply first: 3 × 5 + 2 = 17, so the answer is 17/5.
• Forgetting to simplify: Leaving 22/6 as 3 4/6 is correct but unfinished when the page asks for simplest form; 3 2/3 is the expected answer. The guide on how to simplify fractions covers the fraction part.
• Dropping the remainder: Writing 17/5 as 3 loses two fifths. The remainder is the fraction part, so it must appear.

Edge cases worth a minute

Edge cases come up when the remainder is 0 or the number is negative. An improper fraction with a remainder of 0 is a whole number: 12 ÷ 4 = 3 remainder 0, so 12/4 = 3, and nobody writes 3 0/4.

A negative mixed number puts the minus sign in front of the whole amount. The number −1 1/2 means −(1 + 1/2), which is −3/2, and not −1 + 1/2, which would be −1/2. Convert the positive part, 1 1/2 = 3/2, and put the sign back.

Your checklist and next steps for mixed numbers

• Spot it: a fraction is improper when its top is equal to or bigger than its bottom.
• To a mixed number: divide; the quotient is the whole part and the remainder is the new top.
• To an improper fraction: multiply the whole by the denominator, then add the numerator.
• Same denominator: the size of the pieces never changes in either direction.
• Finish: simplify the fraction part and read the instructions for the form they want.

Division with remainders carries every one of these steps, so short daily practice pays off. MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and daily practice in 33 languages, plus 2–5 player real-time battles for the family. Twenty-two of its games play free in the browser with no sign-up, and 30 printable worksheets come with answer keys.