How to divide mixed numbers

There are five steps. Four of them are identical to multiplying; only the third one is new.

Steps 1 and 2: Convert both mixed numbers to improper fractions. Multiply the whole number by the denominator, add the numerator, and write the total over the same denominator. So 2½ becomes 5/2, because 2 × 2 = 4 and 4 + 1 = 5. Do this to both numbers even if one of them is already a plain fraction.

Step 3: Flip the second fraction and change ÷ to ×. The upside-down version is called the reciprocal. 3/4 flipped is 4/3. The first fraction never moves and never flips; only the one you are dividing by. Some teachers call this "keep, change, flip": keep the first, change the sign, flip the second.

Step 4: Multiply straight across. Numerators together, denominators together. There is no common denominator to find here.

Step 5: Simplify and convert back. Reduce the fraction, then divide the numerator by the denominator to get a mixed number. The quotient is the whole part and the remainder becomes the new numerator.

A worked example: 2½ ÷ ¾

Notice something surprising: the answer is larger than the number you started with. That is normal when you divide by something less than one, and it is a good moment to check that the child understands why rather than assuming they made an error.

Why do you flip the second fraction and multiply?

This is the step most parents cannot explain, because most of us were handed the rule with no reason attached. The reason is genuinely simple, and it starts by changing what you think division means.

Dividing does not mean cutting the first number into pieces. It means asking a counting question: how many of the second number fit inside the first? For 2½ ÷ ¾, the question is "how many three-quarter pieces fit into two and a half?"

Now line them up. Draw a bar two and a half units long, then mark off chunks of ¾. Three full chunks fit, and a bit is left over. The answer is a count of chunks, not a smaller version of 2½.

So where does flipping come from? Ask how many ¾-chunks fit inside one single whole. The answer is 4/3 of them: one full chunk plus a third of another. If one whole holds 4/3 chunks, then two and a half wholes hold two and a half lots of 4/3 chunks. Counting the chunks is multiplication:

2½ ÷ ¾  =  2½ × 4/3  =  3⅓

The reciprocal is not a trick. It is the number of chunks in one whole. Flipping the fraction is how you find that number, and multiplying by it is how you scale it up to however many wholes you actually have. The calculator above draws this bar for any problem you enter, with the chunks marked off.

What the leftover actually means

Here is the second misconception, and it trips up children who have otherwise done everything right. In 2½ ÷ ¾ = 3⅓, that ⅓ is one third of a chunk, not one third of a whole.

Three ¾-chunks account for 2¼ units, which leaves ¼ of a unit on the bar. But ¼ is exactly one third of a ¾-chunk. The answer counts chunks, so the leftover has to be measured in chunks too. A child who answers "3 remainder ¼" has understood the picture but reported it in the wrong unit, which is a very different error from not understanding at all, and worth saying out loud.

Dividing by whole numbers, and other shapes of problem

Mixed number ÷ whole number

Write the whole number as a fraction over 1, then flip it. For 4½ ÷ 3, that is 9/2 ÷ 3/1 = 9/2 × 1/3 = 9/6 = 1½. Set the second whole box to the number and the fraction to 0 over 1.

Fraction ÷ mixed number

Only the mixed number needs converting. For ⅚ ÷ 1¼, that is 5/6 ÷ 5/4 = 5/6 × 4/5 = 20/30 = ⅔. Because you are dividing by something bigger than one, the answer comes out smaller than you started with.

Common mistakes to watch for

• Flipping the wrong fraction. Only the second one flips. Flipping the first gives the reciprocal of the right answer, which often looks plausible enough to go unnoticed.
• Flipping before converting. Convert to improper fractions first, then flip. Turning 1¾ into "1 over 4/3" is not a thing.
• Forgetting to change the sign. Flipping and still dividing undoes itself. Change ÷ to × in the same motion as the flip.
• Assuming the answer must get smaller. Dividing by anything less than one makes the answer bigger. This surprises children who have only ever divided whole numbers.
• Reporting the leftover in the wrong unit. The fraction part of the answer counts chunks, not wholes.