Why do you flip the second fraction when dividing?

The second fraction is flipped because dividing by a number gives the same result as multiplying by its reciprocal. Dividing by 1⁄2 asks how many halves fit, and each whole holds 2 halves, so dividing by 1⁄2 equals multiplying by 2. The same logic holds for every fraction: dividing by 3⁄4 equals multiplying by 4⁄3.

How do you divide a fraction by a whole number?

A fraction is divided by a whole number by writing the whole number over 1, flipping it and multiplying. For 2⁄3 ÷ 4, the 4 becomes 4⁄1, flips to 1⁄4, and 2⁄3 × 1⁄4 = 2⁄12 = 1⁄6. A shortcut gives the same result: multiply the denominator by the whole number and keep the numerator.

How do you divide mixed numbers?

Mixed numbers are divided by first turning each one into an improper fraction, then using keep-change-flip. For 2 1⁄2 ÷ 1 1⁄4, the numbers become 5⁄2 and 5⁄4, the second flips to 4⁄5, and 5⁄2 × 4⁄5 = 20⁄10 = 2. Flipping before converting gives a wrong answer.

Does dividing by a fraction make a number bigger?

Dividing by a fraction between 0 and 1 makes a positive number bigger, because more than one copy of a small piece fits into it. For example, 3 ÷ 1⁄2 = 6 and 3⁄4 ÷ 1⁄8 = 6. Dividing by a fraction greater than 1, such as 5⁄4, makes the number smaller: 2 1⁄2 ÷ 1 1⁄4 = 2.

How to Divide Fractions: Steps and Why It Works

Q: How to divide fractions, step by step?

To divide fractions, keep the first fraction, change the division sign to multiplication, and flip the second fraction upside down. Then multiply the numerators, multiply the denominators and simplify. For example, 3⁄4 ÷ 1⁄8 = 3⁄4 × 8⁄1 = 24⁄4 = 6. Turn mixed numbers into improper fractions first, and write a whole number as a fraction over 1.

Your child's homework says 3 ÷ 1⁄2, the answer key says 6, and the child has written 1 1⁄2. How to divide fractions is one of the few school rules that most adults remember as a chant, "keep, change, flip", without the reason behind it. The method takes four steps, and the reason takes one picture, so dividing fractions can make sense to both of you in an evening.

What does dividing fractions mean?

Dividing fractions means asking how many of one amount fit into another. The sum 3 ÷ 1⁄2 asks "how many halves fit into 3?", and since each whole holds two halves, three wholes hold six, so 3 ÷ 1⁄2 = 6.

The child who wrote 1 1⁄2 answered a different question: what is half of 3? That is 3 × 1⁄2, a multiplication. Reading the division sign as "how many fit into" separates the two questions.

The same reading works for every whole-number division your child already knows: 12 ÷ 3 asks how many threes fit into 12. The guide on how to teach long division builds that "how many fit" meaning for bigger numbers.

Three bars, each labeled 1 and cut into two halves, with the six halves numbered 1 to 6, above the equations 3 ÷ 1⁄2 = 6 and 6 × 1⁄2 = 3.

Why the answer is often bigger

The answer to a fraction division is bigger than the starting number whenever the divisor is less than 1. A small piece fits into an amount more times than a whole does, so 3 ÷ 1⁄2 gives 6, and 3 ÷ 1⁄4 gives 12.

The answer gets smaller only when the divisor is bigger than 1. That surprise is worth saying aloud before any rule, because a child who expects division to shrink a number will "correct" a right answer.

How to divide fractions, step by step

The keep-change-flip method divides any two fractions in four steps, plus one setup step for whole numbers and mixed numbers. Take 2⁄3 ÷ 3⁄4, the example used in Common Core standard 6.NS.A.1.

Write every number as a fraction. A whole number goes over 1, so 4 becomes 4⁄1, and a mixed number becomes an improper fraction. Both numbers in 2⁄3 ÷ 3⁄4 are already fractions.
Keep the first fraction. The 2⁄3 stays exactly as it is. Only the second number, the divisor, changes.
Change the division sign to multiplication. The sum now reads 2⁄3 × something.
Flip the second fraction. The divisor 3⁄4 becomes its reciprocal, 4⁄3. The sum is now 2⁄3 × 4⁄3.
Multiply and simplify. Multiply top by top and bottom by bottom: 2 × 4 = 8 and 3 × 3 = 9, so 2⁄3 ÷ 3⁄4 = 8⁄9. That fraction is already in lowest terms.

The standard itself gives the check: 2⁄3 ÷ 3⁄4 = 8⁄9 "because 3⁄4 of 8⁄9 is 2⁄3". Multiply 3⁄4 × 8⁄9 = 24⁄36, and 24⁄36 simplifies to 2⁄3.

Play free in your browserMissing FactorGrade 3–4

The step people get wrong

The setup step is where most errors start, because a mixed number has to become an improper fraction before anything is flipped. For 2 1⁄2 ÷ 1 1⁄4, rewrite 2 1⁄2 as 5⁄2 (two wholes are 4 halves, plus 1 half) and 1 1⁄4 as 5⁄4.

Then run the steps: 5⁄2 × 4⁄5 = 20⁄10 = 2. You can also cancel before multiplying, since the 5 on top and the 5 on the bottom divide out, which leaves 4⁄2 = 2. The missing factor game trains the "what times this gives that" thinking behind every check.

Reciprocal vs common denominator vs measurement model

Three methods divide fractions correctly: multiplying by the reciprocal is fastest, the common-denominator method shows why the answer is right, and the measurement model draws the meaning.

Method	How it works	Best for	Limits
Reciprocal (keep-change-flip)	3/4 ÷ 1/8 = 3/4 × 8/1 = 24/4 = 6	Speed, any fractions, homework sheets	Easy to apply without understanding
Common denominator	3/4 ÷ 1/8 = 6/8 ÷ 1/8 = 6 ÷ 1 = 6	Showing why the answer is right	Slower when the denominators share no factor
Measurement model (picture)	Draw 3/4 on a bar cut in eighths and count the 1/8 pieces	First lessons, checking that an answer is sensible	Hard to draw with large or awkward denominators

Which method to pick

The measurement model comes first for a child meeting fraction division, because counting pieces on a bar makes the answer visible. Use it with friendly numbers like 3 ÷ 1⁄2 or 2 ÷ 1⁄3.

The reciprocal method comes last, once the child can say what the division asks. The common-denominator method is the bridge between the two, and it is the one to reach for when your child asks "but why do we flip?"

Why keep-change-flip works

Keep-change-flip works because dividing by a fraction gives the same answer as multiplying by its reciprocal. Dividing by 1⁄2 counts halves, every whole holds 2 halves, and so dividing by 1⁄2 is the same as multiplying by 2.

The same holds for 1⁄8: every whole holds 8 eighths, so dividing by 1⁄8 is multiplying by 8. For a divisor like 3⁄4, each whole holds 4 quarters, and every 3 of those quarters make one 3⁄4 piece, so dividing by 3⁄4 is multiplying by 4 and then dividing by 3, which is multiplying by 4⁄3.

The What Works Clearinghouse fractions practice guide makes this its third recommendation: "Help students understand why procedures for computations with fractions make sense." A child who knows the reason can rebuild the rule when the chant is forgotten.

The common-denominator proof

The common-denominator method proves the rule with pieces of the same size. Rename 3⁄4 as 6⁄8, and the sum 3⁄4 ÷ 1⁄8 becomes 6⁄8 ÷ 1⁄8, which asks how many eighths are in six eighths. The answer is 6.

The reciprocal method reaches the same number: 3⁄4 × 8⁄1 = 24⁄4 = 6. Showing both chains side by side, on one bar, is usually the moment the flip stops feeling like a trick.

A bar cut in quarters with three quarters shaded above a bar cut in eighths with six eighths shaded and numbered, followed by the equations 3⁄4 ÷ 1⁄8 = 6⁄8 ÷ 1⁄8 = 6 and 3⁄4 × 8⁄1 = 24⁄4 = 6.

The method also works with harder numbers. For 2⁄3 ÷ 3⁄4, rename both over 12: 8⁄12 ÷ 9⁄12 = 8⁄9, the same answer as keep-change-flip gave. The guide to fractions explained for parents covers the equivalent fractions this step relies on.

How to divide fractions by whole numbers, and whole numbers by fractions

A fraction divided by a whole number gets smaller, and a whole number divided by a fraction less than 1 gets bigger. Both cases use the same rule once the whole number is written over 1.

Fraction ÷ whole number: 2⁄3 ÷ 4 = 2⁄3 × 1⁄4 = 2⁄12 = 1⁄6. Sharing 2⁄3 of a cake among 4 people gives each person 1⁄6 of the cake.
Whole number ÷ fraction: 6 ÷ 2⁄3 = 6⁄1 × 3⁄2 = 18⁄2 = 9. Nine pieces of size 2⁄3 fit into 6.
Mixed number ÷ mixed number: 2 1⁄2 ÷ 1 1⁄4 = 5⁄2 ÷ 5⁄4 = 5⁄2 × 4⁄5 = 2. Two pieces of size 1 1⁄4 make 2 1⁄2.
Fraction ÷ fraction: 3⁄4 ÷ 1⁄2 = 3⁄4 × 2⁄1 = 6⁄4 = 1 1⁄2. One and a half halves fit into 3⁄4.

What grade 5 and grade 6 expect

Fifth grade covers the simplest cases only. Common Core standard 5.NF.B.7 asks students to "divide unit fractions by whole numbers and whole numbers by unit fractions", with examples such as 1⁄3 ÷ 4 = 1⁄12 and 4 ÷ 1⁄5 = 20, and a footnote states that "division of a fraction by a fraction is not a requirement at this grade."

Sixth grade adds fraction ÷ fraction through standard 6.NS.A.1, including the general rule (a⁄b) ÷ (c⁄d) = ad⁄bc. England places the first step in Year 6: the national curriculum for mathematics in England asks pupils to "divide proper fractions by whole numbers", with the example 1⁄3 ÷ 2 = 1⁄6.

A fifth grader who meets 2⁄3 ÷ 3⁄4 is working a year ahead. The skills each year expects are listed on the math game page for 5th graders and the math game page for 6th graders.

Common mistakes when dividing fractions

Four mistakes cause most wrong answers in fraction division, and each one has a quick fix.

Flipping the first fraction: 2⁄3 ÷ 3⁄4 done as 3⁄2 × 3⁄4 gives 9⁄8. Only the divisor flips, and the correct answer is 8⁄9.
Flipping before converting a mixed number: A child who turns 1 1⁄4 into 4 by flipping the 1⁄4 and dropping the whole gets 2 1⁄2 × 4 = 10. Convert to 5⁄4 first, and the answer is 2.
Not simplifying: 20⁄10 is correct but unfinished. Divide top and bottom by their greatest common factor, 10, to get 2.
Expecting a smaller answer: A child who expects division to shrink a number distrusts 3 ÷ 1⁄2 = 6. Ask how many halves fit into 3, and the 6 makes sense.

Division by zero sits outside the rule: a fraction with 0 on top, such as 0⁄5, equals 0 and has no reciprocal, so nothing can be divided by it.

Check the answer by multiplying back

Multiplying the answer by the divisor must give back the starting number, and that one line catches every mistake above. For 3⁄4 ÷ 1⁄8 = 6, check 6 × 1⁄8 = 6⁄8 = 3⁄4.

For the flipped-first error, the check fails at once: 9⁄8 × 3⁄4 = 27⁄32, which is not 2⁄3. The same habit of checking by the opposite operation appears in how to calculate percentages and in decimal division, covered in decimals explained for parents.

Free printable worksheetDivision — 6th GradersWorksheet · PDF

Your dividing fractions checklist

Say what the division asks: how many of the second amount fit into the first.
Convert first: whole numbers over 1, mixed numbers into improper fractions.
Keep, change, flip: keep the first fraction, change ÷ to ×, flip only the second.
Multiply and simplify: top × top, bottom × bottom, then reduce to lowest terms.
Check by multiplying back: the answer times the divisor must equal the starting number.

Fluent whole-number division and multiplication facts make every one of those steps faster, since each simplification is a division question. The division facts game plays free in the browser with no sign-up, and the printable division worksheet for 6th graders comes with an answer key.

The MathIt app, free on iOS and Android, has adaptive difficulty, so daily practice moves up only after the easier facts come out right.