How to Divide Fractions: A Step-by-Step Guide You'll Actually Master

Let's learn a bit of magic today — how to master one of math's most dreaded processes and one of its trickiest, but essential, concepts: dividing fractions.

To divide fractions, flip the second fraction (find its reciprocal) and then multiply the fractions. And multiplying is a much lovelier process than dividing, right? That's the golden division of fractions rule. And once you get it, you'll wonder why it ever seemed tricky in elementary school.

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1. Keep the first fraction exactly as it is.
2. Change the division sign (÷) to the multiplication sign (×).
3. Flip the second fraction upside down (that's the reciprocal).
4. Multiply the numerators together, and multiply the denominators.
5. Simplify your answer if needed.

Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.

See? Not so scary after all. The keep-change-flip method works every time, whether you're dividing proper fractions, working with whole numbers, or tackling mixed numbers.

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Before we dive deeper, let's talk about the meaning of division. To divide is the same as finding an answer to, "How many times does this number fit into that one?"

Here are two word problems to understand the division process better:

1. You have half a pizza left over, and you want to know how many friends you can feed if each person gets a quarter-slice.So you're dividing 1/2 of the pizza into 1/4 pieces. And the answer is 2 — you can feed two friends with a quarter-slice each.
2. You have half a pizza to share with your roommate. You want to split it equally between the two of you and understand how big each piece would be.So you're dividing 1/2 of the pizza by 2. The answer is 1/4 — each of you gets a quarter of the whole pizza.

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As we can see, division and multiplication are inverse operations. Like addition and subtraction.

To add 5 is the same as subtracting -5.

When you divide by a number, you're essentially multiplying by something that "undoes" that number — its reciprocal.

Multiplying by 2 is the same as dividing by its reciprocal, which is 1/2. And vice versa — dividing by 2/3 is the same as multiplying by its reciprocal, 3/2.

Let's work through some examples to illustrate how to divide fractions in various scenarios. Each one follows the same keep-change-flip rule, but with a slightly different setup.

Dividing proper fractions — where the numerator (top number) is smaller than the denominator (bottom number) — is the fundamental skill you need. Every other, more complex case can be broken down into it. So if you master this division problem, you've cracked the code.

Let's divide 3/5 by 2/7:

1. Keep the first fraction: 3/5
2. Change ÷ to ×
3. Flip the second fraction: 2/7 becomes 7/2
4. Multiply: 3/5 × 7/2 = 21/10
5. Simplify: 21/10 = 2 1/10

So the final answer: 2 1/10.

This works for unit fractions like 1/2 or 1/4 and all other proper fractions too.

Here's another one with different denominators: 5/8 ÷ 3/4 = 5/8 × 4/3 = 20/24 = 5/6.

Improper fractions — where the numerator is larger than the denominator — work exactly the same way as proper fractions. The keep-change-flip rule doesn't care whether your fractions are "top-heavy" or not.

Let's divide 7/5 by 9/4:

1. Keep the first fraction: 7/5
2. Change ÷ to ×
3. Flip the second fraction: 9/4 becomes 4/9
4. Multiply: 7/5 × 4/9 = 28/45
5. Simplify: 28/45 is already in lowest terms

Here's another: 11/3 ÷ 5/2 = 11/3 × 2/5 = 22/15 = 1 7/15.

See? No extra steps, no special rules. If you can divide proper fractions, you can divide improper ones too.

Here's a secret: different denominators don't actually matter when you're dividing fractions.

Unlike addition and subtraction (where you need a common denominator), division doesn't care if the denominators match. The keep-change-flip rule handles everything.

Example: 2/3 ÷ 5/8

1. Keep the first fraction: 2/3
2. Change ÷ to ×
3. Flip the second fraction: 5/8 becomes 8/5
4. Multiply: 2/3 × 8/5 = 16/15
5. Simplify: 16/15 = 1 1/15

Another example: 7/10 ÷ 3/4 = 7/10 × 4/3 = 28/30 = 14/15.

See? No need to find common denominators first. Just flip and multiply. The denominators sort themselves out during multiplication.

To divide fractions by a whole number, you must first deal with the whole number and convert it into a fraction by putting it over 1. Then follow the usual steps.

Example: 3/4 ÷ 2

1. Convert 2 to 2/1
2. Keep-change-flip: 3/4 × 1/2
3. Multiply: 3/8
4. Done! No simplification needed here.

Another one: 5/6 ÷ 3 = 5/6 × 1/3 = 5/18.

When a whole number comes first — use the same trick — convert the number to a fraction, then apply the keep-change-flip rule.

Example: 4 ÷ 2/3

1. Convert 4 to 4/1
2. Flip 2/3 to get 3/2
3. Multiply: 4/1 × 3/2 = 12/2 = 6

Try this: 5 ÷ 1/4 = 5/1 × 4/1 = 20.

That's how dividing whole numbers by fractions works — and notice how the answer got bigger, not smaller. That's because dividing by a fraction less than 1 increases your result.

Mixed fractions are tricky to work with when you're dividing. You can't apply the keep-change-flip rule until you convert them to improper fractions (where the numerator is bigger than the denominator) first. But the conversion is straightforward once you know the steps.

Turn the whole number part into a fraction and combine it with the fraction part you already have. Here's the process using an example: 2 1/10.

1. Multiply the whole number by the denominator: 2 × 10 = 20
2. Add the numerator to that result: 1 + 20 = 21
3. Put your answer over the original denominator: 21/10

That's it. Now you have an improper fraction you can work with.

Once all mixed numbers have been converted to improper fractions, you can use our keep-change-flip rule.

Example: 2 1/3 ÷ 1 1/2

1. Convert 2 1/3: (2 × 3 + 1) / 3 = 7/3
2. Convert 1 1/2: (1 × 2 + 1) / 2 = 3/2
3. Now divide: 7/3 ÷ 3/2 = 7/3 × 2/3 = 14/9
4. Simplify: 14/9 = 1 5/9

Negative fractions follow the same keep-change-flip rule, but you need to track the signs carefully. Remember the golden rule of signs:

• Same signs (two positive or two negative) = positive answer.
• Different signs (one positive, one negative) = negative answer. 

Example 1: -2/3 ÷ 4/5

1. Keep the first fraction: -2/3
2. Change ÷ to ×
3. Find the reciprocal of the second fraction: 4/5 becomes 5/4
4. Multiply fractions: -2/3 × 5/4 = -10/12
5. Simplify: -10/12 = -5/6

The result is negative because one of the fractions is negative.

Example 2: -3/4 ÷ (-1/2)

1. Keep: -3/4
2. Change and flip: -3/4 × -2/1
3. Multiply: (-3) × (-2) = 6 and 4 × 1 = 4, so we get 6/4
4. Simplify: 6/4 = 3/2 = 1 1/2

The result is positive because both fractions are negative.

Quick tip: This rule applies whether you're multiplying or dividing fractions.

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People often get confused when dividing fractions because the process feels backward compared to regular division. Here are the pitfalls to watch out for.

1. Flipping the wrong fraction. The most common mistake when dividing fractions is flipping the first fraction instead of the second one. Remember: only the divisor (the second fraction) gets flipped. The first fraction stays put.
2. Forgetting to change division to multiplication. You can't just flip the second fraction and keep the division sign. The flip only works when you multiply. So 1/2 ÷ 1/3 becomes 1/2 × 3/1, not 1/2 ÷ 3/1.
3. Skipping simplification. Your answer might be correct, but if it's not simplified, you're not done yet. Always check if your numerator and denominator share common factors. For example, 6/8 should become 3/4.
4. Multiplying denominators by denominators incorrectly. Once you flip and multiply, make sure you're multiplying straight across — numerator times numerator, denominator times denominator. Don't cross-multiply diagonally like you would when solving proportions.

Avoiding these mistakes in dividing fractions comes down to one thing: slow down and follow the steps in order. Keep. Change. Flip. Multiply. Simplify. Every time.

Want to know beyond just using the keep-change-flip method? Here's the real math behind this process. Let's uncover some magic.

Example: (5/6) ÷ (2/3)

The next step may seem a bit crazy, but it is completely justified. Let's write this division as a complex fraction, where one fraction sits on top of another:

  5/6

  ___

  2/3

This is literally "five-sixths divided by two-thirds" written vertically. The entire fraction 5/6 is the dividend, and the entire fraction 2/3 is the divisor.

Now here's the not-obvious part. We want to simplify this monster, and the easiest way to do that is to eliminate the fraction in the denominator. How can we do that? Multiply both the top and bottom by the reciprocal of the denominator — in this case, 3/2.

  5/6  ×  3/2               

  ___________   

  2/3  ×  3/2                

And look what happens: 2/3 × 3/2 = 6/6 = 1 — The reciprocal of the divisor (the second fraction) cancels out the denominator. When you divide anything by 1, the quotient — the result of division — is your original number. So now we have:

  (5/6 × 3/2)

  ___________  =  5/6 × 3/2

       1

There it is! We've transformed division into multiplication by the reciprocal, and the math proves exactly why this works. The reciprocal cancels out the denominator, leaving us with simple multiplication.

So keep-change-flip isn't just a trick — it's what actually happens when you simplify the complex fraction properly. Pretty elegant, right?

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0.333333333333333 is a repeating decimal, and repeating threes always convert to the fraction 1/3. You can check this by multiplying 1/3 by 1 to see that it equals 0.333… indefinitely. If the decimal repeats without ending, it represents a rational number that can be written as a simple fraction — in this case, one-third.

Yes — 0.999… is exactly equal to 1, not almost equal. The repeating nines never stop, so the distance between 0.999… and 1 is zero. One way to prove it: if 1/3 = 0.333…, then multiplying both sides by 3 gives 1 = 0.999…. They’re two perfectly equivalent forms of the same number.

0.66666666666 is a repeating decimal commonly written as 0.6̅, meaning the 6 repeats forever. This repeating decimal is equal to 2/3. You can confirm it by dividing 2 by 3 on a calculator — the result is 0.666…, extending infinitely. Repeating decimals like this always convert cleanly into fractions.

0.84375 is a terminating decimal, which makes it straightforward to convert. Multiply it until the decimal becomes a whole number: 0.84375 × 100000 = 84375, so the fraction is 84375/100000. Then simplify by dividing the numerator and denominator by their greatest common factor. The fully simplified form of 0.84375 is 27/32.

The KFC rule is a simple mnemonic for dividing fractions: Keep, Flip, Change. Keep the first fraction exactly as it is, flip the second fraction to create its reciprocal, and change the division sign to multiplication. This turns every fraction-division problem into an easier multiplication problem. It’s just another name for keep-change-flip.

KCF stands for Keep, Change, Flip, the standard method for dividing fractions. You keep the first fraction, change the division sign to multiplication, and flip the second fraction upside down. This process transforms a division problem into multiplication, which is much easier to compute accurately and consistently.

0.6 ÷ 0.2 equals 3 because both numbers scale up proportionally when you remove the decimals. Multiply each by 10 to get an equivalent expression: 6 ÷ 2. Since scaling both numbers by the same factor doesn’t change the quotient, the result is 3. This works for any division involving decimals.