Math anxiety is a very real thing. You're probably here because you're staring at a homework sheet, or maybe you're just trying to scale down a recipe that calls for some weird measurements and your brain has decided to go on strike. It happens to everyone. Honestly, the way we're taught fractions in school is kinda tragic. We get handed these arbitrary rules like "Keep, Change, Flip" without ever being told why they work. So, when you're faced with 2/3 divided by 3/4, it feels less like logic and more like a magic trick you forgot the secret to.
Let's just get the answer out of the way first so you can breathe. The result of 2/3 divided by 3/4 is 8/9.
If you just needed the number, there you go. But if you want to actually understand what’s happening in that space between the numbers, stick around. Fractions are basically just divisions that haven't happened yet. When you divide one by another, you're essentially asking: "How many times does this three-quarter-sized chunk fit into this two-third-sized space?" Since 3/4 (which is 0.75) is actually bigger than 2/3 (which is roughly 0.66), you already know the answer has to be less than one. It’s a logic check. If you ended up with something like 5/2, you’d know something went sideways.
The Mechanics of 2/3 divided by 3/4
To solve this, we use the reciprocal method. You've heard it called "invert and multiply." It sounds fancy, but it's just a shortcut.
Take your first fraction: 2/3. Leave it alone. It’s fine.
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Now, look at the second one: 3/4. This is the one we’re going to mess with. You flip it upside down to get 4/3. Now, instead of dividing, you multiply them together.
$$\frac{2}{3} \times \frac{4}{3} = \frac{8}{9}$$
Why do we do this? It's because division and multiplication are inverse operations. Think of it like this: dividing by 2 is the exact same thing as multiplying by 1/2. Dividing by 3/4 is the exact same thing as multiplying by 4/3. It’s a mathematical pivot.
The numerator (top number) is $2 \times 4 = 8$.
The denominator (bottom number) is $3 \times 3 = 9$.
8/9 is your final answer. It can’t be simplified any further because 8 and 9 don't share any factors other than 1. They are "relatively prime" to each other, which is just a nerdy way of saying they’re done.
Why 8/9 makes more sense than you think
Numbers can be dry. Let's talk about something real, like a wooden board or a giant chocolate bar.
Imagine you have a piece of wood that is 2/3 of a yard long. You need to cut it into pieces that are each 3/4 of a yard long. Immediately, you see the problem. Your individual pieces are supposed to be longer than the whole board you started with.
You can’t even get one full piece out of it.
You can only get a fraction of a piece. Specifically, you get 8/9 of a piece. You’re just a tiny bit short of having a full 3/4 yard section. This is where people get tripped up. We are so used to division making things "smaller" (like 10 divided by 2 is 5) that when we divide by a fraction, our brains protest. But when you divide by a number smaller than one, the result actually looks larger than the starting number.
Wait.
Let me rephrase that. 8/9 (0.88) is larger than 2/3 (0.66). This happens because you are measuring your starting amount using a "unit" that is smaller than 1, but in this specific case, our divisor 3/4 is larger than our dividend 2/3, so we get a result that is less than 1 but greater than our original 2/3.
Confused? Don't be. Just remember: division is just counting how many of "Thing B" fits into "Thing A."
The common mistakes people make with these numbers
Most people mess this up in one of two ways.
First, they flip the wrong fraction. They flip the 2/3 instead of the 3/4. If you do that, you're calculating 3/2 times 3/4, which gives you 9/8. That’s a completely different answer. 9/8 is greater than one. If you're trying to fit a 75-cent sized object into a 66-cent sized hole, you aren't going to have "more than one" of them. It’s physically impossible.
Second, people try to find a common denominator before dividing. You can do this, but it’s the long way around the mountain. To do it, you’d turn 2/3 into 8/12 and 3/4 into 9/12.
Then you divide: 8/12 divided by 9/12.
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The 12s cancel out.
You're left with 8 divided by 9.
Which is... 8/9.
It works! But it’s extra work. Most mathematicians are secretly lazy—that’s why they invented the "flip and multiply" shortcut. It skips the whole "finding the 12" step and gets you straight to the 8 and the 9.
Visualizing the 2/3 divided by 3/4 problem
If you were to draw this out, imagine a rectangle divided into three vertical strips. Color in two of them. That's your 2/3.
Now, imagine a rectangle of the same size divided into four horizontal strips. Color in three of them. That's your 3/4.
Now, try to overlay them. You're trying to see how much of that 3/4 area can fit into the 2/3 area. You'll find that the 2/3 area covers exactly 8 out of the 9 "sub-squares" that make up the 3/4 area.
Why does this matter in the real world?
Honestly? It usually matters in construction, cooking, and chemistry.
If a recipe calls for 3/4 cup of flour to make a full batch, but you only have 2/3 cup left in the bag, you need to know what fraction of the recipe you can actually make.
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By calculating 2/3 divided by 3/4, you find out you can make 8/9 of the recipe.
If you’re baking a cake, that’s a useful number. You’ll probably just eyeball it and take a little bit of the eggs and sugar out, but at least you know you aren't making a half-batch or a three-quarter batch. You're making nearly the whole thing.
Moving past the "Keep, Change, Flip" mantra
While "Keep, Change, Flip" is a great mnemonic, it’s better to think about the "Multiplicative Inverse."
Every number has a partner that, when multiplied together, equals 1.
For 2, it's 1/2.
For 3/4, it's 4/3.
When we divide by 3/4, we are performing an operation that is the mirror image of multiplying by 3/4.
It's similar to how subtracting 5 is the same as adding negative 5. In the world of fractions, division is just multiplication’s weird reflection. If you can multiply, you can divide. You just have to turn the second number into its inverse first.
Actionable Steps for Solving Fraction Division
If you're stuck on a similar problem, follow this mental checklist:
- Check the sizes: Is the first number bigger or smaller than the second? This tells you if your answer should be greater than or less than 1. In our case, 2/3 < 3/4, so the answer must be less than 1.
- Flip the second, not the first: The divisor (the number you're dividing by) is always the one that gets inverted.
- Multiply across: Don't cross-multiply like you're solving an equation. Just go straight across the top and straight across the bottom.
- Simplify last: Don't worry about making the numbers smaller until the very end.
For the 2/3 divided by 3/4 problem, the steps are clean, the math is solid, and the result of 8/9 is as precise as it gets. You don't need a calculator for this; you just need to remember that the second fraction is the one that needs to do a handstand.
Once you get the hang of visualizing these as physical portions rather than just ink on a page, the "rules" start to feel like common sense. You're just comparing two different ways of slicing the same pie.