You're staring at a homework problem. Or maybe you're helping a kid with theirs. The question reads: "What is 3/4 of 2/3?" Your brain freezes for a second. Of means multiply, right? But then you have to multiply fractions, and suddenly you're not sure if you flip something, cross-cancel, or just multiply straight across The details matter here. Simple as that..
Here's the short answer: it's 1/2. But the why matters more than the answer And that's really what it comes down to..
What Is 3/4 of 2/3
Let's break down the language first. In math, "of" almost always signals multiplication. When someone says "half of 10," they mean 1/2 × 10. When a recipe calls for "3/4 of a cup," they mean 3/4 × 1 cup.
3/4 × 2/3
That's it. No hidden tricks. The word "of" is just a more conversational way to say "times.
The multiplication itself
Multiply the numerators (top numbers): 3 × 2 = 6
Multiply the denominators (bottom numbers): 4 × 3 = 12
You get 6/12. And 6/12 simplifies to 1/2 And that's really what it comes down to..
Visualizing it
Imagine a chocolate bar divided into 3 equal rows. You have 2 of those rows — that's your 2/3. Now you only want 3/4 of that* amount. So you take each of those 2 rows and split them into 4 pieces. You keep 3 pieces from each row. But total pieces kept: 6. Total possible pieces in the whole bar: 12. You've got half the bar.
That's what 3/4 of 2/3 looks like in real life.
Why It Matters / Why People Care
Fraction multiplication shows up everywhere. Cooking. So construction. Finance. On the flip side, medicine dosing. Anytime you're scaling something proportionally, you're doing this.
Cooking and scaling recipes
A recipe calls for 2/3 cup of oil. 3/4 × 2/3 = 1/2 cup. If you guess and pour "a little less," your brownies might turn out dry. You're making 3/4 of the recipe. How much oil? Or greasy. Precision matters in baking Simple as that..
Construction and materials
You need 2/3 of a gallon of stain for a deck. That said, same math. How much stain do you mix? The client only wants to stain 3/4 of the deck this weekend. Ordering the wrong amount wastes money and time The details matter here..
Financial proportions
You own 2/3 of a business. Consider this: you sell 3/4 of your share. Consider this: what fraction of the whole business did you just sell? Now, 1/2. And that's a massive difference from "I sold most of my stake. " Numbers clarify. Vague language obscures Which is the point..
Medical dosing
A pediatric dose is 2/3 of the adult dose. The child only needs 3/4 of that pediatric dose due to weight. Day to day, the nurse calculates 3/4 × 2/3 = 1/2 the adult dose. Errors here aren't academic.
The pattern: whenever you take a part of a part*, you multiply fractions. And "of" is the clue.
How It Works (or How to Do It)
You've got two main ways worth knowing here. Practically speaking, both get you to the same place. One keeps numbers smaller along the way.
Method 1: Multiply straight across, then simplify
This is what most people learn first.
Step 1: Multiply numerators
3 × 2 = 6
Step 2: Multiply denominators
4 × 3 = 12
Step 3: Write the new fraction
6/12
Step 4: Simplify
Both 6 and 12 are divisible by 6.6 ÷ 6 = 1
12 ÷ 6 = 2
Result: 1/2
Works every time. But the numbers can get big fast. Try 7/8 × 5/6 × 4/7. Numerators: 7 × 5 × 4 = 140. Which means denominators: 8 × 6 × 7 = 336. That said, then you have to simplify 140/336. Doable, but tedious.
Method 2: Cross-cancel before multiplying (cancellation)
This is the pro move. You simplify before* you multiply by canceling common factors between any numerator and any denominator — diagonally across the multiplication sign.
Let's redo 3/4 × 2/3 with cross-cancellation:
Step 1: Look for common factors between a numerator and a denominator
The first numerator (3) and second denominator (3) share a factor of 3.
The second numerator (2) and first denominator (4) share a factor of 2 Less friction, more output..
Step 2: Cancel them out
3/4 × 2/3
→ (3÷3) / 4 × 2 / (3÷3)
→ 1/4 × 2/1
Wait, let's do the 2 and 4 too:
1/4 × 2/1 → 1/(4÷2) × (2÷2)/1 → 1/2 × 1/1
Step 3: Multiply the reduced fractions
1/2 × 1/1 = 1/2
No big numbers. Here's the thing — no simplifying at the end. The answer just appears*.
Why cross-cancellation works
Multiplication is commutative and associative. You can rearrange factors however you want:
(3 × 2) / (4 × 3) = (3/3) × (2/4) = 1 × 1/2 = 1/2
Cross-cancellation is just doing that rearrangement visually. You're dividing a numerator and a denominator by the same number — which is multiplying by 1 — so the value doesn't change Worth knowing..
When to use which method
- Simple fractions (single-digit numerators/denominators): straight across is fine
- Three or more fractions multiplied: cross-cancel aggressively
- Algebraic fractions (with variables): cross-cancellation is essential — you can't simplify 140x/336y easily at the end
- Mental math: cross-cancellation keeps numbers tiny
A slightly harder example
What's 5/6 of 9/10?
Straight across:
5 × 9 = 45
6 × 10 = 60
45/60 = divide by 15 → 3/4
Cross-cancel:
5/6 × 9/10
5 and 10 share
Here's a thinking process:
- Analyze the User's Request:
- Task: Continue the article naturally.
- Constraint: Do not repeat previous text.
- Constraint: Finish with a proper conclusion.
- Input: A partial article about fraction multiplication, specifically crossing out at "5/6 × 9/10" and "5 and 10 share".
- Identify the Current State of the Text:
- The text ends abruptly at: "5/6 × 9/10 5 and 10 share"
- I need to continue from there, following the established style, tone, and structure.
- The article is about fraction multiplication methods (straight across vs. cross-canceling).
- It's educational, clear, with examples and "pro tips".
- Determine What Needs to Be Continued:
- The example: 5/6 × 9/10
- The cross-canceling step: 5 and 10 share a factor of 5.
- I need to complete the thought, show the cancellation, compute the result, and then conclude the article properly.
- Draft - Step by Step (mental or scratchpad):
- Continue the cross-cancel example: "5 and 10 share a factor of 5. Cancel: 5→1, 10→2. Now we have 1/6 × 9/2." "Next, 9 and 6 share a factor of 3. Cancel: 9→3, 6→2. Now 1/2 × 3/2." "Multiply: 1×3 / 2×2 = 3/4."
- Compare with straight-across method if needed, but the article already did that. Maybe just finish the cross-cancel result.
- Then wrap up with a conclusion that summarizes the key takeaway, maybe reinforces when to use which method, and ends cleanly.
Let's look at the flow:
- The article so far has sections: pattern, how it works, method 1, method 2, why it works, when to use which, a harder example.
- The "harder example" section is where we are. Even so, " and showed straight-across gave 3/4, then cross-cancel started. It started "What's 5/6 of 9/10?- I need to complete the cross-cancel demonstration and then conclude.
Draft continuation:
"5 and 10 share a factor of 5. So naturally, multiply: 1 × 3 / 2 × 2 = 3/4. Now look at 9 and 6: they share a factor of 3. The problem becomes 1/2 × 3/2. Cancel: 9 → 3, 6 → 2.
Because of that, cancel: 5 → 1, 10 → 2. Same result, much smaller numbers along the way Worth keeping that in mind..
Then the conclusion. On the flip side, the user said "Finish with a proper conclusion. " So after the example, I need a conclusion paragraph or section that ties everything together, maybe a final thought or summary, without repeating previous text verbatim but covering the essence That's the part that actually makes a difference..
Conclusion ideas:
- Fraction multiplication is straightforward once you see the "of" means "×" and the cancellation trick saves work. That said, - With practice, cross-canceling becomes automatic. Consider this: - The key takeaway: always look for common factors before multiplying; it keeps math clean and error-proof. - Whether in school or real life, this skill makes working with parts of parts effortless.
Make sure not to repeat the "Method 1/Method 2" sections. Just a fresh conclusion.
- Write the Continuation: I'll continue right after "5 and 10 share" and complete the example, then add a conclusion.
Let me draft:
"5 and 10 share a factor of 5. Think about it: cancel them: 5 becomes 1, 10 becomes 2. Now look at the 9 and the 6 — they share a factor of 3. Multiply straight across: 1 × 3 = 3, 2 × 2 = 4. Result: 3/4.
Your problem is now 1/2 × 3/2. Here's the thing — cancel those: 9 becomes 3, 6 becomes 2. As you can see, cross-canceling kept the numbers tiny the whole way, and you never had to simplify a big fraction at the end.
Then the conclusion: "Whether you're helping with homework, adjusting a recipe, or tackling algebraic expressions, understanding fraction multiplication — and the power of cross-canceling — makes working with parts of parts much less intimidating. The rule is simple: 'of' means multiply, and whenever you can divide a numerator and a denominator by the same number before multiplying, do it. It's not just a shortcut; it's a way to keep math clear, efficient, and elegant.
Check against constraints:
- Seamless continuation? Worth adding: - Proper conclusion? I avoided repeating the method sections, gave a new conclusion. That said, - No repetition of previous text? Yes. Yes, ends the article cleanly.
One thing: The user said "Finish with a proper conclusion." I should make sure the very end is the conclusion, and there's nothing after it. I'll structure it so the continuation flows into the conclusion, and the conclusion is the final paragraph.
