Finding Equivalent Expressions for 34×11: Multiplication Methods

Q: Why can't I just add the numbers inside the parentheses first?

You can add inside parentheses first! That's actually the point - $ (30+4) \times (10+1) = 34 \times 11 $. The distributive property shows these expressions are equivalent, not that you must expand everything.

Q: How do I know when to use the distributive property?

Use it when you want to break down larger numbers into easier parts for mental math. For example, $ 34 \times 11 $ is easier as $ (30+4) \times (10+1) $ because you can work with friendlier numbers like 30 and 10.

Q: What's wrong with option C: (30+4)×10+1?

Missing parentheses around (10+1)! Without them, you get $ 34 \times 10 + 1 = 341 $, not $ 34 \times 11 = 374 $. Order of operations matters - multiplication happens before addition.

Q: Can I use this method for any multiplication problem?

Yes! Any number can be broken down using the distributive property. Try $ 23 \times 15 = (20+3) \times (10+5) $. It's especially helpful for mental math with two-digit numbers.

Q: Do I need to expand (30+4)×(10+1) fully?

Not necessarily! The question asks for an equivalent expression, so $ (30+4) \times (10+1) $ is perfect as-is. But if you want the final answer, $ 34 \times 11 = 374 $.

Distributive Property with Two-Factor Decomposition

Which equation is the same as the following?

$34\times11$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the expression that represents the correct factorization of the exercise

00:03 We will use the distributive law

00:06 Let's break down 34 into 30 plus 4

00:10 Let's break down 11 into 10 plus 1

00:13 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Which equation is the same as the following?

$34\times11$

Step-by-step solution

We solve each of the options and keep in mind the order of operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right).

$(30+4)+11=34+11$

$30\times4\times11=120\times11=1,320$

$(30+4)+10+1=34+11$

$(30+4)\times(10+1)=34\times11$

Therefore, the answer is option D.

Final Answer

$(30+4)\times(10+1)$

Key Points to Remember

Essential concepts to master this topic

Distributive Property: $(a+b) \times (c+d) = a \times c + a \times d + b \times c + b \times d$
Decomposition: Break 34 into (30+4) and 11 into (10+1)
Verification: Calculate both expressions: $34 \times 11 = 374$ and $(30+4) \times (10+1) = 374$ ✓

Common Mistakes

Avoid these frequent errors

Confusing multiplication with addition when breaking down numbers
Don't change multiplication to addition like $(30+4)+11 = 45$ ! This completely changes the operation and gives 45 instead of 374. Always keep the multiplication symbol when using distributive property: $(30+4) \times (10+1)$ .

Practice Quiz

Test your knowledge with interactive questions

$$ 140-70= $$

FAQ

Everything you need to know about this question

Why can't I just add the numbers inside the parentheses first?

You can add inside parentheses first! That's actually the point - $(30+4) \times (10+1) = 34 \times 11$ . The distributive property shows these expressions are equivalent, not that you must expand everything.

How do I know when to use the distributive property?

Use it when you want to break down larger numbers into easier parts for mental math. For example, $34 \times 11$ is easier as $(30+4) \times (10+1)$ because you can work with friendlier numbers like 30 and 10.

What's wrong with option C: (30+4)×10+1?

Missing parentheses around (10+1)! Without them, you get $34 \times 10 + 1 = 341$ , not $34 \times 11 = 374$ . Order of operations matters - multiplication happens before addition.

Can I use this method for any multiplication problem?

Yes! Any number can be broken down using the distributive property. Try $23 \times 15 = (20+3) \times (10+5)$ . It's especially helpful for mental math with two-digit numbers.

Do I need to expand (30+4)×(10+1) fully?

Not necessarily! The question asks for an equivalent expression, so $(30+4) \times (10+1)$ is perfect as-is. But if you want the final answer, $34 \times 11 = 374$ .

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