Properties Worksheets - Free Printable

It looks like you've uploaded an image of a math worksheet titled "Applying Properties of Operations", which asks students to apply the distributive property to produce equivalent expressions. The worksheet includes 20 problems, and the answer key is already provided on the right side.

Let me go through the concept and verify that the answers are correct by explaining how the distributive property works and showing how each answer was derived.

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🔷 What is the Distributive Property?

The distributive property states:

> $ a(b + c) = ab + ac $

This means you multiply the term outside the parentheses by each term inside.

Also, sometimes the reverse is used:
> $ ab + ac = a(b + c) $

This is called factoring — pulling out the greatest common factor (GCF).

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Now let's go through the problems and verify the answers.

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✔ Problem-by-Problem Explanation

1. 3(10a + 5)
→ $ 3 \times 10a = 30a $, $ 3 \times 5 = 15 $
→ 30a + 15 ✔️

2. 2(9 + 2g)
→ $ 2 \times 9 = 18 $, $ 2 \times 2g = 4g $
→ 18 + 4g ✔️

3. 4(2b + 3)
→ $ 4 \times 2b = 8b $, $ 4 \times 3 = 12 $
→ 8b + 12 ✔️

4. 40d + 20
This is not in factored form. We need to factor it:
GCF of 40d and 20 is 20:
→ $ 20(2d + 1) $ ✔️

5. 56s + 63
GCF of 56 and 63 is 7:
→ $ 7(8s + 9) $ ✔️

6. 60 + 20w
GCF of 60 and 20 is 20:
→ $ 20(3 + 1w) $ → usually written as $ 20(3 + w) $, but $ 1w $ is acceptable
→ 20(3 + 1w) ✔️

7. 6(6k + 2)
→ $ 6 \times 6k = 36k $, $ 6 \times 2 = 12 $
→ 36k + 12 ✔️

8. p + p + p + p + p + p + p
That’s 7 p's → $ 7p $? Wait...
But the answer says 8p ✘

Let’s count:
p + p + p + p + p + p + p → that’s 7 terms

So correct answer should be 7p

But the answer key says 8p → ✘ Incorrect

🚨 Error in Answer Key at #8

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9. 6(8 + 2y)
→ $ 6 \times 8 = 48 $, $ 6 \times 2y = 12y $
→ 48 + 12y ✔️

10. 3(3 + 8t)
→ $ 3 \times 3 = 9 $, $ 3 \times 8t = 24t $
→ 9 + 24t ✔️

11. f + f + f + f
→ 4f → 4f ✔️

12. 9(4 + 9n)
→ $ 9 \times 4 = 36 $, $ 9 \times 9n = 81n $
→ 36 + 81n ✔️

13. 2(8 + 3e)
→ $ 2 \times 8 = 16 $, $ 2 \times 3e = 6e $
→ 16 + 6e ✔️

14. 3(4 + 4r)
→ $ 3 \times 4 = 12 $, $ 3 \times 4r = 12r $
→ 12 + 12r ✔️

15. 90c + 40
GCF of 90 and 40 is 10:
→ $ 10(9c + 4) $ ✔️

16. 10(6u + 7)
→ $ 10 \times 6u = 60u $, $ 10 \times 7 = 70 $
→ 60u + 70 ✔️

17. 72j + 72
GCF is 72:
→ $ 72(j + 1) $ ✔️

18. 56 + 21h
GCF of 56 and 21 is 7:
→ $ 7(8 + 3h) $ ✔️

19. 70 + 63z
GCF of 70 and 63 is 7:
→ $ 7(10 + 9z) $ ✔️

20. 9 + 18v
GCF is 9:
→ $ 9(1 + 2v) $ ✔️

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✘ Error Found!

As noted earlier:

> #8: p + p + p + p + p + p + p = 7p, not 8p

But the answer key says 8p → this is wrong

There are seven p's added together.

So the correct answer should be 7p

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✔ Summary

| Problem | Correct Answer | Answer Key Says | Correct? |
|--------|----------------|------------------|----------|
| 1 | 30a + 15 | 30a + 15 | ✔️ |
| 2 | 18 + 4g | 18 + 4g | ✔️ |
| 3 | 8b + 12 | 8b + 12 | ✔️ |
| 4 | 20(2d + 1) | 20(2d + 1) | ✔️ |
| 5 | 7(8s + 9) | 7(8s + 9) | ✔️ |
| 6 | 20(3 + w) | 20(3 + 1w) | ✔️ (equivalent) |
| 7 | 36k + 12 | 36k + 12 | ✔️ |
| 8 | 7p | 8p | ✘ ERROR |
| 9 | 48 + 12y | 48 + 12y | ✔️ |
| 10 | 9 + 24t | 9 + 24t | ✔️ |
| 11 | 4f | 4f | ✔️ |
| 12 | 36 + 81n | 36 + 81n | ✔️ |
| 13 | 16 + 6e | 16 + 6e | ✔️ |
| 14 | 12 + 12r | 12 + 12r | ✔️ |
| 15 | 10(9c + 4) | 10(9c + 4) | ✔️ |
| 16 | 60u + 70 | 60u + 70 | ✔️ |
| 17 | 72(j + 1) | 72(j + 1) | ✔️ |
| 18 | 7(8 + 3h) | 7(8 + 3h) | ✔️ |
| 19 | 7(10 + 9z) | 7(10 + 9z) | ✔️ |
| 20 | 9(1 + 2v) | 9(1 + 2v) | ✔️ |

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✔ Final Answer:

✔ All answers are correct except for #8.

> Problem 8: $ p + p + p + p + p + p + p = 7p $
> ✘ Answer key says 8p → This is incorrect

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📝 Recommendation:

If this is a student worksheet, correct problem #8 to reflect 7p instead of 8p.

Let me know if you'd like a corrected version of the worksheet!