Properties Worksheets | Properties of Mathematics Worksheets - Free Printable

Let's go through each question one by one and solve them step by step, explaining the reasoning behind each answer.

---

1) Which property is used in the following expression?

3(6 + 5) = 18 + 15

- Left side: $3(6 + 5)$ → Multiply 3 by the sum of 6 and 5.
- Right side: $18 + 15$ → This is $3×6 + 3×5$

This shows that multiplication is distributed over addition.

✔ Answer: D. Distributive Property

> Explanation: The distributive property states that $a(b + c) = ab + ac$. Here, $3(6 + 5) = 3×6 + 3×5 = 18 + 15$.

---

2) Which property of addition is used in the following?

(7 + 9) + 5 = 7 + (9 + 5)

- The grouping changes: from $(7 + 9)$ first to $(9 + 5)$ first.
- The order of numbers stays the same; only the grouping changes.

✔ Answer: C. Associative Property

> Explanation: The associative property of addition says $(a + b) + c = a + (b + c)$. It deals with how numbers are grouped.

---

3) Which is an example of Identity Property of Addition?

- Identity Property of Addition: $a + 0 = a$

Let’s check each option:

A. $9 + 7 = 7 + 9$ → Commutative
B. $(7 + 8) + 6 = 7 + (8 + 6)$ → Associative
C. $4 + 0 = 4$ → Yes! Adding zero doesn’t change the number
D. $2 × 1 = 2$ → Identity Property of Multiplication

✔ Answer: C. 4 + 0 = 4

---

4) Which of the following does NOT show the Commutative Property of Addition?

Commutative Property: $a + b = b + a$ — order changes, but result is same.

A. $a + b = b + a$ → YES, this is the definition
B. $3 + x = x + 3$ → YES, same idea
C. $ab = ba$ → This is commutative property of multiplication, not addition
D. $3x + 4y = 4y + 3x$ → YES, terms are reordered

So, C is about multiplication, not addition.

✔ Answer: C. ab = ba

---

5) Which operation will NOT change the value of any nonzero number?

Let’s evaluate:

A. Dividing by Zero → Undefined! Not allowed
B. Multiplying by One → $a × 1 = a$ → No change
C. Adding One → $a + 1 ≠ a$ → Changes value
D. Multiplying by Zero → $a × 0 = 0$ → Changes value

Only multiplying by one keeps the number unchanged.

✔ Answer: B. Multiplying by One

---

6) Which Property of Addition does 3 + 0 = 3 illustrate?

- Adding zero to a number gives the same number.

This is the Identity Property of Addition.

✔ Answer: B. Identity Property

---

7) Which of the following does NOT show the Commutative Property?

Commutative Property: $a + b = b + a$ or $a × b = b × a$

A. $xy - 9 = xy$ → This is subtraction and simplification. No commutativity shown
B. $yx = xy$ → Yes, multiplication commutes
C. $x + y = y + x$ → Yes, addition commutes
D. $3 + y = y + 3$ → Yes, same as above

But A is not showing commutativity — it’s just saying $xy - 9 = xy$, which implies $-9 = 0$, which is false.

Wait — actually, let's reevaluate: Is A even a valid statement?

$xy - 9 = xy$ → Subtracting 9 from $xy$ equals $xy$? That would mean $-9 = 0$, which is false.

So likely, this is a trick — it's not an example of any property at all.

But the question asks: Which does NOT show the commutative property?

Even if A is false, it's not showing commutativity — so it's correct to pick A.

But let’s be careful: Commutative property involves switching order.

In A: $xy - 9 = xy$ — no order switch, and it's incorrect.

But B: $yx = xy$ — this does show commutative property of multiplication.

C: $x + y = y + x$ — yes

D: $3 + y = y + 3$ — yes

Only A does not show commutative property.

✔ Answer: A. xy - 9 = xy

> Note: Even though it's mathematically false, the question is asking which does not show the commutative property — and A clearly doesn't.

---

8) Which property is used in the following expression?

(4 × 9) × 8 = 9 × (8 × 4)

Let’s analyze:

Left: $(4 × 9) × 8 = 36 × 8 = 288$
Right: $9 × (8 × 4) = 9 × 32 = 288$

But what properties are involved?

We can see:
- Numbers are rearranged: 4, 9, 8 → 9, 8, 4
- Grouping changed

This uses both commutative and associative properties.

But let’s look at the options:

A. Associative Property of Addition → Incorrect (multiplication here)
B. Associative Property of Multiplication → Involves grouping: $(a×b)×c = a×(b×c)$ — but here, order also changed
C. Distributive Property → No distribution (no addition)
D. Commutative Property of Addition → Again, wrong operation

Wait — none seem perfect?

But notice: Associative Property of Multiplication allows regrouping: $(a×b)×c = a×(b×c)$

But here, we have reordering too: $4×9×8$ becomes $9×8×4$

That requires commutative property.

But the options don’t list "both" — so maybe the best choice?

Let’s rewrite:

$(4 × 9) × 8 = 9 × (8 × 4)$

We can think of it as:
- Use commutative: $4×9 = 9×4$
- Then: $(9×4)×8 = 9×(4×8)$ → associative
- But right side is $9×(8×4)$ — same since $4×8 = 8×4$

So multiple steps.

But the key point: The expression shows both commutative and associative properties, but none of the choices say that.

Look again at options:

A. Associative Property of Addition → ✘
B. Associative Property of Multiplication → ✔ Partially true
C. Distributive → ✘
D. Commutative Property of Addition → ✘

But B is closest — because even though order changes, associative property allows regrouping, and commutative allows swapping.

But the expression has both.

However, if only one must be chosen, then Associative Property of Multiplication is the only plausible one among the choices, because the parentheses are being moved.

But wait — the numbers are reordered.

For example: $(4×9)×8 = 9×(8×4)$

We can write:
- Left: $4×9×8$
- Right: $9×8×4$

Same numbers, different order and grouping.

So it's using commutative and associative together.

But the options don’t include that.

Now, check if any option fits perfectly.

Actually, Associative Property of Multiplication only applies to regrouping, not changing order.

So if the expression were $(4×9)×8 = 4×(9×8)$, that would be associative.

But here, it’s $(4×9)×8 = 9×(8×4)$ — order changed.

So it's not purely associative.

But commutative is not listed for multiplication.

Wait — option B is Associative Property of Multiplication — but that doesn’t explain the order swap.

Hmm.

Alternatively, perhaps the problem expects us to recognize that the associative property is involved in the regrouping, and the commutative allows the order change.

But since commutative property of multiplication is not listed, and associative is, and the expression involves changing grouping, maybe they expect B.

But let’s see: Can we get from $(4×9)×8$ to $9×(8×4)$ using only associative?

No — because associative doesn’t allow you to swap order.

So unless we use commutative first, we can’t.

But commutative property of multiplication is not an option.

Wait — is there a mistake?

Let’s double-check the expression:

(4 × 9) × 8 = 9 × (8 × 4)

Is this true?

Left: $36 × 8 = 288$
Right: $9 × (32) = 288$ → Yes, equal.

But the property used is both commutative and associative.

But among the choices:

A. Associative Addition → ✘
B. Associative Multiplication → ✔ (partially)
C. Distributive → ✘
D. Commutative Addition → ✘

None are perfect.

But maybe the intended answer is B, assuming they’re focusing on the grouping aspect.

Alternatively, could it be D? No — it’s multiplication.

Wait — perhaps the expression is meant to show associative property, but it’s written incorrectly?

Wait — let’s try to rewrite:

$(4 × 9) × 8 = 9 × (8 × 4)$

We can do:

- Use commutative: $4×9 = 9×4$
- So left: $(9×4) × 8 = 9×(4×8)$ → associative
- But right is $9×(8×4)$ → same as $9×(4×8)$ due to commutative

So overall, both properties are used.

But since commutative property of multiplication isn't listed, and associative is, and the expression involves changing grouping, maybe B is the expected answer.

But technically, it's not purely associative.

Wait — is there a better way?

Actually, option B is Associative Property of Multiplication, which is: $(a×b)×c = a×(b×c)$

But here, the right side is $9×(8×4)$, which is not the same as $4×(9×8)$, etc.

So unless we accept that order is changed, it's not pure associative.

But in reality, this expression demonstrates the combination of commutative and associative, but since commutative of multiplication isn't an option, and associative is, and the problem may expect that, we might go with B.

But let’s look at the next questions.

Alternatively, maybe the expression is meant to be:

(4 × 9) × 8 = 4 × (9 × 8) — that would be associative.

But it’s not.

Wait — could it be a typo? Or perhaps we're missing something.

Another thought: Maybe they consider reordering as part of associative?

No — associative is about grouping, not order.

So perhaps the intended answer is B, but it's not fully accurate.

But let’s move on — maybe we’ll find a pattern.

Wait — look at question 9 and 10 — maybe we can infer.

But let’s try to think differently.

Maybe the expression is showing commutative of multiplication, but that’s not listed.

Options are:

A. Associative Addition → ✘
B. Associative Multiplication → ✔ (possible)
C. Distributive → ✘
D. Commutative Addition → ✘

So only B is remotely possible.

And since the expression involves regrouping, and multiplication, and the values are equal, perhaps they want B.

✔ Answer: B. Associative Property of Multiplication

> Even though it's not perfect, given the choices, B is the most appropriate.

---

9) Which property is used in the following?

3 × (6 + 8) = 3 × 6 + 3 × 8

This is classic: multiplication distributed over addition.

✔ Answer: B. Distributive Property

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

---

10) Which Property of Multiplication is shown?

(5 + 8) × 7 = 5 × 7 + 8 × 7

Again, same as above: distribute multiplication over addition.

✔ Answer: C. Distributive Property

---

✔ Final Answers:

| Question | Answer |
|--------|--------|
| 1 | D. Distributive Property |
| 2 | C. Associative Property |
| 3 | C. 4 + 0 = 4 |
| 4 | C. ab = ba |
| 5 | B. Multiplying by One |
| 6 | B. Identity Property |
| 7 | A. xy - 9 = xy |
| 8 | B. Associative Property of Multiplication |
| 9 | B. Distributive Property |
| 10 | C. Distributive Property |

---

🔍 Summary of Key Properties:

- Commutative: Order doesn't matter: $a + b = b + a$, $a × b = b × a$
- Associative: Grouping doesn't matter: $(a + b) + c = a + (b + c)$, $(a × b) × c = a × (b × c)$
- Identity: $a + 0 = a$, $a × 1 = a$
- Distributive: $a(b + c) = ab + ac$

---

Let me know if you'd like this printed out or formatted as a worksheet!