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.

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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, $3(6 + 5) = 3×6 + 3×5 = 18 + 15$.

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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.

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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

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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

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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

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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

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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, if $xy - 9 = xy$, then subtracting $xy$ from both sides gives $-9 = 0$, which is not true. So this equation is false, not a valid example.

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

So even if A is false, it's still not demonstrating commutativity. But let's see: the others do show commutativity.

A is not an example of commutativity at all — it's a false statement involving subtraction.

✔ Answer: A. xy - 9 = xy

> Note: While this equation is invalid, it certainly does not demonstrate the commutative property.

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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 we need to identify the property.

Let’s rearrange:
$(4 × 9) × 8 = 4 × 9 × 8$
And $9 × (8 × 4) = 9 × 8 × 4$

So the numbers are being reordered and regrouped.

But look:
- Order changed: $4×9×8$ vs $9×8×4$
- Grouping changed

This involves both commutative and associative properties.

But the options are:

A. Associative Property of Addition → ✘ (we’re dealing with multiplication)

B. Associative Property of Multiplication → This allows regrouping: $(a×b)×c = a×(b×c)$

But here, order also changes — so it's more than just associative.

But wait — the associative property alone doesn't allow reordering — only regrouping.

To reorder, you need commutative property.

But the expression is:

$(4 × 9) × 8 = 9 × (8 × 4)$

We can rewrite this using commutative and associative properties.

But none of the options say "both" — so we must pick the best match.

Let’s test:

Can we get from left to right using Associative Property of Multiplication?

No — because associative only changes grouping, not order.

For example:
$(4×9)×8 = 4×(9×8)$ — that’s associative.

But to get $9×(8×4)$, we need to swap 4 and 9, and 4 and 8 — that requires commutative.

So neither associative nor commutative alone explains the full transformation.

But wait — perhaps the intended answer is Associative Property of Multiplication?

No — that’s incorrect.

Let’s look again.

Actually, the expression is:
$$
(4 × 9) × 8 = 9 × (8 × 4)
$$

Let’s simplify both sides:

Left: $36 × 8 = 288$

Right: $9 × (32) = 288$

Now, is this associative? No — because the order of numbers changed.

Is it commutative? Well, commutative is $a×b = b×a$, but here multiple operations.

But notice: both sides have the same numbers: 4, 9, 8 — just reordered.

So this is an example of commutative property of multiplication, applied multiple times.

But the options don’t include “Commutative Property of Multiplication” — they have:

A. Associative Property of Addition
B. Associative Property of Multiplication
C. Distributive Property of Multiplication
D. Commutative Property of Addition

Wait — none of these are exactly correct?

Hold on — Option D is "Commutative Property of Addition", but we’re multiplying.

So what’s going on?

But option B is "Associative Property of Multiplication", which allows regrouping but not reordering.

But this expression involves reordering, so it’s not purely associative.

But maybe the problem expects us to recognize that the associative property allows rearrangement when combined with commutative, but that’s not standard.

Alternatively, let’s try to transform:

Start: $(4 × 9) × 8$

Use commutative: $4 × 9 = 9 × 4$, so $(9 × 4) × 8 = 9 × (4 × 8)$ — now use associative

Then $9 × (8 × 4)$ — yes, because $4 × 8 = 8 × 4$ by commutative

So it takes both commutative and associative.

But since only one choice is allowed, and no option says "Commutative Property of Multiplication", perhaps there's a mistake.

Wait — option B is "Associative Property of Multiplication", but that’s not sufficient.

But look at the answer choices:

A. Associative Property of Addition
B. Associative Property of Multiplication
C. Distributive Property of Multiplication
D. Commutative Property of Addition

None of them are Commutative Property of Multiplication.

So the only possible way this makes sense is if the problem has a typo.

But wait — maybe the expression is meant to be $(4 × 9) × 8 = 4 × (9 × 8)$ — that would be associative.

But it's written as $= 9 × (8 × 4)$

That’s different.

Unless we consider that commutative property is implied, but it's not listed.

Wait — option D is "Commutative Property of Addition" — but we’re multiplying.

So no good.

But perhaps the intended answer is B. Associative Property of Multiplication — but that’s incorrect.

Alternatively, maybe the question wants us to see that multiplication is associative, and the equality holds due to associativity, but the order change is confusing.

But let’s think: is there a way to interpret this as associative?

No — because $ (4×9)×8 = 36×8 = 288 $

$ 9×(8×4) = 9×32 = 288 $

But $ (4×9)×8 \neq 4×(9×8) $? Wait — $4×(9×8) = 4×72 = 288$, same.

So $ (4×9)×8 = 4×(9×8) $ — that’s associative.

But $9×(8×4)$ is $9×32 = 288$, same value.

But to get from $ (4×9)×8 $ to $ 9×(8×4) $, we need to commute the factors.

So unless the expression is meant to be $ (4×9)×8 = 4×(9×8) $, it's not purely associative.

But given the choices, maybe the intended answer is B, assuming the student is supposed to recognize that multiplication is associative, and the expression is equivalent.

But that’s not accurate.

Wait — let’s look at question 9 and 10 — maybe we’ll find a pattern.

Alternatively, perhaps the expression is meant to show commutative, but it’s not listed.

But option D is "Commutative Property of Addition" — wrong.

So this seems like a flawed question.

But let’s suppose the intended answer is B. Associative Property of Multiplication, even though it’s not quite right.

But that’s not correct.

Wait — perhaps the expression is:

$$
(4 × 9) × 8 = 9 × (8 × 4)
$$

Let’s write both sides in terms of factors:

Left: $4×9×8$

Right: $9×8×4$

Same numbers, different order.

So it’s commutative property of multiplication, but not listed.

So the only possibility is that the correct answer is not among the choices.

But since it's a multiple-choice, and likely designed for students, perhaps the intended answer is B, but that’s incorrect.

Alternatively, maybe the question is misprinted.

But let’s move on and come back.

Wait — maybe I made a mistake.

Another thought: could this be associative if we rearrange?

No.

Perhaps the question is testing whether students know that multiplication is commutative and associative, and the expression uses both.

But since only one choice is allowed, and no choice says "commutative of multiplication", it's problematic.

But let’s look at the next questions.

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9) Which property is used in the following?

3 × (6 + 8) = 3 × 6 + 3 × 8

This is clearly: multiply 3 by the sum of 6 and 8 = 3×6 + 3×8

This is the distributive property.

✔ Answer: B. Distributive Property

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10) Which Property of Multiplication is shown?

(5 + 8) × 7 = 5 × 7 + 8 × 7

Again, distributing 7 over the sum: $a(b + c) = ab + ac$

This is distributive property.

✔ Answer: C. Distributive Property

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Now, going back to Question 8:

$$
(4 × 9) × 8 = 9 × (8 × 4)
$$

Let’s re-express:

Left: $ (4×9)×8 = 36×8 = 288 $

Right: $9×(8×4) = 9×32 = 288$

But is this associative? Only if the grouping changes without changing order.

But here, the order changes: 4,9,8 → 9,8,4

So it’s commutative.

But commutative property of multiplication is not listed.

The closest is D. Commutative Property of Addition — but that’s for addition.

So none of the options are correct.

But perhaps the intended answer is B. Associative Property of Multiplication, but that’s wrong.

Wait — maybe the expression is meant to be:

$$
(4 × 9) × 8 = 4 × (9 × 8)
$$

Then it would be associative.

But it’s written as $= 9 × (8 × 4)$

So unless there’s a typo, it’s not associative.

But maybe the question is testing whether students realize that multiplication is both associative and commutative, and thus such equalities hold.

But again, the options are poor.

Given that, and since option B is the only one related to multiplication, and associative property allows regrouping, and commutative allows reordering, but associative alone doesn’t explain the reordering, it’s still not right.

But perhaps the intended answer is B, assuming the student sees that multiplication is associative and commutative.

But strictly speaking, this is not associative.

Alternatively, maybe the answer is none of the above, but that’s not an option.

Wait — option A is Associative Property of Addition — irrelevant.

C. Distributive — no.

D. Commutative Property of Addition — no.

So no correct choice.

But let’s assume a typo: perhaps the expression was meant to be:

$$
(4 × 9) × 8 = 4 × (9 × 8)
$$

Then it would be associative property of multiplication.

Given that, and since this is common in worksheets, likely the intended answer is B.

✔ Answer: B. Associative Property of Multiplication (assuming typo or oversight)

But note: this is not strictly correct — it should be commutative.

But based on available choices, B is the most plausible.

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Final Answers:

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 *(with caveat)*
9. B. Distributive Property
10. C. Distributive Property

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Summary of Answers:

| Q# | Answer |
|----|--------|
| 1 | D |
| 2 | C |
| 3 | C |
| 4 | C |
| 5 | B |
| 6 | B |
| 7 | A |
| 8 | B |
| 9 | B |
| 10 | C |

✔ Note: Question 8 may have an error in the expression or answer choices. If the expression were $(4×9)×8 = 4×(9×8)$, then B would be correct. As written, it involves commutative property, which isn't listed. But given the options, B is the best fit.

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