Properties of Real Numbers Worksheets - 15 Worksheets Library - Free Printable

Let's solve each exercise by identifying the property of real numbers being used in each equation. Here's a breakdown of each one with explanations.

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1. (a)(d) = (d)(a)

✔ Commutative Property of Multiplication
> This property states that the order of multiplication does not affect the product:
> $ a \times b = b \times a $

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2. (8+2)+4 = 8+(2+4)

✔ Associative Property of Addition
> This property states that grouping of numbers in addition doesn't change the sum:
> $ (a + b) + c = a + (b + c) $

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3. 8(1) = 8

✔ Multiplicative Identity Property
> Multiplying any number by 1 gives the same number:
> $ a \times 1 = a $

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4. 42 – 6a = 6(7 – a)

✔ Distributive Property
> This is factoring out a common factor. We rewrite:
> $ 42 - 6a = 6 \times 7 - 6 \times a = 6(7 - a) $
> So, it uses the distributive property in reverse:
> $ a(b + c) = ab + ac $ → here, $ 6(7 - a) = 6 \cdot 7 - 6 \cdot a $

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5. f + 0 = f

✔ Additive Identity Property
> Adding zero to any number leaves it unchanged:
> $ a + 0 = a $

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6. –14 + 14 = 0

✔ Additive Inverse Property
> A number added to its opposite equals zero:
> $ a + (-a) = 0 $

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7. 3a + 2b = 2b + 3a

✔ Commutative Property of Addition
> The order of adding terms doesn't matter:
> $ a + b = b + a $

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8. 5(7c) = (5•7)c

✔ Associative Property of Multiplication
> Grouping of multiplication doesn’t change the product:
> $ a(bc) = (ab)c $

> Note: Sometimes this is confused with distributive, but since it's all multiplication and only grouping changes, it’s associative.

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9. 36 + 48 + 14

This is just an expression — no equality or operation shown. But likely, the intent is to recognize that we can reorder or regroup the addends.

But since there’s no equation, perhaps it’s meant to imply commutative or associative when simplifying.

However, as written, no property is demonstrated unless we assume it's being simplified via reordering.

But if we interpret it as just listing numbers to be added, then:

➡️ If we rearrange: $ 36 + 14 + 48 $, that would be commutative
➡️ If we group: $ (36 + 48) + 14 $, that would be associative

But since no equality or transformation is shown, this might be a typo or incomplete.

👉 Best interpretation: Since it's just an expression, maybe it's testing recognition of addition and order, so possibly Commutative Property of Addition (if rearranged).

But without a transformation, it's ambiguous.

Alternatively, if the original problem had something like:
> $ 36 + 48 + 14 = 48 + 36 + 14 $
Then it would be commutative.

Since it's not given, we’ll assume it's a mistake or missing info.

But based on context, likely intended to show commutative or associative.

But let’s say:
✔ Commutative Property of Addition *(assuming reordering)*

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10. xy + 0 = xy

✔ Additive Identity Property
> Adding zero doesn't change the value:
> $ a + 0 = a $

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11. 99 + 0 = 99

✔ Additive Identity Property
> Same as above: $ a + 0 = a $

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12. 4 + 5 + 6 = 6 + 4 + 5

✔ Commutative Property of Addition
> Order of addition doesn’t matter:
> $ a + b + c = c + a + b $

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✔ Final Answers:

| N° | Exercise | Property |
|----|----------------------------------|----------------------------------------|
| 1 | (a)(d)=(d)(a) | Commutative Property of Multiplication |
| 2 | (8+2)+4 = 8+(2+4) | Associative Property of Addition |
| 3 | 8(1) = 8 | Multiplicative Identity |
| 4 | 42 – 6a = 6(7 – a) | Distributive Property |
| 5 | f + 0 = f | Additive Identity |
| 6 | –14 + 14 = 0 | Additive Inverse |
| 7 | 3a + 2b = 2b + 3a | Commutative Property of Addition |
| 8 | 5(7c) = (5•7)c | Associative Property of Multiplication |
| 9 | 36 + 48 + 14 | Commutative Property of Addition |
| 10 | xy + 0 = xy | Additive Identity |
| 11 | 99 + 0 = 99 | Additive Identity |
| 12 | 4 + 5 + 6 = 6 + 4 + 5 | Commutative Property of Addition |

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🔍 Notes:

- Commutative: Order changes
- Associative: Grouping changes
- Identity: Adding 0 or multiplying by 1
- Inverse: Number + opposite = 0
- Distributive: $ a(b + c) = ab + ac $

Let me know if you'd like this formatted as a printable answer sheet! 😊