Chapter 1.3 — Exercise 1.1 — Properties

Question 1 — Naming the Property

Each part gives a true statement about rational numbers; the task is to say which property justifies it.

• 8/5 + 0 = 8/5 = 0 + 8/5 — Additive identity.
• 2(3/5 + 1/2) = 2×3/5 + 2×1/2 — Distributive property over addition.
• 3/7 × 1 = 3/7 = 1 × 3/7 — Multiplicative identity.
• −2/5 × 1 = −2/5 = 1 × (−2/5) — Multiplicative identity (the sign of the number doesn't change which property applies).
• 2/5 + 1/3 = 1/3 + 2/5 — Commutative property of addition.
• 5/2 × 3/7 = 15/14 — Closure property under multiplication: the product of two rational numbers is again rational.
• 7a + (−7a) = 0 — Additive inverse law.
• x × 1/x = 1 (x ≠ 0) — Multiplicative inverse law; the condition x ≠ 0 is essential since 0 has no reciprocal.
• 2×x + 2×6 = 2×(x + 6) — Distributive property over addition.

Question 2 — Additive and Multiplicative Inverses

For any rational number p/q, its additive inverse is −p/q (the two sum to zero) and, provided p/q ≠ 0, its multiplicative inverse is q/p (the two multiply to 1).

• −3/5: additive inverse = 3/5  |  multiplicative inverse = −5/3
• 1: additive inverse = −1  |  multiplicative inverse = 1
• 0: additive inverse = 0  |  multiplicative inverse = does not exist
• 7/9: additive inverse = −7/9  |  multiplicative inverse = 9/7
• −1: additive inverse = 1  |  multiplicative inverse = −1

Zero is the one entry with no multiplicative inverse — a detail Question 13 asks you to state explicitly, so it's worth remembering on its own.

Question 3 — Filling the Blanks

Each blank is solved by recognising which property the equation is illustrating, then working out what value makes both sides match.

• (i) −1/17 + _____ = −12/15 + −1/17 — blank = −12/15 (commutative property of addition: the two terms have simply swapped sides).
• (ii) −2/3 + ________ = −2/3 — blank = 0 (additive identity: adding 0 leaves the number unchanged).
• (iii) 1 × _______ = 9/11 — blank = 9/11 (multiplicative identity: multiplying by 1 gives back the same number).
• (iv) (−12 + 5/6) + 6/7 = −12 + (5/6 + ______) — blank = 6/7 (associative property: regrouping the same three numbers doesn't change which values appear).
• (v) _____ × 1/2 + 1/3 = 3/4 × 1/2 + 3/4 × 1/3 — blank = 3/4 (distributive property: the same factor multiplies both terms of the sum).
• (vi) −16/7 + _______ = −16/7 — blank = 0 (additive identity again).

Notice that four of the six properties from the previous lesson — commutative, identity (both kinds), associative and distributive — all show up in this one question, packed into a compact format that checks the full property list at once.

Worked Examples — Questions 4 to 10

Q4. Multiply 2/11 by the reciprocal of −5/14.
The reciprocal of −5/14 is −14/5. So 2/11 × (−14/5) = (2×14)/(11×5) = 28/55, with the negative sign giving −28/55.

Q5. Which properties simplify 2/5 × 5 × 7/6 + 1/3 × 3 × 4/11?
Regrouping each product using the associative law, then applying the multiplicative inverse law (5 × 1/5 = 1, 3 × 1/3 = 1) and the multiplicative identity, collapses this to 2×7/6 + 1×4/11 = 14/6 + 4/11 = (154+24)/66 = 178/66 — and closure under addition guarantees this final sum is rational.

Q6. Verify 5/4 + (−1/2 + −3/2) = (5/4 + −1/2) + −3/2 and name the property used.
Both sides simplify to −3/4 (LHS: 5/4 + (−2) = −3/4; RHS: 3/4 + (−3/2) = −3/4), confirming the associative property of addition.

Q7. Evaluate 3/5 + 7/3 + (−2/5) + (−2/3) after rearrangement.
Grouping the like-denominator terms using the commutative and associative properties: (3/5 + (−2/5)) + (7/3 + (−2/3)) = 1/5 + 5/3 = 3/15 + 25/15 = 28/15.

Q8. Subtract: (i) 3/4 from 1/3, (ii) −32/13 from 1/3, (iii) −7 from −4/7.
(i) 1/3 − 3/4 = (4−9)/12 = −5/12. (ii) 1/3 − (−32/13) = (13+96)/39 = 109/39 — since 1/3 − (−32/13) means adding 32/13, converted to a common denominator of 39. (iii) −4/7 − (−7) = −4/7 + 7 = 45/7.

Q9. What number should be added to −5/8 to get −3/2?
Required number = −3/2 − (−5/8) = −12/8 + 5/8 = −7/8.

Q10. The sum of two rational numbers is 8; one of them is −5/6. Find the other.
Other number = 8 − (−5/6) = 48/6 + 5/6 = 53/6.

Questions 11 to 13 — Conceptual Checks

• Q11 — Is subtraction associative? No. Taking 3/5, 4/7 and 2/3: one grouping gives 73/105, the other gives −67/105 — unequal, so subtraction fails associativity, consistent with what the properties lesson already predicts.
• Q12 — Verify −(−x) = x. For x = 2/15: −x = −2/15, and −(−2/15) = 2/15 = x. For x = −13/27: −x = 13/27, and −(13/27) reversed again gives −13/27 = x. Both cases confirm that negating a negative rational number returns the original value.
• Q13 — Short-answer facts: the set of numbers with no additive identity is the natural numbers (since 0 is excluded from N); the one rational number with no reciprocal is 0; and the reciprocal of a negative rational number is always another negative rational number, never positive.

Reading an Expression for Its Property

The fastest way to recognise a property inside an unlabelled expression is to look at what changed between the two sides. If the same two numbers appear in reversed order, that's commutativity. If brackets have moved but the numbers and operation stayed the same, that's associativity. If a single number now multiplies two separate terms instead of one sum, that's the distributive law. If a number appears added to 0 or multiplied by 1 and comes back unchanged, that's an identity. And if two numbers combine to give exactly 0 or exactly 1, that's an inverse pair. Learning to ask "what changed?" rather than trying to recall the property's definition from memory is what makes Question 1's nine parts go quickly, and it's the same instinct Questions 4 to 12 rely on when the property isn't named for you at all — there, the property is simply the reason a particular simplification step is valid, even though nobody labels it as such along the way, and spotting it is what turns a long calculation into a short, confident one.

Two Places This Reasoning Reappears Soon

The identity, inverse and distributive reasoning practised here resurfaces almost immediately in two places: simplifying algebraic expressions, and solving linear equations in one variable, both later this same year. If any property here still feels shaky, it's worth returning to Properties of Rational Numbers before moving on to Exercise 1.2, which shifts focus entirely to placing rational numbers on the number line.