Exercise 1.1 — Properties

What Does Exercise 1.1 Cover?

Exercise 1.1 from Chapter 1 — Rational Numbers puts the properties you studied in theory directly to the test. Questions range from identifying which property is being used, to computing answers by applying inverses, identities, and the distributive law smartly. This exercise is frequently assessed in Class 8 board exams across CBSE, Telangana, and Andhra Pradesh. Working through it carefully will sharpen both your speed and accuracy with rational number operations.

Identifying Properties — Question 1

The first question asks you to name the property used in each statement. Here is a clear guide to the answers and the reasoning behind them:

• 8/5 + 0 = 8/5 = 0 + 8/5 — Additive Identity. Adding zero to any rational number leaves it unchanged.
• 2(3/5 + 1/2) = 2×3/5 + 2×1/2 — Distributive Property of multiplication over addition.
• 3/7 × 1 = 3/7 = 1 × 3/7 — Multiplicative Identity. Multiplying by 1 leaves the number unchanged.
• −2/5 × 1 = −2/5 — Multiplicative Identity (applies to negative rational numbers too).
• 2/5 + 1/3 = 1/3 + 2/5 — Commutative Property of addition.
• 5/2 × 3/7 = 15/14 — Closure Property under multiplication (the product is still a rational number).
• 7a + (−7a) = 0 — Additive Inverse Law.
• x × 1/x = 1 (x ≠ 0) — Multiplicative Inverse Law.
• 2×x + 2×6 = 2×(x + 6) — Distributive Property of multiplication over addition.

Additive and Multiplicative Inverses — Question 2

For any rational number p/q, its additive inverse is −p/q (they sum to zero), and its multiplicative inverse (reciprocal) is q/p (they multiply to 1). Here are the key results from this question:

• −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 (you cannot divide by zero)
• 7/9: Additive inverse = −7/9  |  Multiplicative inverse = 9/7
• −1: Additive inverse = 1  |  Multiplicative inverse = −1

Worked Examples — Selected Questions

Q4. Multiply 2/11 by the reciprocal of −5/14.
The reciprocal of −5/14 is −14/5. Multiplying: 2/11 × (−14/5) = −28/55.

Q7. Evaluate 3/5 + 7/3 + (−2/5) + (−2/3) after rearrangement.
The smart move here is to group like-denominator terms first using the commutative and associative properties: (3/5 + (−2/5)) + (7/3 + (−2/3)) = 1/5 + 5/3 = 3/15 + 25/15 = 28/15.

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. Sum of two rational numbers is 8. One number is −5/6. Find the other.
Other number = 8 − (−5/6) = 48/6 + 5/6 = 53/6.

Key Conceptual Questions — Q11 to Q13

• Q11 — Is subtraction associative? No. Using 3/5, 4/7, 2/3 as an example: grouping them differently gives 73/105 on one side and −67/105 on the other — they are not equal, so subtraction is not associative in rational numbers.
• Q12 — Verify −(−x) = x. Taking the negative of a negative rational number brings you back to the original. For x = 2/15: −x = −2/15, and −(−2/15) = 2/15 = x. This holds for all rational numbers.
• Q13 — Conceptual answers: The set of numbers with no additive identity is the set of natural numbers (since 0 is not a natural number). The rational number with no reciprocal is 0. The reciprocal of a negative rational number is always another negative rational number.

What This Exercise Prepares You For

Mastering Exercise 1.1 builds the algebraic thinking needed across the entire Class 8 syllabus. The same properties — identity, inverse, distributive, and associative laws — appear directly in algebraic expressions and linear equations. For a deeper look at the theory behind these questions, revisit the Properties of Rational Numbers lesson and the Introduction to Rational Numbers.