Vardaan
Class 7 Maths • Chapter 09

Rational Numbers

Vardaan Learning Institute • Detailed Notes with Practice Questions

🔢 1. What is a Rational Number?

A rational number is any number that can be written in the form p/q, where p and q are integers and q ≠ 0.

Rational Numbers (p/q) Positive (p>0, q>0) Negative (one of p,q <0) Zero (0/1, 0/5...) Integers (...−2,−1,0,1,2) Non-integers (1/2, 3/4) Natural Numbers 1, 2, 3, 4… Whole Numbers 0, 1, 2, 3… Key Inclusions: • Every integer is a rational number • Natural Numbers ⊂ Whole Numbers • Whole Numbers ⊂ Integers • Integers ⊂ Rational Numbers • -7 = -7/1, which is p/q form ✓ • 0 = 0/1 = 0/5 → rational ✓

⚖️ 2. Equivalent Rational Numbers

Multiply or divide BOTH numerator and denominator by the same non-zero integer to get an equivalent rational number.

Example: 2/3 = 4/6 = 6/9 = 8/12 → all equivalent!

Standard form: A rational number is in standard/lowest form when:
  1. The denominator is positive
  2. There is no common factor between numerator and denominator (other than 1)
-6/9 → HCF(6,9)=3, divide both: -2/3 → denominator positive → standard form = -2/3

📏 3. Rational Numbers on the Number Line

-2 -1 -½ 0 ⅓ ⅔ 1 2 3/2 ← negative rationals | positive rationals →

➕ 4. Operations on Rational Numbers

Operation Same Denominator Different Denominator
Addition/Subtraction Add/subtract numerators, keep denomination: a/q ± b/q = (a±b)/q Find LCM of denominators, convert to equivalent fractions, then add/subtract
Multiplication p/q × r/s = pr/qs (multiply numerators and denominators separately)
Division p/q ÷ r/s = p/q × s/r (multiply by reciprocal: "KCF" Keep, Change, Flip)
Comparing Direct comparison of numerators Cross multiply: a/b vs c/d → compare a×d with b×c
Worked Examples:
• 2/3 + (-5/6) = 4/6 + (-5/6) = -1/6 (LCM=6)
• 3/4 × (-8/9) = -24/36 = -2/3 (simplify)
• (-7/5) ÷ (14/15) = (-7/5) × (15/14) = -105/70 = -3/2
• Which is greater: -3/4 or -5/6? → Cross multiply: (-3)×6 = -18, (-5)×4 = -20. Since -18 > -20, so -3/4 > -5/6

📋 5. Properties of Rational Numbers

Property Addition Multiplication
Closure Sum of any 2 rationals = rational ✅ Product of any 2 rationals = rational ✅
Commutative a + b = b + a ✅ a × b = b × a ✅
Associative (a+b)+c = a+(b+c) ✅ (a×b)×c = a×(b×c) ✅
Identity a + 0 = a (additive identity = 0) a × 1 = a (multiplicative identity = 1)
Inverse Additive inverse of a/b = -a/b; sum = 0 Multiplicative inverse (reciprocal) of a/b = b/a; product = 1
Distributive a × (b + c) = a×b + a×c ✅
🧠 Infinite Rational Numbers Between Two Rationals! Between any two rational numbers, there are infinitely many rational numbers!
Finding rational numbers between a/b and c/d: Convert to same denominator, then find numbers with numerators between them. Or use mean: midpoint = (a/b + c/d) / 2 is always between them.
✏️ Practice Questions — Rational Numbers (30 Questions)

Section A — Basics Easy

Q1. Is −7 a rational number? Write it in p/q form.
Q2. Reduce to standard form: −18/27
Q3. Find 3 equivalent rational numbers for 2/5
Q4. Write additive inverse of 5/7
Q5. Write reciprocal of -3/8
Q6. Which is greater: -3/5 or -4/7?
Q7. 2/3 + (-5/9) = ?
Q8. (3/4) × (-8/15) = ?

Section B — Operations Medium

Q9. (-7/12) + (5/8)
Q10. (-5/6) − (-3/4)
Q11. (2/9) ÷ (-4/27)
Q12. (-3/5) × (-10/9) + 1/3
Q13. Simplify: (−2/3) × (3/4) × (−4/5) × (5/6)
Q14. Find x: x + (−3/8) = 5/4
Q15. Find 5 rational numbers between 1/2 and 3/4
Q16. Arrange in ascending order: 3/4, −1/2, 5/6, −2/3, 1/3

Section C — Properties Medium

Q17. Verify: 2/3 × (3/4 + 1/6) = 2/3 × 3/4 + 2/3 × 1/6 (distributive law)
Q18. Verify: (−5/6) + (3/8) = (3/8) + (−5/6) (commutative)
Q19. What is the additive identity for rational numbers? Give 2 examples.
Q20. Find the multiplicative identity and verify with 7/9.
Q21. If x is rational and x = multiplicative inverse of (−4/11), find x.
Q22. Find x: x × (−7/8) = (−7/8) × (4/9) [use commutative property — identify x]

Section D — Challenge Hard

  1. Q23. Find the rational number exactly in the middle of −3/4 and 1/2.
  2. Q24. Represent −5/3 on a number line accurately. (Describe or draw the division of 1-unit segments into thirds).
  3. Q25. Find: [(3/4 × −2/9) + (−5/6 × 3/10)] ÷ (−7/20)
  4. Q26. The product of two rational numbers is −8/9. One of them is 4/3. Find the other.
  5. Q27. Sum of a rational number and its additive inverse is always ____. Sum of a rational number and its reciprocal is always ____.
  6. Q28. If m/n and p/q are rational with n,q>0, prove that m/n + p/q = (mq+np)/nq.
  7. Q29. Simplify using distributive property: (−3/5) × 7/9 + (−3/5) × (−2/9)
  8. Q30. Find three rational numbers between −2 and −1. Then find three between any two of them.
✅ Key Answers: Q2: -2/3 | Q6: -3/5 > -4/7 | Q7: 1/9 | Q8: -2/5 | Q9: -1/24 | Q10: -1/12 | Q11: -3/2 | Q14: x=13/8 | Q23: midpoint = -1/8 | Q26: -2/3 | Q27: 0, not fixed

📝 Quick Revision

  1. Rational number = p/q where p,q are integers and q ≠ 0
  2. Standard form: positive denominator, no common factor between p and q
  3. Every integer, fraction, whole number, natural number is a rational number
  4. Additive inverse of p/q = −p/q. Their sum = 0
  5. Multiplicative inverse (reciprocal) of p/q = q/p. Their product = 1
  6. Add/subtract: use LCM of denominators. Multiply: straight across. Divide: multiply by reciprocal
  7. Infinite rational numbers lie between any two rational numbers