Operations on Rational Numbers
Rational numbers can be added, subtracted, multiplied, and divided using rules similar to fractions, with extra attention to signs (positive and negative).
Since every rational number is of the form p/q, the four operations follow the same rules as fraction arithmetic, combined with the rules for signs from integer arithmetic.
What is Operations on Rational Numbers - Grade 7 Maths (Rational Numbers)?
Four Operations:
- Addition: a/b + c/d = (ad + bc) / bd
- Subtraction: a/b − c/d = (ad − bc) / bd
- Multiplication: a/b × c/d = ac / bd
- Division: a/b ÷ c/d = a/b × d/c (multiply by reciprocal)
Operations on Rational Numbers Formula
Addition/Subtraction:
a/b ± c/d = (ad ± bc) / bd
Multiplication:
a/b × c/d = ac / bd
Division:
a/b ÷ c/d = a/b × d/c
Sign rules:
- (+) × (+) = (+), (−) × (−) = (+)
- (+) × (−) = (−), (−) × (+) = (−)
- Same rules for division.
Types and Properties
Properties:
- Closure: Rationals are closed under all four operations (except division by 0).
- Commutative: Addition and multiplication are commutative.
- Associative: Addition and multiplication are associative.
- Distributive: Multiplication distributes over addition.
- Additive identity: 0. Multiplicative identity: 1.
- Additive inverse: a/b + (−a/b) = 0. Multiplicative inverse: a/b × b/a = 1 (for a/b ≠ 0).
Solved Examples
Example 1: Addition
Problem: Find: −2/3 + 5/4
Solution:
- LCM(3,4) = 12
- −2/3 = −8/12, 5/4 = 15/12
- −8/12 + 15/12 = 7/12
Answer: 7/12.
Example 2: Subtraction
Problem: Find: 3/5 − (−1/4)
Solution:
- 3/5 − (−1/4) = 3/5 + 1/4
- LCM(5,4) = 20
- 12/20 + 5/20 = 17/20
Answer: 17/20.
Example 3: Multiplication
Problem: Find: (−3/7) × (2/5)
Solution:
- Multiply numerators: (−3) × 2 = −6
- Multiply denominators: 7 × 5 = 35
- Answer: −6/35
Answer: −6/35.
Example 4: Division
Problem: Find: (4/9) ÷ (−2/3)
Solution:
- 4/9 ÷ (−2/3) = 4/9 × 3/(−2) = 4/9 × (−3/2)
- = (4 × −3) / (9 × 2) = −12/18
- Simplify: HCF(12,18) = 6 → −2/3
Answer: −2/3.
Example 5: Mixed Operations
Problem: Find: 1/2 + 3/4 × (−2/3)
Solution:
- Follow BODMAS: multiplication first.
- 3/4 × (−2/3) = −6/12 = −1/2
- 1/2 + (−1/2) = 0
Answer: 0.
Example 6: Finding Multiplicative Inverse
Problem: Find the multiplicative inverse of −5/7.
Solution:
- Multiplicative inverse of a/b is b/a.
- Inverse of −5/7 = −7/5 (or 7/(−5))
- Check: (−5/7) × (−7/5) = 35/35 = 1 ✓
Answer: −7/5.
Real-World Applications
Real-world uses:
- Finance: Calculating profits/losses as fractions of investment.
- Recipes: Scaling ingredients up or down using multiplication/division of fractions.
- Temperature: Adding and subtracting temperature changes.
Key Points to Remember
- Addition/Subtraction: Find common denominator, then add/subtract numerators.
- Multiplication: Multiply numerators and denominators directly.
- Division: Multiply by the reciprocal of the divisor.
- Follow sign rules: like signs give positive, unlike signs give negative.
- Always simplify the answer to standard form.
- Division by zero is undefined.
Practice Problems
- Find: −3/8 + 5/6
- Find: 7/10 − (−3/5)
- Find: (−4/9) × (−3/8)
- Find: 5/6 ÷ (−15/12)
- Find the additive inverse of 7/11.
- Find the multiplicative inverse of −3/4.
Frequently Asked Questions
Q1. How do you add two rational numbers with different denominators?
Find the LCM of the denominators, convert to equivalent fractions with that denominator, then add the numerators.
Q2. How do you divide rational numbers?
Multiply the first number by the reciprocal (flip) of the second. a/b ÷ c/d = a/b × d/c.
Q3. What is the multiplicative inverse?
The number you multiply by to get 1. For a/b, it is b/a. Note: 0 has no multiplicative inverse.
Q4. Is subtraction of rational numbers commutative?
No. a/b − c/d ≠ c/d − a/b in general. Subtraction is not commutative.
Related Topics
- Rational Numbers
- Properties of Rational Numbers
- Addition of Fractions
- Multiplication of Fractions
- Equivalent Rational Numbers
- Standard Form of Rational Number
- Comparing Rational Numbers
- Rational Numbers on Number Line
- Rational Numbers Between Two Numbers
- Additive Inverse of Rational Numbers
- Multiplicative Inverse of Rational Numbers
- Density Property of Rational Numbers
