Comparing Rational Numbers
Comparing rational numbers means deciding which is greater, smaller, or if they are equal. Since rational numbers include negative fractions, the comparison is not always obvious.
There are two main methods: the LCM method (make denominators equal) and the cross-multiplication method.
For example, which is greater: 3/5 or 2/3? Using LCM: 3/5 = 9/15, 2/3 = 10/15. Since 10/15 > 9/15, we get 2/3 > 3/5.
What is Comparing Rational Numbers - Grade 7 Maths (Rational Numbers)?
Comparing rational numbers a/b and c/d:
- LCM method: Find LCM of b and d, convert both to equivalent fractions with this denominator, compare numerators.
- Cross-multiplication: Compare a × d with b × c (when both denominators are positive).
Rules for sign:
- Every positive rational number is greater than 0.
- Every negative rational number is less than 0.
- Every positive rational number is greater than every negative rational number.
Comparing Rational Numbers Formula
Cross-multiplication method (when b, d > 0):
a/b vs c/d → compare a × d with b × c
If a × d > b × c, then a/b > c/d
Types and Properties
Cases:
- Both positive: Use LCM or cross-multiply directly.
- Both negative: The one closer to 0 is greater. Example: −1/3 > −1/2 (because −1/3 is closer to 0).
- One positive, one negative: The positive one is always greater.
- Comparing with 0: Positive > 0 > Negative.
Solved Examples
Example 1: LCM Method
Problem: Compare 3/4 and 5/6.
Solution:
- LCM(4, 6) = 12
- 3/4 = 9/12, 5/6 = 10/12
- 9 < 10, so 9/12 < 10/12
Answer: 3/4 < 5/6.
Example 2: Cross-Multiplication
Problem: Compare 2/7 and 3/8.
Solution:
- Cross-multiply: 2 × 8 = 16, 7 × 3 = 21
- 16 < 21
Answer: 2/7 < 3/8.
Example 3: Comparing Negatives
Problem: Compare −3/5 and −4/7.
Solution:
- LCM(5, 7) = 35
- −3/5 = −21/35, −4/7 = −20/35
- −21 < −20, so −21/35 < −20/35
Answer: −3/5 < −4/7 (or −4/7 > −3/5).
Example 4: Ordering Rational Numbers
Problem: Arrange in ascending order: 1/2, −3/4, 2/3, −1/6.
Solution:
- Negatives first: −3/4 and −1/6. LCM(4,6)=12: −3/4 = −9/12, −1/6 = −2/12. So −3/4 < −1/6.
- Positives: 1/2 and 2/3. LCM(2,3)=6: 1/2 = 3/6, 2/3 = 4/6. So 1/2 < 2/3.
- All negatives < all positives.
Answer: −3/4 < −1/6 < 1/2 < 2/3.
Example 5: Equal Rational Numbers
Problem: Are 2/3 and 8/12 equal?
Solution:
- 2/3 in standard form: 2/3
- 8/12 in standard form: 8÷4/12÷4 = 2/3
Answer: Yes, 2/3 = 8/12.
Real-World Applications
Real-world uses:
- Temperature: Comparing temperatures like −2.5°C and −3.7°C to find which is warmer.
- Banking: Comparing account balances (negative means debt).
- Sports: Comparing averages or scores expressed as fractions.
Key Points to Remember
- Use LCM method or cross-multiplication to compare.
- Every positive rational number > 0 > every negative rational number.
- For negative numbers: the one closer to 0 is greater.
- Convert to same denominator for easy comparison.
- Always ensure denominators are positive before comparing.
Practice Problems
- Compare 3/7 and 4/9.
- Compare −5/8 and −3/4.
- Arrange in ascending order: −1/2, 3/4, −3/8, 1/6.
- Which is greater: 7/11 or 5/8?
- Compare −2/3 and 0.
Frequently Asked Questions
Q1. How do you compare two rational numbers?
Make the denominators equal using LCM and compare numerators. Or use cross-multiplication: compare a×d with b×c (for a/b vs c/d with positive denominators).
Q2. Is −1/3 greater than −1/2?
Yes. −1/3 ≈ −0.33 and −1/2 = −0.5. Since −0.33 is closer to 0, −1/3 > −1/2.
Q3. How do you arrange rational numbers in ascending order?
Convert all to equivalent fractions with the same positive denominator, then arrange by numerator from smallest to largest.
Q4. Is 0 a rational number?
Yes. 0 = 0/1 is a rational number. It is neither positive nor negative.
Related Topics
- Rational Numbers
- Rational Numbers on Number Line
- Equivalent Rational Numbers
- Comparing Fractions
- Standard Form of Rational Number
- Operations on Rational Numbers
- Properties of Rational Numbers
- Rational Numbers Between Two Numbers
- Additive Inverse of Rational Numbers
- Multiplicative Inverse of Rational Numbers
- Density Property of Rational Numbers
