RATIONAL NUMBERS

DECODING RATIONAL NUMBERS

📖 Introduction: Why Do We Need Rational Numbers?

Imagine you and your friend are sharing a pizza 🍕. You take half and your friend takes one-fourth. How can we compare who got more? We need fractions! But what if we also deal with negative numbers, like a loss in business or a drop in temperature? That’s where rational numbers help us.

🎯 Smart Tip to Remember:

“A rational number is simply a fraction where the denominator is not zero!”

🌟 What Are Rational Numbers?

A rational number is any number that can be written in the form:

p/q

where p (numerator) and q (denominator) are integers, and q ≠ 0.

✅ Examples of Rational Numbers:

✔ 3/5 (a proper fraction)
✔ -7/2 (a negative fraction)
✔ 8 (can be written as 8/1)
✔ 0 (can be written as 0/1)
✔ 1.5 (can be written as 3/2)

❌ Non-Rational Numbers:

❌ √2 (cannot be expressed as a fraction)
❌ π (pi = 3.141592… non-repeating and never-ending)

📌 Memory Hack:
“If a decimal terminates (stops) or repeats, it’s rational!”

🌎 Real-Life Examples of Rational Numbers

📌 Shopping Discounts 🛒: If a ₹100 shirt is on a 25% discount, you pay 75/100 = 3/4 of the price.
📌 Cooking Recipes 🍽: If a cake recipe requires 3/4 cup of milk, that’s a rational number!
📌 Speed of a Car 🚗: If a car travels 120 km in 3 hours, its speed is 120/3 = 40 km/hr, which is rational.

➕ Positive and ➖ Negative Rational Numbers

1️⃣ Positive Rational Numbers: Numerator and denominator have the same signs → Example: 3/4, 7/9, (-2)/(-5)
2️⃣ Negative Rational Numbers: Numerator and denominator have opposite signs → Example: -3/4, 5/-9

📌 Memory Hack:
“Same signs = Positive ✅, Different signs = Negative ❌”

📍 Representing Rational Numbers on a Number Line

Example: Plot 3/4 on a Number Line

1️⃣ Draw a number line from 0 to 1.
2️⃣ Divide it into 4 equal parts.
3️⃣ Count 3 parts from 0 → That’s 3/4!

📌 Smart Way to Practice:
Try plotting -2/5, 7/6, and -3/2 on a number line!

📏 Standard Form of a Rational Number

A rational number is in standard form if:
✔ Denominator is positive
✔ Numerator and denominator have no common factors except 1

Example: -8/12 is not in standard form → Simplify it by dividing by HCF (4) to get -2/3.

📌 Quick Trick:
“Always divide by the HCF to get the simplest form!”

🧮 Operations on Rational Numbers

➕ Addition & Subtraction

Case 1: Same Denominator
Example:

3/5+2/5=(3+2)/5=5/5=1

Case 2: Different Denominators (Find LCM)
Example:

1/2+1/3

Step 1️⃣ Find LCM of 2 & 3 = 6
Step 2️⃣ Convert:

1/2=3/6,1/3=2/6​

Step 3️⃣ Add:

3/6+2/6=5/6​

📌 Shortcut:
“LCM helps to get the same denominator!”

✖ Multiplication of Rational Numbers

Multiply numerator with numerator, denominator with denominator!

Example:

2/3×4/5=(2×4)/(3×5)=8/15​

📌 Smart Trick:
“Cross-cancel before multiplying to simplify faster!”

➗ Division of Rational Numbers

Flip the second fraction (reciprocal) and multiply!

Example:

2/3÷4/5=2/3×5/4=10/12=5/6​

📌 Easy Rule:
“Keep, Change, Flip (KCF)!”

• Keep first fraction
• Change division to multiplication
• Flip second fraction

🔢 Decimal Representation of Rational Numbers

1️⃣ Terminating Decimal: Ends after a few digits
Example: 1/2 = 0.5, 3/8 = 0.375

2️⃣ Non-Terminating, Repeating Decimal: Keeps going but follows a pattern
Example: 1/3 = 0.3333…, 5/6 = 0.8333…

📌 Trick to Identify:
“If denominator has only 2 and/or 5 as factors, it’s terminating!”
Example: 1/8 = 0.125 (Denominator = 8 = 2³ ✅)

🔎 Comparing Rational Numbers

Rule: Convert to a common denominator and compare numerators.

Example: Which is greater, 3/4 or 5/7?
LCM of 4 & 7 = 28, so convert:

3/4=21/28,

5/7=20/28

Since 21 > 20, so 3/4 > 5/7

📌 Quick Trick:
“Cross multiply and compare!”
(3×7 = 21, 5×4 = 20, so 3/4 is greater!)

🚀 Key Takeaways

✔ Rational numbers are in p/q form (q ≠ 0)
✔ They can be positive or negative
✔ Operations follow fraction rules
✔ They can be represented on a number line
✔ Their decimal form is either terminating or repeating
✔ Cross multiplication helps compare fractions quickly!

🔥 Challenge Yourself!

1️⃣ Find 5 rational numbers between 1/3 and 1/2
2️⃣ Convert 7/8 into decimal form
3️⃣ Solve:

(2/5+1/3)×3/4​

4️⃣ Find the standard form of -12/18

📝 Practice MCQs (50 Questions)

Basic Concepts (1–10)

1. A rational number is of form:
   A) p+q
   B) p/q
   C) pq
   D) p−q
2. Condition for denominator:
   A) = 0
   B) ≠ 0
   C) > 0 only
   D) < 0
3. Which is rational?
   A) √2
   B) π
   C) 3/5
   D) ∞
4. Integer 5 can be written as:
   A) 5/0
   B) 5/1
   C) 1/5
   D) 0/5
5. Zero as rational number:
   A) 1/0
   B) 0/1
   C) 0/0
   D) None
6. Which is not rational?
   A) 1/2
   B) -3/4
   C) π
   D) 5
7. Terminating decimal is:
   A) Infinite
   B) Ends
   C) Random
   D) None
8. Repeating decimal:
   A) Ends
   B) Pattern repeats
   C) Random
   D) None
9. Rational numbers include:
   A) Integers
   B) Fractions
   C) Both
   D) None
10. 1.5 is equal to:
    A) 2/3
    B) 3/2
    C) 5/2
    D) 1/2

Positive & Negative (11–20)

11. Same signs →
    A) Negative
    B) Positive
    C) Zero
    D) None
12. Different signs →
    A) Positive
    B) Negative
    C) Zero
    D) None
13. (-3)/(-5) is:
    A) Negative
    B) Positive
    C) Zero
    D) None
14. 3/(-4) is:
    A) Positive
    B) Negative
    C) Zero
    D) None
15. Which is positive?
    A) -2/3
    B) 4/5
    C) -5/6
    D) -7/8
16. Which is negative?
    A) 3/4
    B) -2/5
    C) 7/8
    D) 9/10
17. (-7)/2 is:
    A) Positive
    B) Negative
    C) Zero
    D) None
18. (-1)/(-2) =
    A) -1/2
    B) 1/2
    C) 2
    D) 0
19. Sign depends on:
    A) Denominator only
    B) Numerator only
    C) Both
    D) None
20. Rational numbers can be:
    A) Positive
    B) Negative
    C) Zero
    D) All

Operations (21–35)

21. 2/5 + 3/5 =
    A) 1
    B) 5/5
    C) Both
    D) 2/5
22. 1/2 + 1/3 =
    A) 2/5
    B) 5/6
    C) 3/6
    D) 4/6
23. LCM of 2 & 3 =
    A) 6
    B) 5
    C) 4
    D) 3
24. 3/4 − 1/4 =
    A) 2/4
    B) 1/2
    C) Both
    D) 3/4
25. 2/3 × 4/5 =
    A) 6/15
    B) 8/15
    C) 4/15
    D) 2/15
26. Multiply rule:
    A) Add
    B) Multiply
    C) Divide
    D) Subtract
27. Division rule:
    A) Add
    B) Multiply
    C) Flip
    D) Subtract
28. 2/3 ÷ 4/5 =
    A) 8/15
    B) 10/12
    C) 5/6
    D) 2/5
29. KCF means:
    A) Keep Change Flip
    B) Keep Carry Flip
    C) Keep Change Form
    D) None
30. Cross cancel used in:
    A) Addition
    B) Multiplication
    C) Subtraction
    D) Division
31. 5/6 × 3/5 =
    A) 15/30
    B) 1/2
    C) Both
    D) 5/3
32. 7/8 − 3/8 =
    A) 4/8
    B) 1/2
    C) Both
    D) 7/8
33. 1/4 ÷ 1/2 =
    A) 1/8
    B) 2
    C) 1/2
    D) 4
34. Operation first step:
    A) Same denominator
    B) LCM
    C) Multiply
    D) None
35. Simplify before multiplying:
    A) True
    B) False
    C) Maybe
    D) None

Decimals & Comparison (36–50)

36. 1/2 =
    A) 0.2
    B) 0.5
    C) 0.25
    D) 0.75
37. 1/3 =
    A) 0.333…
    B) 0.5
    C) 0.25
    D) 0.75
38. Terminating decimal ends when denominator has:
    A) 2 or 5
    B) 3
    C) 7
    D) 9
39. 1/8 =
    A) 0.125
    B) 0.25
    C) 0.5
    D) 0.75
40. 5/6 =
    A) 0.83…
    B) 0.5
    C) 0.25
    D) 1
41. Which is greater: 3/4 or 5/7?
    A) 3/4
    B) 5/7
    C) Equal
    D) None
42. Compare using:
    A) LCM
    B) Cross multiply
    C) Both
    D) None
43. 2/3 vs 3/5 → bigger:
    A) 2/3
    B) 3/5
    C) Equal
    D) None
44. Standard form means:
    A) Simplest form
    B) Complex
    C) Larger
    D) None
45. -8/12 simplifies to:
    A) -2/3
    B) 2/3
    C) -4/6
    D) None
46. Denominator in standard form:
    A) Negative
    B) Positive
    C) Zero
    D) None
47. Rational numbers on number line are:
    A) Continuous
    B) Discrete
    C) Random
    D) None
48. Between 1/2 & 1/3:
    A) Exists numbers
    B) No number
    C) One number
    D) None
49. Rational numbers are:
    A) Finite
    B) Infinite
    C) Limited
    D) None
50. Which is smallest?
    A) 1/2
    B) 2/3
    C) 3/4
    D) 1/3

✅ Answer Key

1-B, 2-B, 3-C, 4-B, 5-B
6-C, 7-B, 8-B, 9-C, 10-B
11-B, 12-B, 13-B, 14-B, 15-B
16-B, 17-B, 18-B, 19-C, 20-D
21-C, 22-B, 23-A, 24-C, 25-B
26-B, 27-C, 28-C, 29-A, 30-B
31-C, 32-C, 33-B, 34-B, 35-A
36-B, 37-A, 38-A, 39-A, 40-A
41-A, 42-C, 43-A, 44-A, 45-A
46-B, 47-A, 48-A, 49-B, 50-D

🎯 Smart Revision Tips

• 💡 p/q (q ≠ 0)
• 💡 Same sign → Positive
• 💡 KCF → Division trick
• 💡 Terminating vs Repeating decimals
• 💡 Cross multiply to compare