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)
- A rational number is of form:
A) p+q
B) p/q
C) pq
D) p−q - Condition for denominator:
A) = 0
B) ≠ 0
C) > 0 only
D) < 0 - Which is rational?
A) √2
B) π
C) 3/5
D) ∞ - Integer 5 can be written as:
A) 5/0
B) 5/1
C) 1/5
D) 0/5 - Zero as rational number:
A) 1/0
B) 0/1
C) 0/0
D) None - Which is not rational?
A) 1/2
B) -3/4
C) π
D) 5 - Terminating decimal is:
A) Infinite
B) Ends
C) Random
D) None - Repeating decimal:
A) Ends
B) Pattern repeats
C) Random
D) None - Rational numbers include:
A) Integers
B) Fractions
C) Both
D) None - 1.5 is equal to:
A) 2/3
B) 3/2
C) 5/2
D) 1/2
Positive & Negative (11–20)
- Same signs →
A) Negative
B) Positive
C) Zero
D) None - Different signs →
A) Positive
B) Negative
C) Zero
D) None - (-3)/(-5) is:
A) Negative
B) Positive
C) Zero
D) None - 3/(-4) is:
A) Positive
B) Negative
C) Zero
D) None - Which is positive?
A) -2/3
B) 4/5
C) -5/6
D) -7/8 - Which is negative?
A) 3/4
B) -2/5
C) 7/8
D) 9/10 - (-7)/2 is:
A) Positive
B) Negative
C) Zero
D) None - (-1)/(-2) =
A) -1/2
B) 1/2
C) 2
D) 0 - Sign depends on:
A) Denominator only
B) Numerator only
C) Both
D) None - Rational numbers can be:
A) Positive
B) Negative
C) Zero
D) All
Operations (21–35)
- 2/5 + 3/5 =
A) 1
B) 5/5
C) Both
D) 2/5 - 1/2 + 1/3 =
A) 2/5
B) 5/6
C) 3/6
D) 4/6 - LCM of 2 & 3 =
A) 6
B) 5
C) 4
D) 3 - 3/4 − 1/4 =
A) 2/4
B) 1/2
C) Both
D) 3/4 - 2/3 × 4/5 =
A) 6/15
B) 8/15
C) 4/15
D) 2/15 - Multiply rule:
A) Add
B) Multiply
C) Divide
D) Subtract - Division rule:
A) Add
B) Multiply
C) Flip
D) Subtract - 2/3 ÷ 4/5 =
A) 8/15
B) 10/12
C) 5/6
D) 2/5 - KCF means:
A) Keep Change Flip
B) Keep Carry Flip
C) Keep Change Form
D) None - Cross cancel used in:
A) Addition
B) Multiplication
C) Subtraction
D) Division - 5/6 × 3/5 =
A) 15/30
B) 1/2
C) Both
D) 5/3 - 7/8 − 3/8 =
A) 4/8
B) 1/2
C) Both
D) 7/8 - 1/4 ÷ 1/2 =
A) 1/8
B) 2
C) 1/2
D) 4 - Operation first step:
A) Same denominator
B) LCM
C) Multiply
D) None - Simplify before multiplying:
A) True
B) False
C) Maybe
D) None
Decimals & Comparison (36–50)
- 1/2 =
A) 0.2
B) 0.5
C) 0.25
D) 0.75 - 1/3 =
A) 0.333…
B) 0.5
C) 0.25
D) 0.75 - Terminating decimal ends when denominator has:
A) 2 or 5
B) 3
C) 7
D) 9 - 1/8 =
A) 0.125
B) 0.25
C) 0.5
D) 0.75 - 5/6 =
A) 0.83…
B) 0.5
C) 0.25
D) 1 - Which is greater: 3/4 or 5/7?
A) 3/4
B) 5/7
C) Equal
D) None - Compare using:
A) LCM
B) Cross multiply
C) Both
D) None - 2/3 vs 3/5 → bigger:
A) 2/3
B) 3/5
C) Equal
D) None - Standard form means:
A) Simplest form
B) Complex
C) Larger
D) None - -8/12 simplifies to:
A) -2/3
B) 2/3
C) -4/6
D) None - Denominator in standard form:
A) Negative
B) Positive
C) Zero
D) None - Rational numbers on number line are:
A) Continuous
B) Discrete
C) Random
D) None - Between 1/2 & 1/3:
A) Exists numbers
B) No number
C) One number
D) None - Rational numbers are:
A) Finite
B) Infinite
C) Limited
D) None - 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
