Introduction & Types Of Fractions

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Unit: Number System

Chapter: Introduction & Types Of Fractions

Reference: – Introduction to Rational Numbers, Finding Rational Numbers Between Two Rational Numbers, Finding Rational Numbers Between Two Irrational Numbers, Finding Rational Numbers Between a Rational and an Irrational Number, Average Method, Denominator Equalization Method, Formula-Based Method, Fraction Insertion Technique, Decimal Approach, Number Line Method, Finding Multiple Rational Numbers, Density Property of Rational Numbers, Solved Examples, Common Mistakes, Practice Grid

After studying this chapter, you should be able to understand:

• Introduction to Finding Rational Numbers Between Two Numbers
• Methods: Average Method, Denominator Equalization, Formula Method
• Finding Multiple Rational Numbers Between Two Given Numbers
• Rational Numbers Between Rational, Irrational, and Mixed Pairs

Introduction to Finding a Rational Number Between Two Numbers

Definition

Finding a rational number between two given numbers means identifying or constructing a number that is:

1. Rational (can be expressed as p/q, q ≠ 0)
2. Strictly between the two given numbers (greater than the smaller and less than the larger)

This is a fundamental skill based on the Density Property of rational numbers.

When we find a rational number between two numbers, we essentially ask:

“Can I find a fraction that lies between these two values?”

The answer is always YES for any two distinct real numbers — and there are infinitely many such rational numbers.

Importance

• Demonstrates the density property of rational numbers (rationals are dense on the number line)
• Builds foundational skills for limits, sequences, and calculus
• Essential for approximation and interpolation in science/engineering
• Frequently asked in competitive exams (finding n rational numbers between two numbers)
• Helps understand that between any two numbers, no matter how close, there exists another number

Example

Given: 1 and 2
Rational number between them: 1.5 = 3/2

Common Property: 1 < 3/2 < 2 and 3/2 is rational.

So, if someone asked for a rational number between 1/2 and 3/4, we could say 5/8.

Subtopics

1. Concept of Density of Rational Numbers

The set of rational numbers is dense in the set of real numbers.

Density Property:
Between any two distinct real numbers (no matter how close), there exists infinitely many rational numbers.

Key Points:

• Even between 0.123456 and 0.123457, there are rational numbers.
• This is true for rational-rational, rational-irrational, and irrational-irrational pairs.
• The average (mean) of two distinct numbers always lies strictly between them.

1. Finding the Group Basis (Method Selection)

The method you choose depends on the type of numbers given:

Methods to Find a Rational Number Between Two Numbers

Method 1: Average Method

Definition

The average (arithmetic mean) of two distinct numbers always lies between them.
If the two numbers are rational, their average is also rational.

Formula: For numbers a and b (a < b)
Rational number = (a + b) / 2

Example 1 – Two rational numbers:
Find a rational number between 3/4 and 5/6.

Average = (3/4 + 5/6)/2 = (9/12 + 10/12)/2 = (19/12)/2 = 19/24

Check: 3/4 = 0.75, 19/24 ≈ 0.7917, 5/6 ≈ 0.8333

Example 2 – Two integers:
Between 5 and 6: (5+6)/2 = 5.5 = 11/2

Example 3 – Negative numbers:
Between -3 and -2: (-3 + -2)/2 = -2.5 = -5/2

Quick Tip:
The average method is the simplest and most reliable for finding exactly one rational number between two given numbers.

Method 2: Denominator Equalization Method

Definition

To find rational numbers between two rational numbers with different denominators, convert them to equivalent fractions with the same denominator, then find fractions with numerators between.

Steps:

1. Write both fractions with a common denominator (preferably the LCM)
2. Identify integers between the two numerators
3. Write new fractions with same denominator
4.  

Example: Find a rational number between 3/4 and 5/6

Problem: No integer between 9 and 10. What to do?

Solution – Expand further:
Write with denominator 24:
3/4 = 18/24, 5/6 = 20/24
Now numerators: 18 and 20 → integer 19 exists.
Rational number = 19/24

To find multiple rational numbers: Choose an even larger denominator.

Quick Rule:
If you need n rational numbers between a/b and c/d, choose denominator = (n+1) × LCM(b,d) or larger.

Method 3: Formula Method (For Multiple Rational Numbers)

Definition

To find n rational numbers between two rational numbers a and b (where a < b):

Let d = (b – a) / (n + 1)
Then the n rational numbers are:
a + d, a + 2d, a + 3d, …, a + n×d

These numbers are equally spaced between a and b.

Example: Find 3 rational numbers between 2 and 3.

Check: 2 < 9/4=2.25 < 5/2=2.5 < 11/4=2.75 < 3

Example: Find 4 rational numbers between 1 and 2.

d = (2-1)/(4+1) = 1/5 = 0.2
Numbers: 1.2, 1.4, 1.6, 1.8 = 6/5, 7/5, 8/5, 9/5

Method 4: Fraction Insertion Technique

Definition

A special trick: For fractions a/b and c/d (in lowest terms),
a/b < (a+c)/(b+d) < c/d

This is called the mediant or Farey sum.

Example: Between 3/4 and 5/6

Mediant = (3+5)/(4+6) = 8/10 = 4/5 = 0.8

Check: 3/4=0.75 < 0.8 < 5/6≈0.8333

To find more rational numbers:

• Between 3/4 and 4/5 → (3+4)/(4+5)=7/9≈0.7778
• Between 4/5 and 5/6 → (4+5)/(5+6)=9/11≈0.8182

Special Note:
The mediant is always between the two fractions, but not necessarily the average. It works well for finding additional fractions.

Method 5: Decimal Approach

Definition

Convert both numbers to decimals, then choose a terminating or repeating decimal between them. Convert back to fraction.

Example 1 – Rational numbers:
Between 1/3 (0.3333…) and 1/2 (0.5)
Choose 0.4 = 2/5, or 0.45 = 9/20, or 0.375 = 3/8

Example 2 – Rational and Irrational (√2):
Between 1.4 and √2 ≈ 1.4142135…
Choose 1.41 = 141/100

Example 3 – Two Irrationals (π and e):
π ≈ 3.14159, e ≈ 2.71828 — wait, π > e. But careful!
Actually π ≈ 3.1416, e ≈ 2.7183. So e < π.
Between 2.7183 and 3.1416 → choose 3.0 = 3

Quick Tip:
The decimal method works for any pair of real numbers (rational or irrational), but you must round the irrationals carefully.

Special Cases

Case 1: Finding Rational Numbers Between Two Rational Numbers with Same Denominator

Example: Between 5/8 and 7/8

Numerators: 5 and 7 → integer 6 exists
Rational number = 6/8 = ¾

To find multiple: Increase denominator
Write as 10/16, 14/16 → between: 11/16, 12/16=3/4, 13/16

Case 2: Negative Rational Numbers

Example: Between -2/3 and -1/3

With denominator 3, numerators: -2 and -1 → integer ? No integer -1.5? Wait, integers are -2 and -1. There is no integer between -2 and -1. So expand denominator.

Use denominator 6: -2/3 = -4/6, -1/3 = -2/6
Numerators: -4, -3, -2 → integer -3 exists → -3/6 = -1/2

Check: -2/3 ≈ -0.667 < -0.5 < -0.333 ✓

Case 3: Irrationals That Are Square Roots

Example: Find a rational number between √2 and √3

√2 ≈ 1.4142, √3 ≈ 1.7320
Choose 1.5 = 3/2, or 1.6 = 8/5, or 1.7 = 17/10

Check: √2 ≈1.414 < 1.5 < 1.732 ≈ √3

Alternative method (squaring):
Since 2 < (rational)² < 3. Find a perfect square between 2 and 3? None. But 2.25 is between, so √2.25 = 1.5 works. So pick rational = 3/2.

Case 4: Very Close Numbers

Example: Between 0.1234 and 0.1235

Average = (0.1234 + 0.1235)/2 = 0.12345 = 12345/100000 = 2469/20000

Even closer: Between 1/1000 and 1/1001

Average = (1/1000 + 1/1001)/2 = (1001+1000)/(1000×1001×2) = 2001/2002000

Finding Multiple Rational Numbers (General Formula)

To find n rational numbers between two rational numbers p and q (p < q):

Step 1: Compute d = (q – p) / (n + 1)

Step 2: Required numbers: p + d, p + 2d, …, p + n×d

Example: Find 5 rational numbers between 1/2 and 3/4

Check: 1/2 = 12/24 < 13/24 < 14/24 < 15/24 < 16/24 < 17/24 < 18/24 = 3/4

Solved Examples

Example 1: Find one rational number between 2/5 and 3/5.

Solution (Average): (2/5 + 3/5)/2 = (5/5)/2 = 1/2 = 0.5
Solution (Denominator method): Numerator between 2 and 3? None. Expand to denominator 10: 4/10 and 6/10 → 5/10 = ½

Answer: 1/2

Example 2: Find three rational numbers between -1 and 0.

Solution (Formula method): a = -1, b = 0, n = 3
d = (0 – (-1))/(3+1) = 1/4 = 0.25
Numbers: -0.75, -0.5, -0.25 = -3/4, -1/2, -1/4

Answer: -3/4, -1/2, -1/4

Example 3: Find two rational numbers between √5 and √6.

Solution (Decimal method):

√5 ≈ 2.23607, √6 ≈ 2.44949
Choose 2.3 = 23/10 and 2.4 = 12/5

Check: 2.236 < 2.3 < 2.4 < 2.449

Answer: 23/10 and 12/5

Example 4: Find four rational numbers between 2/3 and 5/6.

Solution:
Common denominator: LCM(3,6)=6 → 2/3=4/6, 5/6=5/6 (no integer between 4 and 5)
Use denominator 12: 4/6=8/12, 5/6=10/12 → between numerators 8 and 10 → 9/12=3/4 (only one)
Need 4 numbers → use denominator 30 (since n+1=5, multiply):
2/3 = 20/30, 5/6 = 25/30

Numerators between 20 and 25: 21,22,23,24
Numbers: 21/30=7/10, 22/30=11/15, 23/30, 24/30=4/5

Answer: 7/10, 11/15, 23/30, 4/5

Example 6 – Odd One Out Style Problem:

Examine the five pairs below. In each pair, a rational number is claimed to lie between the two given numbers. Exactly one of these claims is FALSE. Identify the false claim and justify with three independent reasons.

Solution:

(A) Numerical verification (inequality check):

1. 1/4=0.25 < 1/3≈0.333 < 0.5 ✓ True
2. √2≈1.414 < 1.7 < 1.732≈√3 ✓ True
3. 3/5=0.6, 4/5=0.8, 5/8=0.625 → 0.6 < 0.625 < 0.8 ✓ True
4. -2 < -1.5 < -1 ✓ True
5. 0.99 < 1.0 < 1.01 ✓ True