Logical Reasoning | Fourth Grade

Complete Notes & Formulas

1. Find Two Numbers Based on Sum and Difference

Definition: Finding two unknown numbers when you know their sum (what you get when you add them) and their difference (what you get when you subtract one from the other).

📐 Key Formulas:

Larger Number + Smaller Number = Sum

Larger Number - Smaller Number = Difference

📝 Strategy - Guess and Check Method:

Step 1: Start by dividing the sum by 2 (split equally)
Step 2: Check if those two equal numbers give the correct difference
Step 3: If difference is 0, the numbers must be different - adjust!
Step 4: Increase one number and decrease the other by the same amount
Step 5: Keep adjusting until you get the correct difference

✏️ Example 1:

Problem: The sum of two numbers is 24. Their difference is 8. What are the two numbers?

Solution:

Step 1: Split 24 equally → 12 + 12 = 24 ✓

Check difference: 12 - 12 = 0 ✗ (need 8)

Step 2: Adjust - try 16 and 8

Check sum: 16 + 8 = 24 ✓

Check difference: 16 - 8 = 8 ✓

Answer: 16 and 8

✏️ Example 2:

Problem: The sum of two numbers is 18. Their difference is 2. What are the two numbers?

Solution:

Step 1: Split 18 equally → 9 + 9 = 18 ✓

Check difference: 9 - 9 = 0 ✗ (need 2)

Step 2: Increase one by 1, decrease other by 1 → 10 and 8

Check sum: 10 + 8 = 18 ✓

Check difference: 10 - 8 = 2 ✓

Answer: 10 and 8

💡 Quick Tip:

If difference is 0, the numbers are equal!
Each time you adjust by 1, difference changes by 2!

2. Find Two Numbers Based on Sum, Difference, Product and Quotient

Definition: Finding two unknown numbers using any combination of their sum, difference, product (multiplication), and quotient (division).

📐 All Four Operations:

Operation	Formula	Example
Sum	a + b = Sum	5 + 3 = 8
Difference	a - b = Difference	5 - 3 = 2
Product	a × b = Product	5 × 3 = 15
Quotient	a ÷ b = Quotient	15 ÷ 3 = 5

📝 Problem-Solving Strategy:

Read the problem carefully and identify which operations are given
Start with the easiest clue (product or quotient often help first)
If options are given, test each option
Check all conditions must be satisfied

✏️ Example 1 - Product and Sum:

Problem: The product of two numbers is 20. Their sum is 9. What are the two numbers?

Solution:

Find factor pairs of 20: (1,20), (2,10), (4,5)

Check sums:

• 1 + 20 = 21 ✗

• 2 + 10 = 12 ✗

• 4 + 5 = 9 ✓

Answer: 4 and 5

✏️ Example 2 - Quotient and Product:

Problem: The quotient of two numbers is 3. Their product is 48. What are the two numbers?

Solution:

If a ÷ b = 3, then a = 3b (larger is 3 times the smaller)

If a × b = 48, and a = 3b

Then: 3b × b = 48

Try: b = 4 → 3(4) × 4 = 12 × 4 = 48 ✓

Check quotient: 12 ÷ 4 = 3 ✓

Answer: 12 and 4

✏️ Example 3 - Sum and Difference:

Problem: The sum of two numbers is 15. Their difference is 3. What are the two numbers?

Solution:

Start: 15 ÷ 2 = 7.5 (try numbers near this)

Try 9 and 6:

Sum: 9 + 6 = 15 ✓

Difference: 9 - 6 = 3 ✓

Answer: 9 and 6

3. Find the Order (Ranking and Sequencing)

Definition: Determining the position, sequence, or ranking of people or objects based on given information and clues.

📝 Types of Ordering Problems:

Type 1: Position in a Row/Line

Finding someone's position from the left or right end

Total = Position from Left + Position from Right - 1

Type 2: Comparison Ordering

Using clues like "taller than," "faster than," "heavier than"

Type 3: Sequence/Pattern Order

Alphabetical order, numerical order, chronological order

Type 4: Interchange/Swap Positions

Finding new positions after people swap places

✏️ Example 1 - Position in Row:

Problem: Tom is 5th from the left and 8th from the right in a row. How many students are in the row?

Solution:

Use formula: Total = Left + Right - 1

Total = 5 + 8 - 1

Total = 13 - 1 = 12

Answer: 12 students

✏️ Example 2 - Comparison Order:

Problem: Anna is taller than Ben. Ben is taller than Clara. Who is the shortest?

Solution:

Create order from clues:

Anna > Ben (Anna taller)

Ben > Clara (Ben taller)

Complete order: Anna > Ben > Clara

Answer: Clara is shortest

✏️ Example 3 - Alphabetical Order:

Problem: Arrange these words in dictionary order: Dog, Cat, Ant, Bat

Solution:

Look at first letter: A, B, C, D

Alphabetical order: A comes first, then B, then C, then D

Answer: Ant, Bat, Cat, Dog

✏️ Example 4 - Between Two People:

Problem: In a line, there are 3 people between Amy and Jack. Amy is 2nd from the left. What position is Jack from the left?

Solution:

Amy is at position 2

3 people are between them

Count: Amy (2) → Person 1 (3) → Person 2 (4) → Person 3 (5) → Jack (6)

Formula: Jack = Amy's position + 3 + 1 = 2 + 4 = 6

Answer: 6th from the left

💡 Problem-Solving Tips:

Draw a simple diagram or number line
Use symbols like >, <, = for comparisons
Write down all given information first
Build the order step by step from clues
Always verify your answer with all clues

Logical Reasoning Quick Reference Chart

Concept	Key Formula/Strategy
Sum & Difference	Split sum equally, then adjust for difference
Product Clue	Find factor pairs, test which satisfies other conditions
Quotient Clue	If a÷b=n, then a=n×b (larger is n times smaller)
Position Formula	Total = Left Position + Right Position - 1
Between Formula	New position = Start + Number Between + 1
Comparison Order	Use >, <, = symbols to build sequence
Difference Pattern	Adjust one number by 1 → difference changes by 2
General Strategy	Guess-and-Check, Test all conditions, Verify answer