Solved Exercises on Logical Reasoning with Numbers for Grade 3

Master logical reasoning with numbers: comparing, ordering, and reasoning with numerical relationships through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Comparing numbers
Exercise 1
Four friends collected seashells at the beach. Anna collected 124 seashells, Ben collected 132 seashells, Carol collected 118 seashells, and David collected 129 seashells. Who collected the most seashells? Who collected the least?
Definition:

Comparing numbers: Determining which number is greater than, less than, or equal to another number by examining place values.

Number comparison method:
  1. Look at the highest place value first (hundreds, tens, ones)
  2. Compare digits in each place value position
  3. Order numbers from greatest to least or least to greatest
  4. Identify the maximum and minimum values
Numbers
124, 132, 118, 129
Ordered
132, 129, 124, 118
Most/Less
Ben/Most, Carol/Least
Step 1: Examine the hundreds place

All numbers have 1 in the hundreds place, so move to tens place

Step 2: Compare the tens place

Anna: 2, Ben: 3, Carol: 1, David: 2

Ben has the highest tens digit (3), so he has the most

Carol has the lowest tens digit (1), so she has the least

Step 3: Verify with ones place if needed

Between Anna (124) and David (129), both have 2 in tens place

David has 9 in ones place, Anna has 4, so David > Anna

Ben collected the most (132), Carol collected the least (118)
Final answer:

Ben collected the most seashells (132)

Carol collected the least seashells (118)

Order from most to least: Ben (132), David (129), Anna (124), Carol (118)

Applied rules:

• Place Value Comparison: Start with the largest place value

• Left-to-Right Comparison: Compare digits from left to right

• Digit-by-Digit Analysis: Only compare lower places if higher places are equal

2 Ordering numbers
Exercise 2
Arrange these numbers from least to greatest: 245, 254, 240, 250, 248. Explain your reasoning.
Definition:

Ordering numbers: Arranging numbers in ascending (least to greatest) or descending (greatest to least) order based on their value.

Original
245, 254, 240, 250, 248
Ordered
240, 245, 248, 250, 254
Pattern
Ascending
Step 1: Compare hundreds place

All numbers have 2 in the hundreds place, so move to tens place

Step 2: Compare tens place

All numbers have 4 or 5 in the tens place

Numbers with 4 in tens place: 240, 245, 248

Numbers with 5 in tens place: 250, 254

Step 3: Order numbers with same tens digit

For numbers with 4 in tens place (240, 245, 248): compare ones place

0 < 5 < 8, so 240 < 245 < 248

For numbers with 5 in tens place (250, 254): compare ones place

0 < 4, so 250 < 254

Step 4: Combine the ordered groups

240 < 245 < 248 < 250 < 254

From least to greatest: 240, 245, 248, 250, 254
Final answer:

240, 245, 248, 250, 254

Reasoning: All have 2 in hundreds place. Among those with 4 in tens place, order by ones: 0, 5, 8. Among those with 5 in tens place, order by ones: 0, 4.

Applied rules:

• Systematic Comparison: Group numbers by similar digits first

• Place Value Priority: Higher place values determine order

• Sequential Ordering: Order within groups, then combine groups

3 Number relationships
Exercise 3
If A is 25 more than B, and B is 15 less than C, and C is 100, what are the values of A and B? Which number is the greatest?
Definition:

Number relationships: Understanding how numbers relate to each other through addition, subtraction, multiplication, or division.

Given
C = 100
B = C - 15
B = 85
A = B + 25
A = 110
Step 1: Identify what we know

C = 100 (given directly)

Step 2: Find B using the relationship

B is 15 less than C

B = C - 15 = 100 - 15 = 85

Step 3: Find A using the relationship

A is 25 more than B

A = B + 25 = 85 + 25 = 110

Step 4: Compare the values

A = 110, B = 85, C = 100

So A > C > B

A = 110, B = 85, C = 100; A is the greatest
Final answer:

A = 110, B = 85, C = 100

A is the greatest number

Applied rules:

• Relationship Translation: Convert word relationships to mathematical operations

• Sequential Calculation: Solve relationships in the correct order

• Verification: Check that calculated values satisfy all given relationships

Logical Reasoning with Numbers: Complete Guide
\(\text{A} > \text{B} \iff \text{A} - \text{B} > 0\)
Comparison Logic
Greater Than
A > B
A is larger than B
Less Than
A < B
A is smaller than B
Equal To
A = B
A is same as B
Key definitions:

Logical Reasoning: Using facts, rules, and relationships to draw conclusions

Number Comparison: Determining the relationship between two or more numbers

Place Value: The value of a digit based on its position in a number

Relationship: How numbers are connected through mathematical operations

Conclusion: The result drawn from logical analysis of given information

Logical Reasoning Process:
  1. Identify: Find all given information and what is being asked
  2. Analyze: Look for relationships, patterns, or connections
  3. Plan: Determine the logical steps needed to solve
  4. Execute: Carry out the logical steps systematically
  5. Verify: Check that the conclusion makes sense
Tip 1: Always start with the piece of information you know for certain.
Tip 2: Write down the relationships between numbers as equations.
Tip 3: When comparing, focus on the leftmost different digit first.
Tip 4: Use a number line to visualize relationships between numbers.
Tip 5: Check your work by substituting your answer back into the original relationships.
Common errors: Confusing greater than/less than symbols, misreading place values, not following the correct order of operations.
Success strategies: Systematic comparison, writing relationships as equations, verifying conclusions.
Number Comparison Rules:

• More digits = Greater number: 100 > 99

• Same digits, compare left to right: 345 vs 354 → compare hundreds (same), tens (4 vs 5) → 345 < 354

• Relationships: "more than" = addition, "less than" = subtraction

• Transitivity: If A > B and B > C, then A > C

Solution: Exercises 4 to 5
4 Logical deduction
Exercise 4
Three numbers have these relationships: The first number is 10 more than the second number. The third number is 5 less than the first number. If the second number is 45, what are all three numbers? Put them in order from greatest to least.
Definition:

Logical deduction: Using a chain of reasoning to determine unknown values based on given relationships.

Given
Second = 45
First = Second + 10
First = 55
Third = First - 5
Third = 50
Step 1: Identify the known value

The second number is 45 (given directly)

Step 2: Find the first number

The first number is 10 more than the second number

First = Second + 10 = 45 + 10 = 55

Step 3: Find the third number

The third number is 5 less than the first number

Third = First - 5 = 55 - 5 = 50

Step 4: Order from greatest to least

First = 55, Third = 50, Second = 45

So: First > Third > Second

First = 55, Second = 45, Third = 50; Order: 55, 50, 45
Final answer:

First number = 55

Second number = 45

Third number = 50

From greatest to least: 55, 50, 45

Applied rules:

• Chain of Logic: Each step builds on the previous one

• Relationship Translation: Convert verbal descriptions to mathematical operations

• Verification: Check that calculated values satisfy all given relationships

5 Multi-relationship problem
Exercise 5
Four numbers have these relationships: A is twice B. B is 10 more than C. C is half of D. If D is 20, find all four numbers. Which is the greatest and which is the least?
Definition:

Multi-relationship problem: A problem with multiple interconnected relationships between several unknowns.

Given
D = 20
C = D ÷ 2
C = 10
B = C + 10
B = 20
A = B × 2
A = 40
Step 1: Start with the known value

D = 20 (given directly)

Step 2: Find C using the relationship with D

C is half of D

C = D ÷ 2 = 20 ÷ 2 = 10

Step 3: Find B using the relationship with C

B is 10 more than C

B = C + 10 = 10 + 10 = 20

Step 4: Find A using the relationship with B

A is twice B

A = B × 2 = 20 × 2 = 40

Step 5: Compare all values

A = 40, B = 20, C = 10, D = 20

So A is greatest (40) and C is least (10)

A = 40, B = 20, C = 10, D = 20; Greatest: A (40), Least: C (10)
Final answer:

A = 40, B = 20, C = 10, D = 20

Greatest: A (40)

Least: C (10)

Note: B and D are equal (20)

Applied rules:

• Multi-step Deduction: Solve relationships in dependency order

• Sequential Calculation: Each unknown depends on previously found values

• Verification: Ensure all relationships hold true with calculated values

Advanced Logical Reasoning Concepts
\(\text{A} = f(\text{B}, \text{C}, \text{D})\)
Function Relationship
Advanced definitions:

Conditional Relationships: Relationships that depend on certain conditions being met

Proportional Relationships: Relationships involving multiplication or division by a constant

Inverse Relationships: When one number increases, the other decreases proportionally

Compound Relationships: Multiple relationships that connect several numbers

Logical Chains: Sequences of relationships that build upon each other

Advanced Logical Reasoning Methodology:
  1. Relationship Mapping: Create a diagram showing how numbers relate to each other
  2. Dependency Analysis: Identify which values must be calculated first
  3. Equation Formation: Write mathematical equations for each relationship
  4. Sequential Solution: Solve equations in the correct order
  5. Consistency Check: Verify that all relationships are satisfied
  6. Generalization: Look for patterns that apply to similar problems
Tip 1: Draw arrows between numbers to show relationships visually.
Tip 2: When dealing with "twice as much" or "half of", think about multiplication and division.
Tip 3: For complex relationships, make a table to track values as you calculate them.
Tip 4: Always check your final answer by substituting back into the original relationships.
Tip 5: If you get stuck, try working backwards from what you need to find.
Advanced errors: Mixing up "more than" vs "less than", not following the correct order of operations, misinterpreting proportional relationships.
Advanced strategies: Creating systematic approaches, using visual aids, checking work at each step.
Complex Relationship Patterns:

• Direct Proportion: A = k×B (as B increases, A increases by factor k)

• Inverse Proportion: A = k/B (as B increases, A decreases)

• Additive Chain: A = B + c, B = C + d, etc.

• Multiplicative Chain: A = B×c, B = C×d, etc.

• Combination: Mixed additive and multiplicative relationships

Questions & Answers

Question: I get confused when problems say "more than" or "less than." How do I know whether to add or subtract?

Answer: This is a common confusion! Here's how to remember:

  • "A is 5 more than B" means A = B + 5 (add to the smaller number)
  • "A is 5 less than B" means A = B - 5 (subtract from the larger number)
  • Always start with the reference number (after "than") and apply the operation

Example: "Sarah has 10 more than Tom" - if Tom has 20, then Sarah has 20 + 10 = 30.

Think: "more than" = add to the reference, "less than" = subtract from the reference.

Question: My child struggles with multi-step logical reasoning problems. How can I help them break these down?

Answer: Multi-step problems can be overwhelming! Try these strategies:

  • Have them identify what they know for certain first
  • Write down each relationship as a simple equation
  • Circle or highlight the starting point (the number they know)
  • Go step by step, writing down each new value they find
  • Check their work by substituting back into the original statements

Create a simple template: "I know: ___, I need to find: ___, First I'll find: ___"

Question: What if a problem has "twice as much" or "half of"? How do I handle multiplication relationships?

Answer: Multiplication relationships are handled similarly to addition/subtraction:

  • "A is twice as much as B" means A = B × 2 (multiply the reference number by 2)
  • "A is half of B" means A = B ÷ 2 (divide the reference number by 2)
  • "A is 3 times as much as B" means A = B × 3

Example: "John has twice as many toys as Mary" - if Mary has 15 toys, John has 15 × 2 = 30 toys.

Always identify the reference number (what comes after "as" or "of") and apply the operation to it.