Solved Exercises on Multiplicative Comparison Problems (Times as Many/Much) in Grade 4

Master multiplicative comparison problems: understanding 'times as many/much' relationships, solving word problems, and practical applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Multiplicative Comparison
Exercise 1
Sarah has 6 marbles. John has 3 times as many marbles as Sarah. How many marbles does John have?
Definition:

Multiplicative comparison: Comparing quantities by multiplying one by a factor

"Times as many": Indicates multiplication relationship between quantities

Solution method:
  1. Identify the reference quantity (Sarah's marbles)
  2. Identify the multiplier ("3 times")
  3. Multiply the reference quantity by the multiplier
  4. State the answer in context
Reference
Sarah: 6 marbles
Multiplier
3 times
Result
John: 18 marbles
Step 1: Identify the reference quantity

Sarah has 6 marbles (this is our reference)

Step 2: Identify the multiplier

John has "3 times as many" means multiply by 3

Step 3: Set up the equation

John's marbles = 3 × Sarah's marbles

John's marbles = 3 × 6

Step 4: Solve

John's marbles = 18

John has 18 marbles
Final answer:

John has 18 marbles

Applied rules:

• Multiplication Rule: "Times as many" indicates multiplication

• Reference Rule: Identify the base quantity to multiply

• Context Rule: Always state the answer in the problem's context

Tip: Look for keywords like "times as many," "times as much," "multiplied by"
2 Money Multiplicative Comparison
Exercise 2
Emma saved $15. Her brother saved 4 times as much money as Emma. How much did her brother save?
Definition:

Money comparison: Applying multiplicative comparison to monetary values

Multiplier: The factor by which one quantity is multiplied

Reference
Emma: $15
Multiplier
4 times
Result
Brother: $60
Step 1: Identify the reference quantity

Emma saved $15 (this is our reference)

Step 2: Identify the multiplier

"4 times as much" means multiply by 4

Step 3: Set up the equation

Brother's savings = 4 × Emma's savings

Brother's savings = 4 × $15

Step 4: Solve

Brother's savings = $60

Brother saved $60
Final answer:

Her brother saved $60

Applied rules:

• Multiplication Rule: "Times as much" indicates multiplication

• Unit Rule: Keep the same units in your answer

• Reference Rule: Identify the base quantity to multiply

Tip: Always include the correct units in your answer (dollars, marbles, etc.)
3 Finding the Reference Quantity
Exercise 3
A store sold 48 bottles of juice. This was 6 times as many as they sold yesterday. How many bottles did they sell yesterday?
Definition:

Finding the reference: Working backwards to find the original quantity

Division relationship: When the total is known, divide by the multiplier

Total
Today: 48 bottles
Multiplier
6 times
Result
Yesterday: 8 bottles
Step 1: Identify what we know

Today's sales = 48 bottles

Today's sales = 6 × Yesterday's sales

Step 2: Set up the equation

48 = 6 × Yesterday's sales

Step 3: Solve for yesterday's sales

Yesterday's sales = 48 ÷ 6

Step 4: Calculate

Yesterday's sales = 8 bottles

Yesterday they sold 8 bottles
Final answer:

They sold 8 bottles yesterday

Applied rules:

• Division Rule: When the total is known, divide by the multiplier

• Reverse Operation: Multiplication and division are inverse operations

• Equation Rule: Set up an equation and solve for the unknown

Tip: When the total is given, use division to find the original quantity
Multiplicative Comparison Summary
A = n × B
General Formula
Multiplication
×
Indicates "times as many"
Division
÷
Used to find original quantity
Comparison
A = n × B
A is n times B
Key definitions:

Multiplicative comparison: A relationship where one quantity is a multiple of another

Reference quantity: The base amount that is being multiplied

Multiplier: The number that tells how many times larger one quantity is

Keywords: "times as many," "times as much," "multiplied by," "groups of"

Complete methodology:
  1. Read carefully: Identify the quantities and their relationship
  2. Find keywords: Look for "times as many/much" or similar phrases
  3. Identify reference: Determine which quantity is the base amount
  4. Set up equation: Express the relationship mathematically
  5. Solve: Multiply or divide as needed
  6. Check: Verify your answer makes sense in context
Tip 1: Draw pictures or use objects to visualize the relationship.
Tip 2: When finding the original quantity, use division (opposite of multiplication).
Tip 3: Always check if your answer is reasonable in the problem's context.
Tip 4: Write the equation first: Total = Multiplier × Reference.
Common errors: Confusing addition with multiplication, misidentifying the reference quantity, incorrect operation choice.
Memory aid: "Times as many" means multiply; "original amount" means divide.
Comparison rules:

• "A is n times as many as B" means A = n × B

• "A is n times more than B" means A = B + (n × B) or A = (n+1) × B

• When total is known and multiplier is given, use division to find reference

• Always include units in your final answer

Solution: Exercises 4 to 5
4 Two-Step Multiplicative Comparison
Exercise 4
A bookstore sold 24 mystery books. They sold 3 times as many romance books as mystery books, and 2 times as many fantasy books as romance books. How many fantasy books did they sell?
Definition:

Multi-step comparison: Problems requiring multiple multiplicative comparisons

Chain of relationships: Each comparison builds on the previous one

Step 1
Mystery: 24 books
Step 2
Romance: 3 × 24 = 72
Step 3
Fantasy: 2 × 72 = 144
Step 1: Identify the starting quantity

Mystery books = 24

Step 2: Find romance books

Romance books = 3 × Mystery books

Romance books = 3 × 24 = 72

Step 3: Find fantasy books

Fantasy books = 2 × Romance books

Fantasy books = 2 × 72 = 144

Step 4: State the final answer

Fantasy books = 144

Fantasy books: 144
Final answer:

They sold 144 fantasy books

Applied rules:

• Sequential Rule: Solve one comparison at a time in sequence

• Building Rule: Each step uses the result from the previous step

• Multiplication Rule: "Times as many" indicates multiplication

Tip: Organize multi-step problems with clear steps and track your calculations
5 Inverse Multiplicative Comparison
Exercise 5
A company produced 180 t-shirts this month. This was 5 times as many as they produced last month. Next month, they plan to produce twice as many as last month. How many t-shirts will they produce next month?
Definition:

Inverse comparison: Finding the original quantity from a multiple

Forward projection: Using the original quantity to predict future amounts

Current
This month: 180
Inverse
Last month: 180 ÷ 5 = 36
Future
Next month: 2 × 36 = 72
Step 1: Find last month's production

This month = 5 × Last month

180 = 5 × Last month

Last month = 180 ÷ 5 = 36 t-shirts

Step 2: Find next month's planned production

Next month = 2 × Last month

Next month = 2 × 36 = 72 t-shirts

Step 3: State the final answer

They will produce 72 t-shirts next month

Next month: 72 t-shirts
Final answer:

They will produce 72 t-shirts next month

Applied rules:

• Inverse Rule: When total is known, divide by multiplier to find original

• Forward Rule: Use original quantity to calculate future amounts

• Sequence Rule: Solve problems in logical chronological order

Tip: Work backwards first (find original), then forward (calculate future)
Multiplicative Comparison Fundamentals and Practice
A = n × B
Comparison Formula
Key definitions:

Multiplicative comparison: A mathematical relationship where one quantity is a multiple of another

Reference quantity: The base amount that serves as the starting point for comparison

Multiplier: The factor by which the reference quantity is multiplied

Complete methodology:
  1. Identify quantities: Determine what amounts are being compared
  2. Find relationship: Look for keywords indicating multiplication
  3. Set up equation: Express the relationship as A = n × B
  4. Solve: Multiply if finding total, divide if finding original
  5. Verify: Check if the answer makes sense in context
Tip 1: Draw bar models or arrays to visualize the "times as many" relationship.
Tip 2: Remember that multiplication and division are inverse operations.
Tip 3: Always reread the problem to ensure you're answering the correct question.
Tip 4: Use estimation to check if your answer is reasonable.
Common errors: Confusing additive and multiplicative relationships, misidentifying which quantity is the reference.
Memory aid: "A is n times as many as B" → A = n × B; "Find original" → divide.
Comparison rules:

• "Times as many/much" always indicates multiplication

• When total is known and multiplier is given, divide to find original

• "A is n times as many as B" translates to A = n × B

• Always include units and check if answer is reasonable

Multiplicative Comparison Visualization
Understanding Multiplicative Relationships
Visual representation of multiplicative comparisons:
3 × 4 = 12
5 × 6 = 30
2 × 8 = 16

Analysis: The chart shows how one quantity is a multiple of another.

  • 12 is 3 times as many as 4
  • 30 is 5 times as many as 6
  • 16 is 2 times as many as 8

Questions & Answers

Question: How do I know when to multiply and when to divide in multiplicative comparison problems?

Answer: It depends on what you're trying to find:

  • Multiply when: You know the reference quantity and want to find the total
  • Example: "Sarah has 5 apples, John has 3 times as many" → Multiply 5 × 3 = 15
  • Divide when: You know the total and want to find the reference quantity
  • Example: "John has 15 apples, which is 3 times as many as Sarah" → Divide 15 ÷ 3 = 5

Think of it this way: if you're going from small to large, multiply. If you're going from large to small, divide.

Question: What's the difference between "3 times as many" and "3 more than"? They seem similar.

Answer: These are completely different operations:

  • "3 times as many" = MULTIPLICATION: If you have 4 items, 3 times as many = 4 × 3 = 12
  • "3 more than" = ADDITION: If you have 4 items, 3 more than = 4 + 3 = 7

Think of it with an example: If you have 5 toys and someone has "3 times as many," they have 15 toys (5 × 3). If someone has "3 more than you," they have 8 toys (5 + 3).

Always look for these key words: "times as many/much" = multiply, "more than" = add.

Question: How can I check if my answer is correct in multiplicative comparison problems?

Answer: There are several ways to check your answer:

  • Estimation: Does your answer make sense? If you're told someone has 3 times as many books as another person who has 10 books, the answer should be around 30, not 13 or 300.
  • Reverse operation: If you multiplied, try dividing to check. If you divided, try multiplying.
  • Plug back in: Substitute your answer back into the original relationship to see if it works.

For example, if you found that John has 18 marbles because he has 3 times as many as Sarah who has 6: check that 3 × 6 = 18. Yes, it's correct!

Always ask yourself: "Does this answer make sense in the story?"