Solve Comparison Problems Worksheet

Complete worksheet with exercises on solving both multiplicative and additive comparison problems: word problems, equations, and step-by-step solutions.

Comparison Problems 1 to 3
1 Multiplicative Comparison
Exercise 1
Sarah has 6 apples. John has 4 times as many apples as Sarah. How many apples does John have?
Definition:

Multiplicative comparison: Finding the larger quantity when given a smaller quantity and a multiplier

Solution method:
  1. Identify the smaller quantity (6 apples)
  2. Identify the multiplier (4 times)
  3. Multiply the smaller quantity by the multiplier (6 × 4)
  4. State the answer in context
Smaller quantity
6
Multiplier
× 4
Result
24
Step 1: Identify quantities

Sarah has 6 apples, John has 4 times as many

Step 2: Set up multiplication

6 × 4 = ?

Step 3: Calculate

6 × 4 = 24

Step 4: Interpret result

John has 24 apples

John has 24 apples
Final answer:

John has 24 apples.

Applied rules:

• Multiplication principle: Multiply smaller quantity by multiplier to find larger quantity

• Context: Answer must relate back to the problem situation

• Verification: Check: 24 ÷ 6 = 4 ✓

2 Additive Comparison
Exercise 2
Sarah has 15 books. John has 8 more books than Sarah. How many books does John have?
Definition:

Additive comparison: Finding the larger quantity when given a smaller quantity and a difference

Smaller quantity
15
Difference
+ 8
Result
23
Step 1: Identify quantities

Sarah has 15 books, John has 8 more

Step 2: Set up addition

15 + 8 = ?

Step 3: Calculate

15 + 8 = 23

Step 4: Interpret result

John has 23 books

John has 23 books
Final answer:

John has 23 books.

Applied rules:

• Addition principle: Add difference to smaller quantity to find larger quantity

• Context: Answer must relate back to the problem situation

• Verification: Check: 23 - 15 = 8 ✓

3 Division Comparison
Exercise 3
A large box holds 36 toys. A small box holds 9 toys. How many times more toys does the large box hold than the small box?
Definition:

Division comparison: Finding how many times one quantity is larger than another

Larger quantity
36
Smaller quantity
÷ 9
Result
4
Step 1: Identify quantities

Large box: 36 toys, Small box: 9 toys

Step 2: Set up division

36 ÷ 9 = ?

Step 3: Calculate

36 ÷ 9 = 4

Step 4: Interpret result

The large box holds 4 times more toys than the small box

Large box holds 4 times more toys
Final answer:

The large box holds 4 times more toys than the small box.

Applied rules:

• Division principle: Divide larger quantity by smaller to find multiplier

• Context: Answer must relate back to the problem situation

• Verification: Check: 9 × 4 = 36 ✓

Comparison Problem Types
Multiplicative Comparison 📊
Smaller × Multiplier = Larger
6
Sarah's apples
×
4
Times as many
=
24
John's apples
Check: 24 ÷ 6 = 4
Additive Comparison ➕
Smaller + Difference = Larger
15
Sarah's books
+
8
More books
=
23
John's books
Check: 23 - 15 = 8
Division Comparison ➗
Larger ÷ Smaller = Multiplier
36
Large box
÷
9
Small box
=
4
Times more
Check: 36 ÷ 4 = 9
Advanced Comparison Problems 4 to 6
4 Mixed Comparison
Exercise 4
A store sold 18 notebooks last week. This week they sold 3 times as many notebooks as last week, plus 5 more. How many notebooks did they sell this week?
Definition:

Mixed comparison: Problems combining multiplicative and additive comparisons

Solution method:
  1. Calculate multiplicative part: 18 × 3 = 54 notebooks
  2. Add the additional amount: 54 + 5 = 59 notebooks
  3. State the answer in context
Base amount
18
Multiplier
× 3
Add
+ 5
Total
59
Step 1: Calculate multiplicative part

18 × 3 = 54 notebooks

Step 2: Add additional amount

54 + 5 = 59 notebooks

Step 3: Interpret result

They sold 59 notebooks this week

Store sold 59 notebooks this week
Final answer:

The store sold 59 notebooks this week.

Applied rules:

• Mixed operations: Perform multiplication first, then addition

• Order of operations: Multiplication before addition

• Verification: Check: (59 - 5) ÷ 18 = 3 ✓

5 Weight Comparison
Exercise 5
A small dog weighs 12 pounds. A large dog weighs 60 pounds. How many times heavier is the large dog than the small dog?
Definition:

Weight comparison: Comparing masses using division

Large dog weight
60
Small dog weight
÷ 12
Result
5
Step 1: Identify weights

Large dog: 60 pounds, Small dog: 12 pounds

Step 2: Set up division

60 ÷ 12 = ?

Step 3: Calculate

60 ÷ 12 = 5

Step 4: Interpret result

The large dog is 5 times heavier than the small dog

Large dog is 5 times heavier
Final answer:

The large dog is 5 times heavier than the small dog.

Applied rules:

• Division principle: Divide larger weight by smaller weight

• Context: Answer reflects the multiplicative relationship

• Verification: Check: 12 × 5 = 60 ✓

6 Money Comparison
Exercise 6
Emma earned $24 from her job. Her brother earned $8 less than Emma. How much money did her brother earn?
Definition:

Subtractive comparison: Finding the smaller quantity when given a larger quantity and a difference

Emma's earnings
$24
Difference
- $8
Result
$16
Step 1: Identify amounts

Emma: $24, Brother: $8 less

Step 2: Set up subtraction

24 - 8 = ?

Step 3: Calculate

24 - 8 = 16

Step 4: Interpret result

Her brother earned $16

Brother earned $16
Final answer:

Emma's brother earned $16.

Applied rules:

• Subtraction principle: Subtract difference from larger quantity

• Context: Answer reflects the subtractive relationship

• Verification: Check: 16 + 8 = 24 ✓

Detailed Summary & Study Guide
Solving Comparison Problems - Complete Summary
Definition of Comparison Problems:
Comparison problems involve finding the relationship between two quantities. There are different types of comparisons: multiplicative (times as many), additive (more/fewer), and division (how many times larger).
Types of Comparison Problems:

Multiplicative Comparison: Uses multiplication to find a quantity that is a multiple of another (e.g., "times as many")

Additive Comparison: Uses addition/subtraction to find a quantity that differs by a fixed amount (e.g., "more than", "less than")

Division Comparison: Uses division to find how many times one quantity is larger than another (e.g., "how many times more")

Core Comparison Rules:

1. Multiplicative: Smaller × Multiplier = Larger (or Larger ÷ Smaller = Multiplier)

2. Additive: Smaller + Difference = Larger (or Larger - Smaller = Difference)

3. Division: Larger ÷ Smaller = Multiplier (or Smaller × Multiplier = Larger)

Operation Keywords:

Multiplication: "times as many", "times as much", "times larger", "times more"

Addition: "more than", "additional", "plus", "extra"

Subtraction: "less than", "fewer", "difference", "remaining"

Division: "how many times", "times as many", "ratio of", "fraction of"

Simple Examples:

Multiplicative: 5 × 3 = 15 → "5 times as much as 3 is 15"

Additive: 10 + 7 = 17 → "7 more than 10 is 17"

Division: 20 ÷ 4 = 5 → "20 is 5 times as much as 4"

Mixed Problems Examples:

• "3 times as many plus 4 more" → (Smaller × 3) + 4

• "5 times as many minus 6 less" → (Smaller × 5) - 6

• "Twice as many as 10 more than" → 2 × (Something + 10)

Key Tips for Success:

1. Identify Keywords: Look for words that indicate the type of comparison

2. Draw Diagrams: Visualize the relationship between quantities

3. Check Units: Ensure the answer has the same units as the original quantities

4. Verify Answer: Use inverse operations to check your work

5. Context Matters: Always state the answer in the context of the problem

Common Mistakes to Avoid:

1. Wrong Operation: Using multiplication instead of addition or vice versa

2. Order Confusion: Dividing the smaller by the larger instead of larger by smaller

3. Unit Errors: Mixing different units without conversion

4. Ignoring Context: Providing the numerical answer without relating it to the problem

5. No Verification: Not checking if the answer makes sense

Problem-Solving Strategy:

1. Read Carefully: Understand what quantities are being compared

2. Identify Key Information: Find the known quantities and relationships

3. Determine Operation: Recognize whether it's multiplicative, additive, or division comparison

4. Set Up Equation: Write the mathematical relationship

5. Solve and Check: Calculate and verify the answer

Verification Methods:

Multiplicative: Divide the result by the multiplier to get back to the original number

Additive: Subtract the difference from the result to get back to the original number

Division: Multiply the quotient by the divisor to get back to the original number

Questions & Answers

Question: How do I know if I should multiply or add in a comparison problem?

Answer: Look for key words in the problem! If you see words like "times as many," "times as much," or "times larger," you need to multiply. If you see words like "more than," "additional," or "plus," you need to add.

For example:
• "John has 3 times as many apples as Sarah" → Multiply
• "John has 5 more apples than Sarah" → Add

The word "times" is the biggest clue that you need multiplication.

Question: What if a problem has both "times" and "more" in it? How do I solve that?

Answer: These are called mixed comparison problems. You perform operations in order of operations (PEMDAS/BODMAS). Usually, multiplication comes before addition.

For example: "John has 3 times as many apples as Sarah, plus 4 more" means:
• First: Multiply Sarah's apples by 3
• Then: Add 4 to that result

Always perform the multiplicative comparison first, then the additive part.

Question: How can I check if my answer is correct?

Answer: You can always check your answer by using the inverse operation!

For multiplication problems: Divide your answer by the multiplier to see if you get back to the original number.
Example: If 6 × 4 = 24, check: 24 ÷ 4 = 6 ✓

For addition problems: Subtract the added amount from your answer to see if you get back to the original number.
Example: If 15 + 8 = 23, check: 23 - 8 = 15 ✓

For division problems: Multiply your answer by the smaller number to see if you get back to the original number.
Example: If 36 ÷ 9 = 4, check: 4 × 9 = 36 ✓