Solved Exercises on Comparing Amounts of Money in Grade 2

Master money comparison: counting coins, bills, equivalent amounts, greater than less than comparisons through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Comparing Coin Collections
Exercise 1
Compare the amounts: Tom has 2 quarters, 3 dimes, and 4 pennies. Sarah has 1 quarter, 5 dimes, and 2 nickels. Who has more money?
Definition:

Coin values: Penny = $0.01, Nickel = $0.05, Dime = $0.10, Quarter = $0.25

Comparison method:
  1. Count the value of each type of coin
  2. Add up all the values for each person
  3. Compare the total amounts using >, <, or =
Tom's Money
$0.50 + $0.30 + $0.04
Sarah's Money
$0.25 + $0.50 + $0.10
Step 1: Calculate Tom's total

2 quarters = 2 × $0.25 = $0.50

3 dimes = 3 × $0.10 = $0.30

4 pennies = 4 × $0.01 = $0.04

Total = $0.50 + $0.30 + $0.04 = $0.84

Step 2: Calculate Sarah's total

1 quarter = 1 × $0.25 = $0.25

5 dimes = 5 × $0.10 = $0.50

2 nickels = 2 × $0.05 = $0.10

Total = $0.25 + $0.50 + $0.10 = $0.85

Step 3: Compare the amounts

$0.84 < $0.85

So Sarah has more money than Tom.

25¢
25¢
10¢
10¢
10¢
1¢
1¢
1¢
1¢
$0.84
<
25¢
10¢
10¢
10¢
10¢
10¢
5¢
5¢
$0.85
Final answer:

Sarah has more money ($0.85) than Tom ($0.84)

Applied rules:

• Value recognition: Know the value of each coin

• Addition: Add up all amounts correctly

• Comparison: Use <, >, or = symbols appropriately

2 Bill Comparison
Exercise 2
Compare the amounts: $3.45 and $3.54. Which is greater?
Definition:

Decimal place value: Dollars come before decimal, cents come after

Amount 1
$3.45
Amount 2
$3.54
Comparison
$3.45 < $3.54
Step 1: Compare dollars first

$3.45 has 3 dollars

$3.54 has 3 dollars

Same number of dollars, so compare cents

Step 2: Compare cents

$3.45 has 45 cents

$3.54 has 54 cents

45 < 54, so $3.45 < $3.54

Step 3: Final comparison

$3.45 < $3.54

$3.45
<
$3.54
Final answer:

$3.54 is greater than $3.45

Applied rules:

• Place value: Compare dollars first, then cents

• Decimal comparison: Same process as whole numbers

• Symbol usage: Use appropriate comparison symbol

3 Equivalent Amounts
Exercise 3
Show that 2 quarters equal 10 dimes: $0.50 = $0.50
Definition:

Equivalent amounts: Different combinations of coins that have the same value

2 Quarters
2 × $0.25 = $0.50
10 Dimes
10 × $0.10 = $0.50
Result
$0.50 = $0.50
Step 1: Calculate 2 quarters

2 quarters = 2 × $0.25 = $0.50

Step 2: Calculate 10 dimes

10 dimes = 10 × $0.10 = $0.50

Step 3: Compare the amounts

$0.50 = $0.50

25¢
25¢
=
10¢
10¢
10¢
10¢
10¢
10¢
10¢
10¢
10¢
10¢
$0.50 = $0.50
Final answer:

2 quarters equals 10 dimes ($0.50 = $0.50)

Applied rules:

• Equivalence: Different coins can have the same total value

• Calculation: Multiply quantity by value

• Equality: Use = symbol when amounts are the same

Money Comparison Rules and Methods
$1.00 = 100¢
Dollar to Cent Conversion
Coin Values
Penny = 1¢
1 cent
Coin Values
Nickel = 5¢
5 cents
Coin Values
Dime = 10¢
10 cents
Coin Values
Quarter = 25¢
25 cents
Key definitions:

Greater than (>): One amount is larger than another

Less than (<): One amount is smaller than another

Equal (=): Two amounts have the same value

Equivalent amounts: Different combinations with same total value

Complete methodology:
  1. Identify coin/bill types: Recognize each currency piece
  2. Calculate individual values: Multiply quantity by unit value
  3. Sum totals: Add all amounts together
  4. Compare results: Use appropriate comparison symbol
  5. Verify calculations: Double-check addition
Tip 1: Always count the largest coins first (quarters, then dimes, etc.)
Tip 2: Use skip counting for faster calculation (dimes by 10s, nickels by 5s)
Tip 3: Draw pictures of coins to visualize the problem
Tip 4: Check your work by adding amounts in reverse order
Common errors: Forgetting decimal points, miscounting coins, confusing coin values.
Practice tips: Use real coins for hands-on learning, practice with different combinations.
Money comparison formulas:

• Value calculation: Total = Quantity × Unit Value

• Conversion: $1.00 = 100¢

• Comparison: Compare dollars first, then cents

• Equivalence: Different coins can equal same amount

Solution: Exercises 4 to 5
4 Mixed Currency Comparison
Exercise 4
Compare: $1.75 and 7 quarters. Are they equal?
Definition:

Mixed currency: Comparing dollar bills with coin collections

Amount 1
$1.75
Amount 2
7 × $0.25
Result
$1.75 = $1.75
Step 1: Keep $1.75 as is

First amount = $1.75

Step 2: Calculate 7 quarters

7 quarters = 7 × $0.25 = $1.75

Step 3: Compare the results

$1.75 = $1.75

$1
25¢
25¢
25¢
=
25¢
25¢
25¢
25¢
25¢
25¢
25¢
$1.75 = $1.75
Final answer:

Yes, $1.75 equals 7 quarters ($1.75 = $1.75)

Applied rules:

• Mixed comparison: Convert all to same format (dollars)

• Multiplication: Count coins by multiplying quantity by value

• Equivalence: Both represent the same monetary value

5 Complex Comparison
Exercise 5
Compare: 1 dollar, 3 quarters, 2 dimes, and 5 pennies vs 2 dollars, 1 quarter, 1 nickel. Which is greater?
Definition:

Complex comparison: Multiple types of currency in each group

Group 1
$1.00 + $0.75 + $0.20 + $0.05
Group 2
$2.00 + $0.25 + $0.05
Result
$2.00 < $2.30
Step 1: Calculate Group 1

1 dollar = $1.00

3 quarters = 3 × $0.25 = $0.75

2 dimes = 2 × $0.10 = $0.20

5 pennies = 5 × $0.01 = $0.05

Total = $1.00 + $0.75 + $0.20 + $0.05 = $2.00

Step 2: Calculate Group 2

2 dollars = $2.00

1 quarter = $0.25

1 nickel = $0.05

Total = $2.00 + $0.25 + $0.05 = $2.30

Step 3: Compare the totals

$2.00 < $2.30

Step 4: Determine the answer

The second group has more money ($2.30 > $2.00)

$1
25¢
25¢
25¢
10¢
10¢
1¢
1¢
1¢
1¢
1¢
$2.00
<
$1
$1
25¢
5¢
$2.30
Final answer:

Group 2 is greater ($2.30 > $2.00)

Applied rules:

• Systematic counting: Organize by coin type

• Step-by-step calculation: Add each type separately

• Final comparison: Compare total amounts

Money Comparison Laws, Methods, Rules, Definitions
\text{Total Value} = \sum (\text{Quantity} \times \text{Unit Value})
Total Value Calculation
Key definitions:

Currency: Money in any form when in actual use as a medium of exchange

Value: The worth of money expressed in dollars and cents

Comparison: Determining whether one amount is greater than, less than, or equal to another

Equivalence: When different combinations of money have the same total value

Complete methodology:
  1. Recognition: Identify all coins and bills in each collection
  2. Classification: Group items by type (pennies, nickels, dimes, quarters, dollars)
  3. Quantification: Count how many of each type you have
  4. Calculation: Multiply quantity by unit value for each type
  5. Aggregation: Sum all calculated values for each collection
  6. Comparison: Use comparison symbols to determine relationship
  7. Verification: Double-check calculations for accuracy
Tip 1: Always write amounts in standard form ($1.25) to avoid confusion.
Tip 2: Use a chart or table to organize different coin types and their counts.
Tip 3: Practice skip counting (by 5s for nickels, 10s for dimes, 25s for quarters).
Tip 4: Draw circles or boxes to represent each coin when solving problems.
Tip 5: Remember that 4 quarters equal $1.00, which can help with quick estimates.
Common errors: Misidentifying coin values, forgetting to convert cents to dollars, calculation mistakes.
Memory aids: "Penny for your thoughts" (1¢), "Nickel for change" (5¢), "Dime a dozen" (10¢), "Quarter past" (25¢).
Money comparison laws:

• Conservation law: The total value remains the same regardless of arrangement

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

• Reflexivity law: Any amount equals itself (A = A)

• Antisymmetry law: If A ≥ B and B ≥ A, then A = B

• Trichotomy law: For any two amounts A and B, exactly one is true: A > B, A < B, or A = B

Questions & Answers

Question: I get confused when comparing amounts like $0.95 and $1.05. How do I know which is bigger?

Answer: Great question! Here's how to compare $0.95 and $1.05:

  • Step 1: Look at the dollar amounts first. $0.95 has 0 dollars, $1.05 has 1 dollar.
  • Step 2: Since 0 < 1, we know $0.95 < $1.05 without even looking at the cents.

Think of it this way: $0.95 is less than $1.00, while $1.05 is more than $1.00. So $1.05 is definitely bigger!

Remember the rule: Always compare dollars first, then cents if dollars are equal.

Question: Why does 10 dimes equal 1 dollar? I thought dimes were just small coins.

Answer: Excellent observation! Here's why 10 dimes equal 1 dollar:

  • Each dime: Worth 10 cents ($0.10)
  • 10 dimes: 10 × $0.10 = $1.00
  • Conversion: 100 cents = 1 dollar

Even though dimes are small, each one is worth 10 cents. So 10 of them add up to 100 cents, which equals 1 dollar!

This is part of the base-10 system that makes money math easier to work with.

Question: How can I quickly count a lot of coins without making mistakes?

Answer: Here are some great strategies for counting coins efficiently:

  • Sort first: Put coins in separate piles by type (pennies, nickels, dimes, quarters)
  • Count by groups: Use skip counting - dimes by 10s (10, 20, 30...), nickels by 5s, quarters by 25s
  • Start big: Count the highest value coins first (quarters, then dimes, nickels, pennies)
  • Write it down: Record each group's total as you go
  • Double-check: Count again if the amount seems unusual

For example, if you have 3 quarters, 4 dimes, and 2 nickels: 75¢ + 40¢ + 10¢ = $1.25.

Question: What if I have the same number of coins but different types? Like 5 dimes vs 5 nickels?

Answer: That's a smart question! Even with the same number of coins, different types have different values:

  • 5 dimes: 5 × $0.10 = $0.50
  • 5 nickels: 5 × $0.05 = $0.25

So 5 dimes ($0.50) is more than 5 nickels ($0.25), even though you have the same number of coins!

Always remember: It's not just about how many coins you have, but what each coin is worth.

The key is to multiply the quantity by the unit value for each type.

Question: How do I know if two different sets of coins are equal without counting everything?

Answer: Here are some helpful shortcuts for checking if coin sets might be equal:

  • Look for known equivalents: 4 quarters = $1.00, 10 dimes = $1.00, 20 nickels = $1.00
  • Match large coins first: If one set has more quarters, it's probably larger
  • Estimate with dollars: Group coins into dollar amounts to get a quick estimate
  • Pair equivalent values: Remove pairs that are clearly equal (like 2 dimes vs 1 nickel + 1 penny)

However, for accuracy, always count completely. These shortcuts help you predict which might be larger.

Example: 2 quarters and 5 dimes both equal $1.00, so they're equivalent.