Solved Exercises on Comparing Numbers Using Place Value in Grade 2

Master comparing numbers using place value: tens and ones comparison, greater than less than symbols, place value charts through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Comparing 2-digit numbers
Exercise 1
Compare 47 and 39. Which number is greater?
Definition:

Place value: In a 2-digit number, the left digit is the tens place and the right digit is the ones place.

Comparison method:
  1. Compare the tens digits first
  2. If tens are equal, compare the ones digits
  3. Use symbols: > (greater than), < (less than), = (equal to)
Number: 47
4
4
4
4
7
7
7
7
7
7
7
4 tens and 7 ones
Number: 39
3
3
3
9
9
9
9
9
9
9
9
9
3 tens and 9 ones
Step 1: Compare tens digits

47 has 4 tens, 39 has 3 tens

Since 4 > 3, we don't need to compare ones

Step 2: Write the comparison

47 > 39 (47 is greater than 39)

47 > 39
Final answer:

47 is greater than 39 because 4 tens is more than 3 tens.

Applied rules:

• Place value rule: Tens place is worth 10 times the ones place

• Comparison rule: Compare from left to right (highest place value first)

• Symbol usage: > means greater than, < means less than

2 Equal tens comparison
Exercise 2
Compare 53 and 58. Which number is greater?
Definition:

Equal tens comparison: When tens digits are the same, compare the ones digits.

Number: 53
5
5
5
5
5
3
3
3
5 tens and 3 ones
Number: 58
5
5
5
5
5
8
8
8
8
8
8
8
8
5 tens and 8 ones
Step 1: Compare tens digits

53 has 5 tens, 58 has 5 tens

Since 5 = 5, we need to compare ones

Step 2: Compare ones digits

53 has 3 ones, 58 has 8 ones

Since 3 < 8, then 53 < 58

Step 3: Write the comparison

53 < 58 (53 is less than 58)

53 < 58
Final answer:

53 is less than 58 because although they have the same tens (5), 3 ones is less than 8 ones.

Applied rules:

• Equal tens rule: When tens are equal, compare ones digits

• Sequential comparison: Always go from highest to lowest place value

• Symbol precision: Use correct comparison symbol

3 Mixed comparison
Exercise 3
Compare 62 and 62. Are they equal?
Definition:

Equality: Two numbers are equal when both their tens and ones digits are identical.

Number: 62
6
6
6
6
6
6
2
2
6 tens and 2 ones
Number: 62
6
6
6
6
6
6
2
2
6 tens and 2 ones
Step 1: Compare tens digits

62 has 6 tens, 62 has 6 tens

Since 6 = 6, continue to ones

Step 2: Compare ones digits

62 has 2 ones, 62 has 2 ones

Since 2 = 2, both numbers are equal

Step 3: Write the comparison

62 = 62 (62 is equal to 62)

62 = 62
Final answer:

62 equals 62 because both the tens digits (6) and ones digits (2) are the same.

Applied rules:

• Equality rule: Both place values must match for equality

• Verification: Check both tens and ones before concluding equality

• Symbol usage: Use = when numbers are identical

Place Value Comparison Rules and Methods
47 > 39 because 4 > 3 (tens)
Place Value Comparison
Rule 1
Compare tens first
If tens differ, higher wins
Rule 2
If tens equal, compare ones
Higher ones wins
Rule 3
If both equal, numbers are equal
Use = symbol
Key definitions:

Place value: The value of a digit based on its position (tens, ones)

Comparison: Determining if one number is greater than, less than, or equal to another

Symbol >: Greater than (points to the smaller number)

Symbol <: Less than (points to the smaller number)

Symbol =: Equal to (both sides have the same value)

Complete methodology:
  1. Identify place values: Separate tens and ones digits
  2. Compare tens: Look at the leftmost digit first
  3. Compare ones (if needed): Only if tens are equal
  4. Select symbol: Choose >, <, or = based on comparison
Tip 1: Remember: TENS are more important than ONES in comparison.
Tip 2: Think of the > symbol as a hungry mouth that wants the bigger number!
Tip 3: Draw ten sticks and one dots to visualize the numbers.
Tip 4: Always compare from left to right (highest place value first).
Common errors: Comparing ones first instead of tens, forgetting to check both places when tens are equal.
Memory aid: "Tens come first, then ones - just like reading from left to right!"
Rules to remember:

• Step 1: Compare tens digits (leftmost)

• Step 2: If tens are equal, compare ones digits

• Step 3: Use appropriate symbol (> greater than, < less than, = equal)

• Step 4: Always check both place values before finalizing

25 < 31 because 2 < 3 (tens)
Quick Example
Solution: Exercises 4 to 5
4 Advanced comparison
Exercise 4
Compare 74 and 81. Which number is greater?
Definition:

Advanced comparison: When comparing numbers with different tens, the higher tens digit determines the greater number.

Number: 74
7
7
7
7
7
7
7
4
4
4
4
7 tens and 4 ones
Number: 81
8
8
8
8
8
8
8
8
1
8 tens and 1 one
Step 1: Compare tens digits

74 has 7 tens, 81 has 8 tens

Since 7 < 8, we don't need to compare ones

Step 2: Determine the relationship

Even though 74 has more ones (4 > 1), the tens determine the outcome

Step 3: Write the comparison

74 < 81 (74 is less than 81)

74 < 81
Final answer:

74 is less than 81 because 7 tens is less than 8 tens, regardless of the ones digits.

Applied rules:

• Tens dominance: Tens place has more value than ones place

• Place value hierarchy: Higher place values take precedence

• Comparison logic: Don't let smaller place values override larger ones

5 Multiple comparisons
Exercise 5
Arrange these numbers from least to greatest: 45, 38, 45, 52
Definition:

Ordering numbers: Arranging numbers in ascending order by comparing place values systematically.

Number: 45
4
4
4
4
5
5
5
5
5
4 tens and 5 ones
Number: 38
3
3
3
8
8
8
8
8
8
8
8
3 tens and 8 ones
Number: 52
5
5
5
5
5
2
2
5 tens and 2 ones
Step 1: Compare all tens digits

38 has 3 tens, 45 has 4 tens, 45 has 4 tens, 52 has 5 tens

Step 2: Group by tens value

Lowest tens: 38 (3 tens)

Middle tens: 45, 45 (4 tens each)

Highest tens: 52 (5 tens)

Step 3: Within same tens group

Both 45s are equal, so order doesn't matter between them

Step 4: Write final order

38, 45, 45, 52

38 < 45 = 45 < 52
Final answer:

The numbers arranged from least to greatest are: 38, 45, 45, 52

Applied rules:

• Systematic ordering: Group by tens first, then arrange within groups

• Equality handling: Equal numbers stay together in order

• Ascending principle: Smallest to largest values

Place Value Comparison Laws, Methods, and Definitions
Tens > Ones in value
Place Value Hierarchy
Key definitions:

Place value: The value of a digit based on its position in a number (tens place = 10, ones place = 1)

Comparison: The process of determining the relationship between two or more numbers using >, <, or =

Greater than (>): Symbol indicating the left number is larger than the right number

Less than (<): Symbol indicating the left number is smaller than the right number

Equal (=): Symbol indicating both numbers have the same value

Complete methodology:
  1. Analyze the numbers: Identify tens and ones digits in each number
  2. Compare highest place value: Start with tens digits (leftmost)
  3. Compare lower place values (if needed): Only when higher place values are equal
  4. Select appropriate symbol: Based on the comparison result
  5. Verify the answer: Double-check by counting or visualizing
Tip 1: Always start comparing from the left (highest place value) - this is the same direction we read!
Tip 2: Use base-10 blocks or draw tens sticks and ones dots to visualize the comparison.
Tip 3: The > symbol looks like a hungry mouth opening toward the bigger number.
Tip 4: If tens digits are the same, only then look at ones digits to make the decision.
Common errors: Comparing ones digits first instead of tens, forgetting that tens are worth 10 times more than ones, misreading comparison symbols.
Memory aids: "Tens before ones" and "Left side bigger for >" help remember the process.
Essential laws and rules:

• Place value law: Each position to the left is 10 times greater than the position to its right

• Comparison sequence: Always compare from highest to lowest place value

• Tens dominance: Tens digit determines the relationship unless tens are equal

• Ones comparison: Only used when tens digits are identical

• Equality condition: Both tens and ones digits must be the same for equality

57 > 53 because 7 > 3 (ones, since tens equal)
Special Case Example
Visual representation: Drawing tens rods and ones cubes helps students understand why 45 > 39 even though 9 > 5.
Real-world connection: Relate to money: $47 is more than $39 because 4 tens ($40) is more than 3 tens ($30).

Questions & Answers

Question: I don't understand why 39 is less than 47 when 9 is bigger than 7. Shouldn't the bigger ones digit make the whole number bigger?

Answer: Great question! This is a very common confusion. The key is understanding place value:

  • The tens place is worth 10 times more than the ones place
  • In 39: we have 3 tens (which is 30) + 9 ones (which is 9) = 39 total
  • In 47: we have 4 tens (which is 40) + 7 ones (which is 7) = 47 total

So even though 9 ones is more than 7 ones, having 4 tens (40) is much more valuable than having 3 tens (30). It's like having 4 ten-dollar bills versus 3 ten-dollar bills - the number of tens matters more than the number of ones!

Think of it this way: 39 = 30 + 9 and 47 = 40 + 7. Since 40 > 30, then 47 > 39.

Question: When comparing 53 and 58, both have 5 tens, so how do I know which is bigger?

Answer: When the tens digits are the same, you look at the ones digits! Here's how:

  • First: Compare tens digits → 53 has 5 tens, 58 has 5 tens → They're equal!
  • Second: Since tens are equal, compare ones digits → 53 has 3 ones, 58 has 8 ones
  • Third: Compare 3 and 8 → Since 3 < 8, then 53 < 58

Think of it like this: If you have the same number of ten-dollar bills (5), then whoever has more one-dollar bills wins! 58 has 8 one-dollar bills while 53 only has 3.

The rule is: Always go from left to right (highest place value to lowest) when comparing numbers.

Question: How can I help my child remember the greater than and less than symbols? She keeps mixing them up.

Answer: Here are some effective memory strategies for the symbols:

  • The "Hungry Mouth" Method: The symbol > looks like a mouth opening toward the bigger number: 5 > 3 (the mouth wants the bigger food!)
  • The "Less Than" L: The symbol < looks like the letter L, which helps remember "Less Than"
  • The "Point" Method: The point of the symbol always points to the smaller number: 3 < 5 (point to the 3) and 5 > 3 (point to the 3)

Practice with fun activities: have your child make the symbols with their hands, draw the symbols in sand, or play games where they identify which number the "hungry mouth" wants to eat.

Consistent practice with real examples will help the symbols become automatic over time.

Question: What happens when two numbers are exactly the same? Like 45 and 45?

Answer: When two numbers are exactly the same, we say they are EQUAL! Here's what happens:

  • First: Compare tens digits → 45 has 4 tens, 45 has 4 tens → They're the same!
  • Second: Compare ones digits → 45 has 5 ones, 45 has 5 ones → They're the same too!
  • Third: Since BOTH tens AND ones are the same, the numbers are equal!

We write: 45 = 45 (read as "45 equals 45")

Think of it like having the exact same amount of toys as your friend - you both have the same number, so you're equal in that count!

Remember: Only use = when BOTH place values are identical.

Question: Why do we always start comparing from the left side? Can't we start from the right?

Answer: We start from the left because that's where the most important digits are! Here's why:

  • Place value hierarchy: Tens are worth 10 times more than ones, so they matter more
  • Efficiency: Starting with tens often gives us the answer quickly without needing to check ones
  • Consistency: We read numbers from left to right, so comparing follows the same pattern

Think of it like a race: if one runner is ahead by a full lap (tens), it doesn't matter how many steps ahead the other runner is in the current lap (ones). The runner ahead by a full lap is winning!

Starting from the right (ones first) would be like deciding the race winner based on who's ahead in the current lap, even if someone completed more laps overall!