Solved Exercises on Understanding Place Value Up to Hundred-Thousands in Grade 4

Master place value up to hundred-thousands: rules for understanding digit values and place relationships through these 5 detailed exercises with visual representations.

Solution: Exercises 1 to 3
1 Identify place values in large numbers
Exercise 1
In the number 543,210, identify the place value of each digit.
Definition:

Place value: The value of a digit based on its position in a number, determined by its location from right to left

Method for identifying place values:
  1. Start from the rightmost digit (ones place)
  2. Move left, increasing the place value by powers of 10
  3. Identify each position: ones, tens, hundreds, thousands, ten thousands, hundred thousands
Number
543,210
Hundred Thousands
5
Ten Thousands
4
Thousands
3
Step 1: Identify the place value chart positions

From right to left: Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands

Step 2: Assign each digit to its position

543,210: 5=hundred thousands, 4=ten thousands, 3=thousands, 2=hundreds, 1=tens, 0=ones

Step 3: Calculate each digit's value

5×100,000=500,000; 4×10,000=40,000; 3×1,000=3,000; 2×100=200; 1×10=10; 0×1=0

Hundred
Thousands
5
Ten
Thousands
4
Thousands
3
Hundreds
2
Tens
1
Ones
0
Digit Analysis:

• 5 in hundred thousands place = 5 × 100,000 = 500,000

• 4 in ten thousands place = 4 × 10,000 = 40,000

• 3 in thousands place = 3 × 1,000 = 3,000

• 2 in hundreds place = 2 × 100 = 200

• 1 in tens place = 1 × 10 = 10

• 0 in ones place = 0 × 1 = 0

543,210 = 500,000 + 40,000 + 3,000 + 200 + 10 + 0
543,210: 5=hundred thousands, 4=ten thousands, 3=thousands, 2=hundreds, 1=tens, 0=ones
Final answer:

In 543,210: 5 is in hundred thousands place (value 500,000), 4 is in ten thousands place (value 40,000), 3 is in thousands place (value 3,000), 2 is in hundreds place (value 200), 1 is in tens place (value 10), 0 is in ones place (value 0)

Applied rules:

• Place value system: Each position represents a power of 10

• Right-to-left counting: Positions increase by factors of 10 from right to left

• Digit value: Each digit's value = digit × place value

2 Compare place values of digits
Exercise 2
In the number 256,481, compare the value of the digit 2 to the value of the digit 1.
Definition:

Comparing place values: Determining how many times greater or smaller one digit's value is compared to another's based on their positions

Number
256,481
Digit 2
Hundred Thousands
Digit 1
Ones
Ratio
200,000:1
Step 1: Identify positions

Digit 2 is in hundred thousands place, digit 1 is in ones place

Step 2: Calculate values

2 in hundred thousands = 2 × 100,000 = 200,000; 1 in ones = 1 × 1 = 1

Step 3: Compare values

200,000 ÷ 1 = 200,000, so the 2 is 200,000 times greater than the 1

Hundred
Thousands
2
Ten
Thousands
5
Thousands
6
Hundreds
4
Tens
8
Ones
1
Comparison Analysis:

• 2 in hundred thousands place = 2 × 100,000 = 200,000

• 1 in ones place = 1 × 1 = 1

• Ratio: 200,000 ÷ 1 = 200,000

200,000
÷
1
=
200,000
The digit 2 (200,000) is 200,000 times greater than the digit 1 (1)
The digit 2 is 200,000 times greater than the digit 1
Final answer:

The digit 2 in the hundred thousands place (value 200,000) is 200,000 times greater than the digit 1 in the ones place (value 1)

Applied rules:

• Position importance: Each position to the left is 10 times greater than the position to its right

• Value comparison: Divide the larger value by the smaller to find the ratio

• Power relationship: Difference in positions corresponds to powers of 10

3 Write expanded form of large numbers
Exercise 3
Write 472,509 in expanded form.
Definition:

Expanded form: Writing a number as the sum of each digit multiplied by its place value

Standard Form
472,509
Expanded Form
400,000 + 70,000 + 2,000 + 500 + 0 + 9
Step 1: Identify each digit's position

4=hundred thousands, 7=ten thousands, 2=thousands, 5=hundreds, 0=tens, 9=ones

Step 2: Calculate each digit's value

4×100,000=400,000; 7×10,000=70,000; 2×1,000=2,000; 5×100=500; 0×10=0; 9×1=9

Step 3: Write as sum

400,000 + 70,000 + 2,000 + 500 + 0 + 9

Hundred
Thousands
4
Ten
Thousands
7
Thousands
2
Hundreds
5
Tens
0
Ones
9
Expanded Form Analysis:

• 4 in hundred thousands = 4 × 100,000 = 400,000

• 7 in ten thousands = 7 × 10,000 = 70,000

• 2 in thousands = 2 × 1,000 = 2,000

• 5 in hundreds = 5 × 100 = 500

• 0 in tens = 0 × 10 = 0

• 9 in ones = 9 × 1 = 9

472,509 = 400,000 + 70,000 + 2,000 + 500 + 0 + 9
472,509 = 400,000 + 70,000 + 2,000 + 500 + 0 + 9
Final answer:

472,509 = 400,000 + 70,000 + 2,000 + 500 + 0 + 9

Applied rules:

• Expanded form: Each digit multiplied by its place value

• Zero inclusion: Include zeros in expanded form as 0 × place value

• Sum representation: The expanded form adds up to the original number

Rules and methods, laws,...
\(d_n \times 10^n\)
Digit Value Formula
\(\overline{abcdef} = a \times 100,000 + b \times 10,000 + c \times 1,000 + d \times 100 + e \times 10 + f \times 1\)
Six-Digit Number Expansion
Rule 1
\(d \times 10^n\)
Digit value based on position
Rule 2
\(10^{n+1} = 10 \times 10^n\)
Each position is 10 times previous
Rule 3
Expanded form sum
Individual place values add to number
Key definitions:

Place value: The value of a digit based on its position in a number, determined by powers of 10

Hundred thousands place: The sixth position from the right, representing 100,000s

Ten thousands place: The fifth position from the right, representing 10,000s

Thousands place: The fourth position from the right, representing 1,000s

Hundreds place: The third position from the right, representing 100s

Tens place: The second position from the right, representing 10s

Ones place: The first position from the right, representing 1s

Expanded form: Writing a number as the sum of each digit multiplied by its place value

Standard form: Writing a number in its usual numeric form

Digit: Any of the ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) used to make numbers

Complete methodology:
  1. Identify the number: Recognize the digits in the given number
  2. Count positions: Count from right to left to determine each digit's place
  3. Assign place values: Match each digit to its corresponding place value
  4. Calculate values: Multiply each digit by its place value
  5. Express in forms: Write in standard, expanded, or word form as needed
  6. Compare values: Determine relationships between different digits' values
Tip 1: Remember the order: Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands.
Tip 2: Each position to the left is 10 times greater than the position to its right.
Tip 3: Include zeros in expanded form as 0 × place value.
Tip 4: Use commas to separate thousands for easier reading.
Tip 5: The digit's value = digit × place value.
Tip 6: Visual models help understand the magnitude of each place value.
Common errors: Confusing place names, forgetting that each position is 10 times the previous, miscounting positions from right to left, not including zeros in expanded form, mixing up digit value with place value.
Memory aid: "Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands" - think of it as counting up by powers of 10.
Rules to know by heart:

• Position counting: Count from right to left starting with ones

• Value multiplication: Each digit's value = digit × place value

• Power of 10: Each position is 10 times the position to its right

• Expanded form: Sum of each digit multiplied by its place value

• Comma placement: Place commas every three digits from the right

• Zero handling: Zeros maintain place value position even when value is 0

Solution: Exercises 4 to 5
4 Compare large numbers using place value
Exercise 4
Compare 345,678 and 345,876 using place value.
Definition:

Number comparison: Using place value to determine which of two numbers is greater by comparing digits from left to right

First Number
345,678
Second Number
345,876
Comparison
345,678 < 345,876
Step 1: Compare from left to right

Hundred thousands: 3 = 3, continue; Ten thousands: 4 = 4, continue; Thousands: 5 = 5, continue

Step 2: Compare hundreds place

First number: 6 in hundreds place (value 600); Second number: 8 in hundreds place (value 800)

Step 3: Determine the relationship

Since 600 < 800, then 345,678 < 345,876

Hundred
Thousands
3
Ten
Thousands
4
Thousands
5
Hundreds
6
Tens
7
Ones
8
Hundred
Thousands
3
Ten
Thousands
4
Thousands
5
Hundreds
8
Tens
7
Ones
6
Comparison Analysis:

• First number: 345,678 = 300,000 + 40,000 + 5,000 + 600 + 70 + 8

• Second number: 345,876 = 300,000 + 40,000 + 5,000 + 800 + 70 + 6

• Difference: 800 > 600, so 345,876 > 345,678

345,678
<
345,876
345,678 = 300,000 + 40,000 + 5,000 + 600 + 70 + 8
345,876 = 300,000 + 40,000 + 5,000 + 800 + 70 + 6
345,678 < 345,876
Final answer:

345,678 < 345,876 because the hundreds digit in the first number (6) is less than the hundreds digit in the second number (8)

Applied rules:

• Left-to-right comparison: Compare digits starting from the highest place value

• First difference rule: The first differing digit determines the relationship

• Place value dominance: Higher place values have greater influence on the comparison

5 Advanced place value problem
Exercise 5
What is the value of the digit 5 in the number 567,890 compared to the value of the digit 5 in 456,789?
Definition:

Relative digit value: Comparing the actual value of the same digit in different positions across different numbers

First Number
567,890
Second Number
456,789
Comparison
500,000 vs 50,000
Step 1: Identify the digit 5 in first number

In 567,890, the digit 5 is in the hundred thousands place (value 5 × 100,000 = 500,000)

Step 2: Identify the digit 5 in second number

In 456,789, the digit 5 is in the ten thousands place (value 5 × 10,000 = 50,000)

Step 3: Compare the values

500,000 ÷ 50,000 = 10, so the 5 in the first number is 10 times greater than the 5 in the second number

Hundred
Thousands
5
Ten
Thousands
6
Thousands
7
Hundreds
8
Tens
9
Ones
0
Hundred
Thousands
4
Ten
Thousands
5
Thousands
6
Hundreds
7
Tens
8
Ones
9
Relative Value Analysis:

• In 567,890: digit 5 is in hundred thousands place = 5 × 100,000 = 500,000

• In 456,789: digit 5 is in ten thousands place = 5 × 10,000 = 50,000

• Ratio: 500,000 ÷ 50,000 = 10

500,000
÷
50,000
=
10
The digit 5 in 567,890 (500,000) is 10 times greater than the digit 5 in 456,789 (50,000)
The 5 in 567,890 is 10 times greater than the 5 in 456,789
Final answer:

The digit 5 in 567,890 (value 500,000) is 10 times greater than the digit 5 in 456,789 (value 50,000)

Applied rules:

• Position-based value: Same digit has different values based on position

• Relative comparison: Compare actual values of digits in different positions

• Power of 10 relationship: Each position to the left is 10 times greater

Complete Summary: Place Value System Up to Hundred-Thousands
\(d_n \times 10^n\)
Digit Value Formula
\(\overline{abcdef} = a \times 100,000 + b \times 10,000 + c \times 1,000 + d \times 100 + e \times 10 + f \times 1\)
Six-Digit Number Expansion
\(\frac{d_1 \times 10^m}{d_2 \times 10^n} = \frac{d_1}{d_2} \times 10^{m-n}\)
Relative Value Comparison
Key definitions:

Place value: The value of a digit based on its position in a number, determined by powers of 10.

Hundred thousands place: The sixth position from the right, representing 100,000s, with value 10⁵.

Ten thousands place: The fifth position from the right, representing 10,000s, with value 10⁴.

Thousands place: The fourth position from the right, representing 1,000s, with value 10³.

Hundreds place: The third position from the right, representing 100s, with value 10².

Tens place: The second position from the right, representing 10s, with value 10¹.

Ones place: The first position from the right, representing 1s, with value 10⁰.

Expanded form: Writing a number as the sum of each digit multiplied by its place value.

Standard form: Writing a number in its usual numeric form (e.g., 543,210).

Word form: Writing a number using words (e.g., five hundred forty-three thousand, two hundred ten).

Digit: Any of the ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) used to make numbers.

Base-10 system: Our number system based on powers of 10, where each position is 10 times the previous.

Period: Each group of three digits separated by commas (ones period, thousands period, millions period, etc.).

Complete methodology:
  1. Position identification: Count from right to left to determine each digit's place value
  2. Value calculation: Multiply each digit by its corresponding place value (power of 10)
  3. Form conversion: Convert between standard, expanded, and word forms as needed
  4. Comparison: Compare numbers by evaluating digits from highest place value to lowest
  5. Relationship analysis: Determine how many times greater or smaller one digit is compared to another
  6. Verification: Check that expanded form adds up to the original number
Tip 1: Remember the order: Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands.
Tip 2: Each position to the left is 10 times greater than the position to its right.
Tip 3: Include zeros in expanded form as 0 × place value.
Tip 4: Use commas to separate thousands periods for easier reading.
Tip 5: The digit's value = digit × place value (not just the digit alone).
Tip 6: Compare numbers from left to right, starting with the highest place value.
Tip 7: Visual models help understand the magnitude difference between place values.
Tip 8: When comparing, stop at the first differing digit to determine the relationship.
Common errors: Confusing place names, forgetting that each position is 10 times the previous, miscounting positions from right to left, not including zeros in expanded form, mixing up digit value with place value, comparing from right to left instead of left to right, misreading large numbers due to missing commas.
Memory aids: "Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands" - think of it as counting up by powers of 10; "Each place is 10 times the one before" - the base-10 system principle.
Visualization: Draw place value charts or use base-ten blocks to see the magnitude differences between positions.
Real-world connection: Place value is essential for understanding money, measurements, population counts, and large quantities in daily life.
Rules to know by heart:

• Position counting: Count from right to left starting with ones (1st, 2nd, 3rd, 4th, 5th, 6th)

• Value multiplication: Each digit's value = digit × place value (e.g., 5 in hundreds place = 5 × 100 = 500)

• Power of 10: Each position is 10 times the position to its right (10ⁿ = 10 × 10ⁿ⁻¹)

• Expanded form: Sum of each digit multiplied by its place value

• Comma placement: Place commas every three digits from the right to mark periods

• Zero handling: Zeros maintain place value position even when value is 0

• Comparison method: Compare digits from left to right, starting with highest place value

• First difference rule: The first differing digit determines the relationship between numbers

Place Value System Framework

📊
Place Value Hierarchy

Hundred Thousands: 100,000 = 10⁵

Ten Thousands: 10,000 = 10⁴

Thousands: 1,000 = 10³

Hundreds: 100 = 10²

Tens: 10 = 10¹

Ones: 1 = 10⁰

100,000
10,000
1,000
100
10
1
Number Expansion Example

543,210 = 500,000 + 40,000 + 3,000 + 200 + 10 + 0

= 5×100,000 + 4×10,000 + 3×1,000 + 2×100 + 1×10 + 0×1

500,000
40,000
3,000
200
10
0
Comparison Principle

Compare from left to right: 543,210 vs 543,209

Hundred thousands: 5 = 5 → continue

Ten thousands: 4 = 4 → continue

Thousands: 3 = 3 → continue

Hundreds: 2 = 2 → continue

Tens: 1 > 0 → 543,210 > 543,209

Place Value Analysis Process
1
Identify
2
Position
3
Calculate
4
Compare
5
Verify
Place Value Magnitude

Each position to the left is 10 times greater than the position to its right

Master Place Value Concepts!

Questions & Answers

Question: Why is the digit 5 in 500,000 worth so much more than the digit 5 in 50? I thought 5 is always just 5!

Answer: Great question! The digit 5 itself is always 5, but its VALUE depends on WHERE it sits in the number:

  • In 500,000: The 5 is in the hundred thousands place, so its value is 5 × 100,000 = 500,000
  • In 50: The 5 is in the tens place, so its value is 5 × 10 = 50

Think of it like money: a $5 bill in your pocket is worth $5, but if that $5 represents 5 million dollars in a bank account, it's part of a much larger amount. The "5" is the same, but what it represents depends on its position!

This is why we call it "place value" - the PLACE where the digit sits gives it its value. Each position to the left is 10 times greater than the position to its right.

So: 500,000 is 10,000 times greater than 50 because the 5 moved 4 places to the left!

Question: How can I remember which place is which? The names are so confusing!

Answer: Here's a helpful way to remember the order from right to left:

  1. Ones (rightmost digit) - just says the number itself
  2. Tens - groups of 10 (think: "ten fingers")
  3. Hundreds - groups of 100 (think: "a hundred is a big round number")
  4. Thousands - groups of 1,000 (think: "thousands are big numbers")
  5. Ten Thousands - 10,000 (think: "ten times a thousand")
  6. Hundred Thousands - 100,000 (think: "hundred times a thousand")

You can also remember: "Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands" - notice how it goes from small to large, and after hundreds comes thousands.

A memory trick: "O-T-H-T-T-H" (Ones-Tens-Hundreds-Thousands-Ten Thousands-Hundred Thousands).

Practice by writing numbers and labeling each digit's place - the more you practice, the more natural it becomes!

Question: When comparing 456,789 and 456,876, why do we start comparing from the left instead of the right?

Answer: We start comparing from the left because the LEFTMOST digits have the GREATEST value!

In 456,789 and 456,876:

  • Hundred thousands: 4 = 4 (same, so continue)
  • Ten thousands: 5 = 5 (same, so continue)
  • Thousands: 6 = 6 (same, so continue)
  • Hundreds: 7 vs 8 (different! 7 < 8, so we can stop here)

The hundreds place (700 vs 800) has much more impact on the total value than the tens place (80 vs 70) or ones place (9 vs 6). Since 700 < 800, we know that 456,789 < 456,876 without needing to compare the smaller place values.

Think of it like comparing money: $456,789 vs $456,876. You'd look at the biggest bills first (hundreds of dollars) before looking at smaller bills (tens, ones).

This method is faster and more efficient because higher place values dominate the comparison!