Solved Exercises on Reading, Writing, and Comparing Multi-Digit Numbers in Grade 4

Master reading, writing, and comparing multi-digit numbers: rules for understanding large numbers and their relationships through these 5 detailed exercises with visual representations.

Solution: Exercises 1 to 3
1 Read and write multi-digit numbers
Exercise 1
Write 345,678 in word form and expanded form
Definition:

Multi-digit number: A number that has more than one digit, typically referring to numbers with 3 or more digits

Method for reading multi-digit numbers:
  1. Separate the number into periods (groups of three digits)
  2. Read each period separately with its period name
  3. Combine the readings with commas or "and"
Number
345,678
Word Form
Three hundred forty-five thousand, six hundred seventy-eight
Expanded Form
300,000 + 40,000 + 5,000 + 600 + 70 + 8
Step 1: Identify periods

345,678: 345 (thousands period) and 678 (ones period)

Step 2: Read the thousands period

345 = "three hundred forty-five thousand"

Step 3: Read the ones period

678 = "six hundred seventy-eight"

Step 4: Combine the readings

"Three hundred forty-five thousand, six hundred seventy-eight"

3
4
5
6
7
8
Hundred
Thousands
Ten
Thousands
Thousands
Hundreds
Tens
Ones
Word Form:
Three hundred forty-five thousand, six hundred seventy-eight
345,678 = 300,000 + 40,000 + 5,000 + 600 + 70 + 8
Word form: Three hundred forty-five thousand, six hundred seventy-eight
Final answer:

Word form: Three hundred forty-five thousand, six hundred seventy-eight

Expanded form: 300,000 + 40,000 + 5,000 + 600 + 70 + 8

Applied rules:

• Period reading: Read each group of three digits with its period name

• Comma placement: Use commas to separate periods in standard form

• Hyphenation: Use hyphens for compound numbers from 21 to 99 (twenty-one, thirty-two, etc.)

2 Compare multi-digit numbers
Exercise 2
Compare 234,567 and 234,765 using place value
Definition:

Number comparison: Determining which of two numbers is greater, lesser, or equal using place value analysis

First Number
234,567
Second Number
234,765
Result
234,567 < 234,765
Step 1: Compare from left to right

Hundred thousands: 2 = 2, continue; Ten thousands: 3 = 3, continue; Thousands: 4 = 4, continue

Step 2: Compare hundreds place

First number: 5 in hundreds place; Second number: 7 in hundreds place; Since 5 < 7, 234,567 < 234,765

Step 3: Stop at first difference

Since we found a difference in the hundreds place, we can conclude without checking lower place values

2
3
4
5
6
7
2
3
4
7
6
5
234,567
<
234,765
Hundred
Thousands
2
Ten
Thousands
3
Thousands
4
Hundreds
5 vs 7
Tens
6
Ones
7
234,567 < 234,765 because 5 < 7 in the hundreds place
234,567 < 234,765
Final answer:

234,567 < 234,765

Applied rules:

• Left-to-right comparison: Start with the highest place value and work toward the lowest

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

• Higher place value dominance: Digits in higher place values have greater impact on the comparison

3 Arrange multi-digit numbers in order
Exercise 3
Order these numbers from least to greatest: 456,789, 456,879, 456,798, 456,897
Definition:

Number ordering: Arranging numbers in ascending (least to greatest) or descending (greatest to least) order based on their place values

Numbers
456,789, 456,879, 456,798, 456,897
Sorted
456,789, 456,798, 456,879, 456,897
Step 1: Compare from left to right

All numbers start with 456, so we need to compare the remaining digits

Step 2: Focus on hundreds, tens, and ones places

Look at the last three digits: 789, 879, 798, 897

Step 3: Order the last three digits

789 < 798 < 879 < 897, so the full numbers in order are: 456,789 < 456,798 < 456,879 < 456,897

4
5
6
7
8
9
4
5
6
8
7
9
4
5
6
7
9
8
4
5
6
8
9
7
456,789
456,798
456,879
456,897
456,789
<
456,798
<
456,879
<
456,897
456,789 < 456,798 < 456,879 < 456,897
Final answer:

456,789, 456,798, 456,879, 456,897 (from least to greatest)

Applied rules:

• Systematic comparison: Compare digits from highest to lowest place value

• Partial matching: When initial digits match, focus on the first differing digits

• Ascending order: Arrange from smallest to largest value

Rules and methods, laws,...
\(\overline{abcdef} = a \times 100,000 + b \times 10,000 + c \times 1,000 + d \times 100 + e \times 10 + f \times 1\)
Six-Digit Number Expansion
Rule 1
\(a \times 10^n\)
Digit value based on position
Rule 2
Compare from left to right
Start with highest place value
Rule 3
First difference determines relationship
Stop at first differing digit
Key definitions:

Multi-digit number: A number containing more than one digit, typically referring to numbers with 3 or more digits (e.g., 123, 4,567, 234,567)

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

Period: Each group of three digits in a large number, separated by commas (e.g., thousands period, millions period)

Standard form: Writing a number in its normal numeric form (e.g., 345,678)

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

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

Ascending order: Arranging numbers from least to greatest

Descending order: Arranging numbers from greatest to least

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

Comma placement: Placing commas every three digits from the right to separate periods

Number comparison: Determining which of two numbers is greater, lesser, or equal

Leading digits: The digits at the beginning of a number that have the highest place values

Complete methodology:
  1. Reading multi-digit numbers: Separate into periods, read each period with its name, combine with commas
  2. Writing word form: Spell out each period followed by its name (thousand, million, etc.)
  3. Writing expanded form: Express each digit multiplied by its place value and add them together
  4. Comparing numbers: Start from leftmost digit and compare each position until finding a difference
  5. Ordering numbers: Use comparison method to arrange in ascending or descending order
  6. Verification: Check that the reading/writing matches the original number's value
Tip 1: Use commas to separate thousands, millions, billions, etc. for easier reading.
Tip 2: When comparing, start from the left and move right until you find different digits.
Tip 3: The first different digit determines which number is larger.
Tip 4: For word form, hyphenate compound numbers from 21 to 99 (twenty-one, thirty-two, etc.).
Tip 5: In expanded form, include zeros as 0 × place value if needed.
Tip 6: Practice reading numbers aloud to build fluency with large numbers.
Common errors: Misreading large numbers due to missing commas, confusing place names (hundreds vs thousands), forgetting to include commas in standard form, not hyphenating compound numbers in word form, comparing from right to left instead of left to right, adding denominators when adding fractions (unrelated error that might be confused).
Memory aid: "Read from left to right, largest to smallest place values" and "What you see is what you say" for word form.
Rules to know by heart:

• Place value system: Each position is 10 times the position to its right

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

• 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

• Word form rules: Hyphenate compound numbers 21-99, capitalize only the first word

• Expanded form: Each digit multiplied by its place value, then summed

Solution: Exercises 4 to 5
4 Word problem with multi-digit numbers
Exercise 4
A school collected 567,890 books for charity. Write this number in word form and compare it to 567,980.
Definition:

Word problems with large numbers: Real-world scenarios involving multi-digit numbers that require reading, writing, and comparing skills

Number
567,890
Word Form
Five hundred sixty-seven thousand, eight hundred ninety
Comparison
567,890 < 567,980
Step 1: Identify the periods

567,890: 567 (thousands period) and 890 (ones period)

Step 2: Read the thousands period

567 = "five hundred sixty-seven thousand"

Step 3: Read the ones period

890 = "eight hundred ninety"

Step 4: Compare to 567,980

Same in hundred thousands, ten thousands, and thousands; differ in hundreds: 8 < 9, so 567,890 < 567,980

5
6
7
8
9
0
5
6
7
9
8
0
Word Form:
Five hundred sixty-seven thousand, eight hundred ninety
567,890
<
567,980
567,890 < 567,980 because 8 < 9 in the hundreds place
567,890 = Five hundred sixty-seven thousand, eight hundred ninety; 567,890 < 567,980
Final answer:

Word form: Five hundred sixty-seven thousand, eight hundred ninety

Comparison: 567,890 < 567,980

Applied rules:

• Period reading: Read each group of three digits with its period name

• Comparison method: Compare digits from left to right

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

5 Advanced ordering problem
Exercise 5
Arrange these numbers from greatest to least: 123,456, 123,546, 123,465, 123,654, 123,564
Definition:

Advanced ordering: Arranging multiple large numbers in descending order, requiring careful comparison of digits in each position

Numbers
123,456, 123,546, 123,465, 123,654, 123,564
Ordered
123,654, 123,564, 123,546, 123,465, 123,456
Step 1: Notice all numbers start with 123

We only need to compare the last three digits: 456, 546, 465, 654, 564

Step 2: Compare hundreds place of last three digits

654 has 6 in hundreds place (largest), so it's first; 456 and 465 have 4 (smallest), so they come last

Step 3: Order numbers with 5 in hundreds place

Among 546, 564: 564 > 546

Step 4: Order numbers with 4 in hundreds place

Among 456, 465: 465 > 456

1
2
3
6
5
4
1
2
3
5
6
4
1
2
3
5
4
6
1
2
3
4
6
5
1
2
3
4
5
6
123,654
123,564
123,546
123,465
123,456
123,654
>
123,564
>
123,546
>
123,465
>
123,456
123,654 > 123,564 > 123,546 > 123,465 > 123,456
Final answer:

From greatest to least: 123,654, 123,564, 123,546, 123,465, 123,456

Applied rules:

• Descending order: Arrange from largest to smallest value

• Systematic comparison: Compare digits from highest to lowest place value

• Partial matching: When initial digits match, focus on the first differing digits

Complete Summary: Reading, Writing, and Comparing Multi-Digit Numbers
\(\overline{abcdef} = a \times 100,000 + b \times 10,000 + c \times 1,000 + d \times 100 + e \times 10 + f \times 1\)
Six-Digit Number Expansion
\(a\frac{b}{c} = \frac{a \times c + b}{c}\)
Mixed Number Conversion
\(\frac{n}{b} = q\frac{r}{b}\)
Improper Fraction Conversion
Key definitions:

Multi-digit number: A number containing more than one digit, typically referring to numbers with 3 or more digits (e.g., 123, 4,567, 234,567)

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

Period: Each group of three digits in a large number, separated by commas (e.g., thousands period, millions period)

Standard form: Writing a number in its normal numeric form (e.g., 345,678)

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

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

Ascending order: Arranging numbers from least to greatest

Descending order: Arranging numbers from greatest to least

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

Comma placement: Placing commas every three digits from the right to separate periods

Number comparison: Determining which of two numbers is greater, lesser, or equal

Leading digits: The digits at the beginning of a number that have the highest place values

Period naming: The names given to groups of three digits (thousands, millions, billions, etc.)

Hyphenation rule: Using hyphens for compound numbers from 21 to 99 (twenty-one, thirty-two, etc.)

Complete methodology:
  1. Reading multi-digit numbers: Separate into periods, read each period with its name, combine with commas
  2. Writing word form: Spell out each period followed by its name (thousand, million, etc.)
  3. Writing expanded form: Express each digit multiplied by its place value and add them together
  4. Comparing numbers: Start from leftmost digit and compare each position until finding a difference
  5. Ordering numbers: Use comparison method to arrange in ascending or descending order
  6. Verification: Check that the reading/writing matches the original number's value
Tip 1: Use commas to separate thousands, millions, billions, etc. for easier reading.
Tip 2: When comparing, start from the left and move right until you find different digits.
Tip 3: The first different digit determines which number is larger.
Tip 4: For word form, hyphenate compound numbers from 21 to 99 (twenty-one, thirty-two, etc.).
Tip 5: In expanded form, include zeros as 0 × place value if needed.
Tip 6: Practice reading numbers aloud to build fluency with large numbers.
Tip 7: Visualize large numbers using place value charts to understand their magnitude.
Tip 8: When ordering multiple numbers, focus on the first differing digit to make comparisons efficiently.
Common errors: Misreading large numbers due to missing commas, confusing place names (hundreds vs thousands), forgetting to include commas in standard form, not hyphenating compound numbers in word form, comparing from right to left instead of left to right, adding denominators when adding fractions (unrelated error that might be confused), misplacing commas when writing numbers.
Memory aid: "Read from left to right, largest to smallest place values" and "What you see is what you say" for word form.
Visualization: Use place value charts or base-ten blocks to understand the magnitude of each position.
Real-world connection: Large numbers appear in population counts, distances, money, and measurements.
Rules to know by heart:

• Place value system: Each position is 10 times the position to its right

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

• 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

• Word form rules: Hyphenate compound numbers 21-99, capitalize only the first word

• Expanded form: Each digit multiplied by its place value, then summed

• Period naming: Thousand, million, billion, trillion (every three digits)

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

Multi-Digit Number 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 Forms Conversion

Standard: 345,678

Word: Three hundred forty-five thousand, six hundred seventy-eight

Expanded: 300,000 + 40,000 + 5,000 + 600 + 70 + 8

300,000
40,000
5,000
600
70
8
Comparison Strategy

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

Number Processing Workflow
1
Identify
2
Separate
3
Read
4
Compare
5
Verify
Place Value Magnitude

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

Master Multi-Digit Numbers!

Questions & Answers

Question: Why do we need commas in large numbers like 345,678? Isn't it just making the number longer?

Answer: Commas don't make the number longer - they make it MUCH easier to read! Here's why:

  • Grouping: Commas separate numbers into groups of three digits (periods)
  • Identification: They help you quickly identify place values (thousands, millions, etc.)
  • Reading: Compare 345678 to 345,678 - the comma version is much clearer

Without commas, large numbers become hard to read: 1234567 vs 1,234,567. The commas help your brain process the number in chunks.

The commas go every three digits FROM THE RIGHT: 123,456,789 (hundreds, thousands, millions). This system helps everyone read numbers the same way around the world!

Question: When comparing 456,789 and 456,879, why do we start comparing from the left instead of the right?

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

In 456,789 and 456,879:

  • 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 9). Since 700 < 800, we know that 456,789 < 456,879 without needing to compare the smaller place values.

Think of it like comparing money: $456,789 vs $456,879. 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!

Question: How do I know when to hyphenate numbers in word form? Why is it "twenty-one" but not "two hundred"?

Answer: The hyphenation rule is specific and has to do with compound numbers:

  • Hyphenate compound numbers from 21 to 99: twenty-one, thirty-two, forty-three, fifty-four, etc.
  • Don't hyphenate when using "hundred," "thousand," "million," etc.: two hundred, three thousand, four million
  • Hyphenate within periods: For numbers like 123,456: "one hundred twenty-three thousand, four hundred fifty-six"

The reason is that "twenty-one" is a single compound word representing one number (21), while "two hundred" is two separate words describing two different values (2 × 100).

Examples:

  • Correct: forty-five, seventy-eight, ninety-nine
  • Correct: three hundred, five thousand, seven million
  • Correct: four hundred thirty-two thousand, five hundred sixty-seven

Remember: only hyphenate the compound numbers BETWEEN 21 and 99!