Place Value Up to Hundred Millions - Detailed Exercises

Master place value up to hundred millions: number identification, expanded form, comparison, rounding, and word problems through these 5 comprehensive exercises.

Solution: Exercises 1 to 3
1 Place Value Identification
Exercise 1
In the number 472,853,196:
What is the value of the digit in the ten millions place?
Definition:

Place Value: The value of a digit depends on its position in the number. Each position represents a power of 10.

Place Value Method:
  1. Identify the place value positions from right to left: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions, ten millions, hundred millions
  2. Locate the specified place value position
  3. Identify the digit in that position
  4. Calculate the value of that digit in its position
Hundred Millions
4
Ten Millions
7
Millions
2
Hundred Thousands
8
Ten Thousands
5
Thousands
3
Hundreds
1
Tens
9
Ones
6
Step 1: Understand the place value positions

From right to left: Ones (1), Tens (10), Hundreds (100), Thousands (1,000), Ten Thousands (10,000), Hundred Thousands (100,000), Millions (1,000,000), Ten Millions (10,000,000), Hundred Millions (100,000,000)

Step 2: Locate the ten millions place

Counting from the right: 6(ones), 9(tens), 1(hundreds), 3(thousands), 5(ten thousands), 8(hundred thousands), 2(millions), 7(ten millions), 4(hundred millions)

Step 3: Identify the digit in the ten millions place

The digit in the ten millions place is 7

Step 4: Calculate the value of the digit

Value = Digit × Place Value = 7 × 10,000,000 = 70,000,000

The value of the digit in the ten millions place is 70,000,000
Final answer:

The value of the digit in the ten millions place is 70,000,000

Applied rules:

• Place Value Rule: Each position represents a power of 10

• Position Counting: Count positions from right to left starting with ones

• Value Calculation: Digit × Place Value = Actual Value

2 Expanded Form
Exercise 2
Write 847,295,163 in expanded form.
Definition:

Expanded Form: Writing a number as the sum of the value of each digit based on its place value.

Hundred Millions
8
Ten Millions
4
Millions
7
Hundred Thousands
2
Ten Thousands
9
Thousands
5
Hundreds
1
Tens
6
Ones
3
8
× 100,000,000
= 800,000,000
4
× 10,000,000
= 40,000,000
7
× 1,000,000
= 7,000,000
2
× 100,000
= 200,000
9
× 10,000
= 90,000
5
× 1,000
= 5,000
1
× 100
= 100
6
× 10
= 60
3
× 1
= 3
Step 1: Identify each digit and its place value

8 in hundred millions (8 × 100,000,000 = 800,000,000)

4 in ten millions (4 × 10,000,000 = 40,000,000)

7 in millions (7 × 1,000,000 = 7,000,000)

2 in hundred thousands (2 × 100,000 = 200,000)

9 in ten thousands (9 × 10,000 = 90,000)

5 in thousands (5 × 1,000 = 5,000)

1 in hundreds (1 × 100 = 100)

6 in tens (6 × 10 = 60)

3 in ones (3 × 1 = 3)

Step 2: Write the expanded form

847,295,163 = 800,000,000 + 40,000,000 + 7,000,000 + 200,000 + 90,000 + 5,000 + 100 + 60 + 3

Step 3: Verify the expanded form

Add all parts: 800,000,000 + 40,000,000 + 7,000,000 + 200,000 + 90,000 + 5,000 + 100 + 60 + 3 = 847,295,163 ✓

847,295,163 = 800,000,000 + 40,000,000 + 7,000,000 + 200,000 + 90,000 + 5,000 + 100 + 60 + 3
Final answer:

847,295,163 = 800,000,000 + 40,000,000 + 7,000,000 + 200,000 + 90,000 + 5,000 + 100 + 60 + 3

Applied rules:

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

• Place Value Recognition: Each position has a specific value

• Verification: Always check by adding the expanded parts

3 Number Comparison
Exercise 3
Compare 528,374,196 and 528,374,201. Which number is greater?
Definition:

Number Comparison: Start comparing digits from the highest place value to the lowest until a difference is found.

Hundred Millions
5
Ten Millions
2
Millions
8
Hundred Thousands
3
Ten Thousands
7
Thousands
4
Hundreds
1
Tens
9
Ones
6
Hundred Millions
5
Ten Millions
2
Millions
8
Hundred Thousands
3
Ten Thousands
7
Thousands
4
Hundreds
2
Tens
0
Ones
1
Step 1: Compare from left to right (highest place value)

Hundred Millions: 5 = 5 (equal)

Ten Millions: 2 = 2 (equal)

Millions: 8 = 8 (equal)

Hundred Thousands: 3 = 3 (equal)

Ten Thousands: 7 = 7 (equal)

Thousands: 4 = 4 (equal)

Step 2: Continue to lower place values

Hundreds: 1 vs 2 (different!)

Since 1 < 2, we can conclude that 528,374,196 < 528,374,201

Step 3: State the final comparison

528,374,196 < 528,374,201

528,374,196 < 528,374,201
Final answer:

528,374,201 is greater than 528,374,196

Applied rules:

• Comparison Rule: Start from the highest place value and move right

• First Difference Rule: The first differing digit determines the relationship

• Greater Than Rule: The larger digit in the first differing place makes the number greater

Place Value Rules and Methods
Digit × Place Value = Actual Value
Place Value Formula
Place Values
Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, Ten Millions, Hundred Millions
Position Values
Powers of 10
10⁰, 10¹, 10², 10³, 10⁴, 10⁵, 10⁶, 10⁷, 10⁸
Mathematical Representation
Comparison
Compare from highest to lowest place value
Ordering Rule
Key definitions:

Place Value: The value of a digit based on its position in the number

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

Standard Form: Writing a number in its normal numeric form

Place Value Methodology:
  1. Identify positions: Remember the order from right to left
  2. Locate digit: Find the digit in the specified place
  3. Calculate value: Multiply the digit by its place value
  4. Compare systematically: From highest to lowest place value
Tip 1: Use commas to separate periods (millions, thousands, ones) for easier reading.
Tip 2: Remember: Each place value is 10 times the previous one.
Tip 3: When comparing, stop at the first different digit.
Tip 4: Practice with zeros - they still hold place value positions.
Common errors: Miscounting place values, forgetting zeros affect position, mixing up place value with face value.
Memory aid: "Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, Ten Millions, Hundred Millions" - say it aloud repeatedly.
Essential Rules to Remember:

• Position counting: Right to left starting with ones

• Value calculation: Digit × Place Value

• Comparison: Highest to lowest place value

• Expanded form: Sum of all place values

• Rounding: Look at the digit to the right of the target place

Solution: Exercises 4 to 5
4 Rounding Numbers
Exercise 4
Round 634,729,851 to the nearest million.
Definition:

Rounding: Replacing a number with an approximate value that has a simpler representation.

Hundred Millions
6
Ten Millions
3
Millions
4
Hundred Thousands
7
Ten Thousands
2
Thousands
9
Hundreds
8
Tens
5
Ones
1
Step 1: Identify the target place value

We need to round to the nearest million, so we focus on the millions place (4) and the digit to its right (hundred thousands place, which is 7).

Step 2: Apply the rounding rule

If the digit to the right of the target place is 5 or greater, round up. If less than 5, round down.

The digit in the hundred thousands place is 7, which is ≥ 5, so we round up.

Step 3: Adjust the target digit

Round up the millions digit from 4 to 5.

Step 4: Replace digits to the right with zeros

All digits to the right of the millions place become zeros: 635,000,000

Step 5: Verify the result

634,729,851 is closer to 635,000,000 than to 634,000,000 because it's more than halfway between them.

634,729,851 rounded to the nearest million is 635,000,000
Final answer:

634,729,851 rounded to the nearest million is 635,000,000

Applied rules:

• Rounding Rule: Look at the digit immediately to the right of the target place

• Round Up: If right digit ≥ 5, increase target digit by 1

• Round Down: If right digit < 5, keep target digit the same

5 Word Problem
Exercise 5
The population of a country is 347,829,561 people. The government wants to report this number rounded to the nearest ten million for a news release. What number should they report?
Definition:

Word Problem: A real-world scenario requiring mathematical reasoning and place value understanding.

Hundred Millions
3
Ten Millions
4
Millions
7
Hundred Thousands
8
Ten Thousands
2
Thousands
9
Hundreds
5
Tens
6
Ones
1
Step 1: Identify what needs to be done

Round 347,829,561 to the nearest ten million.

Step 2: Locate the target place value

The ten millions place has the digit 4. The digit to its right (millions place) has the digit 7.

Step 3: Apply the rounding rule

Since the digit to the right (7) is ≥ 5, we round up the ten millions digit from 4 to 5.

Step 4: Replace digits to the right with zeros

All digits to the right of the ten millions place become zeros: 350,000,000

Step 5: Answer the question

The government should report 350,000,000 people.

Step 6: Verify the result

347,829,561 is between 340,000,000 and 350,000,000. Since it's closer to 350,000,000, our rounding is correct.

The government should report 350,000,000 people
Final answer:

The government should report 350,000,000 people

Applied rules:

• Problem Analysis: Identify what place value to round to

• Rounding Application: Apply standard rounding rules

• Context Understanding: Interpret the result in the real-world context

Place Value Laws, Methods, Rules, and Definitions
Digit × Place Value = Actual Value
Place Value Formula
Key definitions:

Place Value: The value of a digit based on its position in the number

Face Value: The actual value of the digit itself (ignoring position)

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

Standard Form: Writing a number in its normal numeric form

Word Form: Writing a number using words

Complete Place Value Methodology:
  1. Position Recognition: Remember the order of place values from right to left
  2. Digit Identification: Locate specific digits in their positions
  3. Value Calculation: Multiply digits by their place values
  4. Comparison: Start from highest place value and move right
  5. Rounding: Look at the digit to the right of the target place
Tip 1: Use commas to separate periods (millions, thousands, ones) for easier reading: 123,456,789.
Tip 2: Each place value is 10 times the previous one (tens = 10 × ones, hundreds = 10 × tens, etc.).
Tip 3: When comparing large numbers, work systematically from left to right.
Tip 4: Zeros in a number still hold place value positions even though their value is zero.
Common errors: Confusing place value with face value, miscounting positions, forgetting to round correctly, skipping zeros when reading numbers.
Memorization aid: "Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, Ten Millions, Hundred Millions" - practice saying this sequence regularly.
Essential Formulas and Rules:

• Place Value Position: Right to left starting with ones

• Actual Value: Digit × Place Value

• Comparison Rule: Start from highest place value and move right

• Expanded Form: Sum of all digit × place value products

• Rounding Rule: Look at digit to the right of target place (≥5 round up, <5 round down)

Questions & Answers

Question: I get confused between place value and face value. Can you explain the difference?

Answer: Great question! The difference is fundamental:

  • Face Value: The actual number you see, regardless of position. In 543,210, the face value of 5 is just 5.
  • Place Value: The value of the digit based on its position in the number. In 543,210, the place value of 5 is 500,000 because it's in the hundred thousands place.

Think of it this way: Face value is what the digit "looks like" (its face), while place value is what it's "worth" in its position.

Example: In 7,845,921:

  • The digit 8 has a face value of 8
  • The digit 8 has a place value of 800,000 (because it's in the hundred thousands place)

Question: When rounding to the nearest million, why do I look at the hundred thousands place instead of the millions place?

Answer: The rounding rule is consistent: always look at the digit immediately to the right of your target place value!

  • To round to the nearest million, look at the hundred thousands place (the digit to the right of millions)
  • To round to the nearest thousand, look at the hundreds place (the digit to the right of thousands)
  • This is because the digit to the right determines whether you're closer to the lower or higher multiple of your target place value

Think of it like a seesaw: if the digit to the right is 5 or more, it "tips" the target digit up. If it's less than 5, the target stays the same.

Example: In 47,500,000, to round to the nearest million:

  • Target place: millions (7)
  • Digit to the right: hundred thousands (5)
  • Since 5 ≥ 5, round up: 48,000,000

Question: My child struggles with large numbers like 234,567,890. How can I help them read and understand these numbers better?

Answer: Here are effective strategies to help your child with large numbers:

  • Break it down: Separate into periods using commas: 234,567,890 (234 million, 567 thousand, 890)
  • Read in chunks: Read each group separately then combine: "two hundred thirty-four million, five hundred sixty-seven thousand, eight hundred ninety"
  • Practice place value positions: Have them identify which digit is in each place
  • Use visual aids: Draw place value charts or use manipulatives

Start with smaller numbers and gradually work up. Practice daily with different numbers, and make it fun with games or competitions!

Tip: Once your child masters the pattern of ones, tens, hundreds within each period, large numbers become much more manageable.