Decimal Place Value
You already know that in the whole number 543, the digit 5 is in the hundreds place, 4 is in the tens place, and 3 is in the ones place. Each position has a value that is 10 times the position to its right.
What happens when we move to the right of the ones place? We enter the world of decimals. The place values become fractions: tenths (1/10), hundredths (1/100), and thousandths (1/1000).
The decimal point separates the whole number part from the fractional part. For example, in 23.65, the 2 is in the tens place, 3 in the ones place, 6 in the tenths place, and 5 in the hundredths place.
Understanding decimal place value is essential for reading, writing, comparing, and performing operations with decimals. In NCERT Class 6, this is part of the Decimals chapter.
What is Decimal Place Value?
Definition: Decimal place value tells us the value of each digit in a decimal number based on its position relative to the decimal point.
| Hundreds | Tens | Ones | . (point) | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|
| 100 | 10 | 1 | . | 1/10 | 1/100 | 1/1000 |
| 100 | 10 | 1 | . | 0.1 | 0.01 | 0.001 |
Key terms:
- Decimal point: The dot (.) that separates the whole number part from the fractional part.
- Tenths place: First digit after the decimal point. Value = digit × 1/10.
- Hundredths place: Second digit after the decimal point. Value = digit × 1/100.
- Thousandths place: Third digit after the decimal point. Value = digit × 1/1000.
Important:
- Each place to the right of the decimal is 1/10 of the place to its left.
- The decimal point is NOT a digit — it is a separator.
- The number of digits after the decimal point tells the number of decimal places.
Decimal Place Value Formula
Finding the value of a digit:
Value of digit = Digit × Place Value of its position
Example: In the number 45.273
| Digit | Position | Place Value | Value of Digit |
|---|---|---|---|
| 4 | Tens | 10 | 4 × 10 = 40 |
| 5 | Ones | 1 | 5 × 1 = 5 |
| 2 | Tenths | 0.1 | 2 × 0.1 = 0.2 |
| 7 | Hundredths | 0.01 | 7 × 0.01 = 0.07 |
| 3 | Thousandths | 0.001 | 3 × 0.001 = 0.003 |
Expanded form:
45.273 = 40 + 5 + 0.2 + 0.07 + 0.003
As fractions:
- 45.273 = 40 + 5 + 2/10 + 7/100 + 3/1000
Types and Properties
Types of problems on decimal place value:
1. Identifying place value of a digit:
- "What is the place value of 6 in 3.64?"
- 6 is in the tenths place → place value = 0.1.
- Value of the digit 6 = 6 × 0.1 = 0.6.
2. Writing in expanded form:
- Break the decimal into the sum of each digit times its place value.
- Example: 7.35 = 7 + 0.3 + 0.05.
3. Writing from expanded form to standard form:
- Add the expanded values together.
- Example: 20 + 3 + 0.4 + 0.08 = 23.48.
4. Counting decimal places:
- 3.7 has 1 decimal place (tenths).
- 3.72 has 2 decimal places (hundredths).
- 3.725 has 3 decimal places (thousandths).
5. Relationship between place values:
- Tenths place is 10 times the hundredths place.
- Hundredths place is 10 times the thousandths place.
- Each move to the right divides the place value by 10.
Solved Examples
Example 1: Example 1: Identifying place value
Problem: In the number 6.835, state the place value of each digit.
Solution:
| Digit | Place | Place Value | Value |
|---|---|---|---|
| 6 | Ones | 1 | 6 |
| 8 | Tenths | 0.1 | 0.8 |
| 3 | Hundredths | 0.01 | 0.03 |
| 5 | Thousandths | 0.001 | 0.005 |
Answer: 6 is in ones (value 6), 8 is in tenths (value 0.8), 3 is in hundredths (value 0.03), 5 is in thousandths (value 0.005).
Example 2: Example 2: Expanded form of a decimal
Problem: Write 32.56 in expanded form.
Solution:
- 3 is in the tens place → 30
- 2 is in the ones place → 2
- 5 is in the tenths place → 0.5
- 6 is in the hundredths place → 0.06
32.56 = 30 + 2 + 0.5 + 0.06
As fractions: 32.56 = 30 + 2 + 5/10 + 6/100
Answer: 32.56 = 30 + 2 + 0.5 + 0.06.
Example 3: Example 3: Standard form from expanded form
Problem: Write in standard form: 100 + 40 + 7 + 0.3 + 0.09 + 0.002.
Solution:
- Whole part: 100 + 40 + 7 = 147
- Decimal part: 0.3 + 0.09 + 0.002 = 0.392
- Standard form: 147.392
Answer: The standard form is 147.392.
Example 4: Example 4: Finding the value of a specific digit
Problem: What is the value of the digit 7 in: (a) 0.7 (b) 0.07 (c) 0.007?
Solution:
- (a) In 0.7, the 7 is in the tenths place → value = 7/10 = 0.7
- (b) In 0.07, the 7 is in the hundredths place → value = 7/100 = 0.07
- (c) In 0.007, the 7 is in the thousandths place → value = 7/1000 = 0.007
Observation: The same digit (7) has different values depending on its position. Moving one place to the right makes it 10 times smaller.
Answer: (a) 0.7, (b) 0.07, (c) 0.007.
Example 5: Example 5: Comparing digits in different positions
Problem: In 8.484, the digit 4 appears twice. What is its value in each position?
Solution:
- First 4: in the tenths place → value = 0.4
- Second 4: in the thousandths place → value = 0.004
Comparison:
- 0.4 ÷ 0.004 = 100
- The first 4 is 100 times the second 4.
Answer: The tenths-place 4 has value 0.4 and the thousandths-place 4 has value 0.004. The first is 100 times the second.
Example 6: Example 6: Writing a decimal from words
Problem: Write as a decimal: "Three hundred and five, and forty-seven thousandths."
Solution:
- Whole part: three hundred and five = 305
- Decimal part: forty-seven thousandths = 47/1000 = 0.047
- Combined: 305.047
Verification:
- 305.047 = 300 + 5 + 0.04 + 0.007
- = 300 + 5 + 4/100 + 7/1000
- = 305 + 47/1000 ✓
Answer: 305.047.
Example 7: Example 7: Decimal as a fraction
Problem: Write 0.625 as a fraction.
Solution:
0.625 has 3 decimal places (thousandths):
- 0.625 = 625/1000
Simplify by dividing by HCF (125):
- 625 ÷ 125 = 5
- 1000 ÷ 125 = 8
- = 5/8
Answer: 0.625 = 5/8.
Example 8: Example 8: Number of decimal places
Problem: How many decimal places do the following have? (a) 5.3 (b) 0.42 (c) 12.708 (d) 3.0
Solution:
- (a) 5.3 → 1 decimal place (tenths)
- (b) 0.42 → 2 decimal places (hundredths)
- (c) 12.708 → 3 decimal places (thousandths)
- (d) 3.0 → 1 decimal place (the trailing zero is still a decimal place, though 3.0 = 3)
Answer: (a) 1, (b) 2, (c) 3, (d) 1.
Example 9: Example 9: Expanded form with fractions
Problem: Write 9.205 in expanded form using fractions.
Solution:
- 9 is in the ones place → 9
- 2 is in the tenths place → 2/10
- 0 is in the hundredths place → 0/100 = 0
- 5 is in the thousandths place → 5/1000
9.205 = 9 + 2/10 + 0/100 + 5/1000
Simplified: 9.205 = 9 + 2/10 + 5/1000
Answer: 9.205 = 9 + 2/10 + 5/1000.
Example 10: Example 10: Building a decimal from clues
Problem: I am a decimal number. My ones digit is 4. My tenths digit is twice my ones digit. My hundredths digit is half my ones digit. What number am I?
Solution:
- Ones digit = 4
- Tenths digit = 2 × 4 = 8
- Hundredths digit = 4 ÷ 2 = 2
The number is: 4.82
Verification:
- Ones = 4 ✓
- Tenths = 8 (twice 4) ✓
- Hundredths = 2 (half of 4) ✓
Answer: The number is 4.82.
Real-World Applications
Real-world uses of decimal place value:
- Money: Rs. 45.75 means 45 rupees and 75 paise. The 7 is in the tenths place (7 ten-paise = 70 paise) and 5 is in the hundredths place (5 paise).
- Measurement: A ruler shows millimetres as decimal parts of a centimetre. 3.5 cm = 3 cm and 5 mm (5 tenths of a cm).
- Temperature: Body temperature 98.6°F uses tenths. Weather reports use decimals like 32.5°C.
- Sports: Running times are measured to hundredths: 10.85 seconds. The 8 in the tenths and 5 in the hundredths determine who wins.
- Science: Scientists measure very small quantities: 0.003 grams means 3 thousandths of a gram.
- Petrol prices: Prices like Rs. 96.72 per litre use decimals. The 7 tenths and 2 hundredths matter when you buy many litres.
Key Points to Remember
- The decimal point separates the whole number part from the fractional part.
- Places to the right of the decimal point are: tenths (1/10), hundredths (1/100), thousandths (1/1000).
- Each place to the right is 1/10 of the place to its left.
- The value of a digit = digit × place value of its position.
- The same digit has different values at different positions (e.g., 4 in tenths = 0.4, in hundredths = 0.04).
- Expanded form breaks a decimal into the sum of each digit's value.
- The number of digits after the decimal point is the number of decimal places.
- Trailing zeros after the last non-zero decimal digit do not change the value: 3.50 = 3.5.
- Leading zeros after the decimal point are important: 0.05 ≠ 0.5.
- Decimal place value extends the same base-10 system used for whole numbers.
Practice Problems
- Write the place value of each digit in 72.304.
- Write 56.91 in expanded form.
- Write in standard form: 200 + 8 + 0.6 + 0.03 + 0.001.
- What is the value of the digit 5 in (a) 3.5 (b) 3.05 (c) 3.005?
- How many decimal places does 100.020 have?
- In 4.444, how many times is the tens-digit 4 compared to the thousandths-digit 4?
- Write 'twenty and three hundred eight thousandths' as a decimal.
- Write 0.375 as a fraction in simplest form.
Frequently Asked Questions
Q1. What is decimal place value?
Decimal place value tells you the value of each digit based on its position after the decimal point. The first position after the decimal is tenths (1/10), the second is hundredths (1/100), the third is thousandths (1/1000), and so on.
Q2. What is the tenths place?
The tenths place is the first digit to the right of the decimal point. It represents 1/10 of a whole. In 3.7, the digit 7 is in the tenths place, meaning 7/10 or 0.7.
Q3. What is the difference between 0.5 and 0.05?
0.5 means 5 tenths (5/10 = 1/2). 0.05 means 5 hundredths (5/100 = 1/20). 0.5 is 10 times bigger than 0.05. The zero after the decimal point in 0.05 pushes the 5 to the hundredths place.
Q4. Does adding zeros at the end change a decimal?
No. Adding zeros at the end of a decimal does not change its value. 3.5 = 3.50 = 3.500. However, removing leading zeros after the decimal point does change the value: 0.05 is not the same as 0.5.
Q5. What is the expanded form of a decimal?
The expanded form writes each digit multiplied by its place value. For example, 4.72 = 4 + 0.7 + 0.02, which can also be written as 4 + 7/10 + 2/100.
Q6. How many decimal places can a number have?
A number can have any number of decimal places. In school, you usually work with 1 (tenths), 2 (hundredths), or 3 (thousandths) decimal places. The number pi (3.14159...) has infinitely many decimal places.
Q7. Is the decimal point a digit?
No. The decimal point is a separator, not a digit. It separates the whole number part from the fractional part. It does not have a place value.
Q8. How is decimal place value related to fractions?
Each decimal place represents a fraction with a power of 10 as the denominator. Tenths = 1/10, hundredths = 1/100, thousandths = 1/1000. So 0.375 = 3/10 + 7/100 + 5/1000 = 375/1000.
