Decimal Place Value

Fifth Grade Math - Complete Guide

📚 Understanding Decimals

What is a Decimal Number?

A decimal number is made up of a whole number part and a fractional part separated by a dot called the decimal point.

Example: \(42.75\)

42 = Whole number part | . = Decimal point | 75 = Fractional part

Important Rule

As we move left from the decimal point, each place value becomes 10 times greater.

As we move right from the decimal point, each place value becomes 10 times smaller.

📊 Decimal Place Value Chart

Complete Place Value Chart

Thousands	Hundreds	Tens	Ones	●	Tenths	Hundredths	Thousandths
1,000	100	10	1	.	\(\frac{1}{10}\) = 0.1	\(\frac{1}{100}\) = 0.01	\(\frac{1}{1000}\) = 0.001
3	2	5	6	.	7	4	8

Example: \(3256.748\) = Three thousand two hundred fifty-six and seven hundred forty-eight thousandths

Understanding Each Place Value

Left of Decimal Point (Whole Numbers):

Ones → Tens → Hundreds → Thousands → Ten Thousands

Right of Decimal Point (Fractional Parts):

Tenths → Hundredths → Thousandths → Ten-Thousandths

🔍 Finding Place Values in Decimal Numbers

Formula for Place Value

\[\text{Place Value} = \text{Digit} \times \text{Position Value}\]

💡 Example: Find place values in \(673.258\)

Digit	Position	Place Value	Value
6	Hundreds	\(6 \times 100\)	600
7	Tens	\(7 \times 10\)	70
3	Ones	\(3 \times 1\)	3
2	Tenths	\(2 \times \frac{1}{10}\)	0.2
5	Hundredths	\(5 \times \frac{1}{100}\)	0.05
8	Thousandths	\(8 \times \frac{1}{1000}\)	0.008

🔗 Relationship Between Decimal Place Values

Key Relationships

Moving LEFT (×10):

Each place value is 10 times greater than the one to its right
Tenths → Ones → Tens → Hundreds → Thousands

Moving RIGHT (÷10):

Each place value is 10 times smaller than the one to its left
Thousands → Hundreds → Tens → Ones → Tenths

💡 Examples

1 one = 10 tenths

\(1 = 10 \times 0.1\)

1 tenth = 10 hundredths

\(0.1 = 10 \times 0.01\)

1 hundredth = 10 thousandths

\(0.01 = 10 \times 0.001\)

1 ten = 10 ones

\(10 = 10 \times 1\)

📝 Expanded Form of Decimals

Standard Form vs Expanded Form

Standard Form: The regular way to write a number

Expanded Form: Writing each digit according to its place value and adding them

Method 1: Expanded Form (Decimals)

Example: Write \(83.34\) in expanded form

Step 1: Expand whole number part → \(80 + 3\)
Step 2: Expand decimal part → \(0.3 + 0.04\)
Step 3: Combine → \(80 + 3 + 0.3 + 0.04\)

✓ Answer: \(83.34 = 80 + 3 + 0.3 + 0.04\)

Method 2: Expanded Form Using Fractions

\[\text{Expanded Form} = (\text{Digit} \times \text{Place Value})\]

Example: Write \(1.234\) in expanded form using fractions

Step 1: Ones place → \(1 \times 1 = 1\)
Step 2: Tenths place → \(2 \times \frac{1}{10} = \frac{2}{10}\)
Step 3: Hundredths place → \(3 \times \frac{1}{100} = \frac{3}{100}\)
Step 4: Thousandths place → \(4 \times \frac{1}{1000} = \frac{4}{1000}\)
Step 5: Combine all

✓ Answer: \(1 + \frac{2}{10} + \frac{3}{100} + \frac{4}{1000}\)

💬 Writing Decimals in Words

📝 Steps to Write Decimals in Words

Write the whole number part in words
Say "and" for the decimal point
Write the decimal part as a whole number
Add the place value name of the last digit

💡 Examples

\(32.5\)

Thirty-two and five tenths

\(7.65\)

Seven and sixty-five hundredths

\(124.082\)

One hundred twenty-four and eighty-two thousandths

\(456.8\)

Four hundred fifty-six and eight tenths

🔄 Compose and Decompose Decimals

What Does It Mean?

Compose:

Putting parts together to make a decimal number

Decompose:

Breaking a decimal number into parts in different ways

💡 Example: Decompose \(2.45\) in Multiple Ways

Way 1: By Place Value

\(2 + 0.4 + 0.05\)

Way 2: Using Whole and Decimal

\(2 + 0.45\)

Way 3: All as Tenths

\(2.0 + 0.3 + 0.15\) or \(1 + 1 + 0.45\)

Way 4: Using Fractions

\(2 + \frac{4}{10} + \frac{5}{100}\)

🎯 Rounding Decimals

Rounding Rules

If the digit is 0, 1, 2, 3, or 4:

Round DOWN (keep the rounding digit the same)

If the digit is 5, 6, 7, 8, or 9:

Round UP (add 1 to the rounding digit)

📝 Steps to Round Decimals

Find the place value you want to round to
Look at the digit to the RIGHT of that place
If it's 5 or more → Round UP
If it's 4 or less → Round DOWN
Drop all digits after the rounding place

💡 Examples

Example 1: Round \(3.67\) to the nearest tenth

Step 1: Identify tenths place → 6
Step 2: Look at the digit to the right → 7
Step 3: Since 7 ≥ 5, round UP → \(6 + 1 = 7\)
Step 4: Drop digits after tenths place

✓ Answer: \(3.7\)

Example 2: Round \(0.439\) to the nearest hundredth

Step 1: Identify hundredths place → 3
Step 2: Look at the digit to the right → 9
Step 3: Since 9 ≥ 5, round UP → \(3 + 1 = 4\)
Step 4: Drop digits after hundredths place

✓ Answer: \(0.44\)

Example 3: Round \(12.832\) to the nearest whole number

Step 1: Identify ones place → 2
Step 2: Look at the digit to the right → 8
Step 3: Since 8 ≥ 5, round UP → \(12 + 1 = 13\)
Step 4: Drop the decimal part

✓ Answer: \(13\)

📏 Decimal Number Lines

Understanding Decimal Number Lines

A decimal number line helps us visualize where decimal numbers are located and compare their values.

Example: Number Line from 0 to 1 (Tenths)

0────0.1────0.2────0.3────0.4────0.5────0.6────0.7────0.8────0.9────1

Each mark represents one tenth (0.1)

Example: Number Line from 0 to 0.1 (Hundredths)

0──0.01──0.02──0.03──0.04──0.05──0.06──0.07──0.08──0.09──0.1

Each mark represents one hundredth (0.01)

💡 Key Points

Numbers to the right are greater
Numbers to the left are smaller
The space between marks shows the interval
Count the marks carefully to identify the decimal value

📋 Quick Reference Summary

Concept	Key Formula/Rule
Place Value	\(\text{Digit} \times \text{Position Value}\)
Tenths	\(\frac{1}{10} = 0.1\)
Hundredths	\(\frac{1}{100} = 0.01\)
Thousandths	\(\frac{1}{1000} = 0.001\)
Relationship	Each place = 10 × place to the right
Rounding (≥5)	Round UP (add 1)
Rounding (<5)	Round DOWN (stay same)