Complete Guide to Decimals
1. Understanding Decimals
A decimal number is a way of representing parts of a whole. The decimal point separates the whole number part from the fractional part.
In the number 123.456:
- 123 is the whole number part
- .456 is the decimal or fractional part
- 4 represents 4 tenths (4/10 = 0.4)
- 5 represents 5 hundredths (5/100 = 0.05)
- 6 represents 6 thousandths (6/1000 = 0.006)
Decimal Place Value
Each position in a decimal number has a specific value:
| Thousands | Hundreds | Tens | Ones | Decimal Point | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|---|
| 1,000 | 100 | 10 | 1 | . | 1/10 | 1/100 | 1/1000 |
| 1 × 10³ | 1 × 10² | 1 × 10¹ | 1 × 10⁰ | . | 1 × 10⁻¹ | 1 × 10⁻² | 1 × 10⁻³ |
Note: Moving one place to the left multiplies the value by 10. Moving one place to the right divides the value by 10.
Types of Decimals
- Terminating decimals: Decimals that end (e.g., 0.25 = 1/4)
- Recurring or repeating decimals: Decimals with a digit or group of digits that repeat indefinitely (e.g., 0.333... = 1/3)
- Non-terminating, non-recurring decimals: Decimals that continue infinitely without a pattern (e.g., π = 3.14159...)
2. Converting Fractions to Decimals
To convert a fraction to a decimal, divide the numerator by the denominator.
Example 1: Convert 3/4 to a decimal
Step 1: Divide the numerator by the denominator.
3 ÷ 4 = 0.75
Therefore, 3/4 = 0.75
Example 2: Convert 2/3 to a decimal
Step 1: Divide the numerator by the denominator.
2 ÷ 3 = 0.6666...
Therefore, 2/3 = 0.6666... or 0.6 (recurring)
3. Converting Decimals to Fractions
To convert a decimal to a fraction:
- For terminating decimals, write the decimal as a numerator over the appropriate power of 10
- For repeating decimals, use algebraic methods
Example 1: Convert 0.75 to a fraction
Step 1: Write as a fraction with denominator based on place value.
0.75 = 75/100
Step 2: Simplify the fraction.
75/100 = 3/4 (dividing both numbers by 25)
Therefore, 0.75 = 3/4
Example 2: Convert 0.333... to a fraction
Step 1: Let x = 0.333...
Step 2: Multiply both sides by 10.
10x = 3.333...
Step 3: Subtract the original equation.
10x - x = 3.333... - 0.333...
9x = 3
Step 4: Solve for x.
x = 3/9 = 1/3
Therefore, 0.333... = 1/3
4. Operations with Decimals
Addition and Subtraction of Decimals
To add or subtract decimals:
- Line up the decimal points
- Add or subtract as with whole numbers
- Place the decimal point in the answer directly below the decimal points in the problem
Example: Add 3.45 + 2.6
Step 1: Line up the decimal points.
3.45 + 2.60 ------
Step 2: Add as with whole numbers.
3.45 + 2.60 ------ 6.05
Therefore, 3.45 + 2.6 = 6.05
Multiplication of Decimals
To multiply decimals:
- Multiply as if they were whole numbers (ignore the decimal points)
- Count the total number of decimal places in both factors
- Place the decimal point in the product so that it has the same number of decimal places as the total in both factors
Example: Multiply 2.3 × 4.5
Step 1: Multiply as if they were whole numbers.
23 × 45 = 1035
Step 2: Count the total number of decimal places in both factors.
2.3 has 1 decimal place
4.5 has 1 decimal place
Total: 2 decimal places
Step 3: Place the decimal point in the product.
1035 with 2 decimal places is 10.35
Therefore, 2.3 × 4.5 = 10.35
Division of Decimals
To divide by a decimal:
- Move the decimal point in the divisor to the right until it becomes a whole number
- Move the decimal point in the dividend the same number of places to the right
- Perform the division as with whole numbers
- Place the decimal point in the quotient directly above where it is in the dividend
Example: Divide 4.5 ÷ 1.5
Step 1: Move the decimal point in the divisor to make it a whole number.
1.5 → 15 (moved 1 place right)
Step 2: Move the decimal point in the dividend the same number of places.
4.5 → 45 (moved 1 place right)
Step 3: Perform the division.
45 ÷ 15 = 3
Therefore, 4.5 ÷ 1.5 = 3
Rounding Decimals
To round a decimal to a specific place value:
- Identify the digit in the place value you want to round to
- Look at the digit to the right:
- If it's less than 5, keep the digit in the rounding place the same
- If it's 5 or greater, increase the digit in the rounding place by 1
Example 1: Round 3.427 to the nearest tenth
Step 1: Identify the digit in the tenths place.
3.427 → 4 is in the tenths place
Step 2: Look at the digit to the right.
The digit to the right of 4 is 2
Step 3: Since 2 is less than 5, keep the 4 the same.
3.427 rounded to the nearest tenth is 3.4
Example 2: Round 6.85 to the nearest tenth
Step 1: Identify the digit in the tenths place.
6.85 → 8 is in the tenths place
Step 2: Look at the digit to the right.
The digit to the right of 8 is 5
Step 3: Since 5 is 5 or greater, increase the 8 by 1.
6.85 rounded to the nearest tenth is 6.9
5. Word Problems with Decimals
Example 1: Money Problem
Sarah bought a book for $12.95 and a notebook for $3.49. How much did she spend in total?
Step 1: Add the two amounts.
$12.95 + $3.49 = $16.44
Sarah spent $16.44 in total.
Example 2: Measurement Problem
Tom ran 3.5 km on Monday, 2.75 km on Wednesday, and 4.25 km on Friday. What was the total distance he ran?
Step 1: Add all the distances.
3.5 km + 2.75 km + 4.25 km = 10.5 km
Tom ran a total distance of 10.5 km.
Example 3: Division Problem
A rope measuring 12.6 meters needs to be cut into pieces of 0.9 meters each. How many pieces can be made?
Step 1: Divide the total length by the length of each piece.
12.6 ÷ 0.9 = 14
14 pieces of rope can be made.
6. Comparing and Ordering Decimals
To compare decimals:
- Line up the decimal points
- Start comparing digits from left to right
- The first position where digits differ determines which number is larger
Example: Compare 4.52 and 4.6
Step 1: Line up the decimal points and add zeros if needed.
4.52 and 4.60
Step 2: Compare digits from left to right.
4 = 4 (first digit is the same)
5 < 6 (second digit differs, 5 is less than 6)
Therefore, 4.52 < 4.6