Adding and Subtracting Decimals

What Are Decimals?

Decimals are another way to show parts of a whole, like fractions.

The decimal point (.) separates the whole number from the part that's less than one.

Example: 3.5 = 3 and 5 tenths = 3 5/10

Place Value Review

Understanding place value is KEY to adding and subtracting decimals!

  Ones | Decimal | Tenths | Hundredths
    3  |    .    |   4    |     5

This is 3.45:

• 3 ones
• 4 tenths (4/10)
• 5 hundredths (5/100)

The Golden Rule: Line Up the Decimal Points!

Most important rule: Always line up the decimal points vertically!

Example: 4.3 + 2.6

CORRECT way:

  4.3
+ 2.6
-----
  6.9 ✓

WRONG way:

  4.3
+  2.6
-----
  (decimal points not lined up - will give wrong answer!)

Adding Decimals: Step-by-Step

Example: 5.4 + 3.8

Step 1: Write the numbers vertically, lining up decimal points

  5.4
+ 3.8
-----

Step 2: Add from right to left, just like whole numbers

  5.4
+ 3.8
-----
  9.2

Step 3: Bring the decimal point straight down

  5.4
+ 3.8
-----
  9.2 ✓

Adding with Different Decimal Places

Example: 6.5 + 2.34

Step 1: Line up decimal points

  6.5
+ 2.34
------

Step 2: Add placeholder zeros to help

  6.50  ← Added a zero
+ 2.34
------

Step 3: Add normally

  6.50
+ 2.34
------
  8.84 ✓

Adding with Regrouping (Carrying)

Example: 7.8 + 5.6

    ¹
  7.8
+ 5.6
-----
 13.4 ✓

Steps:

• Tenths: 8 + 6 = 14 tenths = 1.4
• Write 4, carry 1
• Ones: 7 + 5 + 1 = 13
• Answer: 13.4

Subtracting Decimals: Step-by-Step

Example: 8.7 - 3.2

Step 1: Line up decimal points

  8.7
- 3.2
-----

Step 2: Subtract from right to left

  8.7
- 3.2
-----
  5.5 ✓

Step 3: Bring decimal point straight down

Subtracting with Different Decimal Places

Example: 9.3 - 4.67

Step 1: Line up decimals and add zeros

  9.30  ← Added zero
- 4.67
------

Step 2: Subtract

  9.30
- 4.67
------
  4.63 ✓

Subtracting with Regrouping (Borrowing)

Example: 5.4 - 2.8

Step 1: Line up decimals

  5.4
- 2.8
-----

Step 2: Can't subtract 8 from 4, so borrow!

  ⁴¹⁴
  5.4  → Think: 5.4 = 4.14
- 2.8
-----
  2.6 ✓

Borrowing explained:

• 5 ones = 4 ones + 10 tenths
• 4 tenths + 10 tenths = 14 tenths
• Now: 14 tenths - 8 tenths = 6 tenths ✓

Adding Three or More Decimals

Example: 2.5 + 3.8 + 1.4

Line them all up:

  2.5
  3.8
+ 1.4
-----
  7.7 ✓

Real-World Examples: Money

Example: You have $5.75. You earn$3.50 more. How much now?

  $5.75
+ $3.50
-------
  $9.25 ✓

You have $9.25!

Example: You have $10.00. You spend$6.47. How much left?

 $10.00
-  $6.47
--------
  $3.53 ✓

You have $3.53 left!

Real-World Examples: Measurement

Running:

• Monday: 1.3 miles
• Tuesday: 2.7 miles
• Total: 1.3 + 2.7 = 4.0 miles ✓

Rainfall:

• Week 1: 2.8 inches
• Week 2: 1.6 inches
• Difference: 2.8 - 1.6 = 1.2 inches ✓

Using Estimation to Check

Before solving, estimate:

Example: 7.8 + 4.3 ≈ ?

• Round: 8 + 4 = 12
• So answer should be close to 12

Actual: 7.8 + 4.3 = 12.1 ✓ (Close to 12!)

Example: 9.6 - 3.2 ≈ ?

• Round: 10 - 3 = 7
• So answer should be close to 7

Actual: 9.6 - 3.2 = 6.4 ✓ (Close to 7!)

Adding Whole Numbers and Decimals

Example: 5 + 3.7

Add decimal point and zeros:

  5.0  ← 5 = 5.0
+ 3.7
-----
  8.7 ✓

Subtracting from Whole Numbers

Example: 8 - 3.4

Add decimal point and zero:

  8.0  ← 8 = 8.0
- 3.4
-----
  4.6 ✓

Word Problems

Problem 1: Sarah ran 2.5 miles on Saturday and 3.8 miles on Sunday. How many miles did she run in total?

Solution:

  2.5
+ 3.8
-----
  6.3 miles ✓

Problem 2: A bottle holds 1.5 liters. You drink 0.7 liters. How much is left?

Solution:

  1.5
- 0.7
-----
  0.8 liters ✓

Problem 3: You buy a toy for $12.95 and a book for$8.50. You pay with $25. How much change?

Solution:

Step 1: Find total cost
  $12.95
+  $8.50
--------
  $21.45

Step 2: Find change
  $25.00
- $21.45
--------
  $3.55 change ✓

Using a Number Line

Adding 2.3 + 1.5:

0 ―――1―――2―――2.3―――3―――3.8―――4
              ↑        ↑
            Start    +1.5
              └────────→

Start at 2.3, move right 1.5 → Land at 3.8 ✓

Common Mistakes

❌ Not lining up decimal points

WRONG:     RIGHT:
  4.5        4.5
+ 2.34     + 2.34
------     ------

❌ Forgetting to bring decimal point down

• Must keep decimal in same place!

❌ Adding/subtracting decimal points

• The decimal point doesn't move, just goes straight down!

❌ Not adding zeros as placeholders

• 5.3 + 2.47 → Write as 5.30 + 2.47

✅ Always line up the decimal points ✅ Add zeros to help line up columns ✅ Bring decimal straight down ✅ Check your answer with estimation

Practice Strategy

Before you start:

1. Write numbers vertically
2. Line up decimal points
3. Add zeros if needed to make same length

As you work:

1. Add or subtract from right to left
2. Regroup (carry/borrow) if needed
3. Bring decimal point straight down

After you finish:

1. Does your decimal point line up?
2. Does your answer make sense?
3. Check with estimation

Decimal Addition and Subtraction Rules

Key Rules:

1. Always line up the decimal points
2. Add zeros to make decimals the same length
3. Bring the decimal point straight down
4. Add or subtract like you do with whole numbers
5. Check your answer makes sense

Think About the Size

Adding decimals:

• Sum is always BIGGER than each number
• 3.5 + 2.8 should be more than 3.5 ✓

Subtracting decimals:

• Difference is always SMALLER than the first number
• 5.7 - 2.3 should be less than 5.7 ✓

Special Cases

When the answer is a whole number:

• 3.7 + 2.3 = 6.0 = 6 ✓
• Can drop the .0 at the end

When decimals "cancel out":

• 5.8 - 3.8 = 2.0 = 2 ✓

Remember!

The decimal point is like a fence post:

• It stays in one place
• Everything lines up on either side
• It goes straight down in your answer
• Never moves left or right!

Think of money:

• Adding/subtracting decimals is just like adding/subtracting dollars and cents
• $3.50 +$2.75 = $6.25
• Line up the dollar signs (and decimal points!)

✅ Line up decimal points = Success!