Add and Subtract Decimals

Fifth Grade Math - Complete Guide

➕ Adding Decimal Numbers

The Golden Rules

✅ Rule 1: Line Up the Decimal Points

Always write numbers so the decimal points are in a straight vertical line.

✅ Rule 2: Use Placeholder Zeros

Add zeros to make all numbers have the same number of decimal places.

✅ Rule 3: Add Like Regular Numbers

Add from right to left, carrying when needed. Bring decimal point straight down.

📝 Steps to Add Decimals

Write the numbers vertically with decimal points aligned
Add zeros as placeholders to make equal decimal places
Add each column starting from the rightmost digit
Carry over when the sum is 10 or more
Bring down the decimal point in the answer

💡 Examples

Example 1: Add $3.45 + 2.8$

Step 1: Line up decimals and add placeholder zero
  3.45
+ 2.80
------- Step 2: Add each column: 5+0=5, 4+8=12 (write 2, carry 1), 3+2+1=6
  3.45
+ 2.80
-------
  6.25

✓ Answer: $3.45 + 2.8 = 6.25$

Example 2: Add $12.7 + 5 + 3.456$

Step 1: Line up decimals and add placeholder zeros
12.700
+ 5.000
+ 3.456
-------- Step 2: Add: 0+0+6=6, 0+0+5=5, 7+0+4=11 (write 1, carry 1), 2+5+3+1=11, write 11
12.700
+ 5.000
+ 3.456
--------
21.156

✓ Answer: $12.7 + 5 + 3.456 = 21.156$

🔢 Properties to Add Three Decimals

Three Important Properties

1. Commutative Property

Order doesn't matter: $a + b = b + a$
Example: $2.5 + 3.7 = 3.7 + 2.5 = 6.2$

2. Associative Property

Grouping doesn't matter: $(a + b) + c = a + (b + c)$
Example: $(1.2 + 3.5) + 2.8 = 1.2 + (3.5 + 2.8) = 7.5$

3. Identity Property

Adding zero doesn't change the number: $a + 0 = a$
Example: $5.68 + 0 = 5.68$

💡 Strategy: Look for Compatible Numbers

Compatible numbers are pairs that are easy to add mentally (like numbers that sum to whole numbers).

Example: Add $2.7 + 5.5 + 3.3$

Strategy: Notice 2.7 + 3.3 = 6.0 (compatible numbers!)
Step 1: Regroup → $(2.7 + 3.3) + 5.5$
Step 2: Add compatible pair → $6.0 + 5.5$
Step 3: Final sum → $11.5$

✓ Answer: $11.5$ (Easier than adding in order!)

➖ Subtracting Decimal Numbers

The Rules (Same as Addition!)

✅ Rule 1: Line Up the Decimal Points

Always write numbers so the decimal points are aligned vertically.

✅ Rule 2: Add Placeholder Zeros

Add zeros to make equal decimal places. This helps with borrowing!

✅ Rule 3: Subtract and Borrow When Needed

Subtract from right to left, borrowing when needed. Bring decimal straight down.

📝 Steps to Subtract Decimals

Write the larger number on top, decimal points aligned
Add zeros as placeholders if needed
Subtract each column from right to left
Borrow from the left when top digit is smaller
Bring down the decimal point

💡 Examples

Example 1: Subtract $8.65 - 3.42$

Step 1: Line up decimals (already same decimal places)
8.65
- 3.42
------ Step 2: Subtract: 5-2=3, 6-4=2, 8-3=5
8.65
- 3.42
------
5.23

✓ Answer: $8.65 - 3.42 = 5.23$

Example 2: Subtract $15 - 7.48$ (with borrowing)

Step 1: Add decimal and zeros to 15
15.00
- 7.48
------- Step 2: Subtract with borrowing:
• Can't do 0-8, borrow: 10-8=2
• Can't do 9-4 (after borrowing), borrow again: 9-4=5
• 14-7=7
15.00
- 7.48
-------
7.52

✓ Answer: $15 - 7.48 = 7.52$

🧠 Compensation Method (Mental Math)

What is Compensation?

Compensation is a mental math strategy where you adjust numbers to make them easier to work with, then adjust your answer to compensate.

💡 Key Idea:

Round one number to make math easier, then "compensate" by adjusting the final answer!

Compensation for Addition

Round up one number → Add → Subtract the extra amount

Example: $3.8 + 2.95$

Step 1: Round 2.95 up to 3.00 (added 0.05)
Step 2: Add the easier numbers → $3.8 + 3.0 = 6.8$
Step 3: Compensate by subtracting 0.05 → $6.8 - 0.05 = 6.75$

✓ Answer: $6.75$

Compensation for Subtraction

Add the same amount to both numbers → Subtract

Example: $8.3 - 3.97$

Step 1: Add 0.03 to both numbers to make 3.97 become 4.00
$8.3 + 0.03 = 8.33$
$3.97 + 0.03 = 4.00$
Step 2: Subtract the easier numbers → $8.33 - 4.00 = 4.33$

✓ Answer: $4.33$

🎯 Estimate Using Rounding

Why Estimate?

Estimation helps you quickly check if your answer is reasonable without doing exact calculations.

Rounding Rules:

• If digit is 0-4 → Round DOWN
• If digit is 5-9 → Round UP

📝 Steps to Estimate

Round each decimal to the nearest whole number or tenth
Add or subtract the rounded numbers
Compare your estimate with the exact answer

💡 Examples

Example 1: Estimate $12.78 + 7.23$

Round to nearest whole:
• $12.78 \approx 13$
• $7.23 \approx 7$
Estimate: $13 + 7 = 20$
Actual: $12.78 + 7.23 = 20.01$ ✓ Very close!

Example 2: Estimate $25.4 - 8.82$

Round to nearest whole:
• $25.4 \approx 25$
• $8.82 \approx 9$
Estimate: $25 - 9 = 16$
Actual: $25.4 - 8.82 = 16.58$ ✓ Close estimate!

📍 Estimate Using Benchmarks

What are Benchmark Decimals?

Benchmarks are familiar reference points that help us estimate quickly.

Common Benchmark Decimals

$0$ | $0.25$ | $0.50$ | $0.75$ | $1.0$

Rounding to Benchmarks:

• 0.00 to 0.12 → rounds to 0
• 0.13 to 0.37 → rounds to 0.25
• 0.38 to 0.62 → rounds to 0.50
• 0.63 to 0.87 → rounds to 0.75
• 0.88 to 1.00 → rounds to 1.0

💡 Examples

Example 1: Estimate $0.68 + 0.47$

Round to benchmarks:
• $0.68 \approx 0.75$
• $0.47 \approx 0.50$
Estimate: $0.75 + 0.50 = 1.25$
Actual: $0.68 + 0.47 = 1.15$ ✓ Close!

Example 2: Estimate $0.89 - 0.34$

Round to benchmarks:
• $0.89 \approx 1.0$
• $0.34 \approx 0.25$
Estimate: $1.0 - 0.25 = 0.75$
Actual: $0.89 - 0.34 = 0.55$ ✓ Reasonable estimate!

📖 Word Problems

🎯 Steps to Solve

READ the problem carefully
IDENTIFY the decimal numbers
DECIDE if you need to add or subtract
SOLVE using proper steps
CHECK if your answer makes sense

Problem 1: Addition

Maria bought a book for $12.75 and a pen for $3.50. How much did she spend in total?

Step 1: Identify → $12.75 and $3.50
Step 2: Operation → Addition (total = add)
Step 3: Solve → $12.75 + 3.50 = 16.25$

✓ Answer: Maria spent $16.25

Problem 2: Subtraction

Tom had 25.6 meters of rope. He used 12.85 meters for a project. How much rope does he have left?

Step 1: Identify → 25.6 m and 12.85 m
Step 2: Operation → Subtraction (left = subtract)
Step 3: Solve → $25.60 - 12.85 = 12.75$

✓ Answer: Tom has 12.75 meters left

🔢 Number Sequences with Decimals

What is a Pattern?

A sequence follows a rule where you add or subtract the same amount each time.

💡 Examples

Example 1: Find the pattern: 2.5, 3.0, 3.5, 4.0, ___

Step 1: Find the rule → $3.0 - 2.5 = 0.5$
Step 2: Rule = Add 0.5 each time
Step 3: Next number → $4.0 + 0.5 = 4.5$

✓ Answer: 4.5

Example 2: Find the pattern: 10.2, 9.5, 8.8, 8.1, ___

Step 1: Find the rule → $9.5 - 10.2 = -0.7$
Step 2: Rule = Subtract 0.7 each time
Step 3: Next number → $8.1 - 0.7 = 7.4$

✓ Answer: 7.4

📋 Quick Reference Summary

Operation	Key Steps
Addition	Line up decimals → Add zeros → Add → Bring down decimal
Subtraction	Line up decimals → Add zeros → Subtract with borrowing → Bring down decimal
Compensation	Round to easy number → Calculate → Adjust answer
Estimation (Rounding)	Round to nearest whole or tenth → Calculate
Estimation (Benchmarks)	Round to 0, 0.25, 0.50, 0.75, or 1 → Calculate