Addition and Subtraction
Grade 5 Math - Complete Reference Guide
1. Estimate Sums and Differences
What is Estimation?
Estimation means finding an approximate answer rather than an exact one. We round numbers to make calculations easier.
Rounding Rules:
✓ If the digit is 0-4 → Round DOWN
✓ If the digit is 5-9 → Round UP
Steps to Estimate:
- Round each number to the desired place value (tens, hundreds, thousands)
- Add or subtract the rounded numbers
- Write the estimated answer
Examples:
Example 1: Estimate 3,847 + 2,156 (Round to nearest thousand)
3,847 → 4,000
2,156 → 2,000
Estimate: 4,000 + 2,000 = 6,000
(Actual answer: 6,003)
Example 2: Estimate 8,923 - 3,478 (Round to nearest hundred)
8,923 → 8,900
3,478 → 3,500
Estimate: 8,900 - 3,500 = 5,400
(Actual answer: 5,445)
| Place Value | Example Number | Rounded Value |
|---|---|---|
| Nearest 10 | 3,847 | 3,850 |
| Nearest 100 | 3,847 | 3,800 |
| Nearest 1,000 | 3,847 | 4,000 |
2. Estimate Sums and Differences: Word Problems
Steps to Solve Word Problems:
- Read the problem carefully
- Identify the numbers and operation (+ or -)
- Round the numbers to the nearest appropriate place value
- Perform the operation with rounded numbers
- Write the answer with proper units
Example Word Problems:
Problem 1: Addition
A school library has 4,876 fiction books and 3,234 non-fiction books. About how many books are there in total?
Solution:
Round to nearest thousand:
4,876 → 5,000
3,234 → 3,000
Estimate: 5,000 + 3,000 = 8,000 books
Problem 2: Subtraction
A concert hall has 9,568 seats. If 3,847 tickets have been sold, about how many seats are still available?
Solution:
Round to nearest thousand:
9,568 → 10,000
3,847 → 4,000
Estimate: 10,000 - 4,000 = 6,000 seats
💡 Pro Tip:
Always check if your estimate makes sense! If you're adding, the estimate should be close to the actual sum. If subtracting, make sure the larger number is first.
3. Add and Subtract Whole Numbers (Up to Billions)
Place Value Chart (Up to Billions):
| Billions | Hundred Millions | Ten Millions | Millions | Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|---|---|---|---|
| 1,000,000,000 | 100,000,000 | 10,000,000 | 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 |
Steps for Addition:
- Write numbers in columns, aligning place values
- Start adding from the ones place
- Carry over when sum ≥ 10
- Continue left through all place values
Steps for Subtraction:
- Write the larger number on top, aligned by place value
- Start subtracting from the ones place
- Borrow (regroup) from the next place value if needed
- Continue left through all place values
Examples:
Example 1: Addition
4,567,823,419 + 2,345,678,901 _______________ 6,913,502,320
Example 2: Subtraction with Regrouping
8,456,923,000 - 3,289,456,782 _______________ 5,167,466,218
Key Reminder:
Always line up digits by their place value! Use commas every three digits from right to left to make reading easier.
4. Add and Subtract: Word Problems
Key Words to Identify Operations:
| Addition Keywords | Subtraction Keywords |
|---|---|
|
• Total • Sum • Combined • In all • Altogether • Plus • Increased by |
• Difference • Minus • Less than • Fewer • Remaining • Left • Decreased by • How many more |
Example Problems:
Problem 1:
A city's population was 5,678,432 in 2020. By 2025, it increased by 1,234,567 people. What is the population in 2025?
Solution:
Operation: Addition (keyword: "increased by")
5,678,432 + 1,234,567 ___________ 6,912,999
Answer: 6,912,999 people
Problem 2:
A company earned $8,456,789 this year and spent $3,987,654 on expenses. How much profit did they make?
Solution:
Operation: Subtraction (finding what's left after expenses)
8,456,789 - 3,987,654 ___________ 4,469,135
Answer: $4,469,135
Problem 3: Multi-step
A library had 45,678 books. They received 12,345 new books and removed 8,234 damaged books. How many books does the library have now?
Solution:
Step 1: Add new books
45,678 + 12,345 ________ 58,023
Step 2: Subtract damaged books
58,023 - 8,234 ________ 49,789
Answer: 49,789 books
5. Complete Addition and Subtraction Sentences
What are Missing Number Problems?
These problems have an unknown value (often shown as □ or ?) that needs to be found. The unknown can be in any position: the first number, second number, or answer.
Types of Missing Number Problems:
Type 1: Missing Addend
□ + 345 = 890
Solution: Use subtraction → □ = 890 - 345 = 545
Type 2: Missing Sum
567 + 234 = □
Solution: Simply add → □ = 567 + 234 = 801
Type 3: Missing Minuend
□ - 456 = 321
Solution: Use addition → □ = 321 + 456 = 777
Type 4: Missing Subtrahend
890 - □ = 345
Solution: Use subtraction → □ = 890 - 345 = 545
Formulas to Remember:
| If Missing... | Formula |
|---|---|
| Missing Addend | Addend = Sum - Known Addend |
| Missing Sum | Sum = Addend₁ + Addend₂ |
| Missing Minuend | Minuend = Difference + Subtrahend |
| Missing Subtrahend | Subtrahend = Minuend - Difference |
| Missing Difference | Difference = Minuend - Subtrahend |
6. Properties of Addition
1. Commutative Property
Formula: a + b = b + a
The order of addends does not change the sum.
Example: 567 + 234 = 234 + 567 = 801
2. Associative Property
Formula: (a + b) + c = a + (b + c)
The grouping of addends does not change the sum.
Example: (12 + 25) + 8 = 12 + (25 + 8)
37 + 8 = 12 + 33
45 = 45 ✓
3. Identity Property (Zero Property)
Formula: a + 0 = a
When zero is added to any number, the sum is that number.
Example: 8,765 + 0 = 8,765
4. Inverse Property (Opposite Property)
Formula: a + (-a) = 0
Every number has an opposite. When you add opposites, the sum is zero.
Example: 456 + (-456) = 0
7. Add Using Properties
How Properties Make Addition Easier:
Using properties strategically can simplify mental math and make calculations faster!
Strategy 1: Look for Friendly Numbers
Problem: 87 + 56 + 13
Step 1: Use Commutative Property to rearrange
87 + 13 + 56
Step 2: Use Associative Property to group
(87 + 13) + 56
100 + 56 = 156
Why? 87 + 13 = 100 is easier to work with!
Strategy 2: Breaking Numbers Apart
Problem: 48 + 37
Break 37 into 2 + 35:
48 + 2 = 50
50 + 35 = 85
Why? Making 50 first makes addition easier!
Strategy 3: Using Zero Strategically
Problem: 678 + 0 + 234 + 0 + 89
Use Identity Property: Ignore the zeros
678 + 234 + 89 = 1,001
Why? Adding zero doesn't change the value!
Strategy 4: Doubles and Near-Doubles
Problem: 67 + 68
Think: This is close to a double!
67 + 67 = 134
134 + 1 = 135
Why? Doubling is easier than adding different numbers!
Quick Reference Summary
| Topic | Key Formula/Concept |
|---|---|
| Estimation | Round numbers, then add/subtract |
| Addition | Align place values, carry when ≥ 10 |
| Subtraction | Align place values, borrow/regroup when needed |
| Missing Addend | Addend = Sum - Known Addend |
| Missing Minuend | Minuend = Difference + Subtrahend |
| Commutative Property | a + b = b + a |
| Associative Property | (a + b) + c = a + (b + c) |
| Identity Property | a + 0 = a |
| Inverse Property | a + (-a) = 0 |
📝 Study Tips:
- Always line up place values when adding or subtracting
- Check your work by estimating first
- Use properties to make mental math easier
- Read word problems carefully to identify the operation
- Practice regrouping (carrying and borrowing) regularly