Multi-digit addition and subtraction.
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Lesson Objectives
- Add multi-digit numbers accurately, including carrying (regrouping)
- Subtract multi-digit numbers accurately, including borrowing (regrouping)
- Solve word problems involving multi-digit addition and subtraction
- Use estimation to check the reasonableness of answers
- Apply addition and subtraction skills to real-world contexts
Introduction to Multi-Digit Addition and Subtraction
Adding and subtracting multi-digit numbers is a fundamental skill used in everyday life — from calculating totals at the grocery store to balancing a checkbook and measuring distances. In Grade 8, we review and reinforce these skills, focusing on accuracy, efficiency, and problem-solving. Mastering these operations prepares you for more advanced mathematics like algebra, where these skills are used constantly.
Key Concepts
Addition: Combining quantities to find a total (sum).
Subtraction: Finding the difference between two quantities.
Place Value: The value of a digit based on its position (ones, tens, hundreds, etc.).
Regrouping (Carrying/Borrowing): Moving value from one place to another when adding or subtracting.
Addition: Combining quantities to find a total (sum).
Subtraction: Finding the difference between two quantities.
Place Value: The value of a digit based on its position (ones, tens, hundreds, etc.).
Regrouping (Carrying/Borrowing): Moving value from one place to another when adding or subtracting.
Key Definitions:
• Addend: A number being added together with other numbers.
• Sum: The result of addition.
• Minuend: The number from which another number is subtracted.
• Subtrahend: The number being subtracted.
• Difference: The result of subtraction.
• Regrouping (Carrying): When the sum of digits in a column is 10 or more, we "carry" to the next column.
• Regrouping (Borrowing): When a digit in the minuend is smaller than the digit below it, we "borrow" from the next column.
• Addend: A number being added together with other numbers.
• Sum: The result of addition.
• Minuend: The number from which another number is subtracted.
• Subtrahend: The number being subtracted.
• Difference: The result of subtraction.
• Regrouping (Carrying): When the sum of digits in a column is 10 or more, we "carry" to the next column.
• Regrouping (Borrowing): When a digit in the minuend is smaller than the digit below it, we "borrow" from the next column.
Multi-Digit Addition
Adding multi-digit numbers requires careful alignment of place values. Start from the ones column and move left, regrouping (carrying) when the sum of a column is 10 or greater.
Step-by-Step Method for Multi-Digit Addition:
1. Write the numbers vertically, aligning digits by place value (ones under ones, tens under tens, etc.).
2. Start adding from the ones column (rightmost).
3. If the sum is less than 10, write it in that column.
4. If the sum is 10 or greater, write the ones digit and carry the tens digit to the next column.
5. Repeat for each column (tens, hundreds, thousands, etc.).
6. Continue until all columns are added.
1. Write the numbers vertically, aligning digits by place value (ones under ones, tens under tens, etc.).
2. Start adding from the ones column (rightmost).
3. If the sum is less than 10, write it in that column.
4. If the sum is 10 or greater, write the ones digit and carry the tens digit to the next column.
5. Repeat for each column (tens, hundreds, thousands, etc.).
6. Continue until all columns are added.
Example 1: Simple Addition with Carrying
Problem: Add 3,478 + 2,569.
Solution:
3,478
+ 2,569
--------
Step 1: Ones: 8 + 9 = 17 → write 7, carry 1 to tens.
Step 2: Tens: 7 + 6 + 1 (carried) = 14 → write 4, carry 1 to hundreds.
Step 3: Hundreds: 4 + 5 + 1 (carried) = 10 → write 0, carry 1 to thousands.
Step 4: Thousands: 3 + 2 + 1 (carried) = 6 → write 6.
Answer: 6,047
Problem: Add 3,478 + 2,569.
Solution:
3,478
+ 2,569
--------
Step 1: Ones: 8 + 9 = 17 → write 7, carry 1 to tens.
Step 2: Tens: 7 + 6 + 1 (carried) = 14 → write 4, carry 1 to hundreds.
Step 3: Hundreds: 4 + 5 + 1 (carried) = 10 → write 0, carry 1 to thousands.
Step 4: Thousands: 3 + 2 + 1 (carried) = 6 → write 6.
Answer: 6,047
Example 2: Adding Three Numbers
Problem: Add 12,345 + 23,456 + 34,567.
Solution:
12,345
23,456
+ 34,567
--------
Ones: 5 + 6 + 7 = 18 → write 8, carry 1.
Tens: 4 + 5 + 6 + 1 = 16 → write 6, carry 1.
Hundreds: 3 + 4 + 5 + 1 = 13 → write 3, carry 1.
Thousands: 2 + 3 + 4 + 1 = 10 → write 0, carry 1.
Ten-thousands: 1 + 2 + 3 + 1 = 7 → write 7.
Answer: 70,368
Problem: Add 12,345 + 23,456 + 34,567.
Solution:
12,345
23,456
+ 34,567
--------
Ones: 5 + 6 + 7 = 18 → write 8, carry 1.
Tens: 4 + 5 + 6 + 1 = 16 → write 6, carry 1.
Hundreds: 3 + 4 + 5 + 1 = 13 → write 3, carry 1.
Thousands: 2 + 3 + 4 + 1 = 10 → write 0, carry 1.
Ten-thousands: 1 + 2 + 3 + 1 = 7 → write 7.
Answer: 70,368
Example 3: Adding with Different Numbers of Digits
Problem: Add 45,678 + 9,876.
Solution:
45,678
+ 9,876
--------
Align: Write 9,876 as 09,876.
Ones: 8 + 6 = 14 → write 4, carry 1.
Tens: 7 + 7 + 1 = 15 → write 5, carry 1.
Hundreds: 6 + 8 + 1 = 15 → write 5, carry 1.
Thousands: 5 + 9 + 1 = 15 → write 5, carry 1.
Ten-thousands: 4 + 0 + 1 = 5 → write 5.
Answer: 55,554
Problem: Add 45,678 + 9,876.
Solution:
45,678
+ 9,876
--------
Align: Write 9,876 as 09,876.
Ones: 8 + 6 = 14 → write 4, carry 1.
Tens: 7 + 7 + 1 = 15 → write 5, carry 1.
Hundreds: 6 + 8 + 1 = 15 → write 5, carry 1.
Thousands: 5 + 9 + 1 = 15 → write 5, carry 1.
Ten-thousands: 4 + 0 + 1 = 5 → write 5.
Answer: 55,554
Example 4: Addition with Decimals
Problem: Add 12.75 + 8.36.
Solution:
12.75
+ 8.36
-------
Hundredths: 5 + 6 = 11 → write 1, carry 1 to tenths.
Tenths: 7 + 3 + 1 = 11 → write 1, carry 1 to ones.
Ones: 2 + 8 + 1 = 11 → write 1, carry 1 to tens.
Tens: 1 + 0 + 1 = 2 → write 2.
Answer: 21.11
Problem: Add 12.75 + 8.36.
Solution:
12.75
+ 8.36
-------
Hundredths: 5 + 6 = 11 → write 1, carry 1 to tenths.
Tenths: 7 + 3 + 1 = 11 → write 1, carry 1 to ones.
Ones: 2 + 8 + 1 = 11 → write 1, carry 1 to tens.
Tens: 1 + 0 + 1 = 2 → write 2.
Answer: 21.11
Watch Out!
Always align numbers by their place values (right-align for whole numbers, align decimal points for decimals). Forgetting to carry over is a common mistake — double-check each column!
Always align numbers by their place values (right-align for whole numbers, align decimal points for decimals). Forgetting to carry over is a common mistake — double-check each column!
Practice for Multi-Digit Addition
- Add 6,789 + 4,567.
- Add 123,456 + 789,012.
- Add 9,876 + 5,432 + 1,234.
- Add 45.67 + 23.89.
- Add 8,000 + 5,678 + 432.
Multi-Digit Subtraction
Subtracting multi-digit numbers also requires careful alignment of place values. Start from the ones column and move left, regrouping (borrowing) when the digit in the minuend is smaller than the digit below it.
Step-by-Step Method for Multi-Digit Subtraction:
1. Write the numbers vertically, aligning digits by place value (minuend on top, subtrahend below).
2. Start subtracting from the ones column (rightmost).
3. If the top digit is greater than or equal to the bottom digit, subtract and write the difference.
4. If the top digit is smaller, borrow 1 from the next column to the left (add 10 to the current digit).
5. Reduce the digit in the next column by 1 before proceeding.
6. Repeat for each column.
1. Write the numbers vertically, aligning digits by place value (minuend on top, subtrahend below).
2. Start subtracting from the ones column (rightmost).
3. If the top digit is greater than or equal to the bottom digit, subtract and write the difference.
4. If the top digit is smaller, borrow 1 from the next column to the left (add 10 to the current digit).
5. Reduce the digit in the next column by 1 before proceeding.
6. Repeat for each column.
Example 1: Simple Subtraction without Borrowing
Problem: Subtract 5,678 - 3,456.
Solution:
5,678
- 3,456
--------
Ones: 8 - 6 = 2
Tens: 7 - 5 = 2
Hundreds: 6 - 4 = 2
Thousands: 5 - 3 = 2
Answer: 2,222
Problem: Subtract 5,678 - 3,456.
Solution:
5,678
- 3,456
--------
Ones: 8 - 6 = 2
Tens: 7 - 5 = 2
Hundreds: 6 - 4 = 2
Thousands: 5 - 3 = 2
Answer: 2,222
Example 2: Subtraction with Borrowing
Problem: Subtract 5,234 - 2,876.
Solution:
5,234
- 2,876
--------
Ones: 4 - 6 → need to borrow. Borrow from tens: 3 becomes 2, 4 becomes 14. 14 - 6 = 8.
Tens: 2 - 7 → need to borrow. Borrow from hundreds: 2 becomes 1, 2 becomes 12. 12 - 7 = 5.
Hundreds: 1 - 8 → need to borrow. Borrow from thousands: 5 becomes 4, 1 becomes 11. 11 - 8 = 3.
Thousands: 4 - 2 = 2.
Answer: 2,358
Problem: Subtract 5,234 - 2,876.
Solution:
5,234
- 2,876
--------
Ones: 4 - 6 → need to borrow. Borrow from tens: 3 becomes 2, 4 becomes 14. 14 - 6 = 8.
Tens: 2 - 7 → need to borrow. Borrow from hundreds: 2 becomes 1, 2 becomes 12. 12 - 7 = 5.
Hundreds: 1 - 8 → need to borrow. Borrow from thousands: 5 becomes 4, 1 becomes 11. 11 - 8 = 3.
Thousands: 4 - 2 = 2.
Answer: 2,358
Example 3: Subtraction with Multiple Borrowing
Problem: Subtract 8,000 - 3,456.
Solution:
8,000
- 3,456
--------
Ones: 0 - 6 → borrow from tens, but tens is 0, so borrow from hundreds, which is 0, so borrow from thousands.
Thousands: 8 becomes 7, hundreds become 10.
Hundreds: 10 becomes 9 (after lending 1 to tens), tens become 10.
Tens: 10 becomes 9 (after lending 1 to ones), ones become 10.
Ones: 10 - 6 = 4
Tens: 9 - 5 = 4
Hundreds: 9 - 4 = 5
Thousands: 7 - 3 = 4
Answer: 4,544
Problem: Subtract 8,000 - 3,456.
Solution:
8,000
- 3,456
--------
Ones: 0 - 6 → borrow from tens, but tens is 0, so borrow from hundreds, which is 0, so borrow from thousands.
Thousands: 8 becomes 7, hundreds become 10.
Hundreds: 10 becomes 9 (after lending 1 to tens), tens become 10.
Tens: 10 becomes 9 (after lending 1 to ones), ones become 10.
Ones: 10 - 6 = 4
Tens: 9 - 5 = 4
Hundreds: 9 - 4 = 5
Thousands: 7 - 3 = 4
Answer: 4,544
Example 4: Subtraction with Decimals
Problem: Subtract 45.23 - 12.78.
Solution:
45.23
- 12.78
-------
Hundredths: 3 - 8 → borrow from tenths: 2 becomes 1, 3 becomes 13. 13 - 8 = 5.
Tenths: 1 - 7 → borrow from ones: 5 becomes 4, 1 becomes 11. 11 - 7 = 4.
Ones: 4 - 2 = 2
Tens: 4 - 1 = 3
Answer: 32.45
Problem: Subtract 45.23 - 12.78.
Solution:
45.23
- 12.78
-------
Hundredths: 3 - 8 → borrow from tenths: 2 becomes 1, 3 becomes 13. 13 - 8 = 5.
Tenths: 1 - 7 → borrow from ones: 5 becomes 4, 1 becomes 11. 11 - 7 = 4.
Ones: 4 - 2 = 2
Tens: 4 - 1 = 3
Answer: 32.45
Watch Out!
When borrowing, remember that you are borrowing from the next column. Reduce that column by 1 and add 10 to the current column. Be careful when borrowing across multiple zeros — you may need to borrow several times.
When borrowing, remember that you are borrowing from the next column. Reduce that column by 1 and add 10 to the current column. Be careful when borrowing across multiple zeros — you may need to borrow several times.
Practice for Multi-Digit Subtraction
- Subtract 9,876 - 5,432.
- Subtract 100,000 - 34,567.
- Subtract 8,000 - 2,345.
- Subtract 56.78 - 23.89.
- Subtract 45,678 - 29,876.
Word Problems with Addition and Subtraction
Many real-world problems require adding or subtracting multi-digit numbers. Understanding the context helps determine which operation to use.
Step-by-Step Method for Word Problems:
1. Read the problem carefully to understand what is being asked.
2. Identify the key numbers and what they represent.
3. Decide whether to add or subtract based on keywords (total, sum, more than, less than, difference, remaining).
4. Set up the calculation vertically.
5. Perform the calculation accurately.
6. Write the answer in a complete sentence.
1. Read the problem carefully to understand what is being asked.
2. Identify the key numbers and what they represent.
3. Decide whether to add or subtract based on keywords (total, sum, more than, less than, difference, remaining).
4. Set up the calculation vertically.
5. Perform the calculation accurately.
6. Write the answer in a complete sentence.
Example 1: Addition Word Problem
Problem: A school has 1,234 students in primary and 876 students in secondary. How many students are in the school altogether?
Solution:
Add: 1,234 + 876 = 2,110
Answer: The school has 2,110 students altogether.
Problem: A school has 1,234 students in primary and 876 students in secondary. How many students are in the school altogether?
Solution:
Add: 1,234 + 876 = 2,110
Answer: The school has 2,110 students altogether.
Example 2: Subtraction Word Problem
Problem: A car dealer had 5,678 cars in stock. After a sale, 2,345 cars were sold. How many cars remain?
Solution:
Subtract: 5,678 - 2,345 = 3,333
Answer: 3,333 cars remain.
Problem: A car dealer had 5,678 cars in stock. After a sale, 2,345 cars were sold. How many cars remain?
Solution:
Subtract: 5,678 - 2,345 = 3,333
Answer: 3,333 cars remain.
Example 3: Two-Step Word Problem
Problem: A shopkeeper bought goods for ₦45,678 and sold them for ₦56,789. He also paid ₦5,000 for transport. What was his profit?
Solution:
Step 1: Find total cost: 45,678 + 5,000 = 50,678
Step 2: Find profit: 56,789 - 50,678 = 6,111
Answer: The profit was ₦6,111.
Problem: A shopkeeper bought goods for ₦45,678 and sold them for ₦56,789. He also paid ₦5,000 for transport. What was his profit?
Solution:
Step 1: Find total cost: 45,678 + 5,000 = 50,678
Step 2: Find profit: 56,789 - 50,678 = 6,111
Answer: The profit was ₦6,111.
Example 4: Comparison Problem
Problem: City A has a population of 2,345,678. City B has a population of 1,987,654. How many more people live in City A than City B?
Solution:
Subtract: 2,345,678 - 1,987,654 = 358,024
Answer: City A has 358,024 more people.
Problem: City A has a population of 2,345,678. City B has a population of 1,987,654. How many more people live in City A than City B?
Solution:
Subtract: 2,345,678 - 1,987,654 = 358,024
Answer: City A has 358,024 more people.
Watch Out!
Look for keywords that indicate addition (total, sum, altogether, combined, in all) and subtraction (difference, remaining, left, how many more, less than). Some problems require multiple steps!
Look for keywords that indicate addition (total, sum, altogether, combined, in all) and subtraction (difference, remaining, left, how many more, less than). Some problems require multiple steps!
Practice for Word Problems
- A library has 12,345 fiction books and 8,765 non-fiction books. How many books in total?
- A farmer harvested 45,678 kg of rice and sold 23,456 kg. How much rice remains?
- A company earned ₦567,890 in January and ₦678,901 in February. How much more did they earn in February?
- A school spent ₦45,678 on computers and ₦23,456 on furniture. The total budget was ₦100,000. How much money is left?
- Two towns are 345 km apart. A train travels 234 km. How much further does it need to go?
Estimation and Checking Answers
Estimation is a valuable tool for checking whether your answer is reasonable. By rounding numbers before adding or subtracting, you can quickly see if your exact answer makes sense.
Step-by-Step Method for Checking with Estimation:
1. Round each number to the nearest ten, hundred, or thousand.
2. Perform the operation with the rounded numbers.
3. Compare the estimate to your exact answer.
4. If they are close, your answer is likely correct. If they are far apart, check your work.
1. Round each number to the nearest ten, hundred, or thousand.
2. Perform the operation with the rounded numbers.
3. Compare the estimate to your exact answer.
4. If they are close, your answer is likely correct. If they are far apart, check your work.
Example 1: Checking Addition with Estimation
Problem: Calculate 5,678 + 4,321 = 9,999. Is this reasonable?
Check:
Estimate: 5,678 ≈ 6,000, 4,321 ≈ 4,000, sum ≈ 10,000
9,999 is very close to 10,000, so the answer is reasonable.
Problem: Calculate 5,678 + 4,321 = 9,999. Is this reasonable?
Check:
Estimate: 5,678 ≈ 6,000, 4,321 ≈ 4,000, sum ≈ 10,000
9,999 is very close to 10,000, so the answer is reasonable.
Example 2: Checking Subtraction with Estimation
Problem: Calculate 8,765 - 3,456 = 5,309. Is this reasonable?
Check:
Estimate: 8,765 ≈ 9,000, 3,456 ≈ 3,000, difference ≈ 6,000
5,309 is close to 6,000, so the answer is reasonable.
Problem: Calculate 8,765 - 3,456 = 5,309. Is this reasonable?
Check:
Estimate: 8,765 ≈ 9,000, 3,456 ≈ 3,000, difference ≈ 6,000
5,309 is close to 6,000, so the answer is reasonable.
Example 3: Using Inverse Operations to Check
Problem: For subtraction, add the difference to the subtrahend to get the minuend.
Example: 8,765 - 3,456 = 5,309 → Check: 5,309 + 3,456 = 8,765 ✓
For addition, subtract one addend from the sum to get the other addend.
Problem: For subtraction, add the difference to the subtrahend to get the minuend.
Example: 8,765 - 3,456 = 5,309 → Check: 5,309 + 3,456 = 8,765 ✓
For addition, subtract one addend from the sum to get the other addend.
Watch Out!
Estimation gives a rough check, not a guarantee. If your estimate is very different from your answer, recalculate carefully. Also, use inverse operations (addition to check subtraction, subtraction to check addition) for a more reliable verification.
Estimation gives a rough check, not a guarantee. If your estimate is very different from your answer, recalculate carefully. Also, use inverse operations (addition to check subtraction, subtraction to check addition) for a more reliable verification.
Practice for Estimation and Checking
- Estimate 12,345 + 23,456 by rounding to the nearest thousand. Then calculate exactly and compare.
- Estimate 45,678 - 23,456 by rounding to the nearest hundred. Then calculate exactly and compare.
- Use inverse operations to check: 56,789 + 34,567 = 91,356.
- Use inverse operations to check: 78,901 - 34,567 = 44,334.
- If your estimate for an addition problem is 50,000 and your exact answer is 45,000, should you check your work? Why?
Methods & Techniques
Mastering addition and subtraction requires accuracy and efficiency. Use these strategies to improve.
Verification / Checking Strategy:
1. Use estimation: Round numbers and estimate before or after calculating.
2. Use inverse operations: For addition, subtract one addend from the sum. For subtraction, add the difference to the subtrahend.
3. Add in a different order: Add numbers in a different sequence to verify.
4. Use a calculator for verification: If allowed, use a calculator to double-check.
1. Use estimation: Round numbers and estimate before or after calculating.
2. Use inverse operations: For addition, subtract one addend from the sum. For subtraction, add the difference to the subtrahend.
3. Add in a different order: Add numbers in a different sequence to verify.
4. Use a calculator for verification: If allowed, use a calculator to double-check.
Example: Using Inverse Operations to Check Subtraction
Problem: 10,000 - 3,456 = 6,544. Check.
Check:
Add: 6,544 + 3,456 = 10,000 ✓
Problem: 10,000 - 3,456 = 6,544. Check.
Check:
Add: 6,544 + 3,456 = 10,000 ✓
Common Pitfalls & How to Avoid Them:
• Pitfall 1: Misaligning numbers by place value → Solution: Write numbers vertically with digits in the correct columns (right-align).
• Pitfall 2: Forgetting to carry when the sum is 10 or more → Solution: Always check the sum in each column; if it's 10 or more, carry the tens digit.
• Pitfall 3: Forgetting to borrow when the top digit is smaller → Solution: Check each column; if the top digit is smaller, borrow before subtracting.
• Pitfall 4: Borrowing incorrectly across zeros → Solution: Borrow step by step from the first non-zero column to the left.
• Pitfall 5: Making sign errors in word problems → Solution: Identify keywords (total = add, difference = subtract, remaining = subtract).
• Pitfall 1: Misaligning numbers by place value → Solution: Write numbers vertically with digits in the correct columns (right-align).
• Pitfall 2: Forgetting to carry when the sum is 10 or more → Solution: Always check the sum in each column; if it's 10 or more, carry the tens digit.
• Pitfall 3: Forgetting to borrow when the top digit is smaller → Solution: Check each column; if the top digit is smaller, borrow before subtracting.
• Pitfall 4: Borrowing incorrectly across zeros → Solution: Borrow step by step from the first non-zero column to the left.
• Pitfall 5: Making sign errors in word problems → Solution: Identify keywords (total = add, difference = subtract, remaining = subtract).
Place Value Chart Reference
| Millions | Hundred-thousands | Ten-thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|---|
| 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 |
Use this chart to help align numbers correctly when adding or subtracting.
Technique Practice
- Check 12,345 + 67,890 = 80,235 using estimation.
- Check 90,000 - 45,678 = 44,322 using inverse operations.
- Identify the error: 5,678 + 4,321 = 9,889. What went wrong?
- What would you do differently if you kept getting the wrong answer when subtracting across zeros?
Real-World Applications
Addition and subtraction are used daily in finance, shopping, travel, construction, and many other areas.
Application 1: Budgeting
Scenario: Your monthly income is ₦250,000. Expenses are: rent ₦80,000, food ₦45,000, transport ₦15,000, and bills ₦25,000. How much is left?
Problem: Subtract total expenses from income.
Solution:
Total expenses = 80,000 + 45,000 + 15,000 + 25,000 = 165,000
Money left = 250,000 - 165,000 = 85,000
Answer: ₦85,000 remaining.
Scenario: Your monthly income is ₦250,000. Expenses are: rent ₦80,000, food ₦45,000, transport ₦15,000, and bills ₦25,000. How much is left?
Problem: Subtract total expenses from income.
Solution:
Total expenses = 80,000 + 45,000 + 15,000 + 25,000 = 165,000
Money left = 250,000 - 165,000 = 85,000
Answer: ₦85,000 remaining.
Application 2: Distance Traveled
Scenario: A road trip is 1,250 km. You have driven 678 km. How much further to go?
Problem: Subtract distance driven from total distance.
Solution:
1,250 - 678 = 572 km
Answer: 572 km remaining.
Scenario: A road trip is 1,250 km. You have driven 678 km. How much further to go?
Problem: Subtract distance driven from total distance.
Solution:
1,250 - 678 = 572 km
Answer: 572 km remaining.
Application 3: Inventory Management
Scenario: A warehouse has 45,678 units. They receive 12,345 units and ship out 8,765 units. What is the new total?
Problem: Add the received units, then subtract the shipped units.
Solution:
After receiving: 45,678 + 12,345 = 58,023
After shipping: 58,023 - 8,765 = 49,258
Answer: 49,258 units.
Scenario: A warehouse has 45,678 units. They receive 12,345 units and ship out 8,765 units. What is the new total?
Problem: Add the received units, then subtract the shipped units.
Solution:
After receiving: 45,678 + 12,345 = 58,023
After shipping: 58,023 - 8,765 = 49,258
Answer: 49,258 units.
Application 4: Population Change
Scenario: A town had 234,567 people. During the year, 12,345 moved in and 8,765 moved out. What is the new population?
Problem: Add arrivals, subtract departures.
Solution:
Net change = 12,345 - 8,765 = 3,580 increase
New population = 234,567 + 3,580 = 238,147
Answer: 238,147 people.
Scenario: A town had 234,567 people. During the year, 12,345 moved in and 8,765 moved out. What is the new population?
Problem: Add arrivals, subtract departures.
Solution:
Net change = 12,345 - 8,765 = 3,580 increase
New population = 234,567 + 3,580 = 238,147
Answer: 238,147 people.
Application 5: Sports Scores
Scenario: A basketball team scored 98 points in the first game and 112 points in the second. What is the total?
Problem: Add the two scores.
Solution:
98 + 112 = 210 points
Answer: 210 points.
Scenario: A basketball team scored 98 points in the first game and 112 points in the second. What is the total?
Problem: Add the two scores.
Solution:
98 + 112 = 210 points
Answer: 210 points.
Cross-Curricular Connections
- Science: Recording measurements, calculating differences in data
- Economics: Budgeting, profit/loss calculations, inventory
- Geography: Calculating distances, population changes
- Physical Education: Tracking scores, calculating differences in times
Cumulative Practice Exercises
Try these problems on your own. Show all working steps. Use estimation and inverse operations to check your answers.
- Add 45,678 + 34,567.
- Subtract 89,012 - 45,678.
- Add 123,456 + 789,012 + 345,678.
- Subtract 500,000 - 123,456.
- A school has 2,345 students in Grade 8 and 1,876 in Grade 9. How many students in both grades?
- A company had ₦1,234,567 in January sales and ₦987,654 in February sales. How much more in January?
- A tank holds 45,678 litres of water. If 23,456 litres are used, how much remains?
- Add 67.89 + 45.67.
- Subtract 98.76 - 54.32.
- Estimate 78,901 + 34,567 by rounding to the nearest thousand. Then calculate exactly.
- Use inverse operations to check: 56,789 - 34,567 = 22,222.
- A farmer has 12,345 chickens. He buys 5,678 more, then sells 3,456. How many does he have now?
- A car travels 345 km on Monday, 456 km on Tuesday, and 567 km on Wednesday. Find total distance.
- Error analysis: A student calculated 10,000 - 2,345 = 7,755. Is this correct? If not, correct it.
- A library has 45,678 fiction books and 23,456 non-fiction books. How many more fiction books are there?
Answers to Cumulative Exercises
- Problem: 45,678 + 34,567.
Answer: 80,245 - Problem: 89,012 - 45,678.
Answer: 43,334 - Problem: 123,456 + 789,012 + 345,678.
Answer: 1,258,146 - Problem: 500,000 - 123,456.
Answer: 376,544 - Problem: 2,345 + 1,876.
Answer: 4,221 students - Problem: 1,234,567 - 987,654.
Answer: ₦246,913 more in January - Problem: 45,678 - 23,456.
Answer: 22,222 litres - Problem: 67.89 + 45.67.
Answer: 113.56 - Problem: 98.76 - 54.32.
Answer: 44.44 - Problem: 78,901 + 34,567 estimate.
Answer: 79,000 + 35,000 = 114,000; exact = 113,468 - Problem: Check 56,789 - 34,567 = 22,222.
Answer: 22,222 + 34,567 = 56,789 ✓ correct - Problem: 12,345 + 5,678 = 18,023; 18,023 - 3,456 = 14,567 chickens
- Problem: 345 + 456 + 567 = 1,368 km
- Problem: Error analysis: 10,000 - 2,345 = 7,655, not 7,755. Correct answer is 7,655.
- Problem: 45,678 - 23,456 = 22,222 more fiction books
Conclusion & Summary
Multi-digit addition and subtraction are essential skills for everyday life and advanced mathematics. Accuracy requires careful alignment of place values, correct carrying and borrowing, and checking your work using estimation or inverse operations.
Key Takeaways:
1. Addition: Align numbers by place value, add from right to left, carry when sum ≥ 10.
2. Subtraction: Align numbers, subtract from right to left, borrow when top digit is smaller.
3. Borrowing across zeros: Borrow step by step from the first non-zero column to the left.
4. Decimals: Align decimal points vertically before adding or subtracting.
5. Word problems: Identify keywords to determine whether to add or subtract.
6. Checking: Use estimation and inverse operations to verify answers.
Keep practising with larger numbers and real-world problems. Accuracy and speed improve with regular practice!
Video Resource
Watch this video for more examples of multi-digit addition and subtraction with carrying and borrowing.
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