1/15

00:00

E.5 Add multi-digit whole numbers: word problems

Loading questions...

What does it mean to add multi-digit numbers?

Adding multi-digit numbers means finding the total, or sum, of numbers that have more than one digit. These numbers can be in the hundreds, thousands, ten-thousands, or even larger. When we add them, we must carefully combine the values of the digits based on their place value positions. It is like combining groups of ones, tens, hundreds, thousands, and so on.

Examples:

3,251 + 124 is adding a four-digit number and a three-digit number.
12,058 + 3,942 is adding two multi-digit numbers, one in the ten-thousands.
145,782 + 231,104 is adding numbers in the hundred-thousands.

Let's look at how we line them up:

To add 38 + 21, we write the numbers so their ones places line up perfectly.

+ 21

The 8 and 1 are both in the ones place, so they stack on top of each other. The 3 and 2 are both in the tens place, so they stack, too.

Now with a bigger number:

When we add 145,782 + 231,104, every digit must be in the correct column.

145,782

+ 231,104

376,886

The ones, tens, hundreds, thousands, ten-thousands, and hundred-thousands all have their own column. Lining them up correctly is the first and most important step.

Note

Always align numbers by their rightmost digit before adding. Write numbers vertically, lining up the digits by place value helps you see exactly where each digit belongs.

The importance of place value

Place value is the value of a digit based on its position in a number. When adding multi-digit numbers, you can only combine digits that are in the same place value column. For example, you add ones to ones, tens to tens, hundreds to hundreds, and thousands to thousands.

Example: Adding 4,321 and 2,456

Thousands place: 4 thousand + 2 thousand = 6 thousand (4,000 + 2,000 = 6,000)
Hundreds place: 3 hundred + 4 hundred = 7 hundred (300 + 400 = 700)
Tens place: 2 tens + 5 tens = 7 tens (20 + 50 = 70)
Ones place: 1 one + 6 ones = 7 ones (1 + 6 = 7)
The sum is 6,000 + 700 + 70 + 7 = 6,777.

Note

Always write numbers vertically, lining up the digits by place value. The ones place must be directly above the ones place, the tens above the tens, and so on.

The standard algorithm for addition

The standard algorithm is a step-by-step method for adding numbers. You start at the rightmost column (the ones place) and move to the left. If the numbers in a column add up to ten or more, you "regroup" or "carry" the extra value to the next column on the left.

Example: Adding 5,827 and 3,695

Step 1: Write the numbers vertically, aligning the places.

5,827
+ 3,695
Step 2: Add the ones: 7 + 5 = 12. Write 2 in the ones place and carry the 1 ten to the tens column.
Step 3: Add the tens: 2 + 9 + 1 (carry) = 12. Write 2 in the tens place and carry the 1 hundred to the hundreds column.
Step 4: Add the hundreds: 8 + 6 + 1 (carry) = 15. Write 5 in the hundreds place and carry the 1 thousand to the thousands column.
Step 5: Add the thousands: 5 + 3 + 1 (carry) = 9. Write 9 in the thousands place.
The sum is 9,522.

Note

The carried number represents the extra group of ten that needs to be added to the next larger place value.

Regrouping in addition

Regrouping (also called "carrying") happens when the sum of the digits in a column is ten or greater. Because each place value can only hold a single digit (0 through 9), you must regroup the extra tens into the next column to the left.

Example: 3,758 + 1,564

3,758

+ 1,564

5,322

Ones: 8 + 4 = 12. Write 2, carry 1 ten.
Tens: 5 + 6 + 1 (carry) = 12. Write 2, carry 1 hundred.
Hundreds: 7 + 5 + 1 (carry) = 13. Write 3, carry 1 thousand.
Thousands: 3 + 1 + 1 (carry) = 5. Write 5.
Sum: 5,322.

Note

It is possible to regroup multiple times in a single problem. You may need to carry from the ones to the tens, from the tens to the hundreds, from the hundreds to the thousands, and so on.

Adding numbers with different lengths

Sometimes you will add numbers that have a different number of digits, such as adding a three-digit number to a five-digit number. To do this correctly, you must still align the numbers by their place values. It can be helpful to write the shorter number with leading zeros in the empty places to keep your columns straight, though this is not required.

Example: 42,506 + 8,371

Write the numbers with the larger number on top, aligning by the ones place:

42,506
+ 8,371
Notice that 8,371 has no digit in the ten-thousands place. You can think of it as 08,371.
Ones: 6 + 1 = 7
Tens: 0 + 7 = 7
Hundreds: 5 + 3 = 8
Thousands: 2 + 8 = 10. Write 0, carry 1 to the ten-thousands.
Ten-thousands: 4 + 0 + 1 (carry) = 5
The sum is 50,877.

Note

Always make sure the ones places are lined up, even if the numbers have different lengths. This is the most common error students make.

Estimating sums to check your work

Estimation is a useful skill to check if your exact answer is reasonable. To estimate the sum of multi-digit numbers, you can round each number to its highest place value (or another convenient place) and then add the rounded numbers. The estimate should be close to your exact answer.

Example: Estimate 24,718 + 13,456

Round 24,718 to the nearest thousand: 25,000.
Round 13,456 to the nearest thousand: 13,000.
Estimated sum: 25,000 + 13,000 = 38,000.
Now solve exactly: 24,718 + 13,456 = 38,174.
Compare: 38,174 is close to 38,000, so our exact answer is reasonable.

Note

Estimation does not replace solving the problem, but it is a great way to catch big mistakes. If your estimate and exact answer are very different, you should check your work.

Real-world word problems with multi-digit addition

In real life, we add multi-digit numbers all the time: finding the total cost of items, combining distances traveled, counting populations, or calculating total points in a game. Word problems help you practice applying addition skills to situations you might encounter outside the classroom.

Example word problem:

A school library has 15,238 books. The Parent-Teacher Association (PTA) donates 3,497 more books. How many books does the library have now?

Write the numbers: 15,238 + 3,497

15,238

+ 3,497

18,735

Ones: 8 + 7 = 15. Write 5, carry 1.
Tens: 3 + 9 + 1 (carry) = 13. Write 3, carry 1.
Hundreds: 2 + 4 + 1 (carry) = 7. Write 7.
Thousands: 5 + 3 = 8. Write 8.
Ten-thousands: 1 + 0 = 1. Write 1.
The library now has 18,735 books.

Note

When solving word problems, always look for keywords like "total," "altogether," "in all," and "combined" to know that you need to add.

Adding large numbers up to 1,000,000

Fourth graders are expected to add numbers within 1,000,000. This means you might add numbers with digits in the hundred-thousands place. The process is exactly the same as adding smaller numbers—you just have more columns to work through. Be careful to keep your columns aligned and to regroup correctly.

Example: 346,872 + 531,459

346,872

+ 531,459

878,331

Ones: 2 + 9 = 11. Write 1, carry 1.
Tens: 7 + 5 + 1 (carry) = 13. Write 3, carry 1.
Hundreds: 8 + 4 + 1 (carry) = 13. Write 3, carry 1.
Thousands: 6 + 1 + 1 (carry) = 8. Write 8.
Ten-thousands: 4 + 3 = 7. Write 7.
Hundred-thousands: 3 + 5 = 8. Write 8.
The sum is 878,331.

Note

When writing numbers with many digits, use commas to separate thousands, hundred-thousands, and millions. For example, write 1,246,891 instead of 1246891. This makes the number easier to read and helps you keep place values straight.

Common mistakes to avoid

Even careful students sometimes make errors when adding multi-digit numbers. Knowing the most common mistakes can help you avoid them. Watch out for misaligned digits, forgetting to carry, and making simple addition errors within a column.

Mistakes and how to fix them:

Misalignment: Writing 345 + 27 as 345 + 27 instead of aligning the ones (345 on top, 27 below, with the 7 under the 5). This leads to adding hundreds to tens. Fix: Always line up the rightmost digits, for example,

345
+ 27
372
Forgetting to carry: Adding 6 + 7 = 13 and writing 13 in the ones column. Fix: Remember each column can only have one digit. Write the 3 and carry the 1.
Adding the carried number twice: In the tens column, you have 5 (top) + 4 (bottom) + a carried 1. The mistake is adding the carried 1 twice: 5 + 4 = 9, then + carried 1 = 10, then mistakenly adding the same carried 1 again to get 11. Fix: Add the carried number only once. Either add it first (1 + 5 = 6, then 6 + 4 = 10) or add it last (5 + 4 = 9, then 9 + 1 = 10).

Note

Practice makes perfect. The more problems you solve, the more automatic the steps will become.

Mental math strategies for multi-digit addition

While the standard algorithm is reliable, sometimes you can add multi-digit numbers mentally by breaking them apart. This is called mental math or using number sense. These strategies can help you solve problems faster and understand numbers more deeply.

Strategies:

Add by place value: For 4,325 + 2,431, think 4,000 + 2,000 = 6,000; 300 + 400 = 700; 20 + 30 = 50; 5 + 1 = 6; total 6,756.
Make a ten or hundred: For 1,298 + 345, think 1,298 + 2 = 1,300, then add the remaining 343 to get 1,643.
Compensation: For 2,450 + 1,550, think 2,450 + 1,500 = 3,950, then add 50 to get 4,000.

Note

Mental math strategies are great for estimating and for checking your written work, but always use the algorithm for exact answers with large numbers.

Go to the practice →

Common Core alignment: CCSS.MATH.CONTENT.4.NBT.B.4 – Fluently add and subtract multi-digit whole numbers using the standard algorithm.

Notes for teachers

This free lesson is aligned with CCSS.MATH.CONTENT.4.NBT.B.4. Use it for whole-class instruction, independent practice, or homework.

All content is 100% free, student-safe, and designed for classroom and home use. The lesson covers place value, the standard algorithm with regrouping, adding numbers of different lengths, estimating sums, and solving word problems with numbers up to 1,000,000.