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Multi-Digit Addition

What is multi-digit addition?

Addition is one of the four fundamental operations of arithmetic. It is usually denoted by the plus symbol +. The result of every addition process is called the sum, and the numbers being added are called the addends. In a nutshell, addition is a process in which two or more numbers are combined together. 

Multi-digit, from the word itself, is a number having more than one digit. This means that multi-digit addition is simply the addition of any number having more than one digit.

When adding multi-digit numbers, we usually see use the vertical method and the terms as shown in the figure:

  • The addends are the numbers to which the addition process will be applied.
  • The sum is the result of adding the addends. 
  • The carry over is the single-digit that is regrouped from one column of digits to the column of a higher place value.
  • The plus sign is the symbol that denotes the addition process. 

How to do the multi-digit addition?

While one-digit numbers can be easily added, larger numbers or multi-digit numbers require us to split them into columns according to their respective place values. We always begin addition on the right side, in accordance with the place value system. 

Adding multi-digit numbers without regrouping

To add multi-digit numbers, follow these steps:

  1. Write the numbers with the place values lines up. 
  2. Add the digits in the ones place. 
  3. Add the digits in the tens place. 
  4. Continue doing these steps until you have added all the place values. 

Example #1 

What is the sum of 135 and 23?

Solution

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Add the ones digit. Thus, 5 + 3 = 8.
Then, write the sum of the ones digit under the bar.
Add the tens digit. Thus, 3 + 2 = 5.
Then, write the sum of the tens digit under the bar.
Bring down the hundreds digit of 135 since 23 doesn’t have a hundreds digit.
Therefore, the sum of 135 and 23 is 158.

Example #2

What is the sum of 4381 and 418? 

Solution

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Add the ones digit. Thus, 1 + 8 = 9
Then, write the sum of the ones digit below the bar.
Add the tens digit. Hence,8 + 1 = 9
Then, write the sum of the tens digit below the bar.
Add the hundreds digit. Thus, 3 + 4 = 7
Write the sum of the hundreds digits below the bar.
Bring down the thousands digit of 4381 since 418 doesn’t have a thousands digit.
Therefore, the sum of 4381 and 418 is 4799.

Example #3

Find the sum of 2048 and 2021. 

Solution

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Add the ones digit. Thus, 8 + 1 = 9
Then, write the sum of the ones digit below the bar.
Add the tens digit. Thus, 4 + 2 = 6
Then, write the sum of the tens digit below the bar.
Since the hundreds place is both 0, then the sum will always be 0. Hence, 0 + 0 = 0.
Then, write the sum of the hundreds digits below the bar.
Get the sum of the thousands place. Hence, 2 + 2 = 4
Then, write the sum of the hundreds digit below the bar.
Therefore, the sum of 2048 and 2021 is 4069.

Example #4 

What will be the result of 15 346 + 12 631? 

Solution

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Add the ones digit. Thus, 6 + 1 = 7
Then, write the sum of the ones digit below the bar.
Add the tens digit. Thus, 4 + 3 = 7
Then, write the sum of the tens digit below the bar.
Add the hundreds digit. Thus, 3 + 6 = 9
Then, write the sum of the hundreds digit below the bar.
Get the sum of the thousands digit. Hence, 5 + 2 = 7
Then, write the sum of the thousands digit below the bar.
Lastly, get the sum of the ten thousands place. Thus,1 + 1 = 2
Then, write the sum of the ten thousands digit below the bar.
Therefore, the sum of 15 346 and 12 631 is 27 977

Example #5

Determine the sum of 194 765 and 304 212. 

Solution

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Add the ones digit. Thus, 5 + 2 = 7
Then, write the sum of the ones digit below the bar.
Add the tens digit. Thus, 5 + 2 = 7
Then, write the sum of the tens digit below the bar.
Add the hundreds digit. Thus, 7 + 2 = 9
Then, write the sum of the hundreds digit below the bar.
Get the sum of the thousands digit. Thus, 4 + 4 = 8
Then, write the sum of the thousands digit below the bar.
Any number added by 0 will always be the same. Thus,9 + 0 = 9
Then, write the sum of the ten thousands digit below the bar.
Get the sum of the hundred thousand digit. Thus, 1 + 3 = 4
Then, write the sum of the hundred thousand digit below the bar.
Therefore, the sum of 194 765 and 304 212 is 498 977

Adding multi-digit numbers with regrouping

To add multi-digit numbers that need regrouping, follow these steps:

  1. Write the numbers with the place values lines up. 
  2. Add the digits in the ones place. If the sum exceeds 10, then carry over the digit to the tens place.
  3. Add the digits in the tens place. If the sum exceeds 10, then carry over the digit to the hundreds place.
  4. Continue doing these steps until you have added all the place values. If there is a need to regroup, carry over to the next higher place value.

Example #1 

What is the sum of 821 and 98? 

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Get the sum of the ones digit. Hence,1 + 8 = 9
Then, write the sum of the ones digit below the bar
Add the tens digit. Thus, 2 + 9 = 11
Since it exceeds 10, we must carry over 1 to the next higher place value. In this case, it is the hundreds place. 
Add the carry over 1 to the hundreds place of 821. Thus,1 + 8 = 9
Then, write the sum below the bar. 

Therefore, the sum of 821 and 98 is 919

Example #2

Determine the sum of 999 and 23. 

Solution

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Get the sum of the ones digit. Thus, 9 + 3 = 12
Since the sum exceeds 10, then carry over 1 to the tens digit. 
Add the carry over to the tens digit. Thus, 1 + 9 + 2 = 12
Since it also exceeds 10, then, we must carry over 1 to the hundreds place. 
Get the sum of the carry over and the hundreds digit of 999. Thus,9 + 1 = 10
Since it is equal to 10, carry over 1 to the thousands place. 
Bring down the carry over on the same place value of the thousands place. 
Therefore, the sum of 999 and 23 is 1022.

Example #3

What will be the result of 1995 + 345? 

Solution

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Add the ones digit. Hence,5 + 5 = 10
Since the sum is equal to 10, then carry over 1 to the tens place. 
Get the sum of the carry over and the tens digit. Thus, 1 + 9 + 4 = 14
Since the sum exceeds 10, then write 4 below the bar and write the carry over on top of the hundreds place of 1995. 
Add the carry over and the digits on the hundreds place. Thus, 1 + 9 + 3 = 13
Since the sum exceeds 10, then write 3 below the bar and write the carry over on top of the thousands place of 1995.
Add the carry over and the thousands digit of 1995. Thus, 1 + 1 = 2
Then, write the sum below the bar. 
Therefore, the sum of 1995 and 345 is 2340.

Example #4 

Determine the sum of 4875 and 3249.

Solution

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Add the ones digit of the two numbers. Thus, 5 + 9 = 14
Since the sum is 14, write 4 below the bar and put the carry over 1 on the same column of the tens place. 
Add the carry over and the tens digit of the two numbers. Thus, 1 + 7 + 4 = 12 
Since the sum is 12, write 2 below the bar and put the carry over 1 on the same column of the hundreds place.
Get the sum of the carry over and the digits on the hundreds place. Hence, 1 + 8 + 2 = 11
Since the sum is 11, then write 1 below the bar and put the carry over 1 on the thousands place. 
Add all the thousands digit and the carry over. Thus, 1 + 4 + 3 = 8
Then, write the sum below the bar.
Therefore, the sum of 4875 and 3249 is 8124.

Example #5

What is the sum of 23 765 and 87 845?

Solution

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Add the ones digit. Hence, 5 + 5 = 10
Since the sum is 10, write 0 below the bar and put the carry over on the column of the tens place. 
Get the sum of the carry over and the digits on the column of the tens place. Thus, 1 + 6 + 4 = 11
Since the sum is 11, write 1 below the bar and put the carry over on the same column as the hundreds place. 
Add the digits on the hundreds place and the carry over. Thus, 1 + 7 + 8 = 16
Since the sum is 16, write 6 below the bar and put the carry over on the column of the thousands place.
Add the thousands digit and the carry over. Thus, 1 + 3 + 7 = 11
Since the sum is 11, write 1 below the bar and put the carry over on the column of the ten thousands place. 
Get the sum of the ten thousands place and the carry over. Thus,1 + 2 + 8 = 11
Since the sum is 11, then, write 1 below the bar and put the carry over on the column of the hundred thousand place. 
Bring down the carry over on the hundred thousand place. 
Therefore, the sum of 23 765 and 87 845 is 111 610

How to solve problems involving multi-digit addition?

To solve problems involving multi-digit addition, follow these steps: 

  1. List all the information given in the problem.
  2. Analyze the problem. 
  3. Apply the rules of adding multi-digit numbers.

Example #1 

A flower store sells roses and sunflowers only. Based on their inventory, they have 3546 roses and 4582 sunflowers. How many flowers does the flower store have?

Solution

The problem states that there are 3546 roses and 4582 sunflowers. Since we are asked to get the total number of flowers in the store, then we will use the addition operation. 

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Get the sum of the ones digit. Thus, 2 + 6 = 8
Then, write the sum of the ones digit below the bar. 
Add the tens digit. Hence,8 + 4 = 12
Since the sum is 12, write 2 below the bar and put the carry over on the same column as the hundreds place.
Get the sum of the carry over and the digits on the hundreds place. Thus, 1 + 5 + 5 = 11
Since the sum is 11, write 1 below the bar and put the carry over on the same column as the thousands place. 
Add the carry over and the digit on the thousands place. Thus, 1 + 4 + 3 = 8
Then, write it below the bar. 
Therefore, there are 8128 flowers in the flower store. 

Example #2 

Venus has saved up $30,465 in the last 4 years. This year, she was able to save $2,975. How much is the total savings of Venus? 

Solution

In the problem, it is stated that Venus was able to saved up $30, 465 and $2,975. Thus, using the rules of multi-digit addition, 

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Add the units digit of the two numbers. Thus, 5 + 5 = 10
Since the sum is 10, write 0 below the bar and put the carry over on the column of tens place.
Get the sum of the tens digit and the carry over. Thus, 1 + 6 + 7 = 14
Since the sum is 14, write 4 below the bar and put the carry over on the same column as the hundreds place.
Add the hundreds place and the carry over. Hence,1 + 4 + 9 = 14

Since the sum is 14, write 4 below the bar and put the carry over on the column of the thousands place. 
Get the sum of the carry over and the thousands place. Thus, 1 + 0 + 2 = 3
Then, write the sum below the bar. 
Lastly, bring down the 3 in the ten thousands place. 
Therefore, Venus saved up $33 440.

Example #3 

An Asian restaurant earned $15, 365 last week. This week, they earn $23, 499. How much did they earn in two weeks? 

Solution

Given the problem, we can note that the Asian earned $ 15, 365 and $23, 499 for the past two weeks. Thus, using the rules of multi-digit addition, 

Addition ProcessStep-by-Step Explanation
Arrange the numbers in a column form with the place values lined up.
Add the units digit of the two numbers. Thus, 5 + 9 = 14
Since the sum is 14, regroup it and write 4 below the bar and put the carry over on the tens place column.
Get the sum of the tens digit and the carry over. Hence,1 + 6 + 9 = 16
Since the sum is 16, then write 6 below the bar and put the carry over on the hundreds place. 
Add the carry over and the hundreds digit. Thus, 1 + 3 + 4 = 8
Then, write the sum below the bar.
Add the digits in the thousands place. Thus,5 + 3 = 8
Then, write the sum below the bar. 
Add the digits in the ten thousands place. Hence, 1 + 2 = 3
Then, write the sum below the bar. 

Therefore, the Asian restaurant earned $38,864 in the last two weeks. 

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