3-Digit Multiplication

3-Digit Multiplication

In mathematics, 3-Digit Multiplication is the process of multiplying three-digit numbers by two-digit, one-digit, or three-digit numbers by putting the numbers in columns according to their place values. 3-digit multiplication has an advantage over two-digit and one-digit multiplication.

In this article, we will study multiplication with three digits by one digit, multiplication with three digits by two digits, and multiplication with three digits by three digits, as well as solve a few cases to better comprehend the concept.

What is 3-Digit Multiplication?

3-digit multiplication is the process of multiplying three-digit numbers by other numbers. When multiplying three-digit numbers, the numbers are arranged in columns according to their place values. We are aware that three-digit integers are ordered according to their place values as ones, tens, and hundreds. When multiplying two integers, we often place the larger number on top and the smaller number underneath. The number on top becomes the multiplicand, while the number written below becomes the multiplicator. When numbers are organized according to their place values, the multiplier is multiplied with each of the multiplicand’s digits one by one, first with the one’s digit, then the tens digit, and finally the hundreds digit. All of these goods are written in tandem to produce the final output.

For instance, while multiplying 123 3, the multiplicand and multiplier are placed as shown below. The product of these integers when multiplied is 269

3-Digit By 1-Digit Multiplication

There are two possible outcomes when multiplying a three-digit number by a one-digit number.

The first one refers to the multiplication in which the 1-digit number is multiplied by the 3-digit number with no carryovers and the resulting product is obtained. This is multiplication by three digits without regrouping.

The second one refers to the multiplication in which the 3-digit number is multiplied by the 1-digit number, and the extra digit of the product must be carried over to the next column. This is multiplication by three digits using regrouping. Let’s examine both scenarios with the use of illustrations.

3-Digit Multiplication Without Regrouping

The product of a 3-digit number and a 1-digit number is found by multiplying the one-digit number by each of the three-digit number’s digits. If the sum of the 1-digit number and each digit of the number is a single digit, there is no need to carry any numbers over. Let us consider an example.

Example: Multiply 214 × 2

Solution: The following steps show the procedure of multiplying 214 by 2.

Step 1: Arrange the numbers 214 and 2 in columns according to their place values as shown in the figure given below.

Step 2: Now, first we multiply the 1-digit number (2) by each digit of the 3-digit number (214)

When 2 is multiplied by 4, we get 8.

When 2 is multiplied by 1, we get 2.

When 2 is multiplied by 2, we get 4.

Step 3: Therefore, the product that we get is 428.

3 Digit Multiplication With Regrouping

In this section, we’ll multiply a three-digit number by a one-digit number and see how regrouping works. Let’s solve a problem to show how this works.

Example: Multiply 347 by 3.

Solution: Let us multiply 347 by 3 using the steps given below.

Step 1: Arrange the numbers 347 and 3 in columns according to their place values as shown below.

Step 2: Multiply 3 by each digit of 347.

When 3 is multiplied by 7, we get 21. Since 21 is a 2-digit number, we write 1 under the one’s column and carry 2 to the tens column above 4.

When 3 is multiplied by 4, we get 12. Now, we need to add the carry-over (2) to 12 and we get 14. Since 14 is a 2-digit number, we write 4 under the tens column and carry 1 to the hundreds column above 3.

When 3 is multiplied by 3, we get 9. Now, we need to add the carry-over 1 to 9 and we get 10. Since there is no other digit left for multiplication, we write 10.

Step 3: Therefore, we get the product as 1041.

3-Digit by 2-Digit Multiplication

To multiply a three-digit number by a two-digit number, write the three-digit number on top and the two-digit number below it. Let’s talk about how to multiply 3 digits by 2 digits without regrouping and with regrouping in the sections that follow.

3-Digit by 2-Digit Multiplication Without Regrouping

When we multiply a 3-digit number by a two-digit number, we first multiply the multiplier one’s digit by the multiplicand, and then we multiply the multiplier’s tens digit by the multiplicand. Then we add these two things together to get the final thing. Let’s use the following example to talk about the process step by step.

Example: Multiply 411 by 31.

Solution: Let us multiply 411 by 31 stepwise.

Step 1: Arrange the numbers 411 and 31 in columns according to their place values as shown below.

Step 2: Multiply 1 by each digit of 411.

When 1 is multiplied by 1, we get 1.

When 1 is multiplied by 1, we get 1.

When 1 is multiplied by 4, we get 4. So, we have 411 as the first partial product.

Step 3: Now, we place a zero under the first partial product, that is, just before we write the second partial product in the next line. This 0 is placed here because in this step we are actually multiplying 411 by 30.

Step 4: Multiply 3 by each digit of 411.

When 3 is multiplied by 1, we get 3.

When 3 is multiplied by 1, we get 3.

When 3 is multiplied by 4, we get 12. So, we have 12330 as the second partial product.

Step 5: Add these products to obtain the final answer.

Step 6: 411 + 12330 = 12741. Therefore, the final product is 12741.

3-Digit by 2-Digit Multiplication With Regrouping

Now that we’ve multiplied a three-digit number by a two-digit number, let’s try to solve another problem that involves regrouping or carrying.

Example: Multiply 573 by 46.

Solution: Let us multiply 573 by 46 using the following steps:

Step 1: Arrange the numbers 573 and 46 in columns according to their place values as shown below.

Step 2: Multiply 6 by each digit of 573.

When 6 is multiplied by 3, we get 18. Since 18 is a 2-digit number, we write 8 under the ones column and carry 1 to the tens column above 7.

When 6 is multiplied by 7, we get 42. Now, we need to add the carry-over (1) to 42 and we get 43. Since 43 is a 2-digit number, we write 3 in the tens column and carry 4 to the hundreds column above 5.

When 6 is multiplied by 5, we get 30. Now, we will add the carry-over (4) to 30, we get 34. Since there is no other digit left for multiplication, we write 34. So, we have 3438 in the first line (partial product) of the answer.

Step 3: Now, we will place a zero under the first partial product, that is, before writing the second partial product in the next line. This is because in this step we are actually multiplying 573 with 40.

Step 4: Multiply 4 by each digit of 573.

When 4 is multiplied by 3, we get 12. Since 12 is a 2-digit number, we write 2 under the tens column and carry 1 to the tens column above 7.

When 4 is multiplied by 7, we get 28. Now, we will add the carry-over 1 to 28 to get 29. Since 29 is a 2-digit number, we write 9 under the hundreds column and carry 2 to the hundreds column above 5.

When 4 is multiplied by 5, we get 20. Now, we will add the carried-over number 2 to 20 and we get 22. Since there is no other digit left for multiplication, we write 22. So, we have 22920 as the second line of the product.

Step 5: Add these partial products to obtain the final answer.

Step 6: This means 3438 + 22920 = 26358. Therefore, the final product is 26358.

3-Digit By 3-Digit Multiplication

We will learn how to multiply a three-digit number by a three-digit number in this section. This process is like what we talked about in the last few sections. Let’s use the following example to learn how to multiply 3 digits by 3 digits.

Example: Multiply 123 by 456.

Solution: Let us multiply 123 by 456 using the following steps.

Step 1: Arrange the numbers 123 and 456 in columns according to their place values as shown below.

Step 2: Multiply 6 by each digit of 123.

When 6 is multiplied by 3, we get 18. Since 18 is a 2-digit number, we write 8 under ones column and carry 1 to the tens column above 2.

When 6 is multiplied by 2, we get 12. Now, we add the carried-over 1 to 12 and we get 13. Since 13 is a 2-digit number, we write 3 under the tens column and carry 1 to the next column above 1.

When 6 is multiplied by 1, we get 6. Now, add the carried-over 1 to 6 to get 7. Since there is no other digit left for multiplication, we write 7. So, we have 738 in the first line as the partial product.

Step 3: Now, place a zero under this partial product under ones column. This is because in this step we are actually multiplying 123 with 50.

Step 4: Multiply 5 by each digit of 123.

When 5 is multiplied by 3, we get 15. Since 15 is a 2-digit number, we write 5 in the tens column and carry 1 to the next column above 2.

When 5 is multiplied by 2, we get 10. Now, add the carried-over 1 to 10 to get 11. Since 11 is a 2-digit number, we write 1 in hundreds column and carry 1 to the next column above 1.

When 5 is multiplied by 1, we get 5. Now, add the carried-over 1 to 5 to get 6. Since there is no other digit left for multiplication, we write 6. So, we have 6150 in the second line of the partial product.

Step 5: Now, place two zeros (0s) under the ones and tens column under the partial product obtained in the previous step. This is because in this step we are actually multiplying 123 with 400.

Step 6: Multiply 4 by each digit of 123.

When 4 is multiplied by 3, we get 12. Since 12 is a 2-digit number, we write 2 under the hundreds column and carry 1 to the next column above 2.

When 4 is multiplied by 2, we get 8. Now, add the carried-over 1 to 8 to get 9. We write 9 under the next column.

When 4 is multiplied by 1, we get 4. Since there is no other digit left for multiplication, we write 4. So, we have 49200 in the third line as the partial product.

Step 7: Add all the 3 partial products to obtain the final product. This means 738 + 6150 + 49200 = 56088.

Step 8: Therefore, the final product is 56088.

⁕ Related Topics

• 2-Digit Subtraction
• 2-Digit Addition
• 3-Digit Addition
• 3-Digit Subtraction
• 2-Digit Multiplication
• 4-Digit Addition
• 4-Digit Subtraction
• Multiplication and Division of Integers

3-Digit Multiplication Examples

Example 1: Find the product of 712 and 23.

Solution: Let us do this 3-digit multiplication using the following steps.

Multiply 3 by each digit of 712.

Place a zero on one placed below the product obtained above.

Multiply 2 by each digit of 712.

Add the two products to obtain the final answer.

FAQs on 3-Digit Multiplication