3 Digit Multiplication

The multiplication involving a 3-digit number by other numbers is the 3-digit multiplication. To multiply a 3-digit number, first, we arrange the 3-digit number in columns based on the place value.

The multiplicand (larger number) is written on the top, and the multiplier (smaller number) below it. 

For example, when multiplying 235 by 5, 235 is the multiplicand and 5 is the multiplier. So, 235 × 5 = 1175.

Multiplying a 3-digit number by a 1-digit number can be done in two ways: with regrouping and without regrouping. 

• 3-digit multiplication without regrouping: This is when there is no carrying over when multiplying a 3-digit number by a 1-digit number. For example, 122 × 2 = 244.   

• 3-digit multiplication without regrouping: When the product of a digit is greater than or equal to 10, then the extra digits are carried over to the next digit. For example, 236 × 4 = 944.  

When there is a carryover when multiplying two or more numbers is known as multiplication with regrouping. Let’s learn it from an example, 235 by 5, step by step.

Step 1: Arrange the numbers in columns based on their place values. Here, 235 is the multiplicand and 5 is the multiplier. 

Step 2: Now we multiply the multiplicand by the multiplier. So, here we multiply 235 by 5 from right to left.

• 5 × 5 = 25, write 5 in one's place and carry over 2 to the tens place.

• 5 × 3 = 15, adding carried over 2 with 15, 15 + 2 = 17. Write 7 in the tens place and carry over 1 to the hundreds place.

• 5 × 2 = 10, adding carried over 1 with 10, 10 + 1 = 11.

The product of 235 by 5 is 1175.

Multiplication without regrouping is when the product of multiplying the numbers is less than or equal to 9. When multiplying a 3-digit number by a 1-digit number, we simply multiply each digit of the 3-digit number by the 1-digit number. For example, 123 × 2.

Step 1: Arrange the numbers and multiply each digit of the 3-digit number by the 1-digit number, starting from the right. 

Step 2: Multiply each digit by 2 from left to right.

• 2 × 3 = 6, write 6 in the ones place

• 2 × 2 = 4, write 4 in the tens place

• 2 × 1= 2, write 2 in the hundreds place.

The product of 123 and 2 is 246

The 3-digit by 2-digit multiplication is the process of multiplying a 3-digit number by a 2-digit number. Here, the multiplicand is the 3-digit number, and the multiplier is the 2-digit number.

For example, 235 × 23 = 5405, 123 × 11 = 1353. Now let’s see multiplying 3-digit by 2-digit numbers with and without regrouping. 

Regrouping is applicable when the product of multiplying the digits is more than 9. We can learn the 3-digit by 2-digit multiplication with regrouping with an example, 234 × 52

Step 1: Arrange the numbers in order, and multiply 234 by the one's digit of 52, which is 2.

2 × 4 = 8

2 × 3 = 6

2 × 2 = 4

So, 468 is the first partial product. 

Step 2: Multiply 234 by the tens place of 52, that is 5. Place a 0 in the one's place

5 × 4 = 20, as the product is 20, we write 0 in the tens place and carry over 2 in the hundreds place. 

5 × 3 = 15, the product is 12. Adding the carried-over 2 to 15, 15 + 2 = 17. As the result is 17, we write 7 and carry 1. 

5 × 2 = 10, adding the carried-over 1 with 10, 10 + 1 = 11. 

Therefore, the second partial product is 11700

Step 3: Adding the partial products, 468 + 11700 = 12168. 

The 3-digit by 2-digit multiplication with and without regrouping follows the same method. The only difference is there is no carrying is needed in this method. For example, multiply 212 by 21

Step 1: Arrange the numbers in order. Multiply 212 by the one's digit of 21, which is 1. 

• 1 × 2 = 2
   
• 1 × 1 = 1
   
• 1 × 2 = 2

So, the first partial product is 212. 

Step 2: Multiply the 212 by the tens' digit of 21, which is 2. Adding 0 in one's place before writing the second partial product is because here we are multiplying 212 by 20, as 2 is in tens place. 
 

• 2 × 2 = 4
   
• 2 × 1 = 2
   
• 2 × 2 = 4

The second partial product is 4240.

Step 3: Adding the partial product, that is 212 + 4240 = 4452.

Multiplying a 3-digit number by a 3-digit number follows a similar process as multiplying a 3-digit number by a 1-digit or 2-digit number. Here, we will learn 3-digit by 3-digit multiplication with an example, 132 × 243

Step 1: Arrange the digits in order. First, we multiply 132 by the ones digit of 243, which is 3.

3 × 2 = 6

3 × 3 = 9

3 × 1 = 3

Here, the first partial product is 396

Step 2: Multiplying 132 by the tens' digit of 243, which is 4. Place 0 in the ones places.

4 × 2 = 8 

4 × 3 = 12, write 2 and carry over 1

4 × 1 = 4, adding the carried over 1, 4 + 1 = 5

The second partial product is 5280.

Step 3: Multiply 132 by the hundreds digit of 243, which is 2.

Place 00 in the ones and tens place

2 × 2 = 4

2 × 3 = 6

2 × 1 = 2

The third partial product is 26400

Step 4: To find the final product, we add all the partial products.  

Adding the partial products: 396 + 5280 + 26400 = 32076. 

The product of 132 × 243 = 32076

Multiplication is one of the basic arithmetic operations in math. 3-digit multiplication is used in different situations in real life. Here are a few applications: 

• To calculate the cost when shopping, we use the 3-digit multiplication.

• In the construction field, we use the 3-digit multiplication to calculate the area and material estimation. 

• In cooking, to adjust the recipe according to the number of servings, we use multiplication. 

• The 3-digit multiplication is used in traveling to calculate the distance and fuel consumption