3 Digit Multiplication

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

• Write the multiplicand (larger number) on top.
•  Write 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 with 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.  

Multiplication with regrouping occurs when carrying over is required while multiplying numbers. 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. Multiply each digit of 235 by 5 starting from the ones place moving 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 + 11,700 = 12,168. 

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, i.e. 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

To multiply 3-digit numbers with accuracy will take effective thinking skills, calculation strategies, and knowledge of place value. Here are some strategies or tricks to help develop 3-digit multiplication skills:
 

• Understanding Place Value Arrangement: Be aware of numbers place value arrangement to avoid multiplication arrangement errors.
   
• Breaking into Parts: Develop partial products by considering one place value at a time to ease large calculations.
   
• Keep Alignment in Columns: Make sure your columns are aligned to maintain accuracy, especially while adding the partial product answers.
   
• Use Mental Reasoning About the Estimate: Before you work, make an estimate of the product to check for reasonableness afterward.
   
• Practice Verifying Results: After working a number or using reverse operations, recalculate or check your results by using the reverse operation.

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. 
   
• In business and finance, 3-digit multiplication is used to estimate profits, total sales, and annual expenses accurately.