Multiplying Multi-Digit Numbers

Multiplying multi-digit numbers is an essential skill in mathematics that helps us solve many real-world problems. Let's learn how we can multiply more than one-digit numbers in a systematic and effective way.

Understanding the basics of multiplication

Before we get into multi-digit multiplication, let's recap what multiplication means. Multiplication is a way of adding a number to a certain number. For example, if we have 3 groups of 4 objects, we can say that 3 multiplied by 4 equals 12. In mathematical terms, this is:

3 x 4 = 12

Here 3 and 4 are the product numbers and 12 is the product.

Steps to multiply multi-digit numbers

While multiplying large numbers, we follow a step-by-step procedure to get the correct answer.

Step 1: Align the numbers

Write the numbers you want to multiply one above the other, aligning their digits according to place value. Draw a line beneath them. For example, if you want to multiply 23 and 45:

2 3 x 4 5 ------

Here 23 is the multiplicand and 45 is the multiplier.

Step 2: Multiply the unit digits

Start with the digit in the units place of the bottom number. Multiply it by each digit of the top number. Write the result just below the line, aligning it with the units column. In this case, multiply 5 (from 45) by each digit of 23:

2 3 x 4 5 ------
1 5 (5 x 3 = 15, write 5 and carry 1) +1 0 (5 x 2 = 10, add carry 1, total 11) ------ 1 1 5

Step 3: Multiply the tens digit

Next, multiply the tens digit of the bottom number by each digit of the top number. Remember to add a zero in the tens place, since you're now dealing with tens. In this case, multiply the 4 (in 45, but remember that it's 40 because of its position) by each digit of 23:

2 3 x 4 5 ------
1 1 5 9 2 0 (4 x 3 = 12, write 2 and carry 1; 4 x 2 = 8, add carry 1, total 9) ------

Notice the zero in the ones column of the result, to remember that we're working with tens.

Step 4: Add up the results

Finally, add the numbers obtained from steps 2 and 3 to get the final result.

2 3 x 4 5 ------
1 1 5 9 2 0 ------
1 0 3 5

Multiplying 23 by 45 gives the result 1,035.

Analyzing multiplication with place values

Multiplying multi-digit numbers can be simplified by breaking it down into steps that relate to place values.

Let's multiply 234 by 56 using place values:

Step 1: Divide each number by place value

Break each number into its place values.

234 can be divided into 200, 30 and 4
56 can be divided into 50 and 6

Step 2: Multiply each place value

Multiply each part of 234 by each part of 56:

200 x 50 = 10,000

200 x 6 = 1,200

30 x 50 = 1,500

30 x 6 = 180

4 x 50 = 200

4 x 6 = 24

Step 3: Add all the products together

10,000 1,200 1,500 180 200 24 -------
13,104

Thus, multiplying 234 by 56 gives 13,104.

Practice example

Try these practice examples to improve your multi-digit multiplication skills.

Example 1

Multiply 78 by 32:

7 8 x 3 2 ------ 1 5 6 (Multiply 8 by 2 and 7 by 2) 2 3 4 0 (Multiply 8 by 3 and 7 by 3, stick a zero at the end) ------ 2 4 9 6

So multiplying 78 by 32 gives 2,496.

Example 2

Multiply 367 by 45:

3 6 7 x 4 5 ------ 1 8 3 5 (Multiply 7 by 5, 6 by 5, and 3 by 5) 1 4 6 8 0 (Multiply 7 by 4, 6 by 4, and 3 by 4, stick a zero at the end) ------ 1 6 5 1 5

Thus, multiplying 367 by 45 we will get 16,515.

Through practice and understanding, solving multi-digit multiplication problems becomes easier and becomes an essential skill for future mathematical calculations.

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