How Do We Multiply a Three-digit Number by a Single-digit Number?
When we multiply a multi-digit number by a single-digit number, we perform small multiplication steps to arrive at the answer. The single-digit number lies in the one’s place. So first, we place the single-digit number below the ones place of the multi-digit number along with the multiplication symbol, like this:
We multiply the bottom digit by the digit in the ones place, 7. We always start our multiplication from the extreme right and then work our way to the left. Now, 5 times 7 gives us 35.
Similar to addition, we will regroup 35 as 3 tens and 5 ones. We have to carry the first digit, “35”, to the top of the next column, tens. So, the 5 stays down in the ones place while the 3 is carried over to the top in the tens place.
Now, we move to the next column of tens and multiply 5 by 3 in the tens place. 5 times 3 gives us 15 tens, and along with that, we will add the 3 tens on the top. That gives us 18 tens. Again, we will regroup 18 as 1 hundreds and 8 tens. Then, we will carry the 1 of “18” to the top of the hundreds column while keeping the 8 in the answer spot. Now, we move to the last multiplication step.
We will multiply the 5 by 1 in the hundreds place, and that will give us 5 hundreds. But there is a digit in the hundreds column, and we have to add that along with the value that we got in the third multiplication step. So, that gives us 5 + 1 = 6 in the hundreds place. We bring it down beside the other two digits in the answer spot. So, the final answer comes to 685.
Let us now use the partial product method to solve the same problem. In this method, we won’t be carrying over the digits but consider the digits in their ones, tens, and hundreds forms.
100 | 30 | 7 |
|---|---|---|
1 | 3 | 7 |
X | 5 |
Here, we are assuming the digits from the left as 1 as 100, 3 as 30, and 7 as 7. If we add them up, we get 137, as mentioned in the example. Now, let us begin with multiplication. First, we will multiply 5 by 7, which gives us 35. We will write that down.
100 | 30 | 7 |
|---|---|---|
1 | 3 | 7 |
X | 5 | |
3 | 5 |
Now, we move to the next multiplication step, 5 times 3. But instead of doing 5 times 3, we will be multiplying 5 by 30 as the rule of partial products. That will give us 150.
100 | 30 | 7 |
|---|---|---|
1 | 3 | 7 |
X | 5 | |
3 | 5 | |
1 | 5 | 0 |
In the last step of multiplication, we have to multiply 5 by 1, but instead of that, we will multiply 5 by 100. That will give us 500, which we will write down in our chart.
100 | 20 | 7 |
|---|---|---|
1 | 3 | 7 |
X | 5 | |
3 | 5 | |
1 | 5 | 0 |
5 | 0 | 0 |
Now, we will add all of these three values that we got from multiplying. That will give us 685, similar to the answer that we got from the traditional method.
How Can We Multiply a Four-digit Number by a Single-digit Number?
Similar to the three-digit number multiplication, the four-digit number multiplication involves small steps of multiplication. We can use both the partial product method and the regrouping method. Let us have a look at both of them with the following example.
We always begin with multiplication from the right, with the digit in the ones place. 3 times 5 will give us 15. We will regroup it as 1 ten and 5 ones, and thus carry the “1” to the tens column while writing 5 in the answer spot.
We now multiply 3 by 4 in the tens place. That will give us 12 tens. We have to add the 1 ten along with it, which gives us 13 tens. Now, we will regroup 13 as 1 hundred and 3 tens and write them accordingly.
Now, we move on to 3 times 3 which gives us 9 hundreds. Along with that, we will add the 1 hundred that will give us 10 hundreds. We will regroup 10 as 1 thousand and 0 hundreds.
The last step of multiplication is 3 times 2, which will give us 6 thousands. We will add the 1 thousand to it, and that will give us 7 thousands. So, the answer comes to 7035.
Now, we will use the partial products method for the multiplication of four-digit numbers. In this method, we expand the digits of the number as per the place value and then perform multiplication. Let’s take a look at an example.
2000 | 300 | 40 | 5 |
|---|---|---|---|
2 | 3 | 4 | 5 |
X | 3 |
We begin with the ones place. 3 times 5 will give us 15, and we will write it accordingly.
2000 | 300 | 40 | 6 |
|---|---|---|---|
2 | 3 | 4 | 5 |
X | 3 | ||
1 | 5 |
Now, we move to the ten’s place. We will multiply 3 by 4. But instead of that, we will multiply 3 by 40 as the rule of partial products. That will give us 120.
2000 | 300 | 40 | 5 |
|---|---|---|---|
2 | 3 | 4 | 5 |
X | 3 | ||
1 | 5 | ||
1 | 2 | 0 |
Now, let us multiply 3 by 3 in the hundreds place. But instead of 3, we will multiply it by 300. That will give us 900.
2000 | 300 | 40 | 5 |
|---|---|---|---|
2 | 3 | 4 | 5 |
X | 3 | ||
1 | 5 | ||
1 | 2 | 0 | |
9 | 0 | 0 |
In the final step of multiplication, we will multiply 3 by 2 in the thousands place. But instead of 2, we will multiply it by 2000. This will give us 6000.
2000 | 300 | 40 | 5 |
|---|---|---|---|
X | 3 | ||
1 | 5 | ||
1 | 2 | 0 | |
9 | 0 | 0 | |
6 | 0 | 0 | 0 |
Now, we will add up these columns in the partial product to get the final answer. That will give us 7035 as the answer, which is similar to what we got in the regrouping method.
Solved Multiplication of Multi Digit Number Examples
Example 1: Find the product: 2453 x 4
Answer:
Let us use the regrouping method to find the product.
First, we will begin with the ones place and work our way from right to left. 4 times 3 will give us 12, which we can regroup as 1 tens and 2 ones. So, we carry the “1” to the top of the tens column and place “2” in the answer spot.
Now we multiply 4 by 5 in the tens place. That will give us 20 tens. Now we add the 1 tens along with 20 tens to get 21 tens. We can now regroup it as 2 hundreds and 1 tens.
Moving to the hundreds place, 4 times 4 will give us 16 hundreds. We add the 2 hundreds along with it to get 18 hundreds. Now we regroup 18 as 1 thousands and 8 hundreds and carry “1” to the thousands column and put 8 in the answer spot.
Now, we multiply 4 by 2 in the thousands place. 4 times 2 will give us 8 thousands. We will add the 1 thousand along with it. This will give us 9 thousands, which we will put in the answer spot.
Hence, the answer is 9812.
Example 2: Find 422 x 4 and check if it’s reasonable or not.
Answer:
We can estimate this problem as 400 x 4 = 1600. So, if the real product value is close to the estimated product, then the answer will be reasonable.
Let us use the regrouping method to find the answer.
422
x 4
———–
We begin from the right, where we multiply 4 by 2 in the one’s place. This will give us 8. Since we don’t have any digits to carry over, we move to the next multiplication step where we multiply 4 by 2 in the tens place. This will again give us 8 tens. We don’t have any digits to carry over, so we move to the next multiplication step. 4 times 4 will give us 16 ‘hundreds’. The answer is 1688.
If you check with the estimated product, the answer is quite close and thus it is reasonable.
There are quite a few methods for multiplying multi-digit numbers by one-digit numbers. We can use the traditional method along with regrouping. We can use the partial products method as well, where we expand the digits to multiply. There are associative and commutative properties as well for multiplying multi-digit numbers by one-digit numbers.
When we estimate the product of two numbers, we are trying to find a close value and checking whether the estimated product is similar to the real product. For example, in order to find 452 x 3, we can estimate the numbers as 450 x 3 and then find the estimated product. This will give us 1350 as the estimated value. Now, if we find the product for 452 x 3, that will give us 1356 as the product. If you notice, the real product is quite close to the estimated product and thus the answer is reasonable.