Multiplication of multi digit numbers by multi digit

Multiplying multi-digit numbers by multi-digit numbers is done in the same way as multiplying multi-digit numbers by single-digit numbers. Each digit of a multi-digit number is multiplied by each digit of another multi-digit number. The only difference is that at the end there is a kind of staircase of answers that must be added up. Let's look at a few examples to get a good understanding of this.

Example 1. Find the value of the expression 12 x 14

Write down this expression in a column - ones under ones, tens under tens:

Now multiply each digit of 12 by each digit of 14. You should do this one by one, starting with four. As a result of these actions, we come to the multiplication of a multi-digit number by a single-digit number, which we studied earlier:

By multiplying 12 by 4, we got the number 48, which we wrote down so that the unit digit of this number is under the four by which we multiplied 12.

Now multiply 12 by 1:

We multiplied 12 by 1 to get the number 12 and wrote it down so that the unit digit of this number is below the unit by which we multiplied 12.

We got the staircase of answers 48 and 12, which should be added. We add it up and get the answer 168.

In this example, the factor 14 is four ones and one tenth. Then multiplying 12 by 14 can be understood as multiplying 12 by a factor of four and a factor of ten. This explains the appearance of the staircase at the end of the solution. Let's see how it looks at each step:

Let's multiply the number 12 by a factor of four and get the number 48.

Increase the number 12 ten times, and you get the number 120. Write 120 so that the units of this number can be added to the units of 48, and the tens of 120 can be added to the tens of 48

Now add up the resulting staircase of answers. We add ones to ones, tens to tens, and hundreds to hundreds. The result is the final answer:

But most often the multiplier is not grouped with digits, and multiplication is done by multiplying each digit of the multiplier by each digit of the multiplier.

Example 2. Find the value of the expression 25 x 36

Write this expression in columns

Multiply each digit of 25 by each digit of 36.

Multiply 25 by 6:

Multiply 25 by 3:

Now fold the resulting ladder:

We got an answer of 900.

Consider a large and complicated multiplication example: 12305 x 5641. We will stick to the rules we learned earlier.

First, write down the expression in column

Now start multiplying. The number 12305 must be multiplied by each digit of the number 5641.

By multiplying 12305 by 1, we get 12305 and write this number so that the unit digit of this number is below the unit by which we multiplied 12305.

Now we multiply 12305 by the next digit 4:

We multiply 12305 by the next number 6:

By multiplying 12305 by 6, we get 73830, and we write this number down so that the unit digit of this number is below the six by which we multiplied 12305.

Now multiply 12305 by the last digit of 5:

By multiplying 12305 by 5, we got 61525, and we wrote down this number so that the unit digit of this number is under the five by which we multiplied 12305.

The result is a large ladder, which must be added up. We add up:

The final answer is 69412505.

If you understood this example, we can say that you learned multiplication of large numbers very well.

That concludes the lesson on multiplication. Be sure to practice by solving some of the examples given below.

It is important to note that it is not customary to draw all these arrows and detailed solutions as in pictures in "combat conditions". You need to be able to write down the answers right away, doing all the calculations in your mind.

The exception is if a person hasn't done math for a long time or has never done math before. In this case, you can draw arrows and other auxiliary diagrams for a good assimilation of the material.

Exercises 

Task 1. Perform multiplication:

Solution:

Show Solution

Video lesson

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