Multiplication of Decimal Numbers

We will solve the multiplication of decimal numbers using the vertical multiplication method.
We will proceed in the following order:

We will neatly write the multiplication exercise in vertical form – one decimal point under the other decimal point, tenths under tenths, hundredths under hundredths, etc.
We will solve the exercise and, for now, will not pay attention to the decimal point.
We will strictly adhere to the rules of vertical multiplication.
We will review each number in the exercise and find out how many digits there are after the decimal point.
We will count the total number of digits after the decimal point (taking into account both numbers) and that will be the number of digits after the decimal point in the final answer.

Detailed explanation

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$0.1\times0.5=$

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Multiplication of Decimal Numbers

To easily solve exercises involving the multiplication of decimal numbers, you should know how to solve multiplications of whole numbers using the vertical form.
The most recommended method for solving exercises of this type is, clearly, multiplication in vertical form.
Steps to follow:

• We will write the exercise clearly in vertical form – decimal point under the other decimal point, tenths under tenths, hundredths under hundredths, and so on.
• We will solve the exercise without taking the decimal point into account, but remembering that it exists.
• Let's not stop acting according to the rules of multiplication in vertical form, that is, reserving places for zeros, carrying over remainders and remembering them, etc.
• We will count the total number of digits after the decimal point (taking into account both numbers) and in this way determine the number of digits that will be after the decimal point in the result.

Let's apply the solution steps in an exercise, this will help you understand it better:
Given the exercise $0.4\times 0.2=$
We will write it in vertical form making sure that the decimal point is under the other decimal point:

1a - in vertical form making sure that the decimal point is under the other decimal point

Let's solve the exercise as we always do, remembering the rules of multiplication in vertical form.

Notice: for now we ignore the decimal point and do not note it in the result.
We obtained the result $008$
How will we know where to place the decimal point in the result?
Let's look at the numbers in the exercise and add up the number of digits after the decimal point in both numbers.

2a - there will be 2 digits after the decimal point in the result

What does this mean?
Let's look at the first number $0.4$ , there is one digit after the decimal point.
Let's look at the second number $0.2$ , there is one digit after the decimal point.

Let's ask: How many digits after the decimal point are there in total? $2$
$1+1=2$

Therefore, there will be $2$ digits after the decimal point in the result.
We will obtain:

3a - Therefore, there will be 2 digits after the decimal point in the result

And this is the final result.

Multiplication Exercises with Decimal Numbers

Exercise 1

$0.02\times 0.09=$

Solution:
We will write the exercise in vertical form making sure that the decimal point is under the other decimal point:

4a - the decimal point is under the other decimal point

We will solve it as usual, ignoring the decimal point:

5a - Let's add up the number of digits after the decimal point in the numbers of the exercise

Let's add up the number of digits after the decimal point in the numbers of the exercise and apply the result in the final answer:

A6 - apply the result in the final answer

Let's apply this to the result and we will get $0.0018$ :

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Question 1

$0.1\times0.35=$

Question 2

$0.1\times0.999=$

Question 3

$0.1\times0.004=$

Exercise 2

$45.7\times 0.4=$

Solution:
Let's write the exercise clearly in vertical form.
Let's solve it as usual and ignore the decimal point.
We will obtain:

7a - Let's solve it as usual and ignore the decimal point

Let's calculate where to place the decimal point:

A8 - Total digits after the decimal point → 2

Let's apply it to the answer and we will obtain: $18.28$

Exercise 3

$15.06\times 0.01=$

Solutionן:
Let's write the exercise clearly in vertical form.
Let's solve it as usual and ignore the decimal point in the result.
We will obtain:

9b - Let's write the exercise clearly in vertical form

Let's calculate where to place the decimal point:

Let's apply it in the answer and we will obtain:
That is, $0.1506$

Do you know what the answer is?

Question 1

Find the correct place of the decimal point:

$1.35\times2.47=33345$

Question 2

Given the following exercise, find the correct place of the decimal point:

$0.3\times2.15=0645$

Question 3

Given the following exercise, find the correct place of the decimal point:

$2.5\times0.13=0325$

Exercise 4

$13\times 0.24=$

Solution:
Let's write the exercise clearly in vertical form.
Let's solve it as usual and ignore the decimal point in the result.

We will obtain:

11a- Let's write the exercise clearly in vertical form

Let's calculate where to place the decimal point:

Let's apply it in the answer and we will obtain:
$03.12$ That is, $3.12$

Examples and exercises with solutions for multiplying decimal numbers

Exercise #1

Given the following exercise, find the correct place of the decimal point:

$0.3\times2.15=0645$

Video Solution

Step-by-Step Solution

In the number -0.3, there is one digit after the decimal point, 3

In the number 2.15, there are two digits after the decimal point, 15

Therefore, we have three digits after the decimal point.

So, from our result, we will count three decimal places and find that the answer is -0.645

Answer

$0.645$

Exercise #2

Given the following exercise, find the correct place of the decimal point:

$3.5\times2.4=840$

Video Solution

Step-by-Step Solution

In the number -3.5, there is one digit after the decimal point, 5

In the number 2.4, there is one digit after the decimal point, 4

In other words, we have two digits after the decimal point.

Therefore, from our result, we will count three decimal places and find that the answer is 8.40

Answer

$\text{8}.40$

Exercise #3

$4.5\times3.2\times5.6=$

Video Solution

Step-by-Step Solution

According to the order of operations rules, we will solve the exercise from left to right since multiplication is the only operation in it.

We will solve the left exercise vertically to avoid confusion and get:

$4.5\\\times3.2\\=14.40$ It is important to maintain correct positioning of the exercise, with the decimal point serving as an anchor.
Then we can multiply in order, first the ones digit of the first number by the ones digit of the second number,
then the tens digit of the first number by the ones digit of the second number, and so on.

Now we will get the exercise:

$14.40\times5.6=$

Let's remember that:

$14.40=14.4$

We will solve the exercise vertically as well, and remember the rules of keeping the decimal point and multiplying in order (ones, tens, and so on)

And we will get:

$14.4\\\times5.6\\=80.64$

Answer

$80.64$

Exercise #4

$11.2\times5.6\times7.3=$

Video Solution

Step-by-Step Solution

According to the order of operations rules, we will solve the exercise from left to right since multiplication is the only operation in it.

We will solve the left exercise vertically to avoid confusion and get:

$11.2\\\times5.6\\=62.72$ It's important to maintain correct positioning of the exercise, with the decimal point serving as an anchor.
Then we can multiply in order, first the ones digit of the first number by the ones digit of the second number,
then the tens digit of the first number by the ones digit of the second number, and so on.

Now we'll get the exercise:

$62.72\times7.3=$

Let's remember that:

$7.3=7.30$

We will solve the exercise vertically as well, remembering the rules about keeping the decimal point aligned and multiplying in order (ones, tens, and so on)

And we'll get:

$457.856$

Answer

$457.856$

Exercise #5

$2x\times4.65\times6.3=$

Video Solution

Step-by-Step Solution

Let's look at the exercise, and we'll see that we have two "regular" numbers and one number with a variable.
Since this is a multiplication exercise, there's no problem multiplying a number with a variable by a number without a variable.

In fact, it's important to remember that a variable attached to a number represents multiplication by itself, for example in this case: $2\times x$
Therefore, we can use the distributive property to separate the variable, and come back to it later.
We'll solve the exercise from left to right.

We'll solve the left exercise by breaking down the decimal number into an addition problem of a whole number and a decimal number as follows:

$2\times(4+0.65)=$

We'll multiply 2 by each term in parentheses:

$(2\times4)+(2\times0.65)=$

We'll solve each of the expressions in parentheses and get:

$8+1.3=9.3$

Now we'll get the exercise:

$9.3\times6.3=$

We'll solve the exercise vertically to make the process easier for ourselves.

It's important to be careful with the proper placement of the exercise, using the decimal point as an anchor.
Then we can multiply in order, first the ones digit of the first number by the ones digit of the second number,
then the tens digit of the first number by the ones digit of the second number, and so on.

$9.3\\\times6.3\\=58.59$

Don't forget to add the variable at the end, and the answer will be:

$58.59x$

Answer

$58.59x$

Check your understanding

Question 1

Given the following exercise, find the correct place of the decimal point:

$6.13\times2.05=125665$

Question 2

Given the following exercise, find the correct place of the decimal point:

$3.5\times2.4=840$

Question 3

Given the following exercise, find the correct place of the decimal point:

$3.751\times0.5=18755$

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