Multiplication of Decimal Numbers: Did you have an idea that the history of decimals traces back to the ancient days in countries like India and China? Decimals were even adopted by the Egyptians to the counting system. By around \(1500,\) they became an integral part of the counting system worldwide when almost all professional mathematicians accepted decimals.
All the basic operations like addition, subtraction, multiplication, division can be performed on decimal numbers. In this topic, we shall discuss how to do multiplication on decimal numbers.

Learn The Concept Of Decimals

How do we Multiply Decimals?

We multiply the decimal numbers in the same way we multiply the whole numbers. First, we consider the given decimal numbers without taking into account their decimal point(s) and multiply them as whole numbers. Then, we place the decimal point before the number of digits from the rightmost side of the product, which is equal to the sum of the total number of decimal digits in the two given decimal numbers.

Example:
Let us multiply \(3.7\) and \(5.28\)
For this multiplication, we shall consider \(3.7\) as \(37\) and \(5.28\) as \(528.\) Then, we shall multiply \(37\) and \(528\) to obtain \(19536\) as their product.
We observe that \(3.7\) has one digit after the decimal point, and \(5.28\) has two digits after the decimal point. We have \(1 + 2 = 3.\)
In product \(19536,\) we shall place the decimal point before 3 digits from the rightmost side. Thus, we obtain \(19.536\) as the product of \(3.7\) and \(5.28.\)

The basic procedure for the multiplication of decimal numbers is applied for all cases of the multiplication of decimal numbers. Yet, we can still discuss the multiplication of decimal numbers into the following three sub-categories:

a) Multiplication of decimal numbers by \(10,100\) and \(1000\)
b) Multiplication of a decimal number by a whole number
c) Multiplication of two decimal number

Multiplication of Decimal Numbers by 10, 100 and 1000

When we multiply a decimal number by \(10,100\) or \(1000,\) we move the decimal point to the right by as many places as the number of zeroes after \(1\) in the multiplier.
Look at the table below, for example.

Multiplication by 10

When we multiply the decimal number \(4.39\) by \(10,\) we move the decimal point by one place to the right and obtain \(43.9\) because 10 has one zero in it.

Multiplication by 100

When we multiply the decimal number \(4.39\) by \(100,\) we move the decimal point by two places to the right and obtain \(439.00\) or \(439\) (we do not need to write \(0\) alone after the decimal point) because \(100\) has two zeros in it.

Multiplication by 1000

When we multiply the decimal number \(4.39\) by \(1000,\) we move the decimal point by three places to the right and obtain \(4390,\) because \(1000\) has three zeros in it. Note that we have added a zero after \(439\) in order to shift the decimal point by three places at its right.
This situation can also be explained using the basic multiplication rule for decimals. Without considering the decimal point, we treat \(4.39\) as \(439.\) Then, we multiply \(439\) and \(1000\) to obtain \(439000.\) Next, we shall place the decimal point \(2\) places before the right most digit of the product \(439000\) to obtain \(4390.00\) or \(4390.\)

Arithmetic Operation	Rule	Example
Multiply by \(10\left({{{10}^1}} \right)\)	the decimal point moves one place to the right	\(4.39 \times 10 = 43.9\)
Multiply by \(100\left({{{10}^2}} \right)\)	the decimal point moves two places to the right	\(4.39 \times 100 = 439\)
Multiply by \(1000\left({{{10}^3}} \right)\)	the decimal point moves three places to the right	\(4.39 \times 1000 = 4390\)

Practice Exam Questions

Multiplication of Decimals with Whole Numbers

We shall follow the steps mentioned below for multiplying decimal numbers with whole numbers.

1. Step 1: Ignore the decimal point and multiply the two numbers as usual.
2. Step 2: The result or the product obtained after multiplication will have an equal number of decimal places as the given decimal number (because the given whole number do not have a decimal place).
3. Step 3: Locate the decimal point in the obtained product following Step 2.

Example-1: \(2.3\,{\rm{m}}\) of cloth is needed to make one shirt. How many metres of cloth is needed to make \(25\) such shirts?
Here, the cloth needed to make one shirt \( = 2.3\,{\rm{m}}\)

Total quantity of cloth needed to make \(25\) shirts \( = 2.3 \times 25 = 57.5\,{\rm{m}}\)

Example-2: Joey baked \(12\) cookies. If the weight of one cookie is \(0.003\,{\text{kg}},\) what is the weight of all the cookies?
As we discussed, in the multiplication of decimals with the whole numbers, we ignore the decimal point initially. So,\(12 \times 3 = 36\)
In this case, we observe that the whole number \(12\) has \(0\) decimal places, and the decimal number \(0.003\) has \(3\) decimal places. So, the total number of decimal places is \(0 + 3 = 3.\)
But product \(36\) has two digits only. In this case, we shall write one zero before \(36\) and then write the decimal point to obtain \(0.036.\)(If the product has more decimal places than the number of digits, zero(s) can be inserted prior to placing the decimal point in the product.
\( \Rightarrow 12 \times 0.003 = 0.036\)
Therefore, the total weight of all the \(12\) cookies is \(0.036\,{\text{kg}}.\)

Multiplying Two Decimal Numbers

The basic rule for the multiplication of two decimal numbers is explained earlier.

Follow the step given below:

1. Step 1: At first, ignore the decimal point and multiply the two numbers, as usual, considering them as whole numbers.
2. Step 2: After multiplication, add the total number of decimal places in both the numbers. The product obtained after multiplication will have this total number of decimal places placed before the rightmost digit of the product.
3. Step 3: Locate the decimal point in the product obtained following Step 2.

Example-1: The multiplication of two decimals \(3.23 \times 4.3\)

Thus, \(3.23 \times 4.3 = 13.889\)

Example-2: Consider multiplication of two decimal numbers, \(6.35\) and \(1.8.\) Multiplication of \(635\) and \(18\) (without considering their decimal points) resulted in \(11430.\) After placing the decimal point in the product, we obtain \(11.430.\)
We do not need to write 0 at the extreme right after the decimal point. Hence, the product of \(6.35\) and \(1.8\) will be \(11.43.\)

The multiplication is shown in the figure given below.

Thus, \(6.35 \times 1.8 = 11.430 = 11.43.\)

Attempt Mock Tests

Properties of Multiplication of Decimal Numbers

1. The product of two decimal numbers remains the same, even if the order of multiplication is interchanged. For example: \(0.3 \times 0.5 = 0.5 \times 0.3 = 0.15\)
2. The product of \(1\) and any decimal number is the decimal number itself.
For example, \(7.542 \times 1 = 7.542\)
3. The product of any decimal number with zero is \(0.\) For example: \(0.023 \times 0 = 0\)
4. The product of a whole number and the decimal number remains unchanged when the numbers are multiplied in any order. For example, \(1.1 \times 18 = 1.1 \times 18 = 19.8.\)

Solved Examples – Multiplication of Decimal Numbers

Q.1. Multiply 12.625 and 0.816.
Ans:

Therefore, \(12.625 \times 0.816 = 10.302000\). Note that after the decimal point, there are \(6\) digits. We know that the zeroes at the extreme right of the decimal point can be dropped. Hence, we have \(12.625 \times 0.816 = 10.302\)

Q.2. Find the product of 21.37 and 5.9.
Ans:

Therefore, \(21.37 \times 5.9 = 126.083.\) The product has \(3\) digits after the decimal point.

Q.3. The cost of a box of chocolate is \(₹65.75.\) Rupali needs 10 such boxes. How much amount does she have to pay? And if she needs \(100\) boxes, how much amount she has to pay?
Ans: The cost of \(10\) boxes is more than the cost of one box. So it is obvious that we should multiply.
We know that \(6575 \times 10 = 65750.\)
Hence, \(65.75 \times 10 = 657.50\)
Therefore, Rupali paid \(₹657.50\) to purchase \(10\) boxes of chocolate.
Similarly, if Rupali wants to purchase \(100\) boxes, she would have to pay \(₹65.75 \times 100 = ₹6575.\)

Q.4. Find \(5.3145 \times 1000.\)
Ans: The number of zeros in the multiplier \(\left({1000} \right)\) after \(1\) is \(3.\) So the decimal point will move three places to the right.
Hence, \(5.3145 \times 1000 = 5314.5\)

Q.5. Gopi went to the grocery store with his mother. His mother bought 10 bananas costing \(₹2.50\) each. Help Gopi to calculate the amount his mother needs to pay.
Ans: We have the cost of one banana \( = ₹2.50.\)
Thus, the cost of \(10\) bananas \(=₹ 2.50 \times 10.\)
Therefore, the cost of \(10\) bananas is \(₹25.\) (shifting the decimal point to one place to the right)

Summary

In this article, we discussed the multiplication of decimal numbers that includes the multiplication of decimal numbers with \(10,100\) and \(1000.\) We also discussed how to multiply decimal numbers with whole numbers and how to multiply two decimal numbers. We also described a few basic properties of the multiplication of decimal numbers. Going through this article will help students in making a comprehensive understanding of the multiplication of decimal numbers.

Frequently Asked Questions (FAQs) – Multiplication of Decimal Numbers

Let’s look at some of the commonly asked questions about multiplication of decimal numbers:

Q.1. How do you multiply decimals by 100?
Ans: To multiply decimals with 100, shift the decimal point two places towards the right, as there are two zeroes in 100.

Q.2. What is a non-decimal number?
Ans: All the numbers without any fractional part are non-decimal numbers. We do not use a decimal point for representing these. Both negative and positive integers (that include whole numbers and natural numbers) can be called non-decimal numbers.
Example: 47,5963,9235, etc.

Q.3. How do you multiply positive and negative decimals?
Ans: Positive and negative decimals are multiplied similarly as two decimals. Then, a proper sign will be attached to the product following the rule \(”{\text{Negative}} \times {\text{Positive}} = {\text{Negative}}.”\)
Thus, we shall keep the negative sign before the numerical part of the product that is obtained.
Example:\( – 0.8 \times 0.4 = – 0.32\)

Q.4. What are the four rules for the multiplication of decimals?
Ans: The four rules of decimals multiplication are:
1. Carry out multiplication in a similar way to whole numbers.
2. Suppose the product has more decimal places than the number of digits, zeros can be inserted before placing the decimal point in the product such that only one zero will be there at the left of the decimal point, and the decimal places in the product will be equal to the total number of decimal places in both the numbers.
3. The decimal point should be placed in the product so that the product has a number of decimal places equal to the total number of decimal places of the given decimal numbers.
4. The zeros at the extreme right side after the decimal point in the resultant product can be dropped.

Q.5. What does 3 decimal places mean?
Ans: “Three decimal places” is the same as “the nearest thousandth”. It also means that there will be three digits after the decimal point.

Learn About Decimal Numbers

We hope this detailed article on the multiplication of decimal numbers helps you in your preparation. If you get stuck do let us know in the comments section below and we will get back to you at the earliest.