What are Decimals?
Decimal numbers are those numbers whose whole number part is separated from its fractional part by a decimal point. The decimal point is represented by a “dot” in a decimal number.
For example, the fraction “\(\frac{1}{2}\)” can be written as 0.5 in a decimal form, in which the zero is in the ones place, and the five is in the tenths place.
What is the Multiplying Pattern in Decimals by Powers of 10?
While multiplying any decimal number by 10, 100, 1000..\(10^n\), we shift the decimal point to the right by as many places as the number of zeros after 1.
Here, 10, 100, 1000..\(10^n\) are the multipliers. The multiplying pattern shows that as we multiply by increasing powers of 10, the decimal point shifts to the right. For example, have a look at the following table.
What is the Multiplying Pattern in Decimals by 0.1, 0.01, 0.001?
When we multiply by a tenth (0.1), the product becomes smaller and thus the decimal point shifts to the left.
Similarly, while multiplying by 0.01 and 0.001, the decimal point will shift further to the left. For example, have a look at the following table.
What are the Two Methods to Multiply Decimal and Whole Numbers?
While multiplying decimal numbers with whole numbers, we can follow any of these two methods:
- The similar multiplication method as we do with whole numbers and then place the decimal point.
- Multiplication by using place value and then adding the partial products.
We will solve a few problems on these methods to understand how the methods work.
Discuss about commutative and associative properties of multiplication in decimals.
The commutative property of multiplication mentions that the order in which we multiply numbers doesn’t affect as the product will be equal either way.
For example,
2 .5 (6) = 6 (2.5)
The associative property of multiplication mentions that whatever the order in which we group decimal numbers does not affect as the product will be similar. Grouping denotes the use of parentheses or brackets.
For example,
4 (2.5 x 6) = (4 x 2.5) 6
We will solve a few problems on these properties to understand how they work.
Solved Examples on Multiplying Decimals
Example 1: Find \( 0.46 \times 10^3 \)
Answer: Let us use place value concepts for solving this problem. When we multiply a decimal number by 10, each digit in the number moves one position to the left in the place value chart.
\(0.46 \times 1 = 0.46\)
\(0.46 \times 10^1 = 0.46 \times 10 = 4.6\)
\(0.46 \times 10^2 = 0.46 \times 100 = 46\)
\(0.46 \times 10^3 = 0.46 \times 1000 = 460\)
Let us have a look at the place value chart where we will place the digits after multiplying by the increasing powers of 10.
You can notice the pattern where in every product, the number of places the decimal point shifts to the right is similar to the exponent.
Thus, the answer is 460.
Example 2: Find 5.7 x 0.01
Answer: Here, we will use place value concepts to find the answer to the problem. When we multiply any decimal number by \(\frac{1}{10}\) = 0.1, the digits in the number move one position to the right in the place value chart.
5.7 x 1 = 5.7
5.7 x 0.1 = 0.57
5.7 x 0.01 = 0.057
Let us have a look at the place value chart now.
You can notice a pattern where when you multiply by 0.1, the decimal point shifts one place to the left, and shifts two places to the left when multiplied by 0.01.
Thus the answer is 0.057.
Example 3: Amy bought 5 chocolates and each chocolate costs $2.8. Can you find out the total cost of the chocolates?
Answer: In order to multiply a decimal and a whole number, we can try any of the two methods that we have learned. Here, we will exhibit both the methods for your understanding.
First method: Multiply as we do with whole numbers, and then place the decimal point in the result.
Here we can assume 2.8 as 28 x 0.1. So, we will use the whole number, 28 for multiplying it with 5.
Let us now multiply the numbers.
So, now that we have found the result, we will multiply it by 0.1 to get the answer.
140 x 0.1 = 14.0 is the answer or we can say $14 is the total cost of the chocolates.
Second method: Using place value and partial products.
Here, we will multiply the tenths first.
Here, 5 x 8 tenths equals 40 tenths which can be written as 4 ones and 0 tenths. Now, we will move on by multiplying the ones.
Here, 5 x 2 ones equals 10 ones which can be written as 1 tens and 0 ones. Now, we will add the partial products.
Thus the answer is 14.
Amy spent 14 dollars for 5 chocolates.
Example 4: Stephanie bought 40 pencils for herself. She lent 10 pencils to her friend and the rest she will store it in a box. Each pencil has a size of 5.5 cm. Find out the amount of space that the pencils will take up in the box.
Answer: Number of pencils Stephanie bought = 40
Number of pencils Stephanie lent to her friend = 10
Thus, number of pencils Stephanie has with her now = 40 – 10 = 30
Now, we have to find the space taken up by 30 pencils in the box. Each pencil has a size of 5.5 cm, so 30 pencils will have a larger size.
We will multiply as we do with whole numbers, and then place the decimal point in the result. Let us assume 5.5 as 55 x 0.1.
Now, we will multiply the product by 0.1 to find 30 x 5.5 as per the problem. Here, we will shift the decimal point one place to the left.
1650 x 0.1 = 165.0
Thus the space taken by 30 pencils will be 165 cm.
Example 5: Find 0.2 x 0.8
Answer: Let us use the commutative and associative properties of multiplication to solve this problem.
Imagine 2 x 0.1 = 0.2 and 8 x 0.1 = 0.8
0.2 x 0.8 = 2 x 0.1 x 8 x 0.1
= 2 x 8 x 0.1 x 0.1 (commutative property of multiplication)
= (2 x 8) x (0.1 x 0.1) (associative property of multiplication)
= 16 x 0.01
Multiplying 16 by 0.01 shifts the decimal point to two places to the left.
Thus, 16 x 0.01 = 0.16
Example 6: Find the product for \(1.2\times2.8\)
Answer: Let us use the commutative and associative properties of multiplication to solve this problem.
Imagine \(12\times0.1=1.2\) and \(28\times0.1 = 2.8\)
So, \(1.2\times2.8 = 12\times\ 0.1\times28\times0.1\)
\( = 12\times28\times0.1\times0.1\) (commutative property of multiplication)
\( =(12\times28)\times(0.1\times0.1)\) (associative property of multiplication)
\( = 336\times0.01\)
Multiplying 336 by 0.01 shifts the decimal point to two places to the left.
Thus \(336\times0.01 = 3.36\)
The process for multiplying decimals is quite similar to that of multiplication of whole numbers. The only difference is the placement of the decimal point in the product.
When multiplying two decimal numbers, we can multiply them as normal whole numbers and then place the decimal point in the result.