Multiplying Decimals
We often come across a scenario when after a party at our house, there is a carpet that requires deep cleaning. The average carpet cleaning cost in most cities is $0.25 per square foot of carpet.
To know the carpet cleaning cost we must know the area of the carpet. Carpets are mostly rectangular in shape and here’s one such dirty carpet which requires cleaning.
The area of a rectangle can be calculated by multiplying the length by its width. To calculate the area of this rectangular carpet, we will have to perform the following multiplication:
Area = Length \(\times \) Width
\(~~~~~~~~=10.5\times 7=?\)
Wait! There is a problem! We know how to multiply whole numbers, but how do we multiply decimals? Here’s how to tackle this problem.
Multiplication of Decimals
Multiplication is repetitive addition. If we want to find what is \( 2\times 7 \) then this can be found by adding 2 to itself 7 times as shown below:
In a similar way, we can multiply decimals. Decimal multiplication are of two types:
- Multiplication of decimals with whole numbers
- Multiplication of decimals with decimals
Multiplication of Decimals with Whole Numbers
Let us try to find the product of 7 and 10.5
As we have seen in the previous section, there is a way to find this and that is repetitive addition. This can be seen below:
\(10.5\times 7=\)
We cannot always use this method. It is because performing repetitive addition for bigger numbers (like \(10.5\times 70\)) would become cumbersome. We need an efficient way to calculate these products. There is an easy way out, let’s find out:
Consider \(10.5\times 70\)
This can also be represented as :
Step 1: Count the total number of decimal places
Step 2: Multiply as we would with whole numbers by removing the decimal.
Step 3: Insert the decimal in the product. Take the number of decimal places as counted in Step 1 and insert the decimal accordingly in the product obtained in Step 2.
Let us look at some examples to understand this better :
Example 1: Find \(5.2121\times 15\)
Solution:
Count the number of decimal places, multiply as we would do with whole numbers and insert the decimal back in the product.
Example 2: Find \(0.015\times 5\) and also verify your product using estimation.
Solution :
Count the number of decimal places, multiply as we would do with whole numbers and insert the decimal back in the product.
To have three decimal places as asked in the question, we will insert one zero to the left of 75.
Verification Using Estimation:
\(0.015\approx 0\) and \(0\times 5=0\),
As, \(0.075\approx 0\), hence, we can say that our product is verified.
We have explored the multiplication of decimals with whole numbers. Now, let us see the multiplication of decimals with decimals.
Multiplication of Decimals with Decimals
Consider \(10.5\times 0.7\)
This can also be represented as :
Step 1: Count the total number of decimal places
Step 2: Multiply as we would with whole numbers by removing the decimal.
Step 3: Insert the decimal in the product. Take the number of decimal places as counted in Step 1 and insert the decimal accordingly in the product obtained in Step 2.
Let us look at an example to understand this better:
Example 1: Find \(6.306\times 0.084\)
Solution :
Count the number of decimal places, multiply as we would do with whole numbers and insert the decimal back into the product.
Always remember to add zero to the left of the decimal if it does not have any whole number.
Now let us go back to where we started from.
Example 2:
You have a carpet of length 10.5 feet and width 7 feet. The cost of washing this carpet is $0.25 per square foot of carpet. Find the total money spent by you to get this carpet washed.
Solution:
As the cost is given as $0.25 per square foot of carpet, we need to find the area of the carpet.
The area can be calculated as:
\(\text{Area}=\text{Length}\times \text{Width}\)
\(~~~~~~~~~=10.5\times 7\)
This can be solved as follows:
Count the number of decimal places, multiply as we would do with whole numbers and insert the decimal back in the product.
Therefore, the area is 73.5 square foot.
Now to find the cost let’s multiply the area and the cost per square foot.
Cost \(=73.5\times 0.25\)
Again, let’s count the number of decimal places and multiply as we would do with whole numbers.
Now, insert the decimal back in the product.
So, the amount payable by you to get the carpet cleaned is $18.375.
Multiplying decimals is in fact quite similar to multiplying whole numbers. A small difference is about placing the decimal point according to the total number of decimal places.
Decimal values can be converted as fractions and vice versa. For example, 0.5 is the same as \(\frac{1}{2}\) and 0.44 is the same as \(\frac{11}{25}\) .