|Multiplying Whole Numbers and Decimals Using Long Multiplication
Multiplication involving whole numbers and decimals is exactly like multiplying whole numbers together.
The biggest difference is at the end of the problem when we are adding up our result from the long multiplication.
Be sure to line up the correct units places directly underneath each other.
Let's take a look at a problem in action to see this process in motion.
Multiplying Whole Numbers and Decimals by Long Multiplication
A farmer has 16 chickens that each weigh 4.2 pounds.
Help the farmer figure out the total weight of chickens present.
We want to multiply 4.2 and 16 together.
It's helpful, but not necessary if we line up the correct units places above each other (ones above ones, etc).
Starting with the last digit of the bottom number, begin to multiply it by the digits of the top number.
2 × 6 = 12
Record the 2 underneath, and carry the 1 over to be added later.
After multiplying the rest of the digits by 2 we should have a result of 32 so far.
Move on to the next digit in our bottom number and begin to multiply it by the top number.
Be sure to cross out a unit space as we move to the next number.
This is one of the most important steps!
After multiplying the rest of the digits by 4, we should get the result of 34.
Our last step is to add up the result. You'll notice that we get 373.
At this stage, we need to count how many digits are behind the decimal space in our original problem.
Since we have one digit after the decimal space, we will shift the decimal point one space to the left.
Our final answer is 37.3.
The farmer has 37.3 pounds of chickens on his farm.
The more digits we add on the end of a decimal, the more tricky our problems get to solve. Our method for solving them does not change however.
Multiplying Decimals by Decimals
You may be required to multiply two decimal numbers together in many real life situations.
For example, what if we wanted to know the total amount of money we would need to buy school lunch for the semester?
We know that the semester is 12.5 weeks long and that lunch costs $20.2 each week.
To find the total cost, we would need to multiply 12.5 by 20.2.
There are a few important rules to keep in mind when performing this type of multiplication.
Let's solve this problem together and then we can break our rules down.
Example: Multiplying Two Decimals Together
Set up as a long multiplication problem.
Multiply the right most digit of the bottom number by the number above it.
5 × 2 = 10
Since this is a two digit number, we will record the digit on the left below our 5, and carry the one to the next unit place.
We multiply the rightmost digit of the bottom number by the next digit of the top number.
5 × 0 = 0
Since we carried a number over, we add that to our result.
0 + 1 = 1
Record this number in the next unit place of our answer.
Continue multiplying by the rest of the digits of the top number.
5 × 2 = 10
Since this is our last multiplication by 5, record the two digit number at the end.
We will now shift one digit to the left of our bottom number, and repeat the proces of multiplying by all the digits of our top number.
Since we shifted one unit to the left, we need to cross out the rightmost unit space before recording our result.
2 × 2 = 4
We will repeat the process of multiplying each digit.
This is the result after finishing multiplying the top number by 2.
We will cross out another unit place and multiply the leftmost digit of the bottom number by each digit in the top number.
Now that we are done multiplying, we add up all of our numbers from the solution.
Identify how many digits are behind the decimal point in our original solution.
There are exactly 2 digits behind the decimal point.
We can get our final answer by adding the decimal space after the first two digits on the right.
Our total cost for lunch for the semester is 252.50.
Let's summarize some key aspects of long multiplication with decimals:
Multiplying with decimals is very similar to multiplying whole numbers.
Always cross out a unit space as we shift to the left from our bottom number.
After we add up our solution, we need to count the total number of digits behind the decimals in the original question.
Once we know how many digits are behind the decimal, we need to move our decimal that many spaces to the left.