3.04 Multiplying with decimals
We have previously looked at what happens when we multiply decimals by powers of 10, and how we increase the place value of each digit by one place for every $10$10 that we multiply by. We will now look at how we can multiply any decimal numbers, using the same methods we use to multiply whole numbers.
Exploration
Let's follow through the process for the multiplication $4.83\times5.7$4.83×5.7.
Before we even begin to calculate the answer it can be a good idea to have an estimation of the answer, especially when dealing with decimals. This will help us confirm our final answer is of the right magnitude.
In this case, we can round both numbers to the nearest whole, giving us the much simpler calculation $5\times6$5×6, which we can evaluate to get $30$30. With this in mind, we would expect our answer to be close to this value, and we can know for sure that it will have digits in the tens and ones columns.
Now, to start the process, we simply ignore the decimal points. In this case we get $483$483 and $57$57. We then multiply these together with the method we are used to using for whole numbers:
| $4$4 | $8$8 | $3$3 | |||||
| $\times$× | $5$5 | $7$7 | |||||
| $3$3 | $3$3 | $8$8 | $1$1 | (this is $483\times7$483×7) | |||
| $+$+ | $2$2 | $4$4 | $1$1 | $5$5 | $0$0 | (this is $483\times5\times10$483×5×10) | |
| $2$2 | $7$7 | $5$5 | $3$3 | $1$1 |
Now we need to account for the decimal point. To do so, we add the total number of decimal places in the original numbers together.
In this case the original numbers are $4.83$4.83, which has two decimal places, and $5.7$5.7, which has one decimal place. So their product will have $2+1=3$2+1=3 decimal places.
Then to find the final answer, we take the product that we calculated before and insert the decimal point such that there are $3$3 decimal places:
So we have found that $4.83\times5.7=27.531$4.83×5.7=27.531.
This is very close to our original estimate of $30$30, and as expected we have a digit in each of the tens column and ones column. If we ended up with a final value of $275.31$275.31 we know we have written the decimal point in the wrong place.
Why does this work?
Remember that we can represent any finite decimal as a fraction by using a power of $10$10 in the denominator. In this case, $4.83$4.83 is equal to $\frac{483}{100}$483100 and $5.7$5.7 is equal to $\frac{57}{10}$5710.
Let's now multiply these numbers by using their fraction forms instead. To do so, recall that we just multiply their numerators together and their denominators together:
| $\frac{483}{100}\times\frac{57}{10}$483100×5710 | $=$= | $\frac{483\times57}{100\times10}$483×57100×10 |
| $=$= | $\frac{27531}{1000}$275311000 |
The numerator contains the product $483\times57=27531$483×57=27531, which is what we initially calculated above. The denominator then tells us the place value of the number
In this case, the original denominators were $100$100 and $10$10. The final denominator is their product, which is $1000$1000. Dividing by $1000$1000 is the same as decreasing the place value of each digit by three place values, and so we get $27.531$27.531 as our final answer.
The method of adding the decimals in the original numbers to find the number of decimal places in the answers will always work, but we sometimes end up with a $0$0 as our last decimal place, in which case we can remove the trailing zero when writing our final answer.
The area method
We have previously used area to solve multiplication of whole numbers, and we can use the same idea for decimal problems involving multiplication. Starting with a rectangle, we can break it into a series of smaller rectangles, allowing us to work out our problem in smaller steps. Then we can add our answers together at the end.
We can solve something like $3.4\times34$3.4×34 by thinking of it as a rectangle whose length and width measure $3.4$3.4 units and $34$34 units respectively. Then we can break it into smaller rectangles and solve it in four parts. We finally add the individual solutions together to find our overall answer to our original problem.
| $\times$× | $3$3 | $0.4$0.4 |
|---|---|---|
| $30$30 | ||
| $4$4 | ||
| Total |
The video below shows this process.
Other strategies for multiplying
There are other strategies for multiplying decimal numbers including partitioning, or repeated addition. The video below shows some of these methods.
The way we solve these multiplications is no different to working with whole numbers, and we can use exactly the same methods. We just need to think carefully about where the decimal point goes.
Practice questions
Question 1
Find $2.2\times4$2.2×4, giving your answer as a decimal.
Question 2
Find $1.2\times3.2$1.2×3.2, giving your answer as a decimal.
Question 3
In this question, we will use the area model to find the product of a decimal and a whole number.
Find the area of each rectangle below as a decimal.
$4$4 $0.6$0.6 $0.07$0.07 $9$9 $\editable{}$ $\editable{}$ $\editable{}$ Using the answer from part (a), find $4.67\times9$4.67×9 as a decimal.