2.07 Multiplying and dividing decimals
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 reasonable.
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 evaluate 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 evaluate 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 evaluate 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 evaluate 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\cdot4$2.2·4, giving your answer as a decimal.
Question 2
Find $1.2\cdot3.2$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\cdot9$4.67·9 as a decimal.
Dividing with decimals
In this lesson we will look at dividing with decimals. We will look at methods for solving problems including dividing decimal numbers by whole numbers, dividing whole numbers by decimals, and dividing decimals by decimals.
Why do we need to divide decimals?
Sometimes, before you can really understand how to perform some number problems, it helps to understand why you need to do them.
Worked example
question 4
Matilda works at her local supermarket for four hours every Saturday, and earns a total of $\$84.80$$84.80. How much does she earn per hour?
Think: If she works for $4$4 hours, we can divide Matilda's total earnings by $4$4 to find the amount she earns per hour. So we want to find the value of $84.80\div4$84.80÷4.
Do: We can solve this in a number of ways, but let's start by partitioning $84.80$84.80 into tens, units, tenths and hundredths.
We can see that $84.80=80+4+0.80$84.80=80+4+0.80, that is, it is made up of $8$8 tens, $4$4 units, and $8$8 tenths (or $80$80 hundredths). We can now divide each of these parts by $4$4.
- $80\div4$80÷4, or $8$8 tens divided by $4$4, is equal to $20$20 (or two tens).
- $4\div4$4÷4, or $4$4 units divided by $4$4, is equal to $1$1.
- $8$8 tenths divided by $4$4 is equal to $2$2 tenths. We can also think of this as $80$80 hundredths divided by $4$4, which is $20$20 hundredths.
Notice that as we are dealing with currency, it makes sense to refer to the decimals as hundredths, as this relates to cents.
The table below shows each step in the division.
| $84.80\div4$84.80÷4 | $=$= | $\left(80+4+0.80\right)\div4$(80+4+0.80)÷4 |
Partitioning the number $84.80$84.80 |
| $=$= | $80\div4+4\div4+0.80\div4$80÷4+4÷4+0.80÷4 |
Dividing each part by $4$4 separately |
|
| $=$= | $20+1+0.20$20+1+0.20 |
Evaluating each division |
|
| $=$= | $21.20$21.20 |
Evaluating the addition |
This shows that Matilda earns $\$21.20$$21.20 per hour.
Reflect: By partitioning or breaking up the number $84.80$84.80 into smaller pieces we were able to easily divide each part by $4$4, and then combine them back together at the end to get our final answer.
We use the same approach we use for dividing with whole numbers, it's just that we work on places to the right of the decimal point.
Now we’ll look at using short division as a way to evaluate a division with decimals. In this example we need to rename some of our digits, just like we would when dividing whole numbers.
Practice question
Question 5
We want to find $92.1\div3$92.1÷3
Choose the most reasonable estimate for $92.1\div3$92.1÷3
$31$31
A$310$310
B$3.1$3.1
C$0.31$0.31
DComplete the short division to find $92.1\div3$92.1÷3
$\editable{}$ $\editable{}$ $.$. $\editable{}$ $3$3 $9$9 $2$2 $.$. $\editable{}$ $1$1
When we need to rename
Consider the quotient $54.05\div5$54.05÷5. We can break this up by partitioning the decimal into $5$5 tens, $4$4 units, $0$0 tenths and $5$5 hundredths. The $5$5 tens and $5$5 hundredths parts are easily divisible by $5$5, but how shall we try to divide the $4$4 units by $5$5?
We can use the fact that $4$4 units is the same as $40$40 tenths, which is now divisible by $5$5. Now the number $54.05$54.05 is partitioned into $5$5 tens, $0$0 units, $40$40 tenths and $5$5 hundredths. When we renamed $4$4 units to $40$40 tenths, we made use of a zero placeholder.
The process we went through in partitioning and renaming is what gets used behind the scenes when we perform short division, as shown in the video below.
Practice question
Question 6
Evaluate $11.54\div2$11.54÷2 using short division.
$\editable{}$ $\cdot$· $\editable{}$ $\editable{}$ $2$2 $1$1 $1$1 $\cdot$· $\editable{}$ $5$5 $\editable{}$ $4$4
Dividing by a decimal number
But what about when we want to divide by a decimal number?
We've already seen how to divide decimal numbers by whole numbers, so it would be great if we could just keep using this strategy. Using our knowledge of place value, we can!
We know that $10\div5=2$10÷5=2, but what is $100\div50$100÷50? Well, obviously that is just $2$2 as well. We can see that even though both numbers were ten times larger, we ended up with exactly the same answer. We can use the same strategy but with numbers ten times smaller. This means $1\div0.5=2$1÷0.5=2 as well!
Let's look at some examples to show how this strategy works.
Worked examples
question 7
Evaluate $5.6\div0.8$5.6÷0.8
Think: This will be much easier to evaluate if both numbers were whole numbers, so what can we do to make this happen?
Do: Each decimal in the expression has one decimal place, so we can rewrite the expression with whole numbers by multiplying both decimals by $10$10. Since $5.6\times10=56$5.6×10=56, and $0.8\times10=8$0.8×10=8, then $5.6\div0.8=56\div8$5.6÷0.8=56÷8.
Now $56\div8=7$56÷8=7, so we can return to the original expression and say that $5.6\div0.8=7$5.6÷0.8=7.
Reflect: By multiplying both numbers by ten, we were able to turn this in to a division using whole numbers, which could then evaluate easily.
question 8
Evaluate $5.28\div0.04$5.28÷0.04
Think: How can we write this expression with whole numbers?
Do: Now each decimal in the expression has two decimal places, so we can rewrite this expression with whole numbers by multiplying both decimals by $100$100. Since $5.28\times100=528$5.28×100=528, and $0.04\times100=4$0.04×100=4, then $5.28\div0.04=528\div4$5.28÷0.04=528÷4.
We can evaluate $528\div4$528÷4 to get $132$132, which means that the original expression is $5.28\div0.04=132$5.28÷0.04=132.
To divide a decimal by a decimal, we can first multiply both numbers by a suitable power of $10$10 to make them whole numbers. This can make the division easier.
Practice question
Question 9
Evaluate the quotient $1.2\div0.3$1.2÷0.3