Video Transcript
In this lesson, we will learn how
to multiply multidigit decimals. We can multiply decimals using the
same methods for multiplying whole numbers. In this video, we will look at
three different methods, the column method, the grid method, and the lattice
method. However, it’s important to note
that any method for multiplication of whole numbers can also be used for
decimals.
Let’s begin with the definition of
a decimal. A decimal is a number that contains
a decimal point. The digits to the left of the
decimal point represent the whole part of the number. And the digits to the right
represent the decimal part of the number. We can consider this in more detail
using place value columns. In this example, to the left of the
decimal point, we have the thousands, hundreds, tens, and ones columns. To the right of the decimal point,
we have the tenths, hundredths, and thousandths.
If we consider the number 4736.528,
then the digits to the left of the decimal point, 4736, represent the whole
part. The digits five, two, and eight
represent the decimal part. We have five tenths, two
hundredths, and eight thousandths. Before looking at our three
different methods, we will look at some general rules in how to multiply
decimals. Our first step is to remove the
decimal point from the numbers we want to multiply and consider them as whole
numbers. Next, we need to multiply these
whole numbers using our preferred multiplication method.
As previously mentioned, we will
look at examples using the grid method, the lattice method, and the column
method. Finally, we put the decimal point
back into the answer. The answer will have as many
decimal places as the sum of the decimal places in the original numbers. If there are a total of three
numbers after the decimal points in the question, there will be three numbers after
the decimal point in the answer. We will now look at some examples
of multiplying two decimals.
Calculate 0.39 multiplied by
5.6.
We will answer this question using
the grid method. Our first step is to remove the
decimal points. The calculation, therefore, becomes
39 multiplied by 56. Our second step is to multiply the
whole numbers. We will do this as mentioned using
the grid method. We split both of our numbers into
their tens and ones or units component. 39 becomes 30 and nine, and 56
becomes 50 and six. We then need to carry out four
multiplication calculations.
Firstly, we multiply 50 by 30. Five multiplied by three is equal
to 15. Adding the two zeros means that 50
multiplied by 30 is 1500. Next, we’ll multiply 50 by
nine. Five multiplied by nine is equal to
45. Therefore, 50 multiplied by nine is
450. Next, we will multiply six by
30. This is equal to 180, as six
multiplied by three is 18. Finally, six multiplied by nine is
equal to 54. This means that the solution to 39
multiplied by 56 is the sum of 1500, 415, 180, and 54. 1500 plus 180 is 1680. 450 plus 54 is equal to 504. Adding these two values gives us
2184. Therefore, 39 multiplied by 56 is
2184.
Our final step is to put the
decimal points back in. Remember, the answer will have as
many decimal places as the sum of the decimal places in the question. 0.39 had two digits after the
decimal point. 5.6 had one number after the
decimal point. Two plus one is equal to three. Therefore, our answer needs to have
three numbers after the decimal point. The digits one, eight, and four
will all be after the decimal point. We can therefore say that 0.39
multiplied by 5.6 is equal to 2.184.
We will now look at the second
example using a different method.
Calculate 0.71 multiplied by
0.53.
We will solve this calculation
using the lattice method. Our first step when multiplying two
decimals is to remove the decimal points. In this question, our calculation
becomes 71 multiplied by 53. Next, we need to multiply the two
whole numbers. In this case, as previously
mentioned, we will use the lattice method. To use the lattice method, we draw
a grid and write the digits in the numbers to be multiplied along the top and down
the right side of the lattice. In this case, we have seven and one
on the top and five and three on the right.
We draw diagonal lines through each
section of the grid. In each of the grid entries, we
then calculate the product of the digit at the top of the row and the digit on the
right edge. We can begin by multiplying seven
and five. Seven multiplied by five is equal
to 35. We put a three in the top half and
a five in the bottom half. Seven multiplied by three is equal
to 21. Therefore, we have two in the top
half and one in the bottom half.
Next, we need to multiply one and
five. One multiplied by five is equal to
five. And as this is less than 10, we put
a zero in the top half and five in the bottom half. One multiplied by three is equal to
three. So, once again, we have a zero in
the top half and a three in the bottom half. To work out the answer using the
lattice method, we then sum the digits in the lattice along the diagonal strips.
The first strip just has the number
three. The second strip has a five, a
zero, and a one. These numbers sum to six. The third strip has a zero, a five,
and a two. Zero plus five plus two is equal to
seven. The final strip, once again, only
has a three. Reading in the direction of the
arrow, our answer is 3763. We can therefore say that 71
multiplied by 53 is equal to 3763.
Our final step, in order to
calculate 0.71 multiplied by 0.53, is to put the decimal point back in. Remember that our answer will have
as many decimal places as the sum of the decimal places in the question. The number 0.71 had two digits
after the decimal point. The number 0.53 also had two
numbers after the decimal point. As two plus two is equal to four,
our answer will have four digits after the decimal point. All four of the digits three,
seven, six, and three will be after the decimal point. We can therefore conclude that 0.71
multiplied by 0.53 is equal to 0.3763.
From this question comes an
important point when multiplying decimals. When we multiply two decimals that
are less than one, the answer will also be less than one.
We will now look at a third example
using a different method.
Calculate 7.106 multiplied by
0.29.
We will answer this question using
the column method. When multiplying two decimals
together, our first step is to remove the decimal points. In this question, our calculation
will become 7106 multiplied by 29. Our next step is to multiply the
whole numbers. In this case, as mentioned, we will
use the column method.
We begin by multiplying the top
number by the ones digits in the bottom number. We multiply 7106 by nine. Nine multiplied by six is equal to
54. We put the four in the ones column
and carry the five. Nine multiplied by zero is equal to
zero. Adding the five that we carried
gives us five. Nine multiplied by one is equal to
nine.
Finally, nine multiplied by seven
is equal to 63. 7106 multiplied by nine is equal to
63954. Our next step is to multiply 7106
by 20. As we’re multiplying by 20, we can
put a zero in the ones column and then multiply by two. Two multiplied by six is equal to
12. We put a two in the tens column and
carry the one. Two multiplied by zero is equal to
zero. Adding the one we carried gives us
one. Two multiplied by one is equal to
two. And finally, two multiplied by
seven is equal to 14. 7106 multiplied by 20 is equal to
142120.
Our final step to multiply 7106 by
29 is to add these two rows. Four plus zero is equal to
four. Five plus two is equal to
seven. Nine plus one is equal to 10, so we
need to carry the one. Three plus two plus the one we
carried is six. Six plus four is equal to 10. And finally, one plus one is equal
to two. 7106 multiplied by 29 is equal to
206074. Our final step is to put the
decimal points back in.
At this point, it’s important to
remember that the answer will have as many decimal places as the sum of the decimal
places in the question. 7.106 has three digits after the
decimal point. 0.29 has two digits after the
decimal point. This means that our answer will
have five digits after the decimal point. 7.106 multiplied by 0.29 is equal
to 2.06074. Our answer has five decimal
places.
We will now look at a couple of
different situations where we’re given a whole number multiplication.
Given that 225 multiplied by 248 is
equal to 55800, what is 2.25 multiplied by 24.8?
Before trying to answer this
question, we need to recall one of our key facts when multiplying decimals. That is, that the answer to two
decimals multiplied will have as many decimal places as the sum of the decimal
places in the original numbers. The number 2.25 has two digits
after the decimal point. The number 24.8 has one digit after
the decimal point. This means that our answer will
have three digits after the decimal point. The digits in our original
calculation, 225 and 248, are the same as the digits in the new calculation. This means that the digits in the
answer will be five, five, eight, zero, zero.
As already mentioned, we need three
digits after the decimal point. This means that 2.25 multiplied by
24.8 is equal to 55.800. We can rewrite this ignoring the
final zeros. Therefore, our answer is 55.8. An alternative way of looking at
this question would be to consider how we get from our original numbers to the
numbers in the second calculation. 225 divided by 100 is equal to
2.25. 248 divided by 10 is equal to
24.8. This means that we have divided the
calculation by 100 and by 10.
Dividing by 100 and then dividing
by 10 is the same as dividing by 1000. Dividing 55800 by 1000, once again,
gives us 55.800. When dividing by 1000, all the
digits move three places to the right. This confirms that the answer to
2.25 multiplied by 24.8 is 55.8.
This type of problem where we’re
given a whole number multiplication which can be used to solve a decimal
multiplication is often given in examinations. We do this by working out how many
decimal places the answer should have.
We’ll now look at the summary of
the key points of this lesson on multiplying decimals. Firstly, we can multiply decimals
using the same techniques that we use to multiply whole numbers. In this lesson, we have looked at
the column method, the lattice method, and the grid method.
When we multiply decimals, we
firstly consider the numbers as whole numbers by removing the decimal points. Once we’ve multiplied the two
numbers, we put the decimal point back into the answer so that it has the same
number of decimal places as the sum of the decimal places in the original
numbers.
