Transcript When we “ignore” a decimal point, we are really multiplying

Multiplying and
Dividing with
Decimals
by Allison Hendley
Multiplying
• Maybe multiplying with decimal
points seems a little scary!
BUT…
• There is a simple 3-step trick to
multiply decimal point numbers
just as if they were whole
• While we learn the steps, we
will look at the example:
5.8
X
7
Step 1
• Count the number of digits to
the right of the decimal point
5.8
X
7
1 digit
Step 2
• Just ignore the decimal point!
• Multiply just as if they were
normal whole numbers
5
5.8
5
8
X
7
406
Step 3
• Use the number of digits from
Step 1 to count out where to put
the decimal
5.8
• We counted 1 digit
X 7
in Step 1, so we
move the decimal
4 0. 6 .
point by 1 place to the left now
That’s It!
• Our final answer is:
5.8
X
7
4 0.6
What are we Really Doing?
• When we “ignore” a decimal
point, we are really multiplying
by 10 for each digit to the right of
the decimal.
58
5.8
X
7
x 10
What are we Really Doing?
• Now you multiply the numbers
without decimals
58
X
7
406
• BUT DON’T FORGET WE
MULTIPLIED BY 10 EARLIER!
What are we Really Doing?
• So now we divide by 10
58
X
/ 10
7
406
4 0. 6 .
• Moving the decimal to the left
one place is the same as dividing
by 10!
/ 10
• Let’s try it again with more digits
to the right of the decimal point
this time
0.024
X 0.61
• What was Step 1?
• Count the digits to the right of
the decimal point
• From 0.024, how many digits?
3
• What about from 0.61?
2
• That gives a total of:
5
• What was Step 2?
• Ignore the decimal points
• That means that 0.024 becomes:
24
• In the same way, 0.61 is:
61
• Now we just multiply that out
2
24
X 61
24
144 0
1464
• What was Step 3?
• Place the decimal point in our answer
• What did we count in Step 1? Now
move the decimal that many places
to the left
0.024
X 0.61
1 4 6 4.
• But wait! There are only 4 numbers!
• Since we need to move the decimal
point 5 places to the left but only
have 4 numbers, we write the 0 out
in front.
0.024
X 0.61
.0 1 4 6 4 .
• Don’t forget, its the same as
dividing by 10, five times
• For our final answer, we have
0.024
X
0.61
0.01464
Why It Works
• All we are really doing is multiplying
and dividing by powers of 10 as
needed to make the problem easier
• When we turned 0.024 into 24, we
multiplied it by 10 three times
1.) 0.024 x 10 = 0.24
2.) 0.24 x 10 = 2.4
3.) 2.4 x 10 = 24
• Also, 0.61 becomes 61 by
multiplying by 10 twice
• Each time we move the decimal
to the right on place we are
multiplying by 10
• That’s why we count, so we can
divide by 10 the same number
of times later
• Everybody think of a number
• Multiply that number by 10
• Divide that number by 10
• What do you have now?
Canoeing
If a group of
people set out in
14 canoes, and
they each canoe
for 8.6 miles, how
many total miles
were traveled by
the canoes?
120.4 miles
Backpacks
• The canoers weighed
all their packs before
leaving and found that
all together they
weighed 195.2 lbs. If
1 lb = 0.45 kg, how
many kilograms of
gear have the canoers
brought along?
87.84 kgs
•Remember the 3 steps:
• Count the number of digits to the
right of the decimals
• Ignore the decimals and multiply
• Place the decimal in the answer
by moving it back to the left by
the same number of digits as
before
Dividing
• When we divide numbers with
decimal points, we use the
same idea as multiplying them
• Keeping everything lined up is
key!
6.2
21.08
Step 1
• Multiply both the divisor and
dividend by whatever power of 10
we need so that the divisor is a
whole number
6.2
21.08
62
210.8
Step 2
• Divide just like normal
• Ignore the decimal point
34
62
1 1 0 10
210.8
186
248
248
0
Step 3
• Bring up the decimal point
• The decimal point in the quotient is
placed directly above where it is in the
number being divided
3 .4
62
210.8
• This is why keeping numbers
aligned is so important!
Why It Works
• Just like before, when we multiply
and divide by the same number, we
don’t change what we started with
• Look at the division like this:
21.08
21.08 x 10
210.8
=
=
6.2
6.2 x 10
62
• Notice that by multiplying both the
divisor and the dividend by 10, all
we really did is multiply by a
special form of 1
21.08 x
6.2
10
10
=
21.08 x 1
6.2
Backpacks, Again
• If each canoer carries
5.49 kgs of gear while
hiking (out of the total
87.84 kgs), how many
canoers are there total?
Note: some canoes may
have more than one
canoer.
16 canoers
Kilograms
• If 1 lb = 0.45 kg, how many pounds is
each canoer carrying when every
backpack is 5.49 kgs?
12.2 lbs
•3 Steps of Division:
• Multiply both the divisor and
the dividend by a power of 10
• Ignore the decimal and divide
like normal
• Bring up the decimal from its
position in the divisor