1. Using Decimals
Review of comparing, rounding,
adding & subtracting, multiplying
& dividing decimals
created by Alane Tentoni (copyright 2007)
tentoni.weebly.com
2. What is a decimal?
A decimal is a dot that
goes after the ones
column.
It separates the whole
numbers from the
partial numbers.
3. About Decimals
◼ Decimals as
we know
them were
first used by
John Napier in
the late 1500s
in Scotland.
4. About Decimals
In order to use decimals, you have to
understand place value.
1 2 3 4 . 5 6 7 8
To the left of the decimal, all the numbers are
whole numbers. Each column is worth ten times
the column to its right.
5. About Decimals
◼ To the right of the decimal, all the
numbers are like fractions. Each
column is still worth 10 of the column
to the right.
1 2 3 4 . 5 6 7 8
6. Reading Decimals
◼ Zeroes that come at the end of a decimal
don’t add or take away any value.
◼ .4 = .40 = .400 → This is like saying “four
tenths” = “four tenths and no hundredths”
= “four tenths and no hundredths and no
thousandths.”
7. Reading Decimals
◼ HOWEVER – Zeroes that come between
the decimal and the other numbers are
VERY important!
◼ .4 is “four tenths” but .04 is “four
hundredths.” Would you rather have four
dimes or four cents?
8. Comparing Decimals
◼ To tell if one decimal is
bigger than another, you
have to compare the same
column in both numbers.
◼ The length of the
number does NOT
matter at all!!!!
9. Comparing Decimals
Compare these two numbers:
Which is larger?
.6 or .599823
All you need to do is look at the tenths
column. 6 is more than 5, so .6 is more
than .599823, even though .599823 has
more digits!
10. Comparing Decimals
Another comparison
Which is larger? .457 or .49?
The tenth columns are the same (both 4),
but the hundredths columns are
different. 9 is more than 5, so .49 is
more than .457.
11. Rounding Decimals
◼ Rounding means cutting off
unnecessary digits.
◼ Why would you use fewer digits than
you know? Sometimes it is more
convenient to give an approximate
answer.
12. Rounding Decimals
First, decide how many decimal places you
want in your answer.
Just throw away everything behind that
place. . .
Except! You will have to decide whether to
increase the last digit or leave it alone.
13. Rounding Decimals
Let’s round .576 to the nearest hundredth.
.576 is somewhere between .57 and .58. Which
one is it closer to?
To decide, simply look at the digit after the
hundredths place. Is it 5 or more? If so,
round up. If not, leave it the same.
14. Rounding Decimals
◼ In our case, 6 is more than 5, so .576
should be rounded up to .58.
◼ What happens if you have a number
like .398 to round to the nearest
hundredth? (answer: .398 ~.40)
15. Rounding Decimals
◼ Be Careful!! Don’t just
replace the “chopped
off” numbers with
zeroes! When you
round, you are really
reducing the number of
digits behind the
decimal!
16. Rounding Decimals
Here are some numbers to round to the
nearest hundredth.
1.3247 → 1.32
0.987 → 0.99
4.89721 → 4.90
Because we are rounding to
the nearest hundredth, each of
the numbers ends up with two
digits behind the decimal.
What if we had been rounding
to the nearest tenth?
(answer: Rounding to the nearest tenth leaves one decimal place. In the
example: 1.3, 1.0, 4.9)
17. Adding & Subtracting Decimals
◼ When you add decimals, line the decimals
up – one on top of the other.
◼ You have to add the tenths to the tenths,
the hundredths to the hundredths, and so
on – just as when you add whole
numbers, you add ones to ones and tens
to tens.
18. Subtracting Decimals
◼ When you subtract, you may have to
annex zeroes to the larger number so
you can borrow.
◼ Example: 35.7 – 20.94= ?
35.70
- 20.94
14.76
Annex a zero here so
you can borrow.
19. Multiplying Decimals
When you multiply
decimals, you should set
the problem up just as if
you were multiplying
whole numbers –
longest number on top,
shortest on bottom.
20. Multiplying Decimals
◼ After you multiply the numbers, you are
ready to put your decimal in place.
◼ Count the number of digits behind the
decimal in both of the multiplied
numbers.
◼ Put that many total digits behind the
decimal in your answer.
21. Multiplying Decimals
◼ Here’s an example:
1.2 one digit here
x 3.9 one digit here
108
_36_
4.68 two digits here
22. Multiplying Decimals
◼ Another example – same numbers but
with the decimals in different places.
1.2 one digit here
x .39 two digits here
108
_36_
.468 three digits here
23. WHOA!
◼ Hang on! Did that last problem say 1.2 x .39
= .468?
Question: How can you multiply 1.2 by
something and get an answer less than 1.2?
Answer: Anytime you multiply by something
less than 1, the answer is smaller than the
number you started with.
24. Multiplying Decimals
◼ If the answer doesn’t have enough digits,
you will have to put zeroes between the
decimal and the first number.
.12 two digits here
x .39 two digits here
108
_36_
.0468 four digits here
25. Dividing Decimals
◼ Let’s name the parts of a division
problem so we can talk about them.
8 56
7
dividend
divisor
quotient
Notice that the 7
is over the 6, not
the 5.
The quotient goes
over the LAST
digit you are
working with.
26. Dividing Decimals
◼ Dividing decimals is a lot like
dividing whole numbers, but we
need a way to get the decimals
in the right place in the answer.
◼ Before we start dividing
decimals, let’s look at dividing
some whole numbers.
27. Dividing Decimals
42 ÷ 6 = 7 And 420 ÷ 60 = 7
In the second equation, both 42 and 6 have
been multiplied by ten. Because both
numbers were multiplied by the same
thing, the quotient did not change.
28. Dividing Decimals
◼ We can use that trick to divide numbers
with decimals.
◼ Because moving the decimal to the right is
just like multiplying by ten, if we move the
decimal the same number of places in
both numbers, our quotient stays the
same.
29. Dividing Decimals
Here’s an example: .132 ÷ .12:
.12 .132
If these were whole numbers, you would say, “How many
times will 12 go into 13?” But it’s harder to think of .12
and .13.
If you could move the decimal of the divisor (.12) over 2
places, you would have a whole number. You can do that
as long as you move the decimal of the dividend over 2
places as well.
30. Dividing Decimals
So now our problem looks like this:
NOTICE: The decimal moved
straight up from the dividend to
the quotient.
Lining up the number in the
quotient and the dividend is VERY
important because if they are
wrong, your decimal will be in the
wrong place.
12. 13.2
1.1
-12
1 2
-1 2
0
31. ALWAYS Check!
◼ Now that we have an answer, we need to check our
work.
◼ Multiply the quotient by the divisor. You should get
the dividend back.
1.1
x.12
22
11
.132
1 digit
2 digits
3 digits
32. Hang on!
◼ How can we take two
small numbers like
.12 and .132 and
divide them and get
a bigger number?
Doesn’t dividing
always mean you get
a smaller number?
33. Dividing Decimals
◼ Another way to look at .132 ÷ .12 is to
say, “How many groups of .12 does it
take to make .132?”
.12 + .012 = .132
◼ It takes one and a little more, so our
answer of 1.1 looks reasonable.
34. Dividing Decimals
◼ Let’s try another example:
1.25 ÷ .4 .4 1.25
First of all, let’s estimate how many .4’s it would
take to make 1.25
.4 + .4 + .4 = 1.2 so it will take 3 groups of .4 plus
a little more to make 1.25
35. Dividing Decimals
First, move the decimal in the
divisor and the dividend.
4. 12.5
3.1
-12
05
-4
1
In this case, we have pulled
down all our numbers, but we
still have a remainder.
DO NOT tack your remainder
onto the end of your answer!
36. Annexing Zeroes
◼ Remember that adding zeroes at the end of a
number does not change its value.
12.5 = 12.50000
◼ If you need to keep dividing, just annex zeroes,
pull down & keep dividing until you get a
remainder of zero (or until you see a pattern.)
38. Check Your Work!
The original problem was 1.25 ÷ .4.
The quotient was 3.125
Check: 3.125
x .4
1.2500
3 digits
1 digit
4 digits
Since 1.2500 = 1.25, our answer is correct.
39. Dividing Decimals
◼ Sometimes when we divide, the quotient of
the two numbers makes a pattern that
never stops!
◼ This is called a “repeating decimal.”
◼ The kind that does stop is called a
“terminating decimal.” If you can work
your problem to a remainder of zero, you
have a terminating decimal.
40. Dividing Decimals
Tip:
Divisors that have factors of all
twos or fives will definitely
terminate.
(like 2, 4, 5, 8, 10. . .)
Everything else can repeat – it
depends on the dividend.
41. Dividing Decimals
◼ Here is a repeating decimal.
.3 5.56 First, move the decimal.
3. 55.6
.
Put the decimal on the
quotient line.
42. Repeating Decimals
When you’ve pulled
down all your
numbers and you still
have a remainder, you
need to annex zeroes
and keep going.
3. 55.6
18.5
-3
25
-24
16
-15
1
44. Repeating Decimals
◼ To show that a number repeats, place a
bar over all the numbers that form the
pattern.
◼ In our example, only the 3 was
repeating:
18.53
45. Get the “point”?
◼ Decimals are a pretty convenient way to
represent fractional values.
◼ Decimal rules are not difficult, but even
though you know the rules, you must
practice them until they are second nature!
