Basic Operations with Decimals: Division
3.2: Basic operations with decimals:
Division
Quick division
vocabulary review:
◦ Dividend: the first number in a division
problem; the number being divided.
◦ Divisor: the second number in a division
problem; the number the dividend is
divided into.
◦ Quotient: the answer in a division
problem.
Dividend / Divisor = Quotient
Memory aid: Notice that when it is written out
like this, all the names are in alphabetical order! |
First, we’re going to introduce two
tricks to dividing with decimals points:
◦ Using the long division we’ve already
learned to divide using decimal points.
◦ Shifting the decimal points in division to
make the problem easier
Division with decimals is very similar to
division with whole numbers.
Review: 116/2
116/2 = 58 and 1.16/2 = 0.58
We do the long division exactly the same, just line the decimals up. |
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We can shift the decimal points in a
division problem to make it easier.
Notice the following:
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Notice where the decimal point is moving! |
As long as we move the decimal point in
the same direction and the same
number of places for both numbers, we
have an equivalent problem.
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These all are equal to 10. |
All of these expressions are the same.
We’ve only moved the decimal point in each .
So, let’s try an example with the two
things we’ve just learned:
◦ Using the long division we’ve already
learned to divide using decimal points.
◦ Shifting the decimal points in division to
make the problem easier.
Example: 4.56/0.4
◦ (We can get rid of the decimal point in 0.4)
So, now we’re dealing with the problem
45.6 / 4 |
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Using long division, we arrive at the answer, 45.6/4 = 11.4 |
Example: 3.5 / 0.02
This problem becomes much
easier when the divisor gets rid
of its decimal!
Now, we’ve reduced the problem
into an equivalent problem.
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As shown in the long division, the answer is 175. |
Example: 16.48 / 1.6
◦ First step: Let’s get rid of the decimal place
in the divisor (which is 1.6).
Remember: We have to do the same thing to
both numbers!
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Now we have this expression. |
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Don’t forget to keep the decimals in line. |
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RULE: Division of decimal numbers:
◦ Move the decimal point in the divisor to
obtain a whole number.
◦ Move the decimal point the same number of
places in the dividend.
◦ Proceed with division as with whole
numbers.
◦ Line up the decimal point in the long
division with the quotient.
3.2: Basic operations with decimals:
Terminating and nonterminating decimals
Up to this point, in all of our division
problems, the quotient has been a
whole number.
◦ It is not always the case.
You may have learned to use a
“remainder” in division.
◦ Example: 46 / 7 = 6 R4
In practical life, we rarely use this
notation.
When the quotient comes out exactly, it is
called a terminating decimal.
◦ Example: 1 / 2 = 0.5
◦ 0.5 is a terminating decimal.
When the quotient does not end, it is
called a nonterminating decimal.
◦ Example: 10 / 3 = 3.333… |
◦ 3.333… is a nonterminating decimal. |
◦ 3.333… can be written as![]() |
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Since the 3 is repeated forever, it is called a repeating decimal. |
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If we continued this long division forever, we would never finish. We would eventually see that the digits repeat themselves after a certain point. 46 / 7 = 6.571428571428…
Here is another |
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3.2: Basic operations with
decimals: Irrational numbers
Some numbers are nonterminating and
nonrepeating.
◦ They go on forever, and the digits never
repeat.
◦ Examples:
Both of these numbers neither end, nor repeat.
3.3: Percents: Conversions from
percent to decimal
% is the percent sign.
“Percent” means “per hundred”
So, 100% is 100 per 100.
◦ Or, 100% is the “whole thing.”
“Per” tells us to divide.
◦ To change a percent into a decimal, replace
“%” with “divided by 100”
This would mean to shift the decimal point two
places to the left.
Example: 50%
◦ We replace “%” with “divided by 100”
50%→ 50 divided by 100 → 50 / 100
= 0.5
◦ Does this make sense?
We know 50% is half.
0.5 is half.
Example: 100%
◦ We replace “%” with “divided by 100”
100% →100 divided by 100→ 100 / 100
= 1
Examples:
Percent | Expressed as a decimal |
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Let’s prove that our decimal equivalent
of a percent works in real life:
◦ A $10 movie ticket is 50% off.
Replace 50% with 0.5
◦ How much is the ticket?
$10 * 0.5 = $5
◦ We generally work with the decimal
equivalent of a percent in mathematics.
3.3: Percents: Conversions from
decimal to percent
To change any number into a percent,
multiply by 100%.
◦ This would mean shifting the decimal to
places to the right.
Example: Change 0.75 into a percent
◦ Multiply by 100%
0.75 * 100% = 75%
If we ever forget which direction to
move the decimal when converting,
think of it practical terms.
◦ Half?
◦ A half is 50%
◦ A half is also 0.5
Always think of the simplest example you can come up
with if you can’t remember the rule.
A rule is a rule because it always works!
Homework
Exercise 3.2.4:
◦ 1, 2, 4, 5, 11.
Exercise 3.2.7:
◦ 1, 4, 5, 6.
Exercise 3.3.1:
◦ 1ab, 2a, 3a, 4ab.
Exercise 3.3.2:
◦ 1a, 2ab, 3a, 5a.