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Language: en
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>> Welcome to the Cypress College
Math Review on decimal operations.
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Objective 1: Place settings.
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Here we have a table displaying the first
few place settings around the decimal point.
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Notice that each digit after a decimal
point will represent a different fraction,
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with the denominators of
10, 100, 1,000 etc. Example.
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For the number 48.1268, we want to
identify the digit in the hundredth place.
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As we can see from the previous chart, the digit
in the hundredth place will be the second digit
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to the right of the decimal point.
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Hence, in our example, we have the number 2.
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So our answer is 2.
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Here, we have another example.
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Write out the decimal 0.517, as a sum of
fractions representing each decimal place.
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In this case 5 is in the tenths
place, so we have five-tenths.
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One is in the hundredths place,
so we have one-hundredth.
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And the 7 is in the one-thousandths
place, so we have seven-thousandths.
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And then we want to write it as a sum
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of all these three fractions,
and that will be our answer.
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Pause the video and try these problems.
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Objective 2: Adding and subtracting decimals.
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To set up problems that add or subtract
decimals, we need to be sure to line
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up the decimals, and add
or subtract straight down.
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The reason we do this is
to be sure we are adding
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or subtracting numbers that
are in the correct place.
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Tenths place, ones place, hundredths place,
etc. To understand, let us refer to fractions.
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Example. Add 0.3 plus 0.4.
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In this case, we are going
to do that using fractions,
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0.3 is equal to three-tenths,
and 0.4 is equal to four-tenths.
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So if we want to add 0.3 to 0.4 that will be the
same as adding three-tenths plus four-tenths.
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Since we are adding fractions and the
denominators are the same, this case,
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we add the numerators which is
7, and keep the denominator.
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So in this case, we have seven-tenths.
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So now we convert that to decimal.
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That will be 0.7.
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And that will be our answer.
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With fractions, we only add the numerators,
and the denominators stay the same.
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This is why we line up decimals
when adding or subtracting,
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so that we are adding terms
that have the same placeholder.
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And so from a previous example, 0.3 plus 0.4
will then equal to, let's line up the decimals.
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So we add 3 and 4, that will give us 7, and our
decimal, and then 0, and so our answer is 0.7,
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the same as if we were to add
these two decimals, with fractions.
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Example. Add 1.652, plus 20.91.
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This case, we are going to add the
numbers by lining up the decimals.
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So we have 1.652, plus 20.91.
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To this second number, we are
going to add a zero at the end,
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so that it will also have three decimal places,
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and then we just add as we
would add any two numbers.
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We have 2 plus 0 is 2, 5 plus 1 is 6,
6 plus 9 is 15, and hence, we carry 1,
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1 plus 1 plus 0 will give us 2,
and here at the end we have 2 also.
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So a decimal place will go here.
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And our answer will then be 22.562.
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Keep in mind that carrying over and
borrowing will work the same way
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with decimals as they do with whole numbers.
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Example. Subtract 2.83, minus 1.206.
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To subtract these two numbers, we are going
to line up the decimals, 2.83 minus 1.206,
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since the first number only has two decimal
places, we are going to add a third one.
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And then we subtract as we would
subtract any two whole numbers.
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So we are going to have to borrow 1 from 3, 10
minus 6, that's 4, 2 minus 0 is 2, 8 minus 2,
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that's 6, bring down the decimal
point, 2 minus 1, that's 1.
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So 2.83 minus 1.206 is equal to 1.624.
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Pause the video and try these problems.
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Objective 3: Multiplying
and dividing with decimals.
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To set up decimal multiplication problems,
lining up the decimals is not necessary.
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Instead, we will treat the
numbers as regular integers,
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then account for all the
decimal places at the end.
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To understand this, let us refer to fractions.
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So here, we have an example.
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Multiply 0.32 times 0.4.
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Since 0.32 is equal to thirty-two
hundredths, and 0.4 is equal to four-tenths,
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we have 0.32 times 0.4 is equal
to 32 over 100, times 4 over 10,
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and when we multiply fractions,
we multiply the numerators.
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So 32 times 4, that is 128, and we multiply
the denominators, 100 times 10, that is 1,000.
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And so 0.32 times 0.4 is 128 thousandths.
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This will then equal to 0.128,
that represents 128 thousandths.
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Notice that this times we end
up with a different denominator.
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And this is because when we multiply
fractions, we also multiply the denominators.
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So when we multiply decimals, that means that
we will keep track of all the decimal places,
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and move our decimal point over
that many places in the end.
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To show the same idea we see with
fractions, we can line up the numbers
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like regular integer multiplication.
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But just note that there will be three
shifts of the decimal in the final answer.
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From our previous example, 0.32 times 0.4,
is equal to-so now let's line up the numbers
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like regular integer multiplication,
0.32 times 0.4, and then we multiply
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as we would multiply any two integers, 4
times 2 is 8, 4 times 3 is 12, 4 times 0 is 0,
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and then we carried 1, so that will be 1.
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The first number has two decimal places.
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The second number has one decimal place.
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So we have a total of three
decimal places in our answer.
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So we move the decimal place three times.
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And so our answer is 0.128.
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Here we have another example.
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Multiply 819 times 7.756.
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This case, to multiply, we are going to
put the largest number of digits on top,
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regardless of which number is actually larger.
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The number 7.756 is the number
with the largest number of digits.
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That is going to go on top.
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Times 819.
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And just multiply as we would multiply any
two integers, 9 times 6 is 54, 4, carry 5,
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9 times 5 is 45, plus 5, that is 50,
9 times 7 is 63, plus 5, that is 68.
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Again 9 times 7 is 63, plus 6, is 69.
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Now, let's multiply by 1, 1
times 6 is 6, 1 times 5 is 5,
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1 times 7 is 7, and 1 times 7 is 7.
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Now let's multiply by 8, 8 times 6, that's 48, 8
and carry 4, 8 times 5 is 40, plus 4, that's 44,
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4 and carry 4, 8 times 7
is 56, and 4, that's 60.
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Again, 8 times 7, that's 56, and 6, that's 62.
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And now, let's go ahead and add.
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So we have 4, 6, 8, plus 5, that's 13, plus
8, that's 21, 2 plus 9, that's 11, plus 7,
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that's 18, plus 4, that's 22, 2 plus
6 is 8, plus 7, that's 15, 1 plus 2,
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that's 3, and we have 6 here at the end.
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Now, the first number has three decimal places.
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The second number has 0 decimal
places, therefore,
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we have a total of three
decimal places in our answer.
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Let's move the decimal place three
times to the left then, 1, 2, 3.
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Therefore, our answer is going to be 6,352.164.
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Example. Set up division for the
problem, 72.8 divided by 0.26.
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In this case, to set up a long division problem
with decimals, we are going to move the decimal
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of the divisor, the number on the
outside, to the right, until it is gone.
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However far you must move it, do the same
to the dividend, the number on the inside.
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The decimal in your new dividend
will line up with the answer.
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So in this case, we are dividing, 72.8 by 0.26.
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So 0.26 is our divisor.
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So we are going to move the decimal place
of this number to the right until it's gone.
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So 1, 2, so we move the decimal
place twice to the right.
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So we must do the same to the dividend.
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So we'll move the decimal place twice, 1, 2.
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The final setup of this problem
will then be 26 dividing 7280.
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Example. Divide 2.814 by 2.1.
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So we are dividing 2.814 by 2.1.
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This divisor is 2.1, so we are going to
move its decimal place to the right once.
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And we are going to do the same to the
number on the inside, the dividend.
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So its new place is going to be between 8 and 1.
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Our set up will then be 21 dividing 28.14,
then now we can go ahead and divide.
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So 21 goes into 28 once, 1 times 21, is 21,
subtract, this will give us 7, move the 1 down,
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so 21 goes into 71 three
times, 3 times 1, that's 3,
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3 times 2, that's 6, 71 minus 63 will be 8.
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Now we move the 4 down, and 21 will go into 84
four times, 4 times 1 is 4, 4 times 2 that's 8,
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and when we subtract, we can
see that we have a 0 remainder.
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The decimal in the inside should line
up with the decimal in our answer.
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And so 2.814 divided by 2.1 will be 1.34.
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Pause the video and try these problems.