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