We're asked to use long division to find the quotient. But for this example, instead
of using the traditional long division algorithm, we'll use the method of
partial quotients. The first step is to write the quotient and long division
form which is this form here. So we'll write this quotient as four thousand, seven hundred fifty-six divided by eight. When using the
partial quotients method, we'll draw a vertical line to the right to keep our work
organized. We'll write the partial quotients on the right side. Now to begin the process of
partial quotients, we always consider the entire dividend, in this case four thousand, seven hundred fifty-six.
And because we're dividing by eight, we want to determine a certain number of eights that we know for sure in four thousand, seven hundred fifty-six. To
help us do this let's show some work. Let's say we know that five times eight is
equal to forty. We want to find the product. So because this product is less than the
dividend, we know there's at least five eights in our dividend. But we want to find a product that
is much closer to the dividend, that's not more than the dividend. So five
times eight equals forty. We should be able to recognize that fifty times eight is equal
to four hundred. Since fifty is ten times more than five, fifty times eight is ten times more than five times eight.
And if fifty times eight is equal to four hundred, five hundred times eight is equal to four thouand. Notice how four thousand is still
less than the dividend of four thousand, seven hundred fifty-six. And therefore we know the dividend contains
at least five hundred, eights. So let's use five hundred as our first partial quotient. So we'll write five hundred here on the
right. Now we'll multiply by the divisor. We know five hundred times eight is four thousand. We write four thousand
below the dividend, and we subtract.
The difference is seven hundred fifty-six. And now we repeat the process. We want to determine a certain
number of eights that we know for sure in seven hundred fifty-six. But we know that fifty times eight is
equal to four hundred because four hundred is less than seven hundred fifty-six. We know seven hundred fifty-six has at least
fifty, eights. So let's use fifty as the next partial quotient. So write down fifty on the right.
Multiply by the divisor. Fifty times eight is four hundred. We subtract.
So we have six, five, three. So our
difference is three hundred fifty-six. Repeating the process, we need to find a certain number of eights that
we know for sure in three hundred fifty-six. So because fifty times eight
is equal to four hundred, we might recognize that twenty-five times eight would
have to be equal to two hundred. So we could use a partial quotient of twenty-five, but let's try
a different approach. Because we know that four times eight
is equal to thirty-two. Then forty times eight must be equal to three hundred twenty. Notice three hundred twenty is closer to three hundred fifty-six than
two hundred, so let's use forty as the next partial quotient.
So we'll record a forty on the right. Multiply by the divisor. Forty times
eight is three hundred twenty.
We subtract.
Notice the difference is going to be thirty-six, and we repeat the
process. We want to determine a certain number of eights that we know for sure are thirty-six. Well
four times eight is equal to thirty-two. Because thirty-two is less than thirty-six, we know there's at least four, eights in
thirty-six. Let's use four as the next partial quotient.
Four times eight again is thirty-two. We subtract, and
notice how the differences is four. And because four is less than the divisor of eight,
we're done. We just need to interpret the results. This four is a remainder, and the
sum of these partial quotients will give us our quotient. So let's go ahead and find the sum
here on the right.
So we have four in the ones. A nine in the tens. And a five in the hundreds. So our answer
is five hundred ninety-four with the remainder of four. So the nice thing about the partial quotients
method is that there's not only one way to find the quotient. We can use any
known products involving the divisor to find partial quotients, which we can then
use to find the actual quotient.
Notice how here we do have a remainder, which tells us several things. First it tells us eight is
not a factor of four thousand, seven hundred fifty-six. And it also tells us that the quotient of five hundred ninety-four times the
divisor of eight, plus the remainder of four equals the dividend of four thousand, seven hundred fifty-six. Let's
go ahead and simplify the left side.
Five hundred ninety-four times eight is equal to four thousand, seven hundred fifty-two. And notice how, if we add four, we do get four thousand, seven hundred fifty-two.
I hope you find this helpful.