We're asked to use long division to find the quotient. We have a three digit number
divided by a two digit number. 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 in long division form which is this form here.
We have a eight hundred ninety-eight divided by thirty-eight. And because we're using the method of partial
quotients,
we'll draw a vertical to the right. To keep our work organized, we'll write the partial quotients
on the right side of this vertical line.
Before we begin the method of partial
quotients, it's often helpful to find products involving the divisor. Because our
divisor is thirty-eight, let's find some products involving thirty-eight. For example, we know that
one times thirty-eight is thirty-eight. But because our dividend is in the hundreds, we do want to find some
larger products. So we should be able to recognize that if one times thirty-eight equals thirty-eight,
then ten times thirty-eight would be equal to three hundred eighty. Notice how, when multiplying one of the factors as ten, we
can just add a zero to the right of the other factor. Let's find two more products. If one
times thirty-eight equals thirty-eight, we should be able to recognize that two times thirty-eight is equal to thirty-eight plus thirty-eight, or seventy-six.
And then from here we should be able to recognize that twenty times thirty-eight would be ten times as
large as this product. And therefore, twenty times thirty-eight equals seven hundred sixty. So these products
will be helpful when using the method of partial quotients. To begin the method of
partial quotients, we consider the entire dividend of eight hundred ninety-eight, because the divisor
is thirty-eigiht. We begin by determining a certain number thirty-eights that we know for sure are in
eight hundred ninety-eight. Well looking at our products, because three hundred eighty is less than eight hundred ninety-eight, and three hundred eighty is equal to ten
times thirty-eight, we know there's at least ten, thirty-eights in eight hundred ninety-
eight. But notice how seven hundred sixty is also less than eight hundred ninety-eight.
And because seven hundred sixty is equal to twenty times thirty-eight, we know there's at least twenty, thirty-eights in eight hundred ninety-eight. So let's use
the larger partial quotient of twenty. So we place twenty on the right. we multiply by the divisor.
Twenty times thirty-eight is equal to seven hundred sixty. And now we subtract. and notice by subtracting,
we're removing twenty, thirty-eights from eight hundred ninety-eight. And that's why this is called a partial quotient.
So subtracting, notice how we get one hundred thirty-eight. And now we repeat the process. We want to
determine a certain number of thirty-eights that we know for sure are in one hundred thirty-eight. Well looking at our
products, notice both thirty-eight and seventy-six are less than one hundred thirty-eight, so let's use the larger part of seventy-six. Because
seventy-six is less than one hundred thirty-eight, and seventy-six equals two times thirty-eight. We know there's at least two, thirty-eights
in one hundred thirty-eight. Let's use two as the next partial quotient. We multiply by the divisor. Two
times thirty-eight equals seventy-six, and we subtract. Eight minus six is two. Thirteen minus seven is six. And again we repeat the process. We need
to determine a certain number of thirty-eights that we know are in sixty-two. Well thirty-eight is less than sixty two,
so there's at least one, thirty-eight in sixty-two. So let's use one as the next partial quotient. One times
thirty-eight equals thirty-eight.
And we subtract. Twelve minus eight is four. Five minus three is two. Now that the difference is less than the divisor,
we've completed the method of partial quotients. We just need to interpret the
result. This last difference is the remainder, and to find the quotient we
sum the partial quotients on the right. Twenty plus two, plus one equals twenty-three. So we
have a quotient of twenty-three with a
remainder of twenty-four.
So because thirty-eight does not divide evenly into eight hundred ninety-eight, we know thirty-eight is not a factor
of eight hundred ninety-eight. But this does tell us that the quotient of twenty-three times the divisor of thirty-eight, plus the
remainder of twenty-four does equal eight hundred ninety-eight.
Twenty-three times thirty-eight equals eight hundred seventy-four.
Eight hundred seventy-four plus twenty-four does equal eight
hundred ninety-eight. We can also say that eight hundred ninety-eight has twenty-three groups, or copies of thirty-eight with twenty-four
leftover, or twenty-four remaining.
I hope you find this helpful.