We want 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 be using the method of partial quotients. The
first step is to write the division problem in long division form which is this form here.
So we have six hundred twenty-four divided by twenty-six.
When performing division using partial quotients, to keep
our work organized, we'll draw a vertical line to the right of our problem. We'll record their partial
quotients on the right. Before we begin though, it's often helpful to write some
products that we know involving the divisor. In this case let's write some
products involving twenty-six. For example, we know that one times twenty-six is equal to twenty-six.
But because our dividends is in the hundreds, let's also find ten times twenty six. Well
ten times twenty-six is going to be ten times more than one times twenty-six. And therefore, ten
times twenty-six equals two hundred sixty.
Let's find two more products. If one times twenty-six equals twenty-six, then maybe we
can determine that two times twenty-six is equal to twenty-six plus twenty-six, or fifty-two.
And therefore, twenty times twenty-six
would be ten times more than two times twenty-six. And therefore twenty times twenty-six is equal
to five hundred twenty. We should be able to use these products to help us determine this
quotient. So to begin the method of partial quotients we always consider the entire
dividend, in this case six hundred twenty-four. We need to determine a certain number of twenty-sixes that
we know for sure are in six hundred twenty-four. Well, looking at our products, because two hundred sixty is less than six hundred
twenty-four, and two hundrd sixty equals ten times twenty-six, we know there's at least ten, twenty-sixes in six hundred twenty-four. But if you look at
twenty times twenty-six, which equals five hundred twenty, because five hundred twenty is still less than six hundred twenty-four, we know there's
at least twenty, twenty-sixes in six hundred twenty-four. So we can start with a partial quotient of ten or twenty,
but let's use the see larger partial quotients of twenty. So we'll put a twenty here on the right. And now we
multiply by the divisor. Twenty times twenty-six we know is five hundred twenty. So we write down five hundred twenty below the
dividend, and we subtract.
So we have four, zero, one. So the difference is one hundred four. So we just
removed twenty, twenty-sixes from six hundred twenty-four, and we're left with one hundred four.
And now we want to use one hundred four, and determine a certain number of twenty-sixes
that we know for sure are in one hundred four. Looking back at our work,
notice fifty-two is less than one hundred four, and because fifty-two equals two times twenty-
six, we know there's at least two, twenty-sixes in one hundred four. But maybe we also recognize that if two times twenty-six
equals fifty-two,
then four times twenty-six would be fifty-two plus fifty-two, or one hundred four. And notice how because four times twenty-six
equals one hundred four, this tells us there's exactly four, twenty-sixes in one hundred four. So let's use the partial
quotient of four. Again four times twenty-six equals one hundred four. And we subtract. Notice how this
difference is zero, which means the remainder is zero, and to find our quotient we simply
sum the partial quotients. So twenty plus four is twenty-four. So this tells us
six hundred twenty-four divided by twenty-six is exactly twenty-four. And the nice thing about the partial quotients method is that we
don't have to use the same partial quotients in order to find our quotient. So let's look at this
problem again, and use different partial quotients just a show we do get the same
quotient.
Let's say we begin by using this product to determine our first partial
quotient. Because ten times twenty-six equals two hundred sixty, and two hundred sixty is less than six hundred twenty-four, we know there's at least ten
twenty-sixes in six hundred twenty four. Let's begin with a partial quotient of ten. Ten times twenty-six equals two hundred sixty. We
subtract.
Notice after removing ten, twenty-sixes we have three hundred sixty-four and two hundred sixty is
still less than three hundred sixty-four. So we know there's at least ten, twenty-sixes in three hundred sixty-four.
So let's use the partial quotient of ten again. Ten times twenty-six equals two hundred sixty. Subtract.
After
removing another ten, twenty-sixes, we now have one hundred four. Again, let's say we didn't know this
product here, but we did know that two times twenty-six equals fifty-two. Because fifty-two is less than one hundred
four, we know there's at least two, twenty-sixes in one hundred four.
Let's use a partial quotient of two. Two times twenty-six equals fifty-two. We subtract. This difference if fifty-two. And again, because we know
two times twenty-six equals fifty-two, there's exactly two, twenty-sixes in fifty-two. So we use the partial quotient of two again.
Two times twenty-six equals fifty-two. Subtract. And of course we still have a remainder of zero. So because our
remainder is zero, to find our quotient, we sum the partial questions, which once again gives
us twenty-four.
So because twenty-six divides evenly into six hundred twenty-four, we know twenty-six is a factor of six hundred twenty-four. And so is twenty-four.
We also now know that twenty-four times twenty-six equals six hundred twenty-four.
Meaning twenty-four meaning groups or twenty-four copies of
twenty-six equals six hundred twenty-four.
I hope you found this helpful.