We're asked to use long division to find the quotient. For this example though, we'll use the
method of partial quotients rather than the traditional long division algorithm.
The first step is to write the quotient in long division form which would be this
form here. So we can write the given quotient as two thousand, one hundred ten divided by sixty-two. When performing partial
quotients, it's helpful a draw a vertical line here to keep our work organized. We'll record the
partial quotients here on the right side.
Before we start the method of partial quotients though, it's often helpful to find some
products involving the divisor, or in this case, products involving sixty-two.
So for example we know that one times sixty-two equals sixty-two.
But because the dividend is in the thousands, we do want to find some large products. So from
here, we should recognize that ten times sixty-two is equal to six hundred twenty. And then if we know
one times sixty-two, then two times sixty-two would be equal to sixty-two plus sixty-two, which would be one hundred twenty-four.
So if two times sixty-two equals one hundred twenty-four, then twenty times sixty-two would be equal to one thousand, two hundred forty.
Let's go ahead and start with
these products. Now let's begin dividing using partial quotients. When using the method
of partial quotients, we consider the entire dividend and we begin by
determining a certain number of sixty-twos that we know for sure are in two thousand, one hundred ten. Well if we know that
twenty times sixty-two equals one thousand, two hundred forty, and one thousand two hundred forty is less than
two thousand, one hundred ten, this tells us there must be at least twenty, sixty-twos in two thousand, one hundred ten. So twenty is our first partial
quotient. So now we multiply by the divisor. Twenty times sixty-two we already know is one thousand, two hundred forty. And now
we subtract. By subtracting, what we're doing is removing twenty, sixty-twos from two thousand, one hundred ten.
So subtracting, we have a zero in the ones. Here we perform an exchange. Eleven minus four is seven. Another
exchange here.
Ten minus two is eight. And of course one minus one is zero. So the different is eight hundred seventy. And now we repeat the
process. We want to determine a certain number of sixty-twos that we know for sure are in eight hundred seventy.
Looking back in our products, notice how because six hundred twenty is less
than eight hundred seventy, and six hundred twenty is equal to ten times sixty-two, there must be at least ten, sixty--twos in eight hundred seventy. So
let's use ten as our or next partial quotient. And we multiply by the divisor.
Ten times sixty-two is six hundred twenty, and we subtract.
Because the difference
is two hundred fifty, we now need to determine a certain number of sixty-twos that we know for
sure are in two hundred fifty. Going back to our products, notice how one hundred twenty-four is less than two hundred fifty. And because
one hundred twenty-four is equal to two times sixty-two, we know there's at least two sixty-twos in two hundred fifty. But instead of using two for the
next partial quotient, let's see if we can find a product closer to two hundred fifty but not more than two hundred fifty. Well if
two time sixty-two is equal to one hundred twenty-four, let's see if we can determine four times sixty-two. Four times sixty-two would be
twice the product of two time sixty-two. So one hundred twenty-four times two, or one hundred twenty-four plus one hundred twenty-four will be two hundred forty-eight.
So because two hundred forty-eight is still less than two hundred fifty, and two hundred forty-eight equals four times sixty-two, we now know
there's at least four, sixty-twos in two hundred fifty. Again we could've used two but the larger partial
quotients we use,
the less steps we'll have. So let's use the partial quotients of four, multiplying by the
divisor. Four times sixty-two equals two hundred forty-eight, and we subtract. Notice how the difference is equal two,
which is less than the divisor, and therefore we're done. This two is our
remainder, and to find the quotient, we sum the partial quotieints. So here we
have twenty plus ten, plus four, which equals thirty-four. Which means two thousand, one hundred ten divided by sixty-two
equals thirty-four, with the remainder of two. So because we have a remainder, we know that
sixty-two is not a factor of
two thousand, one hundred ten. But this does tell us that the quotient of thirty-four times the divisor of
sixty-two, plus the remainder of two equals the dividend of two thousand, one hundred ten.
Simplifying the left side.
Thirty-four times sixty-two equals two thousand, one hundred eight. And of course two thousand, one hundred eight plus two, does equal two thousand, one hundred ten.
Another way to think of this is two thousand, one hundred ten is equal to thirty-four groups, or thirty-four copies of sixty-two, with two
left over, or with the remainder of two.
I hope you found this helpful.