We want to use long division to find the quotient. We have a four digit number divided by
a one digit number. In this example, instead of using the traditional long division
algorithm, instead of using the traditional long division algorithm, we'll use
the method of partial quotients. The first step is to write the division problem in
long division form, which is this form here.
These two division problems are equivalent. Which means you can write one thousand, three hundred seventy-two divided by seven as one thousand, three hundred seventy two divided by
seven, in this form here. For the method of partial quotients, to keep things
organized, it's helpful to write a vertical line to the right of our
problem, here.
We'll record the partial quotients here at the right. Before we begin though, it's often
helpful to write some known products involving the divisor, or in this case
products involving seven. And because we're trying to determine how many
sevens there are are in one thousand, three hundred seventy-two, we do want to find some large products. Well we know of
course ones time seven is seven,
and if one times seven is seven, then ten time seven is equal to seventy and a hundred
times seven is equal to seven hundred. Let's go ahead and stop here and start our method
of partial quotients. To begin we consider the entire dividend of one thousand, three hundred seventy-two,
and because the divisor is seven, we want to determine at least a certain number of sevens that we
know for sure are in one thousand, three hundred seventy-two.
And because one hundred times seven equals seven hundred. And because seven hundred is less than one thousand, three hundred seventy-two, we know there's
at least one hundred sevens in our dividend. So let's go ahead and begin
with the partial quotient of one hundred. So write one hundred on the right side. Now multiply by
the divisor and write that product below the dividend. One hundred times seven is seven hundred, so we place seven hundred
below the dividend. Then we subtract. So two minus zero is two. Seven minus zero is seven. Thirteen minus seven is six. And then we repeat
the process. We want to determine a certain number of sevens that we know for sure are in
six hundred seventy-two. Well because seven hundred is greater than six hundred seventy-two, there's not one hundred sevens in six hundred seventy-two. So,
we'll have to use a different product. We could use ten time seven equals seventy,
because seventy is less than six hundred seventy-two. But the closer the product is to
six hundred seventy-two, without being greater than six hundred seventy-two the better. So looking at this product here,
because one hundred times seven equals seven hundred, and half of one hundred is fifty. Hopefully
we can recognize that fifty times seven would be half of seven hundred, or three hundred fifty.
And because three hundred fifty is less than six hundred seventy-two, we know there's at least fifty sevens in six hundred seventy-two.
Let's use fifty as our next partial quotient. Fifty times seven equals three hundred fifty, and now
we subtract. So two minus zero is two. Seven minus five is two. And six minus three is three.
And again we repeat the process.
Notice three hundred thirty-two is less than three hundred fifty so we cannot use this product to determine the
next partial quotient. We need to determine a certain number of sevens that we know for sure are
in three hundred twenty-two. Well because four times seven equals twenty-eight, we should be recognized that means forty times
seven would be equal to two hundred eighty. Because two hundred eighty is less than three hundred twenty-two, we know there's at least
forty, sevens in three hundred twenty-two. Let's use forty as the next partial quotient.
We record the forty. Forty times seven equals two hundred eighty.
We subtract.
Two minus zero is two.
Here we perform an exchange. Three changes
to a two. The two in the tens changes to a twelve.
So twelve minus eight is four. And two minus two is zero.
Now we need to determine a certain number of sevens that we know are in forty-two. And
because six times seven is exactly forty two, there's exactly six, sevens in forty-two.
So we record six as a partial quotient. Six times seven is forty-two. We subtract. Notice how because this differences is zero, this
tells us the remainder is zero. So to find our quotient, we add the partial quotients.
So we have six in the ones place value, and nine in the tens place value, and a
one in the hundreds place value. Which means our quotient is one hundred ninety-six.
So because the remainder is zero, this tells us that seven is a factor of one thousand, three hundred seventy-two. It's also
true because one thousand, three hundred seventy-two divided by seven equals one hundred ninety-six.
One hundred ninety-six times seven equals one thousand, three hundred seventy-two.
The quotient times the divisor equals the dividend. The nice
thing about the partial quotients method is that we did not have to use these
partial quotients to find our quotient. We can use any partial quotients we can
find and still determine the correct quotient.
I hope you find this helpful.