We want to use long division to find the quotient. We have a two digit number
divided by a one digit number. And 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.
These two division problems are equivalent, where a is the dividend, b is the divisor, and c
is the quotient. So we'll write eighty-seven divided by six as eight-seven divided by six in this form here. And for the partial
quotients method, we'll draw a vertical line to the right of our problem. We'll record the
partial quotients on the right side. Before we begin though, because the
divisor is six, it's often helpful to list products involving the divisor, or in
this case products involving six. Let's say we know that one times six equals
six,
two time six equals twelve. We could continue but let's jump to, let's say, five times six,
which equals thirty. And ten times six equals sixty. Having some known products involved in the
divisor is often helpful. To begin the partial quotients method, we consider the entire
dividend of eighty-seven. We need to determine a certain number of sixes
that we know for sure are in eighty-seven.
And because ten times six equals sixty, and sixty is less than eighty-seven, we know there's
at least ten sixes in eighty-seven.
So let's use ten at the first partial quotient. We don't have to ten as the first partial
quotient, but the larger partial quotient we use, the less work we'll have to
determine the quotient.
So we'll record the first partial quotient of ten here on the right. And now we'll multiply by the
divisor, ten times six equal sixty. We write sixty below the eighty-seven and we subtract.
Eighty-seven minue sixty is equal to twenty-seven. Now we need to determine a certain number of sixes
we know for sure in twenty-seven. Well because two times six equals twelve. And twelve is less than twenty-seven, we know there's at
least two sixes in twenty-seven. But let's say that we recognize that because two times six equals twelve, we
also know four times six equals twenty-four. Because twenty-four is less than twenty seven this tells
there's at least four sixes in twenty-seven. So let's use four as our next partial quotient. We could
use two, but using four will cut down on the number of steps that we have to show. So we'll
use the partial quotient of four, multiply it by the divisor. Four times six is twenty-four, and
we subtract. Twenty-seven minus twenty-four is equal to three. Because this difference is less than the divisor, we're
done. This three is a remainder, and to find the quotient we sum the partial
quotients.
So ten plus four equals fourteen, which means our quotient is fourteen with a remainder of three. So
because eighty-seven divided by six equals fourteen with the remainder of three, this tells us there
are fourteen, sixes in eighty-seven with the remainder of three, or with three left over. The nice thing about the partial
quotients method is that we can use a variety of partial quotients to find the
quotient of the division problem. This is just one way to find the quotient using
partial quotients. So before we go, let's look at this another way using different partial
quotients.
Just to emphasize, there's more than one way to determine the quotient using this
method. For example let's say we use this product here to find our first partial
quotient. Because five time six equals thirty, and thirty is less than eighty-seven, we know there's at
least five sixes in eight-seven, Let's use five as the first partial quotient. So place a
five on the right. Five times six equals thirty. We subtract. Eighty-seven minus thirty equals fifty-seven. Because
thirty is still less than fifty seven, we know there's at least five sixes in fifty-seven. Let's
use a partial quotient of five again.
So five times six is thirty.
Subtract. Fifty-seven minus thirty equals twenty-seven. Now we need to determine a certain number of sixes that
are in twenty-seven. And let's just say we'll use the fact that we know that two time six equals twelve. Because
twelve is less than twenty-seven. We know there's at least two sixes in twenty-seven. Let's use two as the next partial quotient.
Two time six is twelve. We subtract. Twenty-seven minus twelve is fifteen.
Notice twelve is still less than fifteen, which means there are at
least two sixes in fifteen. Let's use two as the next partial quotient. Two times six is twelve.
We subtract. Fifteen minus twelve equals three. Three is less than the divisor, so once again we're
done, three is the remainder and we'll sum the partial quotients, which of course we
get fourteen again.
Our quotient is fourteen with a remainder of three. So this just demonstrates there's more than one way
to find a quotient using the method of partial quotients. But of course the
larger partial quotients we use, the less steps there will be. I do want to point
out one more thing before we go. Because eighty-seven divided by six equals fourteen with the remainder
of three,
this tells us that fourteen times six, plus three equals eighty-seven. The quotient times
the divisor, plus the remainer equals a dividend. If we simplify the left side, fourteen times six
equals eighty-four. Eighty-four plus three equals eighty-seven.
I hope you found this helpful.