Long Division with Remainders

We’ve already practiced
long division
, but so far our answers have all come out even (in other words,
our last subtraction problem ended in an answer of 0). However, sometimes our division
problems will not come out evenly, and we will have another number (not 0) when
we do the last subtraction problem. This leftover number is called a remainder,
and it is written as part of the quotient. Follow along with this example:

The red circled number at the bottom our remainder. You do not have to circle the
remainder; we just circled ours so that you know which number it is. After you have
your remainder, you write it on top of the division bar, with an r in front of it,
like this: 25 r 3.

When your division ends with a remainder, you must make sure that your remainder
is less than your divisor. If your remainder is more than your divisor, you need
to go back and check your division, because it is incorrect. We can still use our
multiplication method to check our division; you will multiply the quotient (25)
by the divisor (5), and then add our remainder to the answer to the multiplication
problem, like this:

Let’s try that one more time. Here’s a new example:

Our answer to this problem is 23 r 1; note that we always write the remainder after
the quotient, on top of the division bar. Also notice that our remainder (1) is
smaller than our divisor (6).

Now let’s check our work, like this:

There are also several different ways to write remainders. The standard way is shown
above, with an r in front of the number. However, you can also write remainders
as fractions
and as decimals.

Long Division with Remainders as Fractions

Now that you understand the basics of long division, you may be asked to write your
remainder as a fraction. Don’t worry! It’s not hard at all. You’re going to do long
division the same way—divide, multiply, subtract, bring down, and then you’re going
to get a remainder. Instead of writing r and then the number, you are going to take
your remainder and make it the numerator of a fraction. The denominator comes from
the divisor—you use the same number you’re dividing by in your denominator.

Let’s look at the following example:

Notice that you do not use the r at all in front of your remainder when you’re turning
it into a fraction. However, you do still write the fraction as part of the quotient
(answer to your division problem).

Also, you would check this division problem the same way as a normal division problem;
multiply the quotient (23) by the divisor (6) and then add the remainder (1). Do
not do anything with the fraction in order to check this problem.

Long Division with Remainders as Decimals

Another way you may be asked to express a remainder is in the form of a decimal.
When you’re asked to express your remainder as a decimal, you first complete division
as usual, until you get to the point you usually end at, where you have nothing
else to bring down. Instead of stopping here, however, you are going to keep going
with division. You will add a decimal point (.) after the last number given in the
dividend, and you will also place a decimal point in the quotient after the number
you have so far. After the decimal in the dividend, you will add a zero (0) and
continue division. You will keep adding zeroes until your subtraction step results
in an answer of 0 as well. Follow along with this example:

Notice that we added a decimal after the 6 in the dividend, as well as a decimal
after the 5 in our quotient. Then, we started adding zeroes to the dividend. This
time, it only took us one added zero before our remainder was zero.

Now, let’s look at a problem where you’d have to add more than one zero to the dividend:

When you have your quotient with a decimal, you check the answer differently than
if it had a remainder as a fraction or just a remainder written with r. Instead
of adding the remainder separately, you just multiply the quotient (including decimal)
by the divisor, like this:

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