## Remainder definition

The remainder is the portion of the dividend that cannot be fairly divided by the divisor. After dividing whole numbers to find the quotient, you can end up with a portion of the dividend leftover; this is the remainder. It represents a fraction of the dividend and can be written as a decimal or a fraction.

The remainder is the pieces of a whole or parts of a set that cannot be split fairly among the divisor (whatever it represents, like people, cubbyholes, dollars, boxes, days, etc.). It is a portion of the dividend-per-divisor.

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## Division with remainders

The solution or answer to a division problem is the quotient, which can include a remainder.

In division, we divide single items – like a cake – and we divide sets of items – like cupcakes – among a group, hoping each person can get a fair share. Any amount of the dividend that is not divisible by the divisor is the remainder.

Suppose we are celebrating the 19th year of our school’s Math Club. We have a large, delicious sheet cake to share with all $43$ members. How will we cut up the cake, so everyone gets an equal portion?

We could cut the cake to make $44$ pieces by cutting $4$ rows of $11$ columns. We could cut $45$ pieces ($5$ rows of $9$), or $46$ pieces ($2$ rows of $23$), or $48$ pieces (), but we cannot cut exactly $43$ pieces from a rectangle.

No matter how we cut the cake to give everyone a piece, something will be leftover. A remainder will … remain.

If we cut the cake in $5$ rows and $9$ columns, we have $45$ cake – gives an answer of $1$ remainder $2$.

We have $45$ slices of cake to share among $43$ people; everyone gets $1$ slice, and we have $2$ slices remaining.

The first number, $1$, is the quotient. The $2$ is the remainder, the leftovers, the part of the cake that cannot reasonably be split among all $45$ Math Club members. Imagine trying to divide $2$ little cake slices into $45$ pieces!

### How to find the remainder in division

The steps to long division with a remainder are easy to recall using the mnemonic (memory trick): Does McDonald’s Sell Cheese Burgers Daily?:

• Does = Divide (divide the dividend by the divisor)
• McDonald’s = Multiply (multiply the partial quotient times the divisor)
• Sell = Subtract (subtract the product from the first digits of the dividend)
• Cheese = Compare (compare the difference with the divisor; the difference must be smaller)
• Burgers Daily = Bring Down (bring down the next digit of your dividend and begin again)

If a difference remains after completing all five steps with all the digits of the dividend, that difference is the remainder.

### Remainder symbol

To symbolize that you have a remainder in your answer, you write a capitalized letter $\mathrm{R}$, after the quotient, followed by the number that is your remainder. The answer, or quotient, is a whole number, and the remainder, shown with the letter $\mathrm{R}$.

To interpret the remainder, you must look at the units of the dividend, divisor, and quotient. If the dividend is cupcakes and the divisor represents members of the Math Club. Then, the quotient is cupcakes-per-member, with the remainder being cupcakes that could not be fairly split among the members.

Here is what this looks like in a word problem:

$48$ cupcakes are available to be divided fairly among $7$ Math Club members. How many will each member get, and how many will remain?

$48$ cupcakes $÷$ $7$ members cupcakes per member, with $6$ leftover cupcakes that cannot be fairly divided

Written as an equation, it looks like this:

### How to write a remainder

There are different ways you can write the remainder of a division problem. The remainder can be a whole number or faction. You always write the remainder after a capitalized R to signify that it is the remainder.

If you want to express the remainder as something other than a whole number leftover, you can make it into a fraction of the dividend.

Knowing how to write a remainder as a fraction is very easy. The remainder becomes the numerator, and the divisor is the denominator. The unit for the quotient and remainder is named by the dividend and divisor (dividend-per-divisor). Simplify the fraction if possible.

## Remainder example problems

Here is an example of a sheet of brownies that are cut into $44$ individual brownies. You are giving them out to the $7$ members of Math Club if each member signs up a new member from the freshman class. All $7$ members qualify. How will the $44$ brownies be shared?

[insert drawing of large sheet of cut-up brownies]

The $44$ brownies form the dividend, the item or set being divided fairly. The $7$ members form the divisor, the whole number the dividend is being shared among. The answer, the quotient, and the remainder will be dividend-per-divisor or brownies-per-member.

First, perform the long division:

brownies per member with $2$ brownies remaining (a remainder of $2$)

Next, place the remainder as a numerator of a fraction and use the divisor as the denominator:

$2$ brownies to be chopped up among $7$ members

The fractional amount of brownies-per-member cannot be simplified, so the final answer is $6$ brownies

Fractional remainders work with members of a set, too. Suppose you had $190$ raffle tickets to fairly divide among the $14$ members – $7$ veteran members and $7$ new recruits – of Math Club to sell as a fundraiser.

How many tickets must each member get, and what fraction will be the remainder?

Each member must sell $13$ tickets!

Realistically, no member will attempt to sell roughly half a ticket, but the remainder gives every member a goal: each must try to sell $14$ tickets, knowing some will sell only $13$ and others will sell more.

## What you learned:

After working your way through this lesson and video, you have learned:

• The definition of remainder in math
• The symbol for remainder in math
• How to find the remainder from division
• How to write the remainder in different ways
Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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