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.
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 members. How will we cut up the cake, so everyone gets an equal portion?
We could cut the cake to make pieces by cutting rows of columns. We could cut pieces ( rows of ), or pieces ( rows of ), or pieces (), but we cannot cut exactly 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 rows and columns, we have cake – gives an answer of remainder .
We have slices of cake to share among people; everyone gets slice, and we have slices remaining.
The first number, , is the quotient. The is the remainder, the leftovers, the part of the cake that cannot reasonably be split among all Math Club members. Imagine trying to divide little cake slices into pieces!
The steps to long division with a remainder are easy to recall using the mnemonic (memory trick): Does McDonald’s Sell Cheese Burgers Daily?:
If a difference remains after completing all five steps with all the digits of the dividend, that difference is the remainder.
To symbolize that you have a remainder in your answer, you write a capitalized letter , 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 .
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:
cupcakes are available to be divided fairly among Math Club members. How many will each member get, and how many will remain?
cupcakes members cupcakes per member, with leftover cupcakes that cannot be fairly divided
Written as an equation, it looks like this:
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.
Here is an example of a sheet of brownies that are cut into individual brownies. You are giving them out to the members of Math Club if each member signs up a new member from the freshman class. All members qualify. How will the brownies be shared?
The brownies form the dividend, the item or set being divided fairly. The 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 brownies remaining (a remainder of )
Next, place the remainder as a numerator of a fraction and use the divisor as the denominator:
brownies to be chopped up among members
The fractional amount of brownies-per-member cannot be simplified, so the final answer is brownies
Fractional remainders work with members of a set, too. Suppose you had raffle tickets to fairly divide among the members – veteran members and 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 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 tickets, knowing some will sell only and others will sell more.
After working your way through this lesson and video, you have learned:
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