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Meaning of Remainder

Remainder, as its name suggests, is something that "remains" after completing a task.

Assume that you have 15 cookies that you want to share with 3 of your friends, Mary, David, and Jake.

You want to share the cookies equally among your friends and yourself.

You will begin distributing them in the following way.

Here, you can see that there are 3 cookies "remaining" after the distribution.

These 3 cookies cannot be further shared equally among the 4 of you.

Hence, 3 is called the "remainder" here.

We can understand the meaning of the remainder from the following illustration.

You can change the number of things and the number of groups by moving the corresponding sliders and see the remainder.

Definition of Remainder

Remainder is a part of something that is "remaining" after division.

The remainder is left over when a few things are divided into groups with an equal number of things.

Let's recall the scenario we discussed earlier about 15 cookies being shared equally among \(4\) children.

In other words, 15 cookies had to be divided into 4 equal groups.

We were left with 3 cookies and hence, 3 was the remainder.

Let us consider another example.

Let us assume that 8 slices of pizza are to be shared equally among 2 children. 

How many pieces of pizza remain unshared?

You can look at the picture below to understand how we have divided the slices of pizza equally between the two children.

Thus, the remainder is 0 as no pieces of pizza are left unshared.

We cannot always pictorially show how we divide a number of things equally among the groups in order to find the remainder.

Instead, we can find the remainder using the long division method.

For example, the remainder in the above example on cookies can be found using long division as follows:

The remainder of this division is 3.

The remainder in the above example on pizza can be found using long division as follows:

The remainder of this division is 0.

Examples of Remainder

Let us consider the division \(43 \div 2\).

Let us find the remainder of this division using the following illustration.

You can use the "Play" and "Reset" buttons to view this illustration.

Thus, 1 is the remainder.

Here are some other examples of remainders.

Division	Remainder
\(35 \div 6\)	5
\(42 \div 8\)	2
\(121 \div 11\)	0
\(118 \div 12\)	10
\(120 \div 17\)	1

You can check these remainders by doing long division.

Let us divide 7 by 2 using long division and see what the quotient and the remainder are.

We can represent the remainder of division in two ways.

One is by writing the quotient and the remainder with an "R" in between them.

Dividend \(\div\) Divisor = Quotient \(R\) Remainder

For example, \(7 \div 2\) can be written as:

\[7 \div 2 = 3 \, R \, 1\]

Another way to represent the remainder is by showing it as a part of a mixed fraction.

\( \dfrac{ \text{Dividend}}{ \text{Divisor}} = \text{Quotient} \dfrac{ \text{Remainder}}{\text{Divisor}} \)

For example, \(7 \div 2\) can be written as:

\[7 \div 2 = 3 \dfrac{1}{2}\]

We can enter the dividend and the divisor in the following calculator and it will show us the quotient and the remainder.

It also shows the division equation by clicking on the checkbox.

The properties of the remainder are as follows:

• The remainder is always less than the divisor. A remainder that is either greater than or equal to the divisor indicates that the division is incorrect.
   
• If one number (divisor) divides the other number (dividend) completely, then the remainder is 0.
   
• The remainder can be either greater, lesser, or equal to the quotient.

 

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Example 1

 

 

What is the remainder when 3723 is divided by 23?

Check if the answer you got is correct.

Solution:

Let us divide 3723 by 23 using long division.

Therefore,

Remainder = 20

Now, we will check the answer by substituting all the values of dividend, divisor, quotient, and remainder in the following equation:

\[ \begin{align}\text{Dividend }\!\!&= \!\!\text{ Divisor }\!\!\times\!\!\text{ Quotient }\!\!+\!\!\text{ Remainder}\\[0.2cm] 3723 &= 23 \times 161 + 20 \\[0.2cm] 3723 &= 3703+20\\[0.2cm] 3723&=3723 \end{align} \]

Since we have the same values on both the sides, our answer is correct.

Example 2

 

 

The number of days in the year 1996 was 366 as it was a leap year.

If \(1\)^st Jan 1996 was a Monday, what day was it on \(1\)^st Jan 1997?

Solution:

It is given that 1^st Jan 1996 was a Monday.

We know that every day of the week repeats exactly after 7 days.

Hence, we will divide 366 by 7

2 is the remainder of division here.

We can say that 1^st Jan 1997 = Monday + 2 days = Wednesday.

Hence,

1st Jan 1997 was a Wednesday.

Example 3

 

 

A company has a budget of Rs. 10,540 to buy external hard drives.

The cost of each hard drive is Rs. 450.

(i) How many hard drives can the company buy with the given budget?

(ii) How much money will the company be left with after the purchase?

Solution:

Total budget of the company = Rs. 10,540.

Cost of each hard drive = Rs. 450.

(i) We will divide 10,540 by 450 to find the number of hard drives that the company can buy.

The number of hard drives that the company can buy is nothing but the quotient of the division.

Therefore,

Number of hard drives bought = 23

(ii) The amount of money that the company is left with is nothing but the remainder of the above division.

Therefore,

Amount of money left = Rs. 190

Example 4

 

 

A teacher distributes chocolates to 30 students in her class such that each one gets an equal number.

If there are 315 chocolates in total,

(i) How many chocolates did each student get?

(ii) How many chocolates are left with the teacher after the distribution?

Solution:

Total number of students in the class = 30

Total number of chocolates that the teacher has = 315

(i) We will divide 315 by 30.

The quotient of this division indicates the number of chocolates each student gets, which is 10.

Therefore,

Chocolates each student gets = 10

(ii) The remainder of this division indicates the number of chocolates that are left with the teacher.

Chocolates left with the teacher = 15