How to Calculate Remainders: Step-by-Step Methods for Students
According to the Merriam-Webster Dictionary, the remainder is, “a remaining group, part, or trace/ the number left after a subtraction/ the final undivided part after the division that is less or of lower degree than the divisor”. In simple words, the term "remainder" refers to the part that remains after the division process has been completed. If we divide 8 pencils equally among 7 children, we are left with 1 pencil. In this example, the remaining 1 pencil represents the remainder. Also, if we divide 33 by 2, the quotient is 16 and the remainder is 1. The divisor is always greater than the remainder.
Understanding Remainder in Maths in Detail
The remainder refers to the portion of the dividend that the divisor cannot divide fairly. You may end up with a fraction of the dividend left over after dividing whole numbers to determine the quotient; this is the remainder. It is a decimal or a fraction that represents a portion of the dividend. Let us consider one example.
Suppose you have 23 chocolates and you want to divide it equally among 4 of your friends. So, how many will each friend get? How many chocolates will remain after distributing the chocolates equally? The answer is: Each friend will get 5 chocolates, so 20 chocolates can be distributed equally among 4 friends and 3 chocolates will remain which cannot be distributed. So, here, 23 is the dividend, 4 is the divisor, 5 is the quotient and 3 is the remainder.
Remainder Formula
As we know,
Dividend = Divisor × Quotient + Remainder
Accordingly, the remainder formula is given as:
In the above remainder formula,
The dividend is the number or value that is being divided.
The divisor is the number that divides another number.
The quotient is the result that is obtained after the division of two numbers.
The remainder is the value that is left after the division.
How to Find the Remainder After a Division?
We can't always show in pictures how we divide the number of things equally among the groups to find the remainder. Instead, we can use the long division method to find the remainder. Here is a memory trick to always remember the steps of long division:
Does McDonald’s Sell Cheeseburgers Daily?
Step I: Does - Divide the dividend by the divisor.
Step II: McDonald’s - Multiply the partial quotient times the divisor.
Step III: Sell - Subtract the product from the first digits of the dividend.
Step IV: Cheese - Compare the difference with the divisor; the difference must be smaller.
Step V: Burger’s Daily - Bring down the next digit of your dividend and begin again.
Let us understand in simple terms. Consider the following image:
Here, 68 is the dividend, 5 is the divisor, 13 is the quotient and 3 is the remainder.
At first, 6 is divided by 5 and we got the first digit of the quotient 1. We multiplied 1 and 5 and wrote the result below 5. Then, we subtracted 5 (5 x 1 = 5) from 6 and got 1. Then, we brought down 8 and got the number 18. In the next step, we divided 18 with 5 and got the second digit of the quotient i,e 3. So, after multiplying, the number came to 15. On subtracting 15 from 18, we got 3. So, 3 is the remainder.
How to Write Remainders?
There are several ways to express the remainder of a division problem. The remainder can be either a whole number or a faction. One way to write the remainder is to separate the quotient and the remainder with a "R." 7/2 = Q=3 and R=1 is the formula for dividing 7 by 2. In this case, Q=3 is a quotient, and R=1 is a remainder.
Another way to represent the remainder is as a component of a mixed fraction. 7/2 = 3 12 is the formula for dividing 7 by 2.
Properties of Remainders
The following are the properties of the remainder:
The divisor is always less than the remainder. The division is wrong if the remainder is either more than or equal to the divisor.
When one number (divisor) entirely divides the other number (dividend), the remainder is 0.
The remainder can be more than, equal to, or less than the quotient.
Remainder Examples
Here are a few examples of remainders that will help you understand the term remainder in Maths in a better way.
1. A teacher had 315 chocolates. She divides all chocolate evenly among 30 students. Find
How many chocolates did the teacher give to each of the students?
How many chocolates are left with the teacher after distributing them among the student?
Solution:
Total number of students in class = 30
Total number of chocolates teacher has = 315
Now, we will divide 315 by 30
(Image will be uploaded soon)
i) The quotient value of the above division represents the number of chocolate each student gets is 10.
Therefore,
Chocolates each students gets = 10
ii) The remainder value of the above division represents the number of chocolate left with the teacher.
Therefore,
Chocolates left with the teacher after distribution = 15
2. What is the remainder when 53 is divided by 8?
Solution: To find the value of the remainder, when 53 is divided by 8, we consider the multiples of 8.
8 × 1 = 8
8 × 2 = 16
8 × 3 = 24
8 × 4 = 32
8 × 5 = 40
8 × 6 = 48
8 × 7 = 56
As 8 × 6 = 48 and 8 × 7 = 56, only six 8’s can go into 53. Then the value left over is 53 - 48 = 5. Hence, the value of the remainder when 53 is divided by 8 is 5.
Do You Know?
FAQs on Remainder in Maths: Easy Explanation, Formula & Solved Examples
1. What is a remainder in mathematics?
A remainder is the amount 'left over' after performing a division where one number does not divide another number completely. For instance, when you divide 10 by 3, you get a quotient of 3, and 1 is left over. This '1' is the remainder. It represents the part of the dividend that cannot be evenly grouped by the divisor.
2. How is the remainder calculated in a division problem?
To calculate the remainder, you perform the standard division. For example, to find the remainder when 22 is divided by 5:
Find the largest multiple of the divisor (5) that is less than or equal to the dividend (22). In this case, it is 5 × 4 = 20.
Subtract this multiple from the dividend: 22 - 20 = 2.
The result, 2, is the remainder.
3. What is the formula to verify the remainder in a division?
The relationship between the parts of a division problem is given by the Division Algorithm formula: Dividend = (Divisor × Quotient) + Remainder. You can rearrange this to find the remainder directly: Remainder = Dividend – (Divisor × Quotient). This formula is essential for checking if your division and remainder are correct.
4. Can the remainder be zero? If so, what does it mean?
Yes, a remainder can be zero. A remainder of zero signifies that the division is exact or complete. It means the dividend is a perfect multiple of the divisor. For example, when 20 is divided by 4, the quotient is 5 and the remainder is 0, because 4 divides 20 perfectly.
5. What are the most important properties of a remainder?
The remainder in a division operation has several key properties:
The remainder is always a non-negative integer.
The value of the remainder must always be less than the divisor. If it is equal to or greater than the divisor, the division is incomplete.
If a divisor divides a dividend completely, the remainder is 0.
The remainder can be less than, equal to, or greater than the quotient.
6. How is a remainder different from a quotient?
The quotient and the remainder are two different results of a division. The quotient represents how many times the divisor can fully go into the dividend. The remainder is the amount that is left over after this process. For example, in 17 ÷ 5, the quotient is 3 (5 goes into 17 three full times), and the remainder is 2 (2 is left over).
7. Why must the remainder always be smaller than the divisor?
The remainder must be smaller than the divisor because if it were not, it would mean that at least one more group of the divisor could still be 'taken out' of the dividend. The goal of division is to find the maximum number of times a divisor fits into a dividend. If the remainder is equal to or larger than the divisor, it indicates the division process is not yet complete.
8. Can you provide a real-world example of using remainders?
Yes. Imagine you have 25 cookies to share equally among 4 friends. Each friend would get 6 cookies (4 × 6 = 24). After distributing 24 cookies, you would have 1 cookie left over. This leftover cookie is the remainder. It's the part that couldn't be distributed equally among the groups.
9. What is the Remainder Theorem and how is it related to basic division?
The Remainder Theorem is a more advanced concept used in algebra for polynomials, typically introduced in higher classes like Class 9 or 10. It states that if you divide a polynomial, P(x), by a linear factor (x - a), the remainder is equal to P(a). This provides a shortcut to find the remainder without performing long division. It's a direct application of the same fundamental idea of a 'leftover' value, but applied to algebraic expressions instead of just numbers.
