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# How to do division

Calculators are amazing, but depending on them even for the basics will hamper your math skills. Just like addition, subtraction, and multiplication, you should know how to divide manually, at least for small numbers. This can come handy during tests that do not allow a calculator and, at times when you don’t have one.

## How to divide without a calculator?

Don’t worry, it is not hard!

We will you through basic steps so that you understand how to divide without the help of a calculator.

### Things to remember

To get the process right, you must learn the terminology.

• The number below the division sign is the dividend
• The number outside the division sign (left) is the divisor
• The answer you get is the quotient

## Basic steps for manual division

4 is the divisor and 6492 is the dividend.

1. Find out how many times 4 will go into 6 (the first digit of the dividend). So, put 1 in the quotient and write 4 below 6.
2. You have to subtract them for the remainder. You will get 2.
3. Now, bring the second digit of the dividend (4) below and write it beside 2.
4. The number that you get now is 24.
5. Follow the same steps. Find out how many times 4 will go into 24. The answer is 6. Write 6 in the quotient beside 1 and then subtract the numbers. You will now get 0.
6. Bring the third digit down, beside 0. That’s 9.
7. Since 4 will go into 9 2 times, write 2 in the quotient and 8 below 9. Subtract.
8. You get 1.
9. Bring the last digit, 2, down. Write it beside 1.
10. The remainder now is 12.
11. 4 goes into 12, 3 times. Write 3 in the quotient and 12 below 12.
12. Subtract 12-12 to get 0.

If all numbers are done, and the remainder is not zero yet, you will have to add a decimal in the quotient and write 0 beside the remainder. Continue with your division until you get 0 or a recurring number.

Many tests ask for an answer until 2 or 3 decimal places. So you don’t have to worry beyond that.

## What if the divisor has a decimal?

Sometimes, the problem may look like this – 2367/3.5

In such a case, you can remove the decimal from the divisor by adding a 0 to the dividend. After doing this, you can follow the steps mentioned above.

## Some quick rules about division

We would like to help you a little more with long division by telling you about a few basic rules. Here they are:

1. 0 divided by any number is always 0. You may have learned the same about multiplication as well. For example, there are 40 oranges in the basket and 0 oranges were distributed among 3 children. How many does each child get? 0.
2. On the flip side, you will never divide a number by 0. You can’t ask the remainder of the problem 24/0. There’s no answer. If you come across such a problem, just write the infinity symbol as the answer.
3. If you divide any number by itself, the answer will always be 1.
4. Anything divided by 1 is the number itself.
5. Every fraction is a division sum.
6. When you divide any number by 2, you get its half. For example, 20/2 is 10. 10 is half of 20.
7. Divide 1 by any number and the answer will be the same as that number. 4/1 is 4. 2576/1 is 2576.

### BONUS: Can we solve remainder problems without using long division?

We are speaking about this because a lot of students studying for SAT, GRE, and ACT find themselves stuck. They are unable to figure out how to solve remainder problems without using long division.

The good news is that it is possible. You can totally crack these problems without touching long division. But how?

Here, you will need a calculator. However, simply pressing the numbers in will not help you. You will need to work with calculator algorithms to get to the remainder. Below are the steps:

1. Start with the basic division. You will get a decimal number (Example: 2.4 or 7.8).
2. Now use the integral and multiple it with the divisor.
3. The answer you get should be subtracted from the dividend.

When you are doing it for the first time, remember to punch in the numbers correctly. We see many students get confused between a divisor and a dividend or an integer. This is where basics and terminology come handy.

We hope we helped you understand long division. Practice with a few small numbers and move up to get it thoroughly. In math, like in many other things, practice is the key to success!

• sql
• learn sql
• division

The division operator in SQL is used to divide one expression or number by another. This article will show you exactly how to use it and common mistakes to avoid along the way.

The division operator in SQL is considered an arithmetic operator. The arithmetic operators are addition ( + ), subtraction ( – ), multiplication ( * ), division ( / ), and modulus ( % ). This article will focus on the division operator, discussing the rules that must be followed along with some common mistakes to look out for when trying to divide in SQL.

The syntax for the division operator in SQL is as follows:

Note the inclusion of the WHERE clause is entirely optional. The division operator can be used anywhere there is an expression. This means you can use the SQL division operator with:

• SELECT .
• WHERE .
• HAVING .
• ORDER BY .
• Or as part of a function.

If these expressions cause you any confusion, consider this SQL Practice track from LearnSQL.com. It is an efficient and fun method for sharpening your SQL skills.

The simplest example of the division operator is:

You can execute this query, and it will output the result – in this case, 5. However, it is more likely that you will be working with integers that reside in columns as part of your database tables.

Let’s look at such an example. We will use a table called stock , containing typical food items along with their price and quantity .

stock

item price quantity
Eggs 15 12
Milk 3 4
Flour 5 2

LetвЂ™s apply the division operator to an entire column of our table and witness its effect. Consider the following SQL query:

With this query, we are selecting the quantity column then showing the result of dividing the quantity value in our result column which contains the result of quantity / 2 . LetвЂ™s run this query and observe the results:

quantity result
12 6
4 2
2 1
3 1

The results look like what we would expect, except for the final row. Isn’t 3 / 2 = 1.5?! The result shown by SQL Server shows a value of 1. Depending on the variant of SQL you are using, the division operator may handle the integer division differently. Let’s clear this up in the next section!

## How Are Integers Divided in SQL?

The integer divisions may behave differently depending on your choice of SQL database management system. MySQL and Oracle users can expect to see the real number to be shown with the data type being float, e.g., 3 / 2 = 1.5.

However, for SQL Server and PostgreSQL users, the integer division is more complex. You will have to remember certain rules when dividing numbers that do not divide to an integer. LetвЂ™s bring up that previous example again:

When executed in SQL Server or PostgreSQL, the result of this query is 1, while most users would expect 1.5. What is happening here? The division operator only handles the integer part of the result when dividing two integers. The result obtained is called the quotient. The remainder is not dealt with. With the division of two integers, the result will be how many times one number will go into another, evenly.

Consider these examples in SQL Server or PostgreSQL:

Division Query Result
SELECT 10 / 3 3
SELECT 5 / 2 2
SELECT 11 / 6 1

### Change the Operands to a Decimal or Floating-Point Number

This can only be done if the particular division operation allows it. If two column names are involved, this method will not be possible; instead, use the CAST/CONVERT method described below. You need to be able to access one of the numbers directly. Using this method is simple; we simply change one or both of the operands:

Division Query Result
SELECT 10.0 / 3 3.333333
SELECT 5 / 2.0 2.5
SELECT 11.0 / 6.0 1.833333

We can also apply this method to our previous stock example. We have to update our query to the following:

Executing this query produces a similar result set as before with one key difference.

quantity result
12 6.00000
4 2.00000
2 1.00000
3 1.50000

This is ideal if we can access and alter one of the operands directly. However, this is not always the case.

### Using CAST or CONVERT on Columns

Imagine we divide the value of the price column by the quantity column for a particular item. Our query would look like this:

Executing this query shows a result of 2; this is a classic integer division scenario. Let’s get around this by using CAST on one of our columns.

We have to update our query to the following:

Note that you could also use CONVERT instead of CAST. You can read more about converting an integer using CAST and CONVERT here. Executing this query shows a new result:

CAST(price)/quantity
2.50000

There we have it! The CAST operation was successfully applied to the price column, making it a decimal data type for this arithmetic operation. Note that if the arithmetic expression contains more than one operator, the multiplication and division operators are evaluated first, followed by the addition and subtraction operators. When two operators have the same priority, the expression is evaluated from left to right.

You should also be aware of the SQL “division by zero” error regardless of what DBMS you use. Dividing by zero results in infinity, which SQL does not allow. Doing so will result in an error message being shown. This useful SQL cookbook page shows you how to handle the “division by zero” error in SQL. Check it out if you’re interested!

## Divide and Conquer With the SQL Division Operator!

In this article, we dove into the deep end, showing you the intricacies of using the division operator in SQL for the integer division. We learned how to handle integer division for each SQL DBMS from SQL Server to MySQL. We also touched on the pesky SQL “division by zero” error and introduced resources that will allow you to handle this particular error elegantly.

As with anything in life, practice makes perfect! If you would like to strengthen your SQL skills through fun and engaging monthly challenges, consider this month’s SQL Challenge.

## Calculator Use

Divide two numbers, a dividend and a divisor, and find the answer as a quotient with a remainder. Learn how to solve long division with remainders, or practice your own long division problems and use this calculator to check your answers.

Long division with remainders is one of two methods of doing long division by hand. It is somewhat easier than solving a division problem by finding a quotient answer with a decimal. If you need to do long division with decimals use our Long Division with Decimals Calculator.

## What Are the Parts of Division

For the division sentence 487 ÷ 32 = 15 R 7

• 487 is the dividend
• 32 is the divisor
• 15 is the quotient part of the answer
• 7 is the remainder part of the answer

## How to do Long Division With Remainders

From the example above let’s divide 487 by 32 showing the work.

Set up the division problem with the long division symbol or the long division bracket.

Put 487, the dividend, on the inside of the bracket. The dividend is the number you’re dividing.
Put 32, the divisor, on the outside of the bracket. The divisor is the number you’re dividing by.

Divide the first number of the dividend, 4 by the divisor, 32.

4 divided by 32 is 0, with a remainder of 4. You can ignore the remainder for now.

Put the 0 on top of the division bracket.
This is the beginning of the quotient answer.

Next, multiply 0 by the divisor 32 and insert the result 0 below the first number of the dividend inside the bracket.

Draw a line under the 0 and subtract 0 from 4.

Divide 48 by the divisor, 32. The answer is 1. You can ignore the remainder for now.

Note that you could skip all of the previous steps with zeros and jump straight to this step. You just need to realize how many digits in the dividend you need to skip over to get your first non-zero value in the quotient answer. In this case you could divide 32 into 48 straight away.

Put the 1 on top of the division bar, to the right of the 0. Next, multiply 1 by 32 and write the answer under 48.

Draw a line and subtract 32 from 48.

Divide 167 by the 32. See a pattern emerging?

167 ÷ 32 is 5 with a remainder of 7

Put the 5 on top of the division bar, to the right of the 1. Multiply 5 by 32 and write the answer under 167.

Draw a line and subtract 160 from 167.

Since 7 is less than 32 your long division is done. You have your answer: The quotient is 15 and the remainder is 7.

So, 487 ÷ 32 = 15 with a remainder of 7

For longer dividends, you would continue repeating the division and multiplication steps until you bring down every digit from the divdend and solve the problem.

Math is Fun also provides a step-by-step process for long division with Long Division with Remainders.

Cite this content, page or calculator as:

Like in the case of multiplication, division is usually done in the form of either SHORT division, or LONG division.

This section shows examples of how to divide numbers by performing short division.

Examples of Long Division can be seen here .

## How to do Short Division

If we had a sum where we have to divide 484 by 4:

484 is called the ” DIVIDEND “.

4 is called the ” DIVISOR “.

The eventual answer to the division sum, is called the ” QUOTIENT “.

Generally, a division sum is laid out in the following way:
QUOTIENT
DIVISOR DIVIDEND

So for 484 &div 4 we start out with:

Then carry out the division in stages from left to right, to find the final answer, the “QUOTIENT”.

Firstly, &nbsp4 goes into &nbsp4 only once with no remainder, so a 1 goes above the 4 in the “QUOTIENT” section.

1
4 484

Now, &nbsp4 goes into 8 twice with no remainder, so next a 2 goes above the 8.

12
4 484

Lastly &nbsp4 again goes into &nbsp4 only once, so a 1 goes above the final 4.

121
4 484

And we now have the answer.
484 &div 4 = 121

## How to Divide Numbers,Short Division Examples

Sometimes division sums don’t divide exactly, and there are remainders, like in this example.

5 667

5 goes into &nbsp6 one time, with a remainder of &nbsp1.

1 goes above, and the remainder 1, &nbspgets moved to the next number on the right, which here is 6 in the Tens column.

The effect of the &nbsp1 is that it turns &nbsp6 into &nbsp16.

1
5 6 1 67

Now &nbsp5 goes into &nbsp16 three times with remainder &nbsp1.

3 goes above, and the remainder 1 again moves to the right and turns 7 into 17.

1 3
5 6 1 6 1 7

Lastly &nbsp5 goes into &nbsp17 three times, with remainder &nbsp2.

As there is now no more numbers to do division with, our final answer simply has a remainder of 2.

1 3 3 r 2
5 6 1 6 1 7

667 &div 5 = 133 r2

6 156

First, &nbsp6 does not go into &nbsp1, so a 0 goes above the 1.

0
6 156

Now, we treat the &nbsp1 and the &nbsp5 together in the dividend, as &nbsp15.

In which case, &nbsp6 goes into &nbsp15 two times, with remainder &nbsp3.
So 2 goes above, and the remainder 3 moves right to make 36.

0 2
6 1 5 3 6

Finally 6 goes into 36 exactly six times, completing the division.

0 2 6
6 1 5 3 6

156 &div 6 = 26

1. Home ›
2. Arithmetic Math › How to Divide Numbers

Synthetic Division can be used when dividing a polynomial by a linear expression, a linear expression with leading coefficient 1.

For example, an expression such as x + 1 or x2 .

If you had an expression such as 3 x + 2, where the leading coefficent is 3 .

This can be divided by 3, to convert into suitable form, x + \bf<\frac<2><3>> .

## Synthetic Division Steps

Here we’ll go through the Synthetic Division steps with the aid of an example.

Dividing the polynomial 2 x 3 − 5 x 2 − 7 x + 9

by x + 4 .

1)
Draw up half a box, with only the left side and bottom side.

Then write only the coefficients and constant of the polynomial along the top.

So for 2 x 3 − 5 x 2 − 7 x + 9 , we write 2 , –5 , –7 , 9.

Set the divisor equal to 0, and solve for x .

x + 4 = 0 , x = –4

Place this number on the left.

At the same time, bring the leading coefficient down, and place underneath the box.

Now multiply this first coefficient by the number on the left.

Here this is –4 × 2, which results in &nbsp-8.

This resulting number then gets placed on the bottom of the next column.

The next step is to add the two numbers in this next column together, writing the result underneath the box.

Here this is – 5 + – 8 , which results in &nbsp- 13 .

Now again repeat steps 3) &nbspand 4) &nbspfor each following column, up to the final one.

The row underneath the box now gives the solution to the polynomial division sum.

The last number on the very right is the remainder, here –171.

With all the numbers to the left making up the quotient.
The variables in the quotient start one power less than the original polynomial.

So here, x 2 &nbspinstead of x 3 .

Resulting in a quotient part of 2 x 2 − 13 x + 45.

Now when writing out the complete answer, the remainder is written on top of the original divisor, which was x + 4.

=> 2 x 2 − 13 x + 45 − \boldsymbol<\frac<171>>

Though the answer can be sometimes written slightly differently, where the solution is multiplied through by the divisor, so it can look like:

( x + 4)(2 x 2 − 13 x + 45)171

Divide 2 x 3 − 11 x 2 + 17 x6

by x1 .

x1 = 0 => x = 1

=> 2 x 2 − 9 x + 8 − \boldsymbol<\frac<2>>

## Remainder Theorem

Something that should be mentioned at this point is what’s called the Remainder Theorem.

So for (1.1) above, h was 1, and the remainder was 2.

So f (1) should be 2.

f (1) = 2(1) 3 − 11(1) 2 + 17(1) − 6

= 211 + 176 = 2

This can be a handy way of double checking that you have done each step correctly when doing Synthetic Division steps.

Divide 3 x 3 + 2 x 2 + 5

by x + 2 .

x + 2 = 0 => x = –2

=> 3 x 2 − 4 x + 8 − \boldsymbol<\frac<11>>

Divide 4 x 3 + 6 x 2 + 2 x + 1

by 2 x 1.

Before proceeding with the correct synthetic division steps, we need to change the 2 in front of the x to a 1, so we just have an x at the head of the divisor.

Now this will be the divisor.

x − \bf<\frac<1><2>> = 0 => x = \bf<\frac<1><2>>

The divisor that’s written under the remainder though is the changed divisor, x − \bf<\frac<1><2>> .

=> 4 x 2 + 8 x + 6 + \boldsymbol<\frac<4><2>>>

Find the value of h if x 3 + h x 2 − x + 2

divided by x2 has remainder 16.

x2 = 0 => x = 2

4 h + 8 = 16

h + 2 = 4

1. Home ›
2. Polynomials › Synthetic Division Steps

I have trouble with integer division in Dart as it gives me error: ‘Breaking on exception: type ‘double’ is not a subtype of type ‘int’ of ‘c’.’

Here’s the following code:

As you see, I was expecting that the result should be 2, or say, even if division of ‘a’ or ‘b’ would have a result of a float/double value, it should be converted directly to integer value, instead of throwing error like that.

I have a workaround by using .round()/.ceil()/.floor(), but this won’t suffice as in my program, this little operation is critical as it is called thousands of times in one game update (or you can say in requestAnimationFrame).

I have not found any other solution to this yet, any idea? Thanks.

Dart version: 1.0.0_r30798

That is because Dart uses double to represent all numbers in dart2js . You can get interesting results, if you play with that:

Actually, it is recommended to use type num when it comes to numbers, unless you have strong reasons to make it int (in for loop, for example). If you want to keep using int , use truncating division like this:

Otherwise, I would recommend to utilize num type.

/ helped though, thanks!

Integer division is

you could also use

if you want to define how fractions should be handled.

According to the docs, int are numbers without a decimal point, while double are numbers with a decimal point.

Both double and int are subtypes of num .

When two integers are divided using the / operator, the result is evaluated into a double . And the c variable was initialized as an integer. There are at least two things you can do:

/ operator returns an int .

1. Use var c; . This creates a dynamic variable that can be assigned to any type, including a double and int and String etc.

## Truncating division operator

/ to get an integer result from a division operation:

## Division operator

The regular division operator / will always return a double value at runtime (see the docs):

### Runtime versus compile time

You might have noticed that I wrote a loop for the second example (for the regular division operator) instead of 4 / 2 .

The reason for this is the following:
When an expression can be evaluated at compile time, it will be simplified at that stage and also be typed accordingly. The compiler would simply convert 4 / 2 to 2 at compile time, which is then obviously an int . The loop prevents the compiler from evaluating the expression.

As long as your division happens at runtime (i.e. with variables that cannot be predicted at compile time), the return types of the / ( double ) and

/ ( int ) operators will be the types you will see for your expressions at runtime.

See this fun example for further reference.

## Conclusion

Generally speaking, the regular division operator / always returns a double value and truncate divide can be used to get an int result instead.

Compiler optimization might, however, cause some funky results 🙂

How can I implement division using bit-wise operators (not just division by powers of 2)?

Describe it in detail.

The standard way to do division is by implementing binary long-division. This involves subtraction, so as long as you don’t discount this as not a bit-wise operation, then this is what you should do. (Note that you can of course implement subtraction, very tediously, using bitwise logical operations.)

In essence, if you’re doing Q = N/D :

1. Align the most-significant ones of N and D .
2. Compute t = (N – D); .
3. If (t >= 0) , then set the least significant bit of Q to 1, and set N = t .
4. Left-shift N by 1.
5. Left-shift Q by 1.
6. Go to step 2.

Loop for as many output bits (including fractional) as you require, then apply a final shift to undo what you did in Step 1.

Division of two numbers using bitwise operators.

I assume we are discussing division of integers.

Consider that I got two number 1502 and 30, and I wanted to calculate 1502/30. This is how we do this:

First we align 30 with 1501 at its most significant figure; 30 becomes 3000. And compare 1501 with 3000, 1501 contains 0 of 3000. Then we compare 1501 with 300, it contains 5 of 300, then compare (1501-5*300) with 30. At so at last we got 5*(10^1) = 50 as the result of this division.

Now convert both 1501 and 30 into binary digits. Then instead of multiplying 30 with (10^x) to align it with 1501, we multiplying (30) in 2 base with 2^n to align. And 2^n can be converted into left shift n positions.

Here is the code:

Didn’t test it, but you get the idea.

This solution works perfectly.

Implement division without divison operator: You will need to include subtraction. But then it is just like you do it by hand (only in the basis of 2). The appended code provides a short function that does exactly this.

The below method is the implementation of binary divide considering both numbers are positive. If subtraction is a concern we can implement that as well using binary operators.

With the usual caveats about C’s behaviour with shifts, this ought to work for unsigned quantities regardless of the native size of an int.

This is my solution to implement division with only bitwise operations:

Unsigned Long Division (JavaScript) – based on Wikipedia article: https://en.wikipedia.org/wiki/Division_algorithm: “Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder. When used with a binary radix, this method forms the basis for the (unsigned) integer division with remainder algorithm below.”

Function divideWithoutDivision at the end wraps it to allow negative operands. I used it to solve leetcode problem “Product of Array Except Self”

I don’t really understand how modulus division works. I was calculating 27 % 16 and wound up with 11 and I don’t understand why.

I can’t seem to find an explanation in layman’s terms online. Can someone elaborate on a very high level as to what’s going on here?

Most explanations miss one important step, let’s fill the gap using another example.

Given the following:

The modulus function looks like this:

Let’s determine why this is.

First, perform integer division, which is similar to normal division, except any fractional number (a.k.a. remainder) is discarded:

Then, multiply the result of the above division ( 2 ) with our divisor ( 6 ):

Finally, subtract the result of the above multiplication ( 12 ) from our dividend ( 16 ):

The result of this subtraction, 4 , the remainder, is the same result of our modulus above!

The result of a modulo division is the remainder of an integer division of the given numbers.

The simple formula for calculating modulus is :-

Note:

All calculations are with integers. In case of a decimal quotient, the part after the decimal is to be ignored/truncated.

eg: 27/16= 1.6875 is to be taken as just 1 in the above mentioned formula. 0.6875 is ignored.

Compilers of computer languages treat an integer with decimal part the same way (by truncating after the decimal) as well

Maybe the example with an clock could help you understand the modulo.

A familiar use of modular arithmetic is its use in the 12-hour clock, in which the day is divided into two 12 hour periods.

Lets say we have currently this time: 15:00
But you could also say it is 3 pm

This is exactly what modulo does:

You find this example better explained on wikipedia: Wikipedia Modulo Article

The modulus operator takes a division statement and returns whatever is left over from that calculation, the “remaining” data, so to speak, such as 13 / 5 = 2. Which means, there is 3 left over, or remaining from that calculation. Why? because 2 * 5 = 10. Thus, 13 – 10 = 3.

The modulus operator does all that calculation for you, 13 % 5 = 3.

modulus division is simply this : divide two numbers and return the remainder only

27 / 16 = 1 with 11 left over, therefore 27 % 16 = 11

ditto 43 / 16 = 2 with 11 left over so 43 % 16 = 11 too

Very simple: a % b is defined as the remainder of the division of a by b .

See the wikipedia article for more examples.

I would like to add one more thing:

it’s easy to calculate modulo when dividend is greater/larger than divisor

dividend = 5 divisor = 3

but what if divisor is smaller than dividend

dividend = 3 divisor = 5

This is because, since 5 cannot divide 3 directly, modulo will be what dividend is

I hope these simple steps will help:

1. 20 / 3 = 6 ; do not include the .6667 – just ignore it
2. 3 * 6 = 18
3. 20 – 18 = 2 , which is the remainder of the modulo

You can interpret it this way:

16 goes 1 time into 27 before passing it.

So you could say that 16 goes one time in 27 with a remainder of 11.

An other exemple:

Well 3 goes 6 times into 20 before passing it.

To add-up to 20 we need 2 so the remainder of the modulus expression is 2.

Easier when your number after the decimal (0.xxx) is short. Then all you need to do is multiply that number with the number after the division.

You do 32/12=2.666666667 Then you throw the 2 away, and focus on the 0.666666667 0.666666667*12=8

The only important thing to understand is that modulus (denoted here by % like in C) is defined through the Euclidean division.

For any two (d, q) integers the following is always true:

As you can see the value of d%q depends on the value of d/q . Generally for positive integers d/q is truncated toward zero, for instance 5/2 gives 2, hence:

However for negative integers the situation is less clear and depends on the language and/or the standard. For instance -5/2 can return -2 (truncated toward zero as before) but can also returns -3 (with another language).

In the first case:

but in the second one:

As said before, just remember the invariant, which is the Euclidean division.

Have you ever wished you could divide numbers quickly and easily in your head? Believe it or not, you can! Over the next two weeks, we’ll be learning my top 5 tips to help you become a mental division maestro.

If you’re anything like me, you don’t exactly love doing long division. Which is exactly why I avoid it as much as I can. Of course, one way to avoid doing division the old fashioned way with paper and pen is by using a calculator. Most of the time, that’s exactly what I do.

But the truth is that sometimes calculators—or phones with calculators—are inconvenient. And sometimes you need to do division right there on the spot in your head. How can you do it? Keep on reading to learn 5 simple things that you can do to take your mental division skills to the next level..

## Tip #1: Approximate If You Can

The first thing you can do to speed up a lot of the mental division problems you’ll encounter is to stop and think about just how accurate you need the answer to be. Sometimes you need an exact answer, or perhaps an answer that’s accurate to two decimal places, or three decimal places, or something else specific like that. But a lot of the time you really just need a ballpark estimate.

If you only need an approximate answer, don’t waste time figuring out the exact answer…make a quick and dirty estimate instead.

In those cases where you only need an approximate answer, don’t waste your time by figuring out the exact answer. Instead, make a quick and dirty estimate. How? Well, let’s say you work in a coffee shop and you want to figure out the average amount spent by your customers. So far you’ve collected $164 from 26 people. What’s the average—or more technically the mean—bill? Well,$164 is pretty close to $150, and 26 people is pretty close to 25 people. So instead of calculating$164 / 26 people, let’s start by calculating something that’s close to that: $150 / 25 people. That’s a much easier problem to solve! It says that the answer is roughly$6 / person.

No, this technique doesn’t give exact values, but if you estimate wisely it’ll give you answers that are pretty close…and with a fraction of the work.

## Tip #2: Simplify Before You Start

The rest of today’s tips are all things that you can do when an approximate answer just isn’t good enough. When faced with such problems, the first thing you should do is.

In response to the last article on modular arithmetic , math fan Jeff left a comment on the Math Dude’s Facebook page saying that his daughter had been given the problem of figuring out what day of the week it would be a certain number number of days from today. After starting to count off all the days one by one, Jeff introduced her to modular arithmetic …and she was very excited since it made the problem much easier to solve. But what exactly about modular arithmetic made this so much easier? Well, keep on reading because today we’re going to begin to figure this out.

## Recap: What is Modular Arithmetic?

Before we answer this question, let’s take a few minutes to finish off the introduction to modular arithmetic that we began in the last article. As you’ll recall, modular arithmetic is a form of arithmetic for integers in which the number line that we count on is wrapped around into a circle whose length is given by a number called the modulus. For example, in arithmetic modulo 12, like what we have when adding numbers on a normal 12-hour clock, a problem like 10 + 5 (mod 12) has the answer 3 (and not 15) since once we count up to 12 we start over at 1 again.

Okay, that’s how addition works in modular arithmetic. But there’s a lot more to arithmetic than just addition, so let’s now take a look at how the other arithmetic operations work in modular arithmetic too.

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Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.

What is division?

Division is breaking a number up into an equal number of parts.

20 divided by 4 = ?

If you take 20 things and put them into four equal sized groups, there will be 5 things in each group. The answer is 5.

20 divided by 4 = 5.

Signs for Division

There are a number of signs that people may use to indicate division. The most common one is ÷, but the backslash / is also used. Sometimes people will write one number on top of another with a line between them. This is also called a fraction.

Example signs for “a divided by b”:

Dividend, Divisor, and Quotient

Each part of a division equation has a name. The three main names are the dividend, the divisor, and the quotient.

• Dividend – The dividend is the number you are dividing up
• Divisor – The divisor is the number you are dividing by
• Quotient – The quotient is the answer

Dividend ÷ Divisor = Quotient

In the problem 20 ÷ 4 = 5

Dividend = 20
Divisor = 4
Quotient = 5

Special Cases

There are three special cases to consider when dividing.

1) Dividing by 1: When dividing something by 1, the answer is the original number. In other words, if the divisor is 1 then the quotient equals the dividend.

20 ÷ 1 = 20
14.7 ÷ 1 = 14.7

2) Dividing by 0: You cannot divide a number by 0. The answer to this question is undefined.

3) Dividend equals Divisor: If the dividend and the divisor are the same number (and not 0), then the answer is always 1.

20 ÷ 20 = 1
14.7 ÷ 14.7 = 1

If the answer to a division problem is not a whole number, the “leftovers” are called the remainder.

For example, if you were to try and divide 20 by 3 you would discover that 3 does not divide evenly into 20. The closest numbers to 20 that 3 can divide into are 18 and 21. You pick the closest number that 3 divides into that is smaller than 20. That is 18.

18 divided by 3 = 6, but there are still some leftovers. 20 -18 = 2. There are 2 remaining.

We write the remainder after an “r” in the answer.

12 ÷ 5 = 2 r 2
23 ÷ 4 = 5 r 3
18 ÷ 7 = 2 r 4

Division is the Opposite of Multiplication

Another way to think of division is as the opposite of multiplication. Taking the first example on this page:

You can do the reverse, replacing the = with a x sign and the ÷ with an equal sign:

12 ÷ 4 = 3
3 x 4 = 12

21 ÷ 3 = 7
7 x 3 = 21

Using multiplication is a great way to check your division work and get better scores on your math tests!

Long division is often considered one of the most challenging topics to teach. Luckily, there are strategies that we can teach to make multi-digit division easier to understand and perform.

The Box Method, also referred to as the Area Model, is one of these strategies. It is a mental math based approach that will enhance number sense understanding. Students solve the equation by subtracting multiples until they get down to 0, or as close to 0 as possible.

If you plan on teaching the partial quotients strategy in your classroom (which I highly recommend) the Box Method is a great way to get started. It uses the same steps as partial quotients, but is organized a bit differently.

Let’s learn how to perform the Box Method/Area Model for long division!

Below, I have included both a video tutorial and step-by-step instructions.

VIDEO TUTORIAL

STEP-BY-STEP INSTRUCTIONS

Suppose that we want to solve the equation 324÷2.

Step 1:
First we draw a box. We write the dividend inside the box, and the divisor on the left side.

Step 2:
We want to figure out how many groups of 2 can be made from 324. We will do this in parts to make it easier. We could start by making 100 groups of 2, since we know that we have at least this many groups. So we multiply 100×2 to make 200, and then take that 200 away from 324. Now we have 124 left.

Step 3:
We make another box and carry the 124 over to it. Now let’s take away another easy multiply of 2. How about 50 groups of 2? We know that we can take out another 50 groups of 2 from 124. 50×2=100, so we take 100 from 124. Now we have 24 left.

Step 4:
We make another box and carry the 24 over to it. We know that 12 groups of 2 makes 24, so let’s write a 12 on top and take away 24 from the 24. Now we end up with 0, so we know that we are finished our equation.

Step 5:
Now we add the “parts” from the top of the boxes to find our quotient. 100+50+12=162, so we know that 324÷2=162.

ONE MORE EXAMPLE (WITH A REMAINDER)

Let’s take a look at one more example. In this example, we will solve 453÷4.

1. First we wrote our dividend inside the box, and our divisor on the left side.
2. We took out 100 groups of 4 first. This made 400. We subtracted 400 from 453 and were left with 53.
3. We carried the 53 to the next box, and then took out another 10 groups of 4 to make 40. We took the 40 away from the 53 and were left with 13.
4. We carried the 13 over to the next box, and then took out 3 groups of 4 to make 12. We took the 12 away from the 13 and were left with 1.
5. We cannot take any more groups of 4 out, so our remainder is 1. To find our final quotient, we add 100+10+3+remainder 1 to make 113 R1.

HELPFUL RESOURCES FOR THE BOX STRATEGY/AREA MODEL FOR DIVISION

I would love to help you teach the box strategy for long division in your classroom. You may find the following resources helpful:

FREE MINI COURSE

Register here for free Multi-Digit Multiplication and Division Mini Course. It will only take you about an hour to complete and you’ll leave with tons of new ideas, a strategic plan of action, free resources, a PD certificate, and more!

These task cards give students the opportunity to practice the box method/area model for long division in a variety of different ways. Students will calculate quotients, solve division problems, figure out missing dividends and divisors, think about how to efficiently solve an equation using the box method, and more. See the Box Method Task Cards HERE or the Big Bundle of Long Division Task Cards HERE.

THE LONG DIVISION STATION

The Long Division Station is a self-paced, student-centered math station for long division. Students gradually learn a variety of strategies for long division, the box method being one of them. One of the greatest advantages to this Math Station is that is allows you to target every student and their unique abilities so that everyone is appropriately challenged. See The Long Division Station HERE.

is equivalent to .

## Basic Examples (7)

Enter in 2D form using :

Divide reduces fractions to lowest terms:

Force a numerical result by including a decimal point in the input:

/ is applied sequentially:

is always converted to products and powers:

Enter ÷ using div :

## Applications (2)

Successive ratios in a list:

## Properties & Relations (3)

Cancel common factors in rational expressions:

Split into partial fractions:

## Possible Issues (2)

Pattern matching relies on FullForm :

## Neat Examples (2)

Array of possible fractions:

Integers that divide exactly:

• Arithmetic ▪
• Operators
• Wolfram Language Syntax ▪
• Arithmetic Functions
• An Elementary Introduction to the Wolfram Language: Introducing Functions
• NKS|Online (A New Kind of Science)

Introduced in 1988 (1.0) | Updated in 1996 (3.0)

Today, we’re going to learn how to divide with 3-digit numbers. Before we start, we’re going to do a quick review on the parts of a division problem. Do you remember what a dividend is? And what about the remainder? I’ve prepared a diagram for you to help jog your memory.

Good, now we’re going to explain how to solve a 3-digit division problem in 5 simple steps.

#### Step #1. Observe how many digits the divisor has.

In our example, the divisor has 3 digits.

#### Step #3. Compare the 3 digits from the dividend and divisor.

• If the dividend’s number is greater , we can begin dividing. In our example, we see that 389 is greater than 125, so let’s get started!
• If the dividend’s number is less than the divisor, we must use another digit from the dividend. We’ll look at an example of this situation at a different time.

#### Step #4. Divide the dividend’s numbers by the divisor’s numbers.

Divide the dividend’s first number (which is 3 in our example) by the divisor’s first number (1). 3 divided by 1 is 3. Then, we multiply our divisor (125) by 3 and see that it fits in (in other words, that it’s less than) the dividend’s 3 numbers. So, we place the result underneath the dividend’s 3 digits and subtract.

#### Step #5. Bring down the dividend’s next number and divide as you did in the last step until there are no more numbers left.

After bringing down the 2, now we need to divide 142 by 125. The first digit of each number is 1, so we go with 1 and write 1 in the quotient. Then, we multiply 125 x 1 and get 125; 125 fits in 142, so we can move on to subtract it.

Now we don’t have any more numbers left from the dividend to bring down. And just like that, we divided 3892 by 125.

The answer is 31, which is the quotient, and we have a remainder of 17.

Quickly, we’re going to check our answer. Do you know how to check a division problem?

Dividend = divisor x quotient + remainder

Let’s put it to the test. What is 125 x 31 + 17? 3892. That’s just the same value as the dividend.

Checked and successful!

Be careful! If we don’t get the same answer when you check the problem, we have to retrace our steps because it means that something went wrong in our calculations.

I hope this helped you learn how to divide with 3-digit numbers.

If you want to learn lots of math, try out Smartick free.

Today we’re going to look at another, more complicated, example. Let’s start and divide by 3 digit numbers!

### 3. We compare the 3 digits in the dividend with the 3 digits in the divisor.

Since 385 is greater than 125 we can start dividing.

### 4. We divide the first digits of the dividend and the divisor.

3 divided by 1 is 3. We need to multiply 125 by 3 and see if it goes into 385.

We get 375 so we know it fits. We put the 3 in the quotient.

### 5. We bring down the next digit of the dividend.

We’ve brought down the 3 but now 125 doesn’t go into 103. So, how can we continue?

When this happens, we have to add a 0 to the quotient and bring down the next number in the dividend.

Now we can keep dividing.

First we divide 12 by 1 to see what number to put in the quotient. 12 divided by 1 is 12, and since it is greater than 10, we keep 9, the greatest one-digit number.

And 1125 doesn’t go into 1035.

Let’s try it with the next smallest number, 8.

That’s it!

We’ve finished the division.

38,535 divided by 125 gives us a quotient of 308 with a remainder of 35.

The only thing left to do, as always, is to check our work:

divisor x quotient + remainder = dividend

125 x 308+ 35 = 38,535

Now we know how to divide by 3 digits. If you want to keep learning more primary mathematics, register with Smartick and try it for free.

The division is the process of splitting a group of objects into equal parts. It is one of the basic operations of arithmetic. Subtraction is also an arithmetic operation. The repeated subtraction is a method of solving the division. Here, we will discuss the repeated subtraction meaning, how to divide two numbers using repeated subtraction and example questions in the following sections.

## What is Repeated Subtraction?

Subtraction is a process used to find the difference between any two numbers. Repeated subtraction means finding the difference between the numbers continuously. The division problems can be solved using multiplication or long division or short division or repeated subtraction. The parts of division are dividend, divisor, quotient and remainder. Get the detailed process for Divide by Repeated Subtraction in the below segments of this page.

### How to Divide Using Repeated Subtraction?

Follow the steps that are listed to find the division with the repeated subtraction process.

• Get dividend (the number which is divided by the divisor) and divisor (number by which divided is to be divided).
• Subtract divisor from the dividend.
• Again subtract the divisor from the obtained difference.
• Repeat the subtraction process until you get a number that is less than the divisor.
• The number of times the subtraction process is completed is called the quotient.
• The number that is left at the end of subtraction is called the remainder.

### Divide using Repeated Subtraction Method Examples

Example 1:
Solve 81 ÷ 9.
Solution:
Given that,
Divisor = 81
Dividend = 9
Subtract 9 from 81 repeatedly.
81 – 9 = 72
72 – 9 = 63
63 – 9 = 54
54 – 9 = 45
45 – 9 = 36
36 – 9 = 27
27 – 9 = 18
18 – 9 = 9
9 – 9 = 0
Here, 81 is subtracted 9 times from the number 9 and get the remainder 0.
Hence, 81 ÷ 9 = 9, 9 is the quotient.

Example 2:
Divide 120 ÷ 15.
Solution:
Given that,
Dividend = 120
Divisor = 15
Subtract 15 from 120 repeatedly.
120 – 15 = 105
105 – 15 = 90
90 – 15 = 75
75 – 15 = 60
60 – 15 = 45
45 – 15 = 30
30 – 15 = 15
15 – 15 = 0
Here, 120 is subtracted 8 times from the number 15 and get the remainder 0.
Hence, 120 ÷ 15 = 8, 8 is the quotient.

Example 3:
Divide 20 ÷ 4.
Solution:
Given that,
Dividend = 20
Divisor = 4
Subtract 4 from 20 repeatedly.
20 – 4 = 16
16 – 4 = 12
12 – 4 = 8
8 – 4 = 4
4 – 4 = 0
Here, 20 is subtracted 5 times from the number 4 and get the remainder 0.
Hence, 20 ÷ 4 = 5, 5 is the quotient.

### FAQs on Division Using Repeated Subtraction

1. How do you divide with subtraction?
Subtraction is one of the arithmetic operations that is used to calculate the difference between two numbers. To divide the numbers, we can use the repeated subtraction method. Repeated subtraction is nothing but subtract divisor from the dividend repeatedly. The number of times subtraction done is called the quotient.

2. Explain dividend, divisor, quotient and remainder?
The dividend is the number that is o be divided and a divisor is a number which is dividend is to be divided. The output of the division process is the quotient. The number leftover after division is the remainder.
The division formula is Dividend ÷ divisor = quotient + remainder

3. What is a repeated subtraction example?
The repeated subtraction means performing the subtraction operation continuously. The example is divided 48 persons into 8 groups. Subtract 8 from 48.
48 – 8 = 40 – 8 = 32 – 8 = 24 – 8 = 16 – 8 = 8 – 8 = 0. Here, the subtraction is done 6 times. So, the each group has 6 members.

In this post, you are going to learn how to do 3-digit division. Before beginning to divide, it is important that you know the multiplication tables (1×1 through 9×9) because you need them to solve division.

Once you know the multiplication tables, you can begin to do 3-digit division. The steps that you must follow are:

1. Since the divisor has 3 digits, we must start with the first 3 digits of the dividend.
2. We compare the 3 digits of the dividend with the 3 digits of the divisor:
• If the number of the 3 digits of the dividend is greater than the number of the divisor, you can begin to divide.
• If the number of the 3 digits of the dividend is less than the number of the divisor, you have to start with the first 4 digits in the dividend.

Since 459 is greater than 438, you can begin to do the division

3. Divide the digits of the dividend by the digits of the divisor.

In order to divide 459 by 438, we take the first digit of each number and divide them: 4 ÷ 4 = 1

Write the 1 in the quotient and multiply it by the divisor: 438 x 1 = 438.

Now subtract: 459 – 438 = 21

4. Now bring down the next digit from the dividend and repeat the process.

Divide 219 by 438. Since 438 goes into 219 zero times, put a 0 in the quotient and bring down another digit from the dividend .

Now divide 2190 by 438, which equals 5.

The division ends when there are no more digits in the dividend to bring down.

Write 5 in the quotient and multiply by the divisor: 438 x 5 = 2190

Subtract: 2190 – 2190 = 0

Since we do not have more digits to bring down, we are finished with the division:

45990 ÷ 438 = 105; remainder = 0

This has been an example of 3-digit division, but you can find more examples and a more detailed explanation in this previous post:

If you liked this post, share it with your friends and colleagues so that they can also learn.

How does division occur inside digital computers? What is the algorithm for it?

I have searched hard in google but haven’t got satisfactory results. Please provide a very clear algorithm/flowchart for division algorithm with a sample illustration.

Division algorithms in digital designs can be divided into two main categories. Slow division and fast division.

I suggest you read up on how binary addition and subtraction work if you are not yet familiar with these concepts.

Slow Division

The simplest slow methods all work in the following way: Subtract the denominator from the numerator. Do this recursively with the result of each subtraction until the remainder is less than the denominator. The amount of iterations is the integer quotient, and the amount left over is the remainder.

1. $$7-3=4$$
2. $$4-3=1$$
3. $$1 Thus the answer is 2 with a remainder of 1. To make this answer a bit more relevant, here is some background. Binary subtraction via addition of the negative is performed e.g.: 7 – 3 = 7 + (-3). This is accomplished by using its two’s complement. Each binary number is added using a series of full adders: Where each 1-bit full adder gets implemented as follows: Fast Division While the slower method of division is easy to understand, it requires repetitive iterations. There exist various “fast” algorithms, but they all rely on estimation. Consider the Goldschmidt method: I’ll make use of the following:$$Q = \frac$$This method works as follows: 1. Multiply N and D with a fraction F in such a way that D approaches 1. 2. As D approaches 1, N approaches Q This method uses binary multiplication via iterative addition, which is also used in modern AMD CPUs. Hardware for floating point division is part of a logic unit that also does multiplication; there is a multiplier hardware module available. Floating point numbers, say A and B, are divided (forming A/B) by 1. decomposing the floating point numbers into sign (+1 or -1), mantissa (“a” and “b”, and (binary integer type) exponents 2. the sign of the result is (+1) iff both signs are the same, else (-1) 3. exponents are subtracted (exponent of B subtracted from exponent of A) to form the exponent of the result mantissas (the binary digits of the numbers) are a fixed-point binary number between 1/2 and 1; that means that the first digit after the binary point is ‘1’, followed by zeroes and ones. as a first step, a lookup table finds the reciprocal accurate to six bits (there are only 32 possibilities, it’s a small table) to begin to compute a/b, do two multiplications$$ = <\over b * reciprocal(b)> $$and note that six-bit accuracy implies that the denominator of the result is very near 1 (to five or more binary places). • Now note that for very-near-one denominators, ‘d’, we can see that defining$$ d == 1 +\epsilon  d * (2-d) = ( 1+ \epsilon) \times (1-\epsilon) = 1 – \epsilon ^2  This implies that our five-bit accurate ‘one’ in the denominator will become ten-bit accurate after one more pair of multiplications, twenty-bit accurate after two, and forty-bit accurate after three. Do as many iterations of multiplying numerator and denominator by (2 – denominator) as your result precision requires.
• The numerator, now that the denominator is exactly ‘1’, is the mantissa of the result, and can be combined with the previously computed sign and exponent.
• IEEE floating point allows some exceptions (denormalized numbers, NAN; those have to be handled by other logical operations.
• Interestingly, the old Pentium divide bug (very newsworthy in 1994) was caused by a printing error that made faulty reciprocal-table values for step (4). An early paper, “A Division Method Using a Parallel Multplier”, Domenico Ferrari, IEEE Trans. Electron. Comput. EC-16/224-228 (1967), describes the method, as does “The IBM System/360 Model 91: Floating-Point Execution Unit” IBM J. Res. Dev. 11: 34-53 (1967).

The Numpy divide function is a part of numpy arithmetic operations. There are basic arithmetic operators available in the numpy module, which are add, subtract, multiply, and divide. The significance of the python divide is equivalent to the division operation in mathematics.

## What does Numpy Divide Function do?

The numpy divide function calculates the division between the two arrays. It calculates the division between the two arrays, say a1 and a2, element-wise. The numpy.divide() is a universal function, i.e., supports several parameters that allow you to optimize its work depending on the specifics of the algorithm.

## Parameters of Numpy Divide

 Parameter Mandatory or Not a1 Mandatory a2 Mandatory / Not-Mandatory out Not-Mandatory * Not-Mandatory where Not-Mandatory casting Not-Mandatory order Not-Mandatory dtype Not-Mandatory subok Not-Mandatory unfunc Not-Mandatory
• a1: [arrayLike]
1st Input array for calculating the division.
• a2: [arrayLike]
2nd input array for calculating the division.
• out: [ndarray, None, or tuple of ndarray and None, optional]
out will be the location where the result is to be stored. have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned.
• where: [array_like, optional]
If the value of where is true, it indicates to calculate the unfunc at that position, whereas if the value is false, then it denotes to leave the value in output only.

## Return Value of Numpy Divide

The divide function returns the division between a1 and a2. The divide() function can be scalar of nd-array. It depends on the a1 and a2. If a1 and a2 are scalar, than numpy.divide() will return a scalar value. Else it will return an nd-array.

Note: The input a1 and a2 must be broadcastable to a common shape (which becomes the shape of the output).

## Examples of Numpy Divide Function

Let’s go through the examples of Numpy divide() function and see how it works.

Output:

### Explanation

In this simple first example, we just divided two numbers and get the result. Let’s take a look at each step and know what happens in each stage. First of all, we imported the numpy module as np it’s obvious because we are working on the numpy library. After that, we have taken two pre-defined inputs ’24’, ’13’, and stored them in variables ‘a1’, ‘a2’ respectively. We printed our inputs to check whether they are specified properly or not. Then the main part comes where we will find the division between the two numbers.

Herewith the help of the np.divide() function, we will calculate the division between a1 and a2. This division operation is identical to what we do in mathematics.

So, we will get the division between the number 24 and 13 which is 11.

Output:

### Explanation

From this example, things get Lil bit tricky; instead of numbers, we have used arrays as our input value.
We can now see we have two input arrays a1 & a2 with array inputs [20, 21, 5, 9] and [13, 17, 6, 11], respectively. The divide() function will find the division between a1 & a2 array arguments, element-wise.

So, the solution will be an array with the shape equal to input arrays a1 and a2. The division between a1 and a2 will be calculated parallelly, and the result will be stored in the ad variable.

Output:

### Explanation

The third example in this divide() function tutorial is slightly similar to the second example which we have already gone through. What we have done here in this example is instead of a simple array we have used a multi-dimensional array in both of our input values a1 and a2.

Make sure both the input arrays should be of the same dimension and same shapes. The numpy.divide() function will find the Division between array arguments, element-wise.

## Can We Find Division Between Two Numpy Arrays With Different Shapes?

In simple words, No, we can’t find division or use the numpy divide function in two numpy arrays that have different shapes.

Let’s look it through one example,

Output:

### Explanation

If the shape of two numpy arrays will be different than we will get a value error. The value error will say something like for example.

Here in this example, we get a value error because the a2 input array has a different shape than the a1 input array. In order to get the division without any value error, make sure to check the shape of arrays.

## What’s Next?

NumPy is very powerful and incredibly essential for information science in Python. That being true, if you are interested in data science in Python, you really ought to find out more about Python.

You might like our following tutorials on numpy.

## Conclusion

The numpy divide() is a compelling and essential function available in the numpy module, which can be very useful and highly recommended by many experts while finding the division between very large data sets.

If you still have any questions regarding the NumPy divide function. Leave your question in the comments below.

Happy Pythonning!

In this article I explain how to teach long division in several steps. Instead of showing the whole algorithm to the students at once, we truly take it “step by step”.

Before a child is ready to learn long division, he/she has to know:

• multiplication tables (at least fairly well)
• basic division concept, based on multiplication tables
(for example 28 ÷ 7 or 56 ÷ 8)
• basic division with remainders (for example 54 ÷ 7 or 23 ÷ 5)

## One reason why long division is difficult

Long division is an algorithm that repeats the basic steps of
1) Divide; 2) Multiply; 3) Subtract; 4) Drop down the next digit.

Of these steps, #2 and #3 can become difficult and confusing to students because they don’t seemingly have to do with division —they have to do with finding the remainder. In fact, to point that out, I like to combine them into a single “multiply & subtract” step.

To avoid the confusion, I advocate teaching long division in such a fashion that children are NOT exposed to all of those steps at first. Instead, you can teach it in several “steps” :

• Step 1: Division is even in all the digits. Here, students practice just the dividing part.
• Step 2: A remainder in the ones. Now, students practice the ” multiply & subtract ” part and connect that with finding the remainder.
• Step 3: A remainder in the tens. Students now use the whole algorithm, including “dropping down the next digit”, using 2-digit dividends.
• Step 4: A remainder in any of the place values. Students practice the whole algorithm using longer dividends.

## Step 1: Division is even in all the digits

We divide numbers where each of the hundreds, tens, and ones digits are evenly divisible by the divisor. The GOAL in this first, easy step is to get students used to two things:

1. To get used to the long division “corner” so that the quotient is written on top.
2. To get used to asking how many times does the divisor go into the various digits of the dividend.

Example problems for this step follow. Students should check each division by multiplication.

In this step, students also learn to look at the first two digits of the dividend if the divisor does not “go into” the first digit:

4 does not go into 2. You can put zero in the quotient in the hundreds place or omit it. But 4 does go into 24, six times. Put 6 in the quotient.

The 2 of 248 is of course 200 in reality. If you divided 200 by 4, the result would be less than 100, so that is why the quotient won’t have any whole hundreds.

But then you combine the 2 hundreds with the 4 tens. That makes 24 tens, and you CAN divide 24 tens by 4. The result 6 tens goes as part of the quotient.

Check the final answer: 4 × 62 = 248.

More example problems follow. Divide. Check your answer by multiplying the quotient and the divisor.

## Step 2: A Remainder in the ones

Now, there is a remainder in the ones (units). Thousands, hundreds, and tens digits still divide evenly by the divisor. First, students can solve the remainder mentally and simply write the remainder right after the quotient:

4 does not go into 1 (hundred). So combine the 1 hundred with the 6 tens (160).

4 goes into 16 four times.

4 goes into 5 once, leaving a remainder of 1.

8 does not go into 3 of the thousands. So combine the 3 thousands with the 2 hundreds (3,200).

8 goes into 32 four times (3,200 ÷ 8 = 400)
8 goes into 0 zero times (tens).
8 goes into 7 zero times, and leaves a remainder of 7.

Next, students learn to find the remainder using the process of “multiply & subtract”. This is a very important step! The “multiply & subtract” part is often very confusing to students, so here we practice it in the easiest possible place: in the very end of the division, in the ones colum (instead of in the tens or hundreds column). Of course, this assumes that students have already learned to find the remainder in easy division problems that are based on the multiplication tables (such as 45 ÷ 7 or 18 ÷ 5).

In the problems before, you just wrote down the remainder of the ones. Usually, we write down the subtraction that actually finds the remainder. Look carefully:

When dividing the ones, 4 goes into 7 one time. Multiply 1 × 4 = 4, write that four under the 7, and subract. This finds us the remainder of 3.

Check: 4 × 61 + 3 = 247

When dividing the ones, 4 goes into 9 two times. Multiply 2 × 4 = 8, write that eight under the 9, and subract. This finds us the remainder of 1.

Check: 4 × 402 + 1 = 1,609

Here are some example problems. Now, the students check the answer by multiplying the divisor times the quotient, and then adding the remainder.

## Step 3: A remainder in the tens

In this step, students practice for the first time all the basic steps of long division algorithm: divide, multiply & subtract, drop down the next digit. We use two-digit numbers to keep it simple. Multiply & subtract has to do with finding the remainder , and after finding a remainder, we combine that with the next unit we are getting ready to divide (dropping down the digit).

Two goes into 5 two times, or 5 tens ÷ 2 = 2 whole tens — but there is a remainder!

To find it, multiply 2 × 2 = 4, write that 4 under the five, and subtract to find the remainder of 1 ten.

Next, drop down the 8 of the ones next to the leftover 1 ten. You combine the remainder ten with 8 ones, and get 18.

Divide 2 into 18. Place 9 into the quotient.

Multiply 9 × 2 = 18, write that 18 under the 18, and subtract.

The division is over since there are no more digits in the dividend. The quotient is 29.

## Step 4: A remainder in any of the place values

After the previous step has been mastered, students then practice long division with three- and four-digit numbers where they will have to go through the basic steps several times.

Two goes into 2 one time, or 2 hundreds ÷ 2 = 1 hundred.

Multiply 1 × 2 = 2, write that 2 under the two, and subtract to find the remainder of zero.

Next, drop down the 7 of the tens next to the zero.

Divide 2 into 7. Place 3 into the quotient.

Multiply 3 × 2 = 6, write that 6 under the 7, and subtract to find the remainder of 1 ten.

Next, drop down the 8 of the ones next to the 1 leftover ten.

Divide 2 into 18. Place 9 into the quotient.

Multiply 9 × 2 = 18, write that 18 under the 18, and subtract to find the remainder of zero.

There are no more digits to drop down. The quotient is 139.

These ideas are also explained in the YouTube video below:

## Why long division works

I feel the long division algorithm AND why it works presents quite a complex thing for students to learn, so in this case I don’t see a problem with students first learning the algorithmic steps (the “how”), and later delving into the “why”. Trying to do both simultaneously may prove to be too much to some.

However, once the student has a basic mastery of how to do long division, it is time to also study what it is based on. To learn more about that, please see:

Why long division works (based on repeated subtraction)

### Worksheets

Long division worksheets
Create an unlimited supply of worksheets for long division (grades 4-6), including with 2-digit and 3-digit divisors. The worksheets can be made in html or PDF format – both are easy to print. You can also customize them using the generator.

Question from MICHELLE, a student:

DIVIDE 538 BY 14 IN BASE 2, 3, 4 & 5

I’m going to illustrate the procedure by dividing 821 by 17 in base 7. I am going to first convert 821 and 17, both written in base 10, to base 7. This procedure is illustrated in my response to an earlier question. Here is my conversion of 821 to base 7

821 ÷ 7 = 117, remainder of 2
117 ÷ 7 = 16, remainder 5
16 ÷ 7 = 2, remainder 2
2 ÷ 7 = 0, remainder 2

Thus 821 = 22527. Similarly 17 = 237.

Before I actually start the division I am going to calculate 1× 237, 2 × 237, 3× 237 4× 237 5× 237 and

and so on. My resulting table is

Now I can start the division.

I can see from my multiplication that 237 divided 2257. 6 times. Thus I get, after subtraction

Again using of the table I see that 237 divided 2127 6 times and thus

Thus 22527 divided by 237 is 667 with a remainder of 57 .

Here we will show you step-by-step with detailed explanation how to calculate 10 divided by 4 using long division.

Before you continue, note that in the problem 10 divided by 4, the numbers are defined as follows:

10 = dividend
4 = divisor

Step 1:
Start by setting it up with the divisor 4 on the left side and the dividend 10 on the right side like this:

 4 ⟌ 1 0

Step 2:
The divisor (4) goes into the first digit of the dividend (1), 0 time(s). Therefore, put 0 on top:

 0 4 ⟌ 1 0

Step 3:
Multiply the divisor by the result in the previous step (4 x 0 = 0) and write that answer below the dividend.

 0 4 ⟌ 1 0 0

Step 4:
Subtract the result in the previous step from the first digit of the dividend (1 – 0 = 1) and write the answer below.

 0 4 ⟌ 1 0 – 0 1

Step 5:
Move down the 2nd digit of the dividend (0) like this:

 0 4 ⟌ 1 0 – 0 1 0

Step 6:
The divisor (4) goes into the bottom number (10), 2 time(s). Therefore, put 2 on top:

 0 2 4 ⟌ 1 0 – 0 1 0

Step 7:
Multiply the divisor by the result in the previous step (4 x 2 = 8) and write that answer at the bottom:

 0 2 4 ⟌ 1 0 – 0 1 0 8

Step 8:
Subtract the result in the previous step from the number written above it. (10 – 8 = 2) and write the answer at the bottom.

 0 2 4 ⟌ 1 0 – 0 1 0 – 8 2

You are done, because there are no more digits to move down from the dividend.

The answer is the top number and the remainder is the bottom number.

Therefore, the answer to 10 divided by 4 calculated using Long Division is:

2
2 Remainder

Long Division Calculator
Enter another problem for us to explain and solve:

If you enter 10 divided by 4 into a calculator, you will get:

The answer to 10 divided by 4 can also be written as a mixed fraction as follows:

Note that the numerator in the fraction above is the remainder and the denominator is the divisor.

How to calculate 10 divided by 5 using long division
Here is the next division problem we solved with long division.

## Key Question: How do you divide fractions by fractions and fractions with whole numbers?

Learn to divide fractions using 3 easy steps.

Welcome to this free step-by-step guide to dividing fractions. This guide will teach you how to use a simple three-step method called Keep-Change-Flip to easily divide fractions by fractions (and fractions by whole numbers as well).

Below you will find several examples of how to divide fractions using the Keep-Change-Flip method along with an explanation of why the method works for any math problem that involves dividing fractions. Additionally, this free guide includes an animated video lesson and a free practice worksheet with answers!

Are you ready to get started?

Before you learn how to divide fractions using the Keep-Change-Flip method, you need to make sure that you understand how to multiply fractions together (which is even easier than dividing!).

Since multiplying fractions is typically taught before dividing fractions, you may already know how to multiply two fractions together. If this is the case, you can skip ahead to the next section.

However, if you want a quick review of how to multiply fractions, here is the rule:

Multiplying Fractions Rule: Whenever multiplying fractions together, multiply the numerators together, then multiply the denominators together as follows…

### Solutions to 6 Common Reasons Kids Have Trouble with Division

Many kids have difficulty with division. Their difficulties often stem from one of these 6 reasons. Use this tool to see if any of these situations apply to your child. Then, get solutions to help your child become a division super star!

First, let’s brush up on our math vocabulary. The dividend is the number that is being divided. The divisor is the number of groups going into the dividend and the quotient represents how many units are in each group.

dividend ÷ divisor = quotient

The divisor and the quotient can be reversed and the equation remains true. 12÷2=6 and 12÷6=2. 12 can have two groups of six or six groups of two.

Reason #1 Kid Struggle with Division. Many kids don’t understand what division actually means. It means making equal groups. Taking 12 apples and making one group of 6 apples, one group of 1 apple, and one group of 5 apples is not dividing the apples. here’s another way to think of it: the dividend is the whole and the divisor ia the parts. Some people call this “decomposing the dividend” into multiples. At Mathnasium we prefer calling it “wholes and parts” We also talk about division being “How many of these are in that?” For example: 12 ÷ 6 = means “How many 6’s are in 12?” Likewise, 4 ÷ 1/2 = means “How many 1/2’s are in 4?”

Solution: Explain that, like multiplication, division must have equal groups. Then give the child lots of opportunities to work with actual objects and divide them into equal groups. Have the child write the division equation they just represented with objects.

Reason #2 Kids Have Trouble with Division. Children often forget the steps for long division. This happens when they don’t understand why it works. They are relying on memorizing a series of steps, the algorithm, that seem nonsensical to them. Memory is a poor substitute for understanding because it cannot be relied upon as well.

Solutions: Encourage your child to create another method of solving division problems with multiple digits. The method they come up with might make more sense to them than the one they learn in school. If your child understands why long division works, they are more likely remember each step. Long and short division are simply different algorithms for solving a division problem. Long division is the more commonly taught method in schools in the U.S.

Consider bringing your child into our center, Mathnasium of Littleton. We specialize in making math make sense.

Take a look at ways division is taught around the world. Maybe the notation system of Brazil will make more sense to your child.

Have your child try short division instead of long division. It can be a great tool for solving division problems when long division makes it overly complex. Although it has been traditionally left out of the U.S school curriculum, it is a useful tool. If you are interested in learning more about how to do short division here are two resources. One is a video and the other is a written explanation to learn more about short division. Beware that a child should have a firm understanding of simple multiplication and division concepts before learning short or long division.

Reason #3 Kids Find Division Challenging. They have always struggled with math. This is just the latest concept that is giving them difficulties.

Solution: Sometimes you have to go backwards in order to go forward. A child who is missing a foundational skill will find division difficult because it is related to previous concepts. Division is repeated subtraction and the opposite of multiplication. It is related to counting, wholes and parts, and proportional thinking.

Call Suzie at 303-979-9077 to schedule an assessment and we’ll look at whether your child is missing skills from earlier grades.

Reason #4 Children Have a Hard Time with Division. They never developed multiplication fluency. Children can use repeated subtraction and backward skip counting to solve a division problem, but it is a laborious process. Children using those methods will be prone to making mistakes. Since division is the inverse of multiplication a child with multiplication fluency will be able to do division quickly and accurately.

Solution: Enroll your child in our multiplication fluency program or our New! Multiplication Boot Camp.
Your child will become so good at multiplication division will get that much easier.

Reason #5 Kids Struggle with Division. They don’t understand remainders, the way they are written and what they mean. Remainders are a simplified method for expressing a quotient. They are used when the divisor is not divisible into the dividend using only whole numbers. 47÷3=15.66666… or 47÷3=15 2/3. That works if you are dividing objects that can be infinitely broken down into smaller parts. Some objects don’t break apart easily. If you have 47 balls that you are going to split 3 ways, you will end up with 3 groups of 15 and then have two balls left over, or remaining. So 47 ÷3 = 15 remainder 2. Educators often tell children in 3rd-5th grade to use quotients with remainders rather than fractions or decimals. This tendency exacerbates children’s confusion about the meaning of a remainder.

Solution: Use money, units of measurement, pizza, and other objects to explore the how different things are divided. Discuss situations where it does not make sense for the quotient to have a remainder, such as dividing cars, people, or balls. Also discuss situations where it makes sense for the quotient to use a fraction or a decimal, such as calculating a rate. If the division problem is not part of a real life situation or word problem, there is no way to know how the quotient should be expressed.

Reason #6 Kids Find Division Hard. Kids get confused with the symbols because division can be written in different formats.

Solution: Acknowledge the different ways of writing division can be confusing. Talk about how reading has some of the same challenges. Sounds (phonemes) can be made with different letter combinations (such as /x/, and /cks/, or /ew/ and /oo/). Assure your child that if they learned to read, they will soon learn the different ways of writing division. It takes practice. Help them create a cheat sheet showing one simple division problem three ways.

12÷4 = 3 12/4= 3

If your child is experiencing difficulty with division for another reason, give us a call. We’d love to hear about it and see if we can help. We usually can. 303-979-9077 or click the button below for more information.