How to teach long division with remainders?

 

Master Long Division Practice Workbook: Improve Your Math Fluency Series (Volume 8)

Basic Math and Pre-Algebra Workbook For Dummies 

Division, Ages 7-12 (Workbook w/Music CD) 

 

Solution for 76999 ÷ 123 - with remainder

Step 1 Long division works from left to right. Since 123 will not go into 7, a grey 0 has been placed over the 7 and we combine the first two digits to make 76. In this case, 76 is still too small. A further 0 is added above 6 and a third digit is added to make 769. Note the other digits in the original number have been turned grey to emphasise this. The closest we can get to 769 without exceeding it is 738 which is 6 × 123. These values have been added to the division, highlighted in red.

0 0 6  rem 1 123 7 6 9 9 9 7 3 8

123 × table 1 × 123 = 123 2 × 123 = 246 3 × 123 = 369 4 × 123 = 492 5 × 123 = 615 6 × 123 = 738 7 × 123 = 861 8 × 123 = 984 9 × 123 = 1107

Step 2 Next, work out the remainder by subtracting 738 from 769. This gives us 31. Bring down the 9 to make a new target of 319.	6  rem 1 123 7 6 9 9 9 7 3 8 3 1 9
123 × table 1 × 123 = 123 2 × 123 = 246 3 × 123 = 369 4 × 123 = 492 5 × 123 = 615 6 × 123 = 738 7 × 123 = 861 8 × 123 = 984 9 × 123 = 1107

Step 3 With a target of 319, the closest we can get is 246 by multiplying 123 by 2. Write the 246 below the 319 as shown.	6 2  rem 1 123 7 6 9 9 9 7 3 8 3 1 9 2 4 6
123 × table 1 × 123 = 123 2 × 123 = 246 3 × 123 = 369 4 × 123 = 492 5 × 123 = 615 6 × 123 = 738 7 × 123 = 861 8 × 123 = 984 9 × 123 = 1107

Step 4 Next, work out the remainder by subtracting 246 from 319. This gives us 73. Bring down the 9 to make a new target of 739.	6 2  rem 1 123 7 6 9 9 9 7 3 8 3 1 9 2 4 6 7 3 9
123 × table 1 × 123 = 123 2 × 123 = 246 3 × 123 = 369 4 × 123 = 492 5 × 123 = 615 6 × 123 = 738 7 × 123 = 861 8 × 123 = 984 9 × 123 = 1107

Step 5 With a target of 739, the closest we can get is 738 by multiplying 123 by 6. Write the 738 below the 739 as shown.	6 2 6  rem 1 123 7 6 9 9 9 7 3 8 3 1 9 2 4 6 7 3 9 7 3 8
123 × table 1 × 123 = 123 2 × 123 = 246 3 × 123 = 369 4 × 123 = 492 5 × 123 = 615 6 × 123 = 738 7 × 123 = 861 8 × 123 = 984 9 × 123 = 1107

Step 6 Finally, subtract 738 from 739 giving 1. Since there are no other digits to bring down, 1 is therefore also the remainder for the whole sum. So 76999 ÷ 123 = 626 rem 1	6 2 6  rem 1 123 7 6 9 9 9 7 3 8 3 1 9 2 4 6 7 3 9 7 3 8 1
123 × table 1 × 123 = 123 2 × 123 = 246 3 × 123 = 369 4 × 123 = 492 5 × 123 = 615 6 × 123 = 738 7 × 123 = 861 8 × 123 = 984 9 × 123 = 1107

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How to do long division of large numbers?

How to divide larger numbers and do long division?

This is video for how to divide large numbers by doing long division with easy methods. This video describes that dividing the "Dividend" with "Divisor" to get the "Quotient" and "Remainder" if they exist. In the given example first "25" taken as the Dividend and "5" is will be the Divisor. Divide this value the multiplication is needed. For above example "5*5=25" is the result so, the "Quotient" is "5".If the value is large for example "506" of dividend and "8" of divisor means, first we should take the first two digit of the dividend value for division, the value will be "6*8=48". Now, we will get the Quotient of "6" first and the remainder will be "2", now the third digit of the dividend "6" will be suffixed with remainder, now the value will be "26".Then "3*8=24" will be the value for second division. So now finally we will get the "Quotient" as "63" and the "Remainder" as "2".

Basic procedure for long division by longhand

1. When dividing two numbers, for example, n divided by m, n is the dividend and m is the divisor; the answer is the quotient.
2. Find the location of all decimal points in the dividend and divisor.
3. If necessary, simplify the long division problem by moving the decimals of the divisor and dividend by the same number of decimal places, to the right, (or to the left) so that the decimal of the divisor is to right of the last digit.
4. When doing long division, keep the numbers lined up straight from top to bottom under the tableau.
5. After each step, be sure the remainder for that step is less than the divisor. If it is not, there are three possible problems: the multiplication is wrong, the subtraction is wrong, or a greater quotient is needed.
6. In the end, the remainder, r, is added to the growing quotient as a fraction, r/m.

Long division does not use the slash (/) or obelus (÷) signs, instead displaying the dividend, divisor, and (once it is found) quotient in a tableau. An example is shown below, representing the division of 500 by 4 (with a result of 125).

 125     (Explanations)
   4)500
     4        (4 ×  1 = 4)
     10       (5 -  4 = 1)
      8       (4 ×  2 = 8)
      20     (10 -  8 = 2)
      20      (4 ×  5 = 20)
       0     (20 - 20 = 0)

The process is begun by dividing the left-most digit of the dividend by the divisor. The quotient (rounded down to an integer) becomes the first digit of the result, and the remainder is calculated (this step is notated as a subtraction). This remainder carries forward when the process is repeated on the following digit of the dividend (notated as 'bringing down' the next digit to the remainder). When all digits have been processed and no remainder is left, the process is complete.

Here is an example of the process not producing an integer result:

  31.75     
   4)127
     12         (12-12=0 which is written on the following line)                    
      07        (the seven is brought down from the dividend 127) 
       4       
       3.0      (3 is the remainder which is divided by 4 to give 0.75)
       2.8      (7 × 4 = 28)
         20     (an additional zero is brought down)
         20     (5 × 4 = 20)
          0

In this example, the decimal part of the result is calculated by continuing the process beyond the units digit, 'bringing down' zeros as being the decimal part of the dividend.

This example also illustrates that, at the beginning of the process, a step that produces a zero can be omitted. Since the first digit 1 is less than the divisor 4, the first step is instead performed on the first two digits 12. Similarly, if the divisor were 13, one would perform the first step on 127 rather than 12 or 1.

Source: Wikipedia

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How to do long division of polynomials?

• Divide 3x³ – 5x² + 10x – 3  by  3x + 1

This division did not come out even. What am I supposed to do with the remainder? Think back to when you did long division with plain numbers. Sometimes there would be a remainder; for instance, if you divide 132 by 5:

...there is a remainder of 2. Remember how you handled that? You made a fraction, putting the remainder on top of the divisor, and wrote the answer as "twenty-six and two-fifths":

The first form, without the "plus" in the middle, is how "mixed numbers" are written, but the meaning of the mixed number is actually the addition. We do the same thing with polynomial division. Since the remainder is –7 and since the divisor is 3x + 1, then I'll turn the remainder into a fraction (the remainder divided by the original divisor), and add this fraction to the polynomial across the top of the division symbol. Since the division looks like this:

...then the answer is this:   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved
 

Warning: Do not write the polynomial "mixed number" in the same format as numerical mixed numbers! If you just append the fractional part to the polynomial part, this will be interpreted as polynomial multiplication, which is not what you mean!

Note: Different books format the long division differently. When writing the expressions across the top of the division, some books will put the terms above the same-degree term, rather than above the term being worked on. In such a text, the long division above would be presented as shown here: The only difference is that the terms across the top are shifted to the right. Otherwise, everything is exactly the same. You should probably use the formatting that your instructor uses.

• Divide 2x³ – 9x² + 15  by  2x – 5

First off, I note that there is a gap in the degrees of the terms of the dividend: the polynomial 2x³ – 9x² + 15 has no x term. My work could get very messy inside the division symbol, so it is important that I leave space for a x-term column, just in case. I can create this space by turning the dividend into 2x³ – 9x² + 0x + 15. This is a legitimate mathematical step: since I've only added zero, I haven't actually changed the value of anything. Now that I have all the "room" I might need for my work, I'll do the division:

I need to remember to add the remainder to the polynomial part of the answer:

• Divide 4x⁴ + 3x³ + 2x + 1  by x² + x + 2

I'll add a 0x² term to the dividend (inside the division symbol) to make space for my work, and then I'll do the division in the usual manner:

Then my answer is:
 

To succeed with polyomial long division, you need to write neatly, remember to change your signs when you're subtracting, and work carefully, keeping your columns lined up properly. If you do this, then these exercises should not be very hard; annoying, maybe, but not hard.

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How to do long division without a calculator?

How to teach long division without a calculator?

In the olden days, knowing how to divide large numbers was important. Basic long division is still good to know, so the following examples will show you how to divide a one-digit divisor into another number, and then how to find a remainder.

Recall that the divisor in a division problem is the number that you’re dividing by. When you’re doing long division, the size of the divisor is your main concern: Small divisors are easy to work with, and large ones are a royal pain. So here you’ll work with a nice, small, one-digit divisor. Suppose you want to find 860 5. Start off by writing the problem like this:
Unlike the other Big Four operations, long division moves from left to right. In this case, you start with the number in the hundreds column (8). To begin, ask how many times 5 goes into 8 — that is, what’s 8 5? The answer is 1 (with a little bit left over), so write 1 directly above the 8. Now multiply 1 5 to get 5, place the answer directly below the 8, and draw a line beneath it:
Subtract 8 – 5 to get 3. (Note: After you subtract, the result should always be smaller than the divisor. If not, you need to write a higher number above the division symbol.) Then bring down the 6 to make the new number 36:
These steps are one complete cycle, and to complete the problem you just need to repeat them. Now ask how many times 5 goes into 36 — that is, what’s 36 5? The answer is 7 (with a little left over). Write 7 just above the 6, and then multiply 7 5 to get 35; write the answer under 36:
Now subtract to get 36 – 35 = 1; bring down the 0 next to the 1 to make the new number 10:
Another cycle is complete, so begin the next cycle by asking how many times 5 goes into 10 — that is, 10 5. The answer this time is 2. Write down the 2 in the answer above the 0. Multiply to get 2 5 = 10, and write this answer below the 10:
Now subtract 10 – 10 = 0. Because you have no more numbers to bring down, you’re finished, and here’s the answer (that is, the quotient):
So 860 5 = 172.
This problem divides evenly, but many don’t. The following instructions tell you what to do when you run out of numbers to bring down.
Division is different from addition, subtraction, and multiplication in that having a remainder is possible. A remainder is simply a portion left over from the division.

The letter r indicates that the number that follows is the remainder.

For example, suppose you want to divide seven candy bars between two people without breaking any candy bars into pieces (too messy). So each person receives three candy bars, and one candy bar is left over. This problem shows you the following:
7 2 = 3 with a remainder of 1, or 3 r 1
In long division, the remainder is the number that’s left when you no longer have numbers to bring down. The following equation shows that 47 3 = 15 r 2:
Note that when you’re doing division with a small dividend and a larger divisor, you always get a quotient of 0 and a remainder of the number you started with:
1 2 = 0 r 1
14 23 = 0 r 14
2,000 2,001 = 0 r 2,000

Source: Mark Zegarelli, dummies dot com

How to do long division step by step?

How to long division with remainder? How to divide a three digit number by a one digit number (e.g 416 ÷ 7)?

Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps.

• Place the divisor before the division bracket and place the dividend (416) under it.

     
7)416

• Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.

   5 
7)416

• Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.

   5 
7)416
  35

• Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.

   5 
7)416
  35
   66

• Divide 66 by 7 and place that answer above the division bracket to the right of the five.

   59
7)416
  35
   66

• Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.

   59 R 3
7)416
  35
   66
   63
    3

How to long division without remainder? How to divide a three digit number by a one digit number (e.g 413 ÷ 7)?

• Place the divisor before the division bracket and place the dividend (413) under it.

     
7)413

• Examine the first digit of the dividend(4). It is smaller than 7 so it can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.

   5 
7)413

• Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.

   5 
7)413
  35

• Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 3 from the 413 and place it to the right of the 6.

   5 
7)413
  35
   63

• Divide 63 by 7 and place that answer above the division bracket to the right of the five.

   59
7)413
  35
   63

• Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 63 under the dividend. Subtract 63 from 63 to give an answer of 0. This indicates that there is nothing left over and 7 can be evenly divided into 413 to produce a quotient of 59.

   59
7)413
  35
   63
   63
    0

More books about long division for kids

How to do Long Division with Remainders?

When we are given a long division to do it will not always work out to a whole number. Sometimes there will be numbers left over. These are known as remainders. Taking an example similar to that on the Long Division page it becomes more clear: 435 ÷ 25. If you feel happy with the process on the Long Division

page you can skip the first bit.
 

	4 ÷ 25 = 0 remainder 4	The first number of the dividend is divided by the divisor.
		The whole number result is placed at the top. Any remainders are ignored at this point.
	25 × 0 = 0	The answer from the first operation is multiplied by the divisor. The result is placed under the number divided into.
	4 – 0 = 4	Now we take away the bottom number from the top number.
		Bring down the next number of the dividend.
	43 ÷ 25 = 1 remainder 18	Divide this number by the divisor.
		The whole number result is placed at the top. Any remainders are ignored at this point.
	25 × 1 = 25	The answer from the above operation is multiplied by the divisor. The result is placed under the last number divided into.
	43 – 25 = 18	Now we take away the bottom number from the top number.
		Bring down the next number of the dividend.
	185 ÷ 25 = 7 remainder 10	Divide this number by the divisor.
		The whole number result is placed at the top. Any remainders are ignored at this point.
	25 × 7 = 175	The answer from the above operation is multiplied by the divisor. The result is placed under the number divided into.
	185 – 175 = 10	Now we take away the bottom number from the top number.
		There is still 10 left over but no more numbers to bring down.
		With a long division with remainders the answer is expressed as 17 remainder 10 as shown in the diagram

How to explain long division to children?

Solution for 531219 ÷ 579 - with remainder

Step 1 Long division works from left to right. Since 579 will not go into 5, a grey 0 has been placed over the 5 and we combine the first two digits to make 53. In this case, 53 is still too small. A further 0 is added above 3 and a third digit is added to make 531. Note the other digits in the original number have been turned grey to emphasise this. The closest we can get to 531 without exceeding it is 5211 which is 9 × 579. These values have been added to the division, highlighted in red.

0 0 0 9  rem 276 579 5 3 1 2 1 9 5 2 1 1

579 × table 1 × 579 = 579 2 × 579 = 1158 3 × 579 = 1737 4 × 579 = 2316 5 × 579 = 2895 6 × 579 = 3474 7 × 579 = 4053 8 × 579 = 4632 9 × 579 = 5211

Step 2 Next, work out the remainder by subtracting 5211 from 5312. This gives us 101. Bring down the 1 to make a new target of 1011.	9  rem 276 579 5 3 1 2 1 9 5 2 1 1 1 0 1 1
579 × table 1 × 579 = 579 2 × 579 = 1158 3 × 579 = 1737 4 × 579 = 2316 5 × 579 = 2895 6 × 579 = 3474 7 × 579 = 4053 8 × 579 = 4632 9 × 579 = 5211

Step 3 With a target of 1011, the closest we can get is 579 by multiplying 579 by 1. Write the 579 below the 1011 as shown.	9 1  rem 276 579 5 3 1 2 1 9 5 2 1 1 1 0 1 1 5 7 9
579 × table 1 × 579 = 579 2 × 579 = 1158 3 × 579 = 1737 4 × 579 = 2316 5 × 579 = 2895 6 × 579 = 3474 7 × 579 = 4053 8 × 579 = 4632 9 × 579 = 5211

Step 4 Next, work out the remainder by subtracting 579 from 1011. This gives us 432. Bring down the 9 to make a new target of 4329.	9 1  rem 276 579 5 3 1 2 1 9 5 2 1 1 1 0 1 1 5 7 9 4 3 2 9
579 × table 1 × 579 = 579 2 × 579 = 1158 3 × 579 = 1737 4 × 579 = 2316 5 × 579 = 2895 6 × 579 = 3474 7 × 579 = 4053 8 × 579 = 4632 9 × 579 = 5211

Step 5 With a target of 4329, the closest we can get is 4053 by multiplying 579 by 7. Write the 4053 below the 4329 as shown.	9 1 7  rem 276 579 5 3 1 2 1 9 5 2 1 1 1 0 1 1 5 7 9 4 3 2 9 4 0 5 3
579 × table 1 × 579 = 579 2 × 579 = 1158 3 × 579 = 1737 4 × 579 = 2316 5 × 579 = 2895 6 × 579 = 3474 7 × 579 = 4053 8 × 579 = 4632 9 × 579 = 5211

Step 6 Finally, subtract 4053 from 4329 giving 276. Since there are no other digits to bring down, 276 is therefore also the remainder for the whole sum. So 531219 ÷ 579 = 917 rem 276	9 1 7  rem 276 579 5 3 1 2 1 9 5 2 1 1 1 0 1 1 5 7 9 4 3 2 9 4 0 5 3 2 7 6
579 × table 1 × 579 = 579 2 × 579 = 1158 3 × 579 = 1737 4 × 579 = 2316 5 × 579 = 2895 6 × 579 = 3474 7 × 579 = 4053 8 × 579 = 4632 9 × 579 = 5211

External Link

• How to do long division step by step? (video)
• How to do long division for kids

Read more

• How to do long division step by step?
• How to do long division with remainders?
• How do you do long division with decimals?
• Understanding long division as repeated subtraction
• Teaching Long Division
• Double Division with 3 Digit Divisors
• How to do long division with two digit quotients?
• Long Division of Polynomials Step by Step
• How to do long division with 2 digit divisor?
• How to do long division with two digit quotients?

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