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Division with Decimal Results
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Introduction

In this lesson, you will learn how to divide multiple-digit numbers that have a whole number or a decimal as an answer. Unfortunately, answers to division problems are not always whole numbers.

People around the world use different division algorithms or methods. This lesson will teach one method. Feel free to use the method that best works for you.

These videos illustrate the lesson material below. Watching the videos is optional.

Dividing by a Single Whole Number

You will learn how to do division with multiple digits. This process is applicable for numbers that divide evenly, and those with decimal answers. If you are familiar with another algorithm or pattern, you are welcome to use it. The important thing is to understand what you are doing and why so you can apply it to real-world situations.

Steps for division:

  1. Put the number being divided (the dividend) under the box and the other number (the divisor) to the left of the box.
  2. Starting on the left, determine how many times the divisor can go into the first digit of the dividend.
  3. Write that number above the digit and subtract the product of that number and the divisor from the dividend.
  4. Repeat steps 2 and 3, moving to the right.

The same process of division is used, even with a decimal answer. Remember, you can always add 0’s after the decimal place on a number (for example: \(2 = 2.0 = 2.00\)). The following instructions will show how to divide with decimals.

Note: This process may be slow at first. Take your time learning to divide accurately. Studying and memorizing your 1-digit multiplication facts will speed up the process and help you as you follow the division steps.

Example 1
\(364\div 7\)

The goal is to divide 364 into 7 equal pieces. Start by creating a box to help you organize the problem. Put 364 under the box, and 7, the divisor, outside of the box: \(\nequire{enclose}7\enclose{longdiv}{364}\)

Start with the first number on the left. In this case, 3. How many sevens can go into three? The answer is zero, so either put the zero here or just leave it blank. Normally, you leave it blank.

Since seven does not go into three, look at the next number over. Now you’re looking at the number 36. How many times does 7 go into 36, or what multiplied by 7 is close to 36? From your multiplication facts, you know that \(7\times5=35\). Because 35 is close to 36 while still being less than 36, you can put a 5 in the tens column.

The incomplete equation shows 364 with the number 26 being divided by 7, which equals to 5.

Figure 1

\(7\times5=35\), so subtract 35 from 36, which leaves 1.

The incomplete equation shows 364 divided by 7, which equals to 5, 5 is multiplied to 7 which is equals to 35. The number 35 is placed under the numbers 3 and 6 which was divided to 7 earlier. 35 is then subtracted to 36, which results to 1.

Figure 2

Carry down the number in the next column to the right, which is four. Now you are working with 14.

The equation shows 364 with the number 36 being divided by 7, which is equals to 5, 5 is then multiplied to 7, which is equals to 35, 35 is then subtracted to 36 which is equals to 1, the number 4 is now brought down next to 1, which now shows 14.

Figure 3

How many times does 7 go into 14? Your multiplication facts tell you that \(2\times7=14\), so the answer is two. \(14-14=0\), and because you have 0, you know that you’re done with the problem. The answer is 52: \(364\div7=52\).

the equation shows 364 with the number 36 being divided by 7 which is equals to 5, 5 is now multiplied to 7 which is equals to 35. 35 is then subtracted to 36 which is equals to 1, the number 4 is now brought down next to number one, which now reads as number 14, the number 14 is now divided by 7, which is equals to 2, now 2 is multiplied to 7 which is equals to 14. Now 14 is subtracted to 14, which results to 0. The final answer to this entire equation shows the number 52.

Figure 4

Consider why this works. When you multiply 5 and 7, it’s actually 50 multiplied by 7 because the 5 is in the tens column. When you subtract, you’re actually subtracting 350 from 364. You couldn’t use a number larger than five here because anything larger than a five in the tens place would give you too large of a number.

You want to know how much is left over, so you subtract 350 from 364. In the first example, you just brought down the four. You can do this because there’s really a zero under the four. The remaining 14 means there’s 14 that still needs to be divided by 7.

Example 2
\(\nequire{enclose}8\enclose{longdiv}{984}\)

\(984\div8\). Start with the column farthest to the left, which in this case is the hundreds column. 8 goes into 9 one time. \(1\times8=8\), and \(9-8=1\).

The equation shows the number 984 with the number 9 is divided by 8 which results to 1, the number 1 is now multiplied to 1 which also equals to 1, now 8 is subtracted to 9 which is equals to 1.

Figure 5

Bring the remaining one down the next number to the right.

The equation shows the number 984 with the number 9 being divided by 8 which is is equals to 1, now 1 is multiplied to 8 which equals to 8, the number 8 is then subtracted from 9, which results to 1, now the number 8 from 984 is brought down next to number one and now reads as number 18.

Figure 6

After bringing down the 8, you have 18. How many times does 8 go into 18? \(8\times2=16\), which is less than 18 but close to it. Put a 2 in the tens place, subtract 16 from 18, and bring the next number down.

the equation shows 984 with the number 8 divided by 8 which equals to 1, and 1 multiplied to 8 which is equals to 8, 8 is subtracted from 9 which equals to 1. The number 8 from 984 is brought down next to number 1 and now reads as number 18, number 18 is now divided by 8 which equals to 2, 2 is now multiplied to 8 which equals to 16. The number 16 is subtracted from 18 which equals to 2.

Figure 7

When you bring the next number down, you have 24.

The number 4 from 984 is brought down next to number 2 and now reads as 24.

Figure 8

\(8\times3=24\). Subtract 24 from 24, and you get a remainder of zero, meaning the answer is a whole number. \(984\div8=123\)

The number 24 is divided by 8 which equals to 3, 3 is multiplied to 8, which equals to 24, 24 is then subtracted from 24 which results to zero.

Figure 9

Division Algorithm

How can you find the answer to a division problem when the answer is not a whole number? If the answer is not a whole number, it will be a decimal.

Example 3
\(17\div5\)

From your multiplication facts, you know that \(5\times3=15\), and \(5\times4=20\). This means that \(17\div5\) is going to have an answer somewhere between three and four, but it’s not exactly three or four. The answer will have a decimal in it.

Start by creating a little box: \(\nequire{enclose}5\enclose{longdiv}{17}\)

Use the same division algorithm used in the previous section. Five does not go into one, so move on to the next number: 5 does go into 17. \(3\times5=15\), which is ideal because it is close to 17 but still smaller than 17. Now subtract: \(17-15=2\).

the equation shows 17 being divided by 5 which equals to 3, 3 is multiplied to 5 which equals to 15, 15 is then subtracted from 17 which results to 2.

Figure 10

The answer means that \(17\div5\) gives you an answer of three with two pieces remaining.

Now you must figure out how to put the two remaining pieces into decimal form. Put a decimal after 17 because 17 is the same as 17.0. If you put a zero here, you can bring a zero down and continue doing the algorithm as before. Keep in mind that if you add a decimal to the number you are dividing, you must put a decimal in the final answer as well.

A decimal point is added next to the number 17 and a zero followed it. A decimal point is added next to the answer as well. The zero is then brought down next to number 2 and now reads as number 20.

Figure 11

\(5\times4=20\), so put a four in the answer after the decimal place.

The number 20 is then divided by 5 which equals to 4, the number 4 is added next to the decimal point. The number 4 is multiplied to 5 which equals to 20. 20 is then subtracted from 20 which results to zero.

Figure 12

You estimated at the beginning that the answer would be somewhere between three and four, and it is.

Repeating Decimals

There are some division problems where the answer will continue to repeat forever.

Example 4
\(1\div3\) or \(\nequire{enclose}3\enclose{longdiv}{1}\)

If you solve this equation using the regular division algorithm, you know that three doesn’t go into one, so you put a decimal in the answer and next to the one. Now you pretend like the answer is 10. Three goes into 10 three times.

The equation shows how to divide a number that has a decimal point.

Figure 13

\(3\times3=9\), and when you subtract, you get one left over. Bring down another zero. Three goes into 10 three times, and \(3\times3=9\). When you subtract 9 from 10, you have 1 remaining. If you repeat the process again, you get the same answer, so you find that there’s a pattern going on.

the equation shows that there is a pattern going on.

Figure 14

You could continue doing this forever, and you would get a never-ending stream of threes. In this case, the answer is 0.333… The little dots after it mean that the number goes on forever. Another way you can show that the answer repeats is by putting a line over the three: \(0.\overline3\)

Example 5
\(\nequire{enclose}6\enclose{longdiv}{5}\)

Sometimes it’s not immediately obvious that the answer will repeat forever.

Six doesn’t go into five, so you put a decimal point and a zero: \(\nequire{enclose}6\enclose{longdiv}{5.0}\)

Six can go into 50. \(6\times8=48\), so there’s two left over from 50.

the equation shows that the repetition won't be obvious immediately.

Figure 15

Add another zero to the decimal point and bring it down.

Another zero is added to the decimal point and brought down next to number 2.

Figure 16

\(3\times6=18\), so you know that six goes into 20 three times. \(20-18=2\), so you’ll bring down another zero.

The equation shows that 3 x 6 = 18, 18 is then subtracted from 20 which results to 2. Another zero is added to the decimal point and brought down next to 2.

Figure 17

You didn’t see repetition in the first digit, but you see it in the second and third digits. If you continued on, you’d see the repetition forever, so you can write the answer with a bar over the three to show that the three repeats from here on out: \(0.8\overline{3}\)

Example 6
\(1\div7\)

A calculator will help demonstrate this example. You get a huge number when you plug this equation into the calculator: 0.1428571428571428571….

The calculator will pull up a huge number when you plug in the 1 divided by 7 equation.

Figure 18

It doesn’t look like it’s repeating initially. However, you see that the string of numbers repeats the sequence 142857.

When you run across problems like this, it’s efficient to round to the nearest hundredth. In the example above, because the two in the thousandths position indicates rounding down, the answer would be 0.14.

Things to Remember


  • Steps for division:
    1. Put the number being divided (the dividend) under the box and the other number (the divisor) to the left of the box.
    2. Starting on the left, determine how many times the divisor can go into the first digit of the dividend.
    3. Write that number above the digit and subtract the product of that number and the divisor from the dividend.
    4. Repeat steps 2 and 3, moving to the right.
  • If you get to the end of the dividend and the problem isn't complete, add a decimal to the end with zeros so you can bring down additional numbers as needed.
  • When adding a decimal to the dividend, a decimal must also be added to the answer.
  • When rounding to the nearest hundredth place, if the number in the thousandths place is five or greater, round up. If the number is lower than five, round down.

Practice Problems

Evaluate the following expressions. Round to the nearest hundredth.
  1. \(77 ÷ 2 = ?\) (
    Solution
    x
    Solution: 38.5
    Details:
    Start by dividing in parts. First, divide 7 by 2.
    This image shows a division symbol of a vertical line and a horizontal line coming together as though they are making the top left corner of a rectangle. Inside this imaginary rectangle is the number 77. To the left of the vertical line and on the same horizontal plane as the 77 is the number 2. This means 77 is being divided by the 2.

    Place your answer above the line and directly above the 7. The answer will be 3, because 2 goes into 7 about 3 times.
    This image is the same as the previous one except now there is a 3 above the division symbol horizontal line and in the same column as the leftmost 7 in the number 77. In this case this means the 3 is in the tens place just as the leftmost seven is in the tens place in 77.

    Place the result of \(2 \times 3\) below the 7 and subtract.
    This image is the same as the previous one except now there is a 6 in the tens column below the leftmost 7 in 77. There is also a subtraction symbol to the left of the 6. A horizontal line is below the 6 indicating we will put our new number below that line.

    The difference between 7 and 6 is 1. Place the answer directly below the 6.
    This image is the same as the previous one except now there is a 1 below the subtraction horizontal line directly below the 6. This is the solution of 7-subtract-six from the previous image.

    Next step is to bring down the next number. Bring the second 7 down and place it next to the 1.
    This image is the same as the previous image except now there is a downward arrow from the rightmost 7 in 77 and a 7 is written to the right of the 1. This makes the new number below the subtraction line 17.

    Divide 17 by 2. The answer is about 8.
    This image is the same as the previous image except now, above the division symbol, to the right of the three is a number 8. This indicates that 2 goes into 17 eight times.

    Place the result of \(8 \times 2\), 16, below the 17 and subtract.
    This image is the same as the previous image except now there is a 16 directly below the 17, a new subtraction symbol to the left of the 16, and a new horizontal line below it as well.

    The difference between 17 and 16 is 1. Place the 1 below the 6.
    This image is the same as the previous one except now there is a 1 directly below the 7 and 6 in 17 and 16 respectively. This 1 is also below the newest horizontal line.

    At this point, there’s a remainder. Add a decimal and a zero after the 77.
    This image is the same as the previous one except now there is a decimal point and a zero written to the right of the 77. This makes it seventy-seven-point-zero. An arrow is used to draw attention to this new part of the number.

    Continue by bringing the new zero down.
    This image is the same as the previous one except now there is a downward arrow from the 0 in seventy-seven-point-zero. It points to a zero to the right of the 1 below the most recent horizontal line. This makes the newest number we will be dividing a 10.

    Divide 10 by 2 and place the answer, 5, next to the 8 above the new 0 in the solution.
    This image is the same as the previous one except now there is a 5 above the division symbol and to the right of the 8. This makes the number above the division symbol three-eight-five from left to right.

    Place the results of \(2 \times 5\) below the 10 and subtract. The difference between 10 and 10 is zero. When you reach 0 you stop dividing.
    This image is the same as the previous one except that below the 10 at the bottom of our problem there is another 10. They are written directly one over the other. There is also a subtraction symbol to the left and a new horizontal line below the new 10. Below this horizontal line is zero-zero. This is because the solution to 10 subtract 10 is zero.

    Place the decimal point between 38 and 5. Notice that the decimal point in the answer is placed directly above the decimal point in 77.0.
    This image is the same as the previous one except there is a decimal point in the number above the division symbol. The decimal point is between the 8 and the 5. There is an arrow pointing it out. This makes the solution to the division problem thirty-eight-point-five.

    The answer is 38.5.
    )
  2. \(339 ÷ 4 = ?\) (
    Solution
    x
    Solution: 84.75
    Details:
    Start by dividing 33 by 4.
    This image shows a division symbol of a vertical line and a horizontal line coming together as though they are making the top left corner of a rectangle. Inside this imaginary rectangle is the number 339. To the left of the vertical line and on the same horizontal plane as the 339 is the number 4. This means 339 is being divided by the 4.

    4 goes into 33 about 8 times.
    This image is the same as the previous one except now there is an 8 above the division symbol horizontal line and in the same column as the rightmost 3 in the number 339. In this case this means the 8 is in the tens place just as the rightmost three is in the tens place in 339.

    Place the answer of \(8 \times 4\), 32, below 33 and subtract.
    This image is the same as the previous one except now there is a 33 below the 33 in 339. There is also a subtraction symbol to the left of the 32. A horizontal line is below the 32 indicating we will put our new number below that line.

    There is a remainder of 1. Place it below the line and bring down the 9.
    This image is the same as the previous one except now there is a 1 below the subtraction horizontal line directly below the 32. This is the solution of 33-subtract-32 from the previous image.

    Divide 19 by 4.
    This image is the same as the previous image except now there is a downward arrow from the 9 in 339 and a 9 is written to the right of the 1. This makes the new number below the subtraction line 19.

    Four goes into 19 about 4 times, because of \(4 \times 4 = 16\). Place the new 4 next to the 8 in the answer.
    This image is the same as the previous image except now, above the division symbol, to the right of the 8 is a number 4. This indicates that 4 goes into 19 four times.

    Place the product of \(4 \times 4\), 16, below the 19 and subtract.
    This image is the same as the previous image except now there is a 16 directly below the 19, a new subtraction symbol to the left of the 16, and a new horizontal line below it as well.

    The difference between 19 and 16 is 3. Place the 3 below the right-most column.
    This image is the same as the previous one except now there is a 3 directly below the 9 and 6 in 19 and 16 respectively. This 3 is also below the newest horizontal line.

    Next, add a decimal point after the 9 in the dividend (339) and include two zeros.
    This image is the same as the previous one except now there is a decimal point and two zeros written to the right of the 339. This makes it three-hundred-thirty-nine-point-zero. An arrow is used to draw attention to this new part of the number.

    Continue the same pattern. Bring the next number down. In this case, the zero. Divide 30 by 4.
    This image is the same as the previous one except now there is a downward arrow from the 0 in 339.00. It points to a zero to the right of the 3 below the most recent horizontal line. This makes the newest number we will be dividing a 30.

    4 goes into 30 about 7 times.
    This image is the same as the previous one except now there is a 7 above the division symbol and to the right of the 84. This makes the number above the division symbol eight-four-seven from left to right.

    Place the product of \(4 \times 7\), 28, below the 30 and subtract.
    This image is the same as the previous one except that below the 30 at the bottom of our problem there is 28. They are written directly one over the other. There is also a subtraction symbol to the left and a new horizontal line below the new 30.

    The difference between 30 and 28 is 2. Place the 2 below the line. Notice you had to do some regrouping in order to do this subtraction.
    This image is the same as the previous one, except that the 3 in 30 is crossed out and under it is written 2. The 0 is 30 is also crossed out and under it is written a 10. There is a new horizontal line on the bottom, below it is written 2. This indicates that 2 is the result when 28 is subtracted from 30.

    Continue by bringing the last 0 down, next to the 2. Divide 20 by 4.
    This image is the same as the previous one, except that the second zero in 339.00 is brought down below the bottom horizontal line, right next to the number 2. The new number under the bottom horizontal line is 20.

    The number 4 goes into 20 exactly 5 times, because \(4 \times 5 = 20\). Subtract to get a zero remainder.
    This image is the same as the previous one, except that there's a 5 written on the top right next to 7. Now the top line consists of the numbers 8, 4, 7, and 5.

    This image is the same as the previous one, except that there's another 20 written below the 20, with a subtraction sign to its right. There's a new horizontal line on the bottom, below which two zeros are written, indicating that 20 subtracted from 20 equals zero.

    The last step is to place the decimal point in the answer. Place the decimal point between 84 and 75, lined up with the decimal point in the dividend: 339.00.
    This image is the same as the previous one, except that a decimal point is placed on the top line between numbers 4 and 7. This makes the final answer eighty-four-point-seventy-five.

    The final answer: 84.75.
    )
  3. \(21 ÷ 8 = ?\) (
    Video Solution
    x
    Solution: 2.63, when rounded to the nearest hundredth.
    Details:

     | Transcript)
  4. \(37 ÷ 3 = ?\) (
    Solution
    x
    Solution: 12.33, when rounded to the nearest hundredth.
    )
  5. \(559 ÷ 6 = ?\) (
    Video Solution
    x
    Solution: 93.17, when rounded to the nearest hundredth.
    Details:

     | Transcript)
  6. \(258 ÷ 7 = ?\) (
    Solution
    x
    Solution: 36.86, when rounded to the nearest hundredth.
    )
Multiplication and Division with Negative Numbers
Rules of Exponents