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Dividing by a Decimal
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Introduction

In this lesson, you will learn how to divide by a decimal. Previously, you have worked with the dividend as a decimal, and now you will work with the divisor as a decimal. Dividing by a decimal can look difficult, but there’s actually only one simple step you need to do before you can divide by a decimal number.

Ways to Show Division

The division symbol (÷) is not the only way to show division. As learned previously in the lesson about how to use Excel as a calculator, division can be symbolized with a slash (/) too. You may see this on some calculators as well. This course will often use the / symbol to represent division.


This video illustrates the lesson material below. Watching the video is optional.


Dividing by a Decimal

When the divisor (the number you are dividing by) has a decimal, follow these steps:

  1. Count the decimal places in the divisor.  
  2. Move the decimal in the dividend as many places as you counted in the divisor.  If the divisor does not have a decimal, add an extra place to the end of the number and fill it with 0.  Place the decimal for your answer directly above the decimal place you counted in the dividend.
  3. Divide, ignoring the decimals.

Example 1

\(12\div1.5\)

The dividend is the original amount that you start with, in this case, 12. The divisor in this example is a decimal: 1.5. Start by counting the decimal places in the divisor.

In this example, the decimal would be moved over one space, turning 1.5 into 15. Since you had to move the decimal over one place in the divisor, you must move the decimal over one space in the dividend as well, turning 12 into 120.

\begin{align*} 12&\div 1\underrightarrow{.5} &\color{red}\small\text{Count decimal places in the divisor}\\\\ 12\underrightarrow{.0}&\div 15 &\color{red}\small\text{Move the decimal in the dividend}\\\\ 120&\div 15 &\color{red}\small\text{Divide}\end{align*}

The answer is 8.

\(12\div 1.5\) will give you the same answer as \(120\div15\). Why does this work?

Example 2
\(6\div2\)

\(6\div2\) is represented with the following squares.

The image shows 6 squares.

Figure 1

If you divide the six squares into two groups, or two sets, how many would be in each set?

The 6 squares are shown to be divided into two groups.

Figure 2

Each set has three inside of it because \(6\div2=3\).

What if you multiply both the dividend and divisor by ten? \(6\times10=60\) and \(2\times10=20\). Now, instead of \(6\div2\), you have \(60\div20\). What is the answer?

\(60\div20\) is represented with sixty squares. To divide them into twenty different sets, you want to find out how many would be in each set.

The image shows how 60 divided by 20 is represented with sixty squares.

Figure 3

Notice there are 20 rows of squares. To divide by 20, you can simply select each row, making 20 sets. There are 3 squares in each set, which means \(60\div20=3\).

The image shows 20 rows of squares. There are 3 squares in each set.

Figure 4

This demonstrates that as long as you multiply both dividend and divisor by the same number, in this case 10, the answer will still be the same.

Remember Example 1: \(12\div1.5\). You moved both numbers one decimal place over, making \(120\div15\). In other words, you multiplied both dividend and divisor by 10.

\begin{align*} \frac{12}{1.5} = \frac{12 \color{red}\times 10}{1.5 \color{red}\times 10} & = \frac{120}{15}\\ \end{align*}

Example 3
\(8.52 \div 2.13\)

  1. Count the decimal places in the divisor. In this case, there are two decimal places.
  2. Count the decimal in the dividend two places to the right. Put the decimal for the answer directly above where you stopped counting.

The image two decimal places in the divisor, and two decimal places to the right for the dividend, putting the decimal for the answer directly above where you stopped counting.

Figure 5

Divide as if there are no decimals

The image shows that you should divided as if there are no decimals.

Figure 6

Thus, \(8.52 \div 2.13=4\)

Example 4
\(43.875 \div 2.1\)

  1. Count the decimal places in the divisor. In this case, there is one decimal place.
  2. Count the decimal in the dividend one place to the right, then put the decimal for the answer directly above where you stopped counting.

There is one decimal place for the divisor. There is one decimal place in the dividend where you should put the decimal for the answer directly above where you stopped counting.

Figure 7

Divide as if there are no decimals.

The number 2 is multiplied to 21 which equals to 42, then 42 is subtracted to 23 which equals to 1. The number 8 is then brought down next to number 1.

Figure 8

The number 20 is multiplied to 21 which equals to zero, zero is then subtracted from 18 which equals to 18. the number 7 is then brought down next to number 18.

Figure 9

The number 8 is multiplied to 21 which equals to 168, 168 is then subtracted from 187 which equals to 19. Number 5 is then brought down next to number 19.

Figure 10

Number 9 is multiplied to 21 which equals to 189, it is then subtracted from 195 which equals to 6.

Figure 11

Thus, \(438.75 \div 21\), rounding to the nearest hundredth, is 20.89.


Things to Remember


  • When doing the division algorithm with a decimal in the divisor:
    1. Count the decimal places in the divisor.
    2. Move the decimal in the dividend as many times as you counted in the divisor. Move the decimal up to your answer.
    3. Divide.

Practice Problems

Evaluate the following expressions:
  1. \(54 \div 1.2 = \text{?}\) (
    Solution
    x
    Solution:
    45
    )
  2. \(69 \div 2.4 = \text{?}\) (
    Solution
    x
    Solution: 28.75
    Details:
    Move the decimal to the right one space in the divisor and dividend. (This is the same as multiplying both by 10.)

    The numbers two-point-four and sixty-nine-point-zero are written on the same imaginary horizontal line but they are separated by the division symbol. The division symbol is a short vertical line connected to a long horizontal line as though they are making the top left corner of a rectangle. The decimal points in each number are highlighted and there is a curved arrow below each pointing over to the right one digit.

    The divisor is now a whole number, 24, and the dividend is 690.

    This is the same as the previous image except the arrows are gone and the decimal points have moved. The numbers are now twenty four and six hundred ninety. They are still separated by the division symbol.

    Divide 69 by 24. The answer is about 2, since \(2 \times 24\) is 48. Place the 48 below the 69 and subtract.

    This is the same as the previous image except there is a 2 above the division symbol and directly above the 9 in 690. There is also the number 48 written directly below the 6 and 9 respectively. There is a subtraction symbol written to the left of 48 and a horizontal line below the 48.

    The difference between 69 and 48 is 21. The arrow shows the next step, to bring down the zero.

    This is the same as the previous image except there is the number 21 written directly below the new horizontal line. This is the solution to 69 subtract 48. There is also an arrow pointing downward from the zero in 690.

    To continue dividing, divide 210 by 24.

    This is the same as the previous image except there is an arrow pointing straight down from the 0 in 690 and a new 0 to the right of the 21 making it 210.

    The answer is about 8.

    This is the same as the previous image except there is an 8 to the right of the 2 in the solution area, directly above the division symbol. This new 8 is in the same imaginary column as the 0 in 690 but above it and the division symbol.

    \(8 \times 24\) is 192. Place 192 below 210 and subtract.

    This is the same as the previous images except at the bottom, under the number 210, is the number 192. They are stacked according to place value. There is a subtraction symbol to the left and a horizontal line stretching across the very bottom of the numbers.

    The difference between 210 and 192 is 18. Some regrouping was required to do the subtraction.

    This is the same as the previous image except now the subtraction of 210 minus 192 is taking place. To do this, the 1 in 210 was crossed out and replaced with a 0 and the 0 in 210 was replaced with 10. Below the ones column, under the solution line is now the number 8. In order to subtract the tens column, the 2 in the hundreds column is also crossed out and replaced by a 1, and the current 0 in the tens column is now a 10. Below the tens column in the solution is the number 1. The solution for this subtraction step is the 18 below the horizontal line at the bottom.

    There is still a remainder but there are not any more digits to bring down in the dividend. Place a decimal and place a 0 in the tenths place of the dividend to have a 0 to bring down to continue the division process.

    This is the same as the previous image except now there is a point 0 to the right of the 690 in the original number under the division symbol. There is also an arrow pointing directly down from this new 0 to the right of the 18 from the previous step and a new 0 written there making it 180.

    180 divided by 24 is about 7. Place the 7 in the top next to the 8, and leave a space for the decimal point which will be inserted at the end.

    This is the same as the previous images except there is a 7 written above the division symbol, directly above the new 0 to the far right of the number 690 point 0.

    \(24 × 7 = 168\)

    Place 168 below 180 and subtract.

    This is the same as the previous image except there is now the number 168 directly under the 180 at the bottom of the problem. There is also a subtraction symbol to the left and a new horizontal line stretching under the 168.

    The difference between 180 and 168 is 12.

    This is the same as the previous image except that the 8 in 180 has been crossed out and replaced with a 7. The 0 in 180 has also changed to 10. Under the newest horizontal line, under the 10 subtract 8 is the number 2. Under the 7 subtract 6 is the number 1. This makes the newest number at the very bottom of our columns of numbers the number 12.

    To continue, place another zero in the hundredths place of the dividend, The number will now show as 690.00. Bring down the new zero.

    This image is the same as the previous image another 0 has been added to the right side of the original number 690 point 0 making it 690 point zero zero. There is also an arrow pointing straight down from this new 0 and just to the right of the 12 from the previous step. There is a new 0 there making the bottom most number 120.

    Now divide 120 by 24.

    24 goes into 120 exactly 5 times. Place a 5 next to the 7.

    Place 120 below the 120 and subtract. The difference is zero, or no remainder, so you know to stop the division process.

    This is the same as the previous image except there is now a 5 to the right of the 7 above the division symbol. There is also the number 120 written directly under the other 120 and a subtraction symbol, and a horizontal line. Below this new horizontal line are the numbers zero zero zero signifying that 120 minus 120 equals 0 for all the columns.

    The answer isn’t complete yet. Notice there is a space between 28 and 75. Place the decimal point in this space. The answer to this problem is 28.75. There’s an arrow pointing from the decimal point in 690.00 to 28.75 to indicate where to place the decimal point.

    This is the same as the previous image except there is now an arrow from the decimal point in 690 point zero zero pointing up to a new decimal point directly above it in the solution area between the 8 and the 7. This makes the final solution 28 point 75.

    Note: It is important to keep all the numbers lined up neatly in their columns when doing division problems in order to keep track of the decimal point and bring down the correct digits.
    )
  3. \(8.4 \div 6.5 = \text{?}\) (Round your answer to the nearest hundredth.) (
    Video Solution
    x
    | Transcript)
  4. \(24.7 \div 3.9 =\text{?}\) (Round your answer to the nearest hundredth.) (
    Video Solution
    x
    | Transcript)
  5. \(62.5 \div 1.75 =\text{?}\) (Round your answer to the nearest hundredth.) (
    Solution
    x
    Solution:
    35.71
    )
  6. \(25 \div 4.35 = \text{?}\) (Round your answer to the nearest hundredth.) (
    Solution
    x
    Solution: 5.75
    Details:
    Since the divisor in the problem has digits out to the hundredths place value, you need to multiply both the divisor and dividend by 100. This will make the divisor an integer but will not change the final answer of the division problem.

    There are three lines of equations. The top line is 25 divided by 4.35. The middle line is (100 times 25) divided by (100 times 4.35). The third line is 2500 divided by 435. On this third row, under the number 2500 is an arrow showing the decimal point for 25 moved two digits to the right and was filled in by two zeros. It also shows an arrow from the original location of the decimal point of 4.35, pointing two spaces to the right making it 435.

    You now have 2500 divided by 435.

    The long division symbol of a long horizontal line and a short vertical line intersecting to create the upper left hand corner of a rectangle is in this picture. Under the horizontal line is the number 2500. To the right of the symbol is the number 435. They are separated by the vertical line of the symbol.

    435 does not go into 2 or 25 or even 250.

    You need to go all the way out to 2500 before you can start making sets of 435.

    When dealing with larger numbers, sometimes you have to guess and then change your guess in order to find the correct numbers for the solution.

    You are looking for a number, when multiplied to 435, that is close to 2500 but not more than it.

    Guess #1: 7

    \(435 \times 7 = 3045\)

    Since 3045 is greater than 2500, 7 is too big. You need to guess a smaller number.

    Guess #2: 6

    \(435 \times 6 = 2610\)

    This is still more than 2500, but it is very close to it. You probably just need to go down 1 more number.

    Guess #3: 5

    \(435 \times 5 = 2175\)

    This is less than 2500 but still close to it.

    Place the 5 in the answer location above the 0 in the ones place of 2500.

    This is the same as the previous image except there is now a 5 above the division symbol and directly over the rightmost 0 of the number 2500.

    Multiply \(5 \times 435 = 2175\).

    Place 2175 below 2500 and subtract.

    This is the same as the previous image except there is now the number 2175 directly under the corresponding place value digits of the number 2500. There is also a subtraction symbol to the left and a horizontal line stretching below them.

    \(2500 - 2175 = 325\)

    Some regrouping is needed to do this subtraction.

    This is the same as the previous image except now the 5 and two zeros of 2500 are all crossed out and replaced with 4, 9, and 10 respectively. Below the new horizontal line is the number 325. It is lined up so the 3 is under the hundreds column of the previous numbers, the 2 is under the tens column, and the 5 is under the ones column.

    This means you have 325 left over, or remaining. This isn’t enough to make a group of 435 so you place a decimal point and another zero in the dividend and bring it down.

    This is the same as the previous image except there is now point 0 added to the right hand side of the original number 2500. There is also an arrow pointing directly down from this new 0 pointing to a new zero written to the right of the number 325 making the new number at the bottom of the rows 3250.

    Now find the number of times 435 goes into 3250. Again, use the guessing method.

    Guess #1: 7

    You guessed this before and know that \(435 \times 7 = 3045\). This is close to 3250 but still less than it.

    Place a 7 in the answer location next to the 5 and above the new 0.

    This is the same as the previous image except there is now a 7 above the division symbol just to the right of the 5.

    Multiply \(7 \times 435 = 3045\) and place the 3045 below the 3250 and subtract. Do any regrouping needed.

    This is the same as the previous image except now the number 3045 is positioned below the number 3250 according to place value. There is also a subtraction symbol to the left and a horizontal line below these numbers. The subtraction for this has taken place. The numbers 5 and 0 of 3250 were crossed out during the subtraction process and replaced with the numbers 4 and 10 respectively. The number 205 is now positioned according to place value, below the horizontal line, according to place value.

    You now have 205 remaining. You need another 0 to continue the division process. Place another 0 in the hundredths place of the dividend and bring it down next to the 205 making it 2050.

    This is the same as the previous image except now there is another 0 to the right of the decimal point making the original number under the division symbol 2500 point zero zero. There is also a line pointing straight down from this new 0 to a 0 at the end of our previous subtraction solution. This makes the number at the bottom of the image 2050.

    This time you guess that 435 goes into 2050 4 times.

    Place 4 above the new 0 in the dividend and next to the 7.

    Multiply \(4 \times 435 = 1740\). Place 1740 below the 2050 and subtract.

    This is the same as the previous image except now there is a 4 in the solution area above the division symbol, directly above the new 0 from the previous step. This makes the numbers in the current solution area, from left to right, 5 7 4. At the bottom, under the 2050, is the number 1740. There is a subtraction symbol to the left and a horizontal line below this new number. The 2 in the number 2050 has been crossed out and replaced by a 1 and the 0 in the hundreds column has been changed into a 10. Below the new horizontal line is the number 310.

    You still have a remainder so the process continues.

    Bring another 0 down. Guess a close multiple of 435. Multiply that guess by 435. Subtract.

    This is the same as the previous image except there is a new 0 to the far right of the original number being divided making it 2500 point zero zero zero. There is a line pointing straight down from it to a new zero at the end of the previous subtraction solution at the bottom, making it 3100. A 7 is now to the far right of the solution area making the digits in the solution area above the division symbol 5 7 4 7. Back down at the bottom, below the 3100 is the number 3045. There is a subtraction symbol to the left of this and a horizontal line below it. Under this new horizontal line is the number 65.

    Again, you still have a remainder left over. If you need to continue the division problem, then continue, but at some point it is usually sufficient to stop and round the final answer.

    In this problem, place the decimal point between the 5 and 7.

    This means the solution is close to 5.747…. The places after the decimal point continue beyond what you have solved.

    This is the same as the previous image except there is now an arrow pointing up from the decimal place in the number 2500 point zero zero zero. It is pointing to a decimal place in the solution area between the 5 and the first 7 in the numbers 5 7 4 7. This makes the solution 5 point 7 4 7.

    You have found enough digits in the solution to round to the nearest hundredth.

    5.747… rounded to the nearest hundredth is 5.75.

    The final solution is approximately: 5.75.
    )

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