Dividing Decimals
Learning Outcomes
- Divide a decimal by a whole number
- Divide a decimal by another decimal
- Divide a whole number by a decimal
Remember, a multiplication problem can be rephrased as a division problem. So we can write
We can think of this as "If we divide 8 tenths into four groups, how many are in each group?" The number line below shows that there are four groups of two-tenths in eight-tenths. So .
![A number line is shown with 0, 0.2, 0.4, 0.6, 0.8, and 1. There are braces showing a distance of 0.2 between each adjacent set of 2 numbers.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/277/2017/04/24221612/CNX_BMath_Figure_05_02_001.png)
![A division problem is shown. 0.8 is on the inside of the division sign, 4 is on the outside. Above the division sign is 0.2.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/277/2017/04/24221614/CNX_BMath_Figure_05_02_004_img.png)
Divide a decimal by a whole number.
- Write as long division, placing the decimal point in the quotient above the decimal point in the dividend.
- Divide as usual.
example
Divide: SolutionWrite as long division, placing the decimal point in the quotient above the decimal point in the dividend. | ![]() |
Divide as usual. Since does not go into or we use zeros as placeholders. | ![]() |
try it
[ohm_question]146600[/ohm_question]example
In everyday life, we divide whole numbers into decimals—money—to find the price of one item. For example, suppose a case of water bottles cost $3.99. To find the price per water bottle, we would divide $3.99 by , and round the answer to the nearest cent (hundredth). Divide: $3.99\div 24Answer: Solution
$3.99\div 24 | |
Place the decimal point in the quotient above the decimal point in the dividend. | ![]() |
Divide as usual. When do we stop? Since this division involves money, we round it to the nearest cent (hundredth). To do this, we must carry the division to the thousandths place. | ![]() |
Round to the nearest cent. | $0.166\approx $0.17 |
$3.99\div 24\approx $0.17 |
try it
[ohm_question]145993[/ohm_question]Divide a Decimal by Another Decimal
So far, we have divided a decimal by a whole number. What happens when we divide a decimal by another decimal? Let’s look at the same multiplication problem we looked at earlier, but in a different way.Remember, again, that a multiplication problem can be rephrased as a division problem. This time we ask, "How many times does go into Because , we can say that goes into four times. This means that divided by is .
![A number line is shown with 0, 0.2, 0.4, 0.6, 0.8, and 1. There are braces showing a distance of 0.2 between each adjacent set of 2 numbers.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/277/2017/04/24221612/CNX_BMath_Figure_05_02_001.png)
We multiplied the numerator and denominator by and ended up just dividing by . To divide decimals, we multiply both the numerator and denominator by the same power of to make the denominator a whole number. Because of the Equivalent Fractions Property, we haven’t changed the value of the fraction. The effect is to move the decimal points in the numerator and denominator the same number of places to the right. We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign. It may help to review the vocabulary for division:
![a divided by b is shown with a labeled as the dividend and b labeled as the divisor. Then a over b is shown with a labeled as the divided and b labeled as the divisor. Then a is shown inside a division problem with b on the outside with a labeled as the dividend and b labeled as the divisor.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/277/2017/04/24221619/CNX_BMath_Figure_05_02_026_img.png)
Divide decimal numbers
- Determine the sign of the quotient.
- Make the divisor a whole number by moving the decimal point all the way to the right. Move the decimal point in the dividend the same number of places to the right, writing zeros as needed.
- Divide. Place the decimal point in the quotient above the decimal point in the dividend.
- Write the quotient with the appropriate sign.
example
Divide:Answer: Solution
Determine the sign of the quotient. | The quotient will be negative. |
Make the divisor the whole number by 'moving' the decimal point all the way to the right. 'Move' the decimal point in the dividend the same number of places to the right. | ![]() |
Divide. Place the decimal point in the quotient above the decimal point in the dividend. Add zeros as needed until the remainder is zero. | ![]() |
Write the quotient with the appropriate sign. |
try it
[ohm_question]146601[/ohm_question]example
Divide:Answer: Solution
The signs are the same. | The quotient is positive. |
Make the divisor a whole number by 'moving' the decimal point all the way to the right. 'Move' the decimal point in the dividend the same number of places. | ![]() |
Divide. Place the decimal point in the quotient above the decimal point in the dividend. | ![]() |
Write the quotient with the appropriate sign. |
try it
[ohm_question]146604[/ohm_question]example
Divide:Answer: Solution
The signs are the same. | The quotient is positive. |
Make the divisor a whole number by 'moving' the decimal point all the way to the right. Move the decimal point in the dividend the same number of places, adding zeros as needed. | ![]() |
Divide. Place the decimal point in the quotient above the decimal point in the dividend. | ![]() |
Write the quotient with the appropriate sign. |