6.6: Division of Decimals (2024)

The Logic Behind the Method

As we have done with addition, subtraction, and multiplication of decimals, we will study a method of division of decimals by converting them to fractions, then we will make a general rule.

We will proceed by using this example: Divide 196.8 by 6.

\(\begin{array} {r} {32\ \ \ } \\ {6\overline{)196.8}} \\ {\underline{18\ \ \ \ \ }} \\ {16\ \ \ } \\ {\underline{12\ \ \ }} \\ {4\ \ \ } \end{array}\)

We have, up to this point, divided 196.8 by 6 and have gotten a quotient of 32 with a remainder of 4. If we follow our intuition and bring down the .8, we have the division \(4.8 \div 6\).

\(\begin{array} {rcl} {4.8 \div 6} & = & {4 \dfrac{8}{10} \div 6} \\ {} & = & {\dfrac{48}{10} \div \dfrac{6}{1}} \\ {} & = & {\dfrac{\begin{array} {c} {^8} \\ {\cancel{48}} \end{array}}{10} \cdot \dfrac{1}{\begin{array} {c} {\cancel{6}} \\ {^1} \end{array}}} \\ {} & = & {\dfrac{8}{10}} \end{array}\)

Thus, \(4.8 \div 6 = .8\).

Now, our intuition and experience with division direct us to place the .8 immedi­ately to the right of 32.

From these observations, we suggest the following method of division.

A Method of Dividing a Decimal by a Nonzero Whole Number

Method of Dividing a Decimal by a Nonzero Whole Number
To divide a decimal by a nonzero whole number:

Write a decimal point above the division line and directly over the decimal point of the dividend.
Proceed to divide as if both numbers were whole numbers.
If, in the quotient, the first nonzero digit occurs to the right of the decimal point, but not in the tenths position, place a zero in each position between the decimal point and the first nonzero digit of the quotient.

A Method of Dividing a Decimal By a Nonzero Decimal

Now that we can divide decimals by nonzero whole numbers, we are in a position to divide decimals by a nonzero decimal. We will do so by converting a division by a decimal into a division by a whole number, a process with which we are already familiar. We'll illustrate the method using this example: Divide 4.32 by 1.8.

Let's look at this problem as \(4 \dfrac{32}{100} \div 1 \dfrac{8}{10}\).

\(\begin{array} {4 \dfrac{32}{100} \div 1 \dfrac{8}{10}} & = & {\dfrac{4 \dfrac{32}{100}}{1 \dfrac{8}{10}}} \\ {} & = & {\dfrac{\dfrac{432}{100}}{\dfrac{18}{10}}} \end{array}\)

The divisor is \(\dfrac{18}{10}\). We can convert \(\dfrac{18}{10}\) into a whole number if we multiply it by 10.

\(\dfrac{18}{10} \cdot 10 = \dfrac{18}{\begin{array} {c} {\cancel{10}} \\ {^1} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{10}} \end{array}}{1} = 18\)

But, we know from our experience with fractions, that if we multiply the denomina­tor of a fraction by a nonzero whole number, we must multiply the numerator by that same nonzero whole number. Thus, when converting \(\dfrac{18}{10}\) to a whole number by multiplying it by 10, we must also multiply the numerator \(\dfrac{432}{100}\) by 10.

\(\begin{array} {rcl} {\dfrac{432}{100} \cdot 10 = \dfrac{432}{\begin{array} {c} {\cancel{100}} \\ {^{10}} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{10}} \end{array}}{1}} & = & {\dfrac{432 \cdot 1}{10 \cdot 1} = \dfrac{432}{10}} \\ {} & = & {43 \dfrac{2}{10}} \\ {} & = & {43.2} \end{array}\)

We have converted the division \(4.32 \div 1.8\) into the division \(43.2 \div 18\), that's is,

\(1.8\overline{)4.32} \to 18 \overline{)43.2}\)

Notice what has occurred.

If we "move" the decimal point of the divisor one digit to the right, we must also "move" the decimal point of the dividend one place to the right. The word "move" actually indicates the process of multiplication by a power of 10.

Method of Dividing a Decimal by a Decimal NumberTo divide a decimal by a nonzero decimal,

Convert the divisor to a whole number by moving the decimal point to the position immediately to the right of the divisor's last digit.
Move the decimal point of the dividend to the right the same number of digits it was moved in the divisor.
Set the decimal point in the quotient by placing a decimal point directly above the newly located decimal point in the dividend.
Divide as usual.

Calculators

Calculators can be useful for finding quotients of decimal numbers. As we have seen with the other calculator operations, we can sometimes expect only approximate results. We are alerted to approximate results when the calculator display is filled with digits. We know it is possible that the operation may produce more digits than the calculator has the ability to show. For example, the multiplication

\(\underbrace{0.12345}_{\text{5 decimal places}} \times \underbrace{0.4567}_{\text{4 decimal places}}\)

produces \(5 + 4 = 9\) decimal places. An eight-digit display calculator only has the ability to show eight digits, and an approximation results. The way to recognize a possible approximation is illustrated in problem 3 of the next sample set.

Dividing Decimals By Powers of 10

In problems 4 and 5 of Practice Set B, we found the decimal representations of \(8,162.41 \div 10\) and \(8,162.41 \div 100\). Let's look at each of these again and then, from these observations, make a general statement regarding division of a decimal num­ber by a power of 10.

\(\begin{array} {r} {816.241} \\ {10 \overline{)8162.410}} \\ {\underline{80\ \ \ \ \ \ \ \ \ \ \ }} \\ {16\ \ \ \ \ \ \ \ \ } \\ {\underline{10\ \ \ \ \ \ \ \ \ }} \\ {62 \ \ \ \ \ \ \ } \\ {\underline{60\ \ \ \ \ \ \ }} \\ {24\ \ \ \ \ } \\ {\underline{20\ \ \ \ \ }} \\ {41\ \ \ } \\ {\underline{40\ \ \ }} \\ {10\ } \\ {\underline{10\ }} \\ {0\ } \end{array}\)

Thus, \(8,162.41 \div 10 = 816.241\)

Notice that the divisor 10 is composed of one 0 and that the quotient 816.241 can be obtained from the dividend 8,162.41 by moving the decimal point one place to the left.

\(\begin{array} {r} {81.6241} \\ {100 \overline{)8162.4100}} \\ {\underline{800\ \ \ \ \ \ \ \ \ \ \ }} \\ {162\ \ \ \ \ \ \ \ \ } \\ {\underline{100\ \ \ \ \ \ \ \ \ }} \\ {624 \ \ \ \ \ \ \ } \\ {\underline{600\ \ \ \ \ \ \ }} \\ {241\ \ \ \ \ } \\ {\underline{200\ \ \ \ \ }} \\ {410\ \ \ } \\ {\underline{400\ \ \ }} \\ {100\ } \\ {\underline{100\ }} \\ {0\ } \end{array}\)

Thus, \(8,162.41 \div 100 = 81.6241\).

Notice that the divisor 100 is composed of two 0's and that the quotient 81.6241 can be obtained from the dividend by moving the decimal point two places to the left.

Using these observations, we can suggest the following method for dividing decimal numbers by powers of 10.

Dividing a Decimal Fraction by a Power of 10
To divide a decimal fraction by a power of 10, move the decimal point of the decimal fraction to the left as many places as there are zeros in the power of 10. Add zeros if necessary.

Exercises

For the following 30 problems, find the decimal representation of each quotient. Use a calculator to check each result.

For the following 5 problems, find each quotient. Round to the specified position. A calculator may be used.

For the following 9 problems, find each solution.

Calculator Problems
For the following problems, use calculator to find the quotients. If the result is approximate (see Sample Set C) round the result to three decimal places.