- The Logic Behind the Method
- A Method of Dividing a Decimal By a Nonzero Whole Number
- A Method of Dividing a Decimal by a Nonzero Decimal
- Dividing Decimals by Powers of 10
Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to divide decimals. By the end of the module students should understand the method used for dividing decimals, be able to divide a decimal number by a nonzero whole number and by another, nonzero, decimal number and be able to simplify a division of a decimal by a power of 10.
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.
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
Thus,
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.
To divide a decimal by a nonzero whole number:
Find the decimal representations of the following quotients.
Thus,
Check: If
Place zeros in the tenths and hundredths positions. (See Step 3.)
Thus,
Find the following quotients.
61.5
1.884
0.0741
0.000062
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
The divisor is
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
We have converted the division
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.
To divide a decimal by a nonzero decimal,
Find the following quotients.
Thus,
Check:
Thus,
Checking by multiplying 2.1 and 0.513 will convince us that we have obtained the correct result. (Try it.)
This is now the same as the division of whole numbers.
Checking assures us that
Find the decimal representation of each quotient.
2.96
4.58
40,000
816.241
81.6241
8.16241
0.816241
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
produces
Find each quotient using a calculator. If the result is an approximation, round to five decimal places.
Display Reads | ||
Type | 12.596 | 12.596 |
Press | ÷ | 12.596 |
Type | 4.7 | 4.7 |
Press | = | 2.68 |
Since the display is not filled, we expect this to be an accurate result.
Display Reads | ||
Type | .5696376 | 0.5696376 |
Press | ÷ | 0.5696376 |
Type | .00123 | 0.00123 |
Press | = | 463.12 |
Since the display is not filled, we expect this result to be accurate.
Display Reads | ||
Type | .8215199 | 0.8215199 |
Press | ÷ | 0.8215199 |
Type | 4.113 | 4.113 |
Press | = | 0.1997373 |
There are EIGHT DIGITS — DISPLAY FILLED! BE AWARE OF POSSIBLE APPROXIMATIONS.
We can check for a possible approximation in the following way. Since the division
This multiplication produces
Find each quotient using a calculator. If the result is an approximation, round to four decimal places.
2.98
0.005
3.5197645 is an approximate result. Rounding to four decimal places, we get 3.5198
In problems 4 and 5 of Section 8, we found the decimal representations of
Thus,
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.
Thus,
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.
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.
Find each quotient.
Since there are 2 zeros in this power of 10, we move the decimal point 2 places to the left.
Since there are 4 zeros in this power of 10, we move the decimal point 4 places to the left. To do so, we need to add three zeros.
Find the decimal representation of each quotient.
18.25
1.825
0.1825
0.01825
6.4618
0.021926
For the following 30 problems, find the decimal representation of each quotient. Use a calculator to check each result.
1.6
3.7
6.04
9.38
0.46
8.6
2.4
6.21
2.18
1.001
4
14
111
643.51006
0.064351006
For the following 5 problems, find each quotient. Round to the specified position. A calculator may be used.
Actual Quotient | Tenths | Hundredths | Thousandths |
Actual Quotient | Tenths | Hundredths | Thousandths |
1.81 | 1.8 | 1.81 | 1.810 |
Actual Quotient | Tenths | Hundredths | Thousandths |
Actual Quotient | Tenths | Hundredths | Thousandths |
Actual Quotient | Tenths | Hundredths | Thousandths |
4.821 | 4.8 | 4.82 | 4.821 |
Actual Quotient | Tenths | Hundredths | Thousandths |
Actual Quotient | Tenths | Hundredths | Thousandths |
Actual Quotient | Tenths | Hundredths | Thousandths |
0.00351 | 0.0 | 0.00 | 0.004 |
For the following 9 problems, find each solution.
Divide the product of 7.4 and 4.1 by 2.6.
Divide the product of 11.01 and 0.003 by 2.56 and round to two decimal places.
0.01
Divide the difference of the products of 2.1 and 9.3, and 4.6 and 0.8 by 0.07 and round to one decimal place.
A ring costing $567.08 is to be paid off in equal monthly payments of $46.84. In how many months will the ring be paid off?
12.11 months
Six cans of cola cost $2.58. What is the price of one can?
A family traveled 538.56 miles in their car in one day on their vacation. If their car used 19.8 gallons of gas, how many miles per gallon did it get?
27.2 miles per gallon
Three college students decide to rent an apartment together. The rent is $812.50 per month. How much must each person contribute toward the rent?
A woman notices that on slow speed her video cassette recorder runs through 296.80 tape units in 10 minutes and at fast speed through 1098.16 tape units. How many times faster is fast speed than slow speed?
3.7
A class of 34 first semester business law students pay a total of $1,354.90, disregarding sales tax, for their law textbooks. What is the cost of each book?
For the following problems, use calculator to find the quotients. If the result is approximate (see Sample Set C Example 8) round the result to three decimal places.
1.94
29.120
0.173
1.111
((Reference)) Convert
((Reference))
((Reference)) Find the sum.
((Reference)) Round 0.01628 to the nearest ten-thousandths.
((Reference)) Find the product (2.06)(1.39)
2.8634
"Used as supplemental materials for developmental math courses."