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Chapter 4—Decimals

4-7 Dividing Decimals

 

When you complete the work for this section, you should be able to:
  • Demonstrate an ability to divide decimal numbers without making errors.
Say 'Good-Bye' to Remainders. Recall that you work with dividing whole numbers turned up remainder terms.  You would say, for example, that 17 5 = 3 R 2, or "seventeen divided by five equals three with a remainder of two." In this section, you are going to find that the remainders can be converted into decimal fractions. This way, the result of such division operation is a far more useful number value than a whole number with a remainder.  You will find, for example, that 17 5 = 3.4, or "seventeen divided by five equals three-point-four."

fig050702.jpg (8934 bytes)
Review the names of the parts of a division operation.

The procedures for dividing decimal fractions is almost the same as dividing whole numbers. The only difference is that we have to deal with the decimal points.  Examples of whole numbers are 2, 4, 5, 89, 150. But when dividing two whole numbers, the result (quotient)  is not always a whole number. Dividing 5 by 2, for example is 2, and dividing 2 by 6 is 1/3. You should know that already from your work with fractions; however, you should also be able to express the results in terms of decimals.

Of course you could take the fractions that turn up in the quotient and convert them to decimals:

From your work with fractions, you should know that 5 2 = 2
Then you should know how to convert the mixed number to a decimal value: 2 = 2.5
So the problem can be expressed this way:
5 2 = 2.5

Examples

Make sure you understand the answers in this examples.

10 4 = 2.5 9 4 = 2.25 2 )25 = 12.5 33 8 = 4.124 8 )49 = 6.125

Placing the Decimal Point -- Divisor is a Whole Number

One of the important rules for dividing decimal values is about locating the decimal point in the quotient, or answer.

Rule

The decimal point in the quotient for division problems is always located directly over the decimal point in the divisor.

This applies only when the divisor is a whole-number value.

Examples: 

     6.1
5 )30.5
     42.5
9 )382.5
       18.02
25 )450.5
            3.105
350 )1086.720    
(rounded to the nearest thousandth)

In those instances when there there is no decimal point in the divisor, you can actually do the long division, totally ignoring any decimal point in the dividend. This is exactly like dividing whole number values. But when you've completing the basic division, remember to fix the decimal point in the answer directly over the decimal point in the dividend.

Note

We are not showing every step involved in doing these long-division problems. You have already mastered that process in an earlier  lesson, and there no need for illustrating all those steps here. The important issue here is placement of the decimal point.

Examples and Exercises

Placing the Decimal Point

Rewrite the solution to the given division problems, showing the location of the decimal point.

Placing the Decimal Point -- Divisor Has a Decimal Part

Here is a multiplication problem where the dividend has no decimal part. In the previous topic in this lesson, you know you proceed with the division operations, irrespective of the decimal points, then place the decimal point in your answer directly over the one in the dividend. Simple.

       18.02
25 )450.5

But what if the divisor in this example is changed to 2.5? The quotient will actually have the number combination 1802 , but where does the decimal point go?

2.5 )450.5

The "secret" to dividing by a decimal  value is straightforward: Convert the divisor to a whole-number value and then divide.

And exactly how do you go about doing that? In this example, multiplying the divisor by a factor of 10 eliminates the decimal part:

2.5 x 10 = 25

But you know it is essential to keep the numbers balanced in such problems, so in this instance you also have to multiply the divident by a factor of 10:

450.5 x 10 = 4505

Now the original problem looks like this:

25 )4505

You know what do to from here:

       180.2
25 )4505

Examples

  1. Suppose the divisor is 34.5. Multiplying by 10, changes it to 345 -- a whole-number value
     
  2. Suppose the divisor is 10.455. Multiplying by 10 three times (same as multiplying by 1000) changes 10.455  to 1455
     
  3. Suppose the divisor is 0.00025. Multiplying by 10 five times (as a multiplying by 100,000) changes  0.00025 to 25

You probably know by now that you cannot do something to one value in an expression without doing something else to compensate for the change. In this instance, whenever you multiply the divisor by some multiple of ten, you must  do the same to the divisor.

Rule

After converting the divisor to a whole-number value, you must multiply the dividend by the same value.

This step preserves the ratio and does not affect the answer.

Examples

55.4 )830.52

Multiply by 10:

554 )8305.2

     9.09 )382.5

Multiply by 100:

 909 )38250.

       0.0025 )4.505

Multiply by 10,000:

25 )45050.

           35.0 )1086.72

There is no need to adjust the dividend because 35.0 is the same as 35. However there is nothing technically wrong with multiplying by 10:

350 )10867.2

 

Procedure

Step 1: Multiply both the divisor and dividend by 10 until the divisor is a whole-number value.
Step 2: Perform the division operations without paying any attention to the decimal points.
Step 3: Place the decimal point for the quotient directly above the decimal point in the dividend.

 

Examples and Exercises

Preparing for Division

Use these interactive examples and exercises to strengthen your understanding and build your skills: Prepare these problems for division by converting the divisor to a whole-number value.

 

David L. Heiserman, Editor

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All Rights Reserved

Revised: June 06, 2015