386 CHAPTER 5.

5.4 Dividing Decimals

In this and following sections we make use of the terms divisor, dividend, quo- tient,andremainder.

Divisor, Dividend, Quotient, and Remainder. This schematic reminds readers of the position of these terms in the division process. quotient divisor) dividend ... remainder

Now that these terms are defined, we begin the discussion of division of . Suppose that we wish to divide 637 by 100. We could do this in form, change the result to a mixed fraction, then the mixed fraction to decimal form. 637 37 =6 =6.37 100 100 We can also arrange the division much as we would the division of two whole numbers. 6.37 100)637.00 600 37 0 30 0 700 700 0 Note that adding two zeros after the decimal point in the dividend does not change the value of 637. Further, note that we proceed as if we are dividing two whole numbers, placing the decimal point in the quotient directly above the decimal point in the dividend. These observations lead to the following algorithm.

Dividing a Decimal by a Whole . To divide a decimal number by a whole number, proceed as follows: 1. Set up the long division as you would the division of two whole numbers. 2. Perform the division as if the numbers were both whole numbers, adding zeros to the right of the decimal point in the dividend as necessary to complete the division. 3. Place the decimal point in the quotient immediately above the decimal point in the dividend. 5.4. DIVIDING DECIMALS 387

You Try It!

EXAMPLE 1. Divide 23 by 20. Solution. Arrange as if using long division to divide whole numbers, adding enough zeros to the right of the decimal point in the dividend to complete the division. 1.15 20)23.00 20 30 20 100 100 0 Hence, 23 divided by 20 is 1.15. 

Adding Zeros to the Right of the Decimal Point. Usually, one does not immediately see how many zeros to the right of the decimal point in the dividend are needed. These zeros are usually added at each step of the division, until the division is completed or the user is willing to terminate the process and accept only an estimate of the quotient.

You Try It!

EXAMPLE 2. Divide: 155.2 ÷ 25. Divide: 42.55 ÷ 23 Solution. Arrange as if using long division to divide whole numbers, and begin. 6.2 25)155.2 150 52 50 2 We still have a nonzero remainder. Adding another zero does no good. 6.20 25)155.20 150 52 50 20 388 CHAPTER 5. DECIMALS

However, if we add one more additional zero, the division completes with a zero remainder. 6.208 25)155.200 150 52 50 200 200 0

Answer: 1.85 Thus, 155.2 divided by 25 is 6.208. 

Decimal Divisors When the divisor contains a decimal point, we have a little work to do before we begin the division process. Suppose that we wish to divide 1.25 by 2.5. In fraction form, we could start with 1.25 , 2.5 then clear the decimals from the denominator by multiplying both numerator and denominator by 10. Note: Recall that multiplying by 10 moves the decimal point one place to the right. 1.25 1.25 · 10 = 2.5 2.5 · 10 12.5 = 25 Thus, dividing 1.25 by 2.5 is equivalent to dividing 12.5 by 25. This we know how to do. 0.5 25)12.5 12 5 0 Thus, 1.25 divided by 2.5 is 0.5.

Writing Mathematics. Never write .5. Always add the in the ones place and write 0.5.

Instead of working in fraction form, we can take care of positioning the decimal point in the long division framework. Start with: 5.4. DIVIDING DECIMALS 389

2.5)1.25 Move the decimal point in the divisor to the end of the divisor, then move the decimal point in the dividend an equal number of places.

2.5 )1.2 5 Thus, the division becomes

25)12.5 and we proceed as above to find the quotient. This discussion motivates the following algorithm.

Dividing by a Decimal Divisor. If the divisor contains a decimal, proceed as follows: 1. Move the decimal to the end of the divisor. 2. Move the decimal in the dividend an equal number of places.

We can then complete the division using the rules for dividing a decimal by a whole number. You Try It!

EXAMPLE 3. Divide: 0.36)4.392 Divide: 0.45)36.99 Solution. Move the decimal in the divisor to the end of the divisor. Move the decimal in the dividend an equal number of places (two places) to the right.

0.36 )4.39 2

Now we can follow the algorithm for dividing a decimal number by a whole number. 12.2 36)439.2 36 79 72 72 72 0 390 CHAPTER 5. DECIMALS

Answer: 82.2 Thus, 4.392 divided by 0.36 is 12.2. 

Dividing Signed Decimal Numbers The rules governing division of signed decimal numbers are identical to the rules governing division of .

Like Signs. The quotient of two decimal numbers with like signs is positive. That is: (+) (−) =+ and =+ (+) (−) Unlike Signs. The quotient of two decimal numbers with unlike signs is negative. That is: (+) (−) = − and = − (−) (+)

You Try It!

Divide: −0.0113 ÷ 0.05 EXAMPLE 4. Divide: −0.03 ÷ 0.024. Solution. First, divide the magnitudes. Move the decimal in the divisor to the end of the divisor. Move the decimal in the dividend an equal number of places (three places) to the right. Note that this requires an extra trailing zero in the dividend.

0.024 )0.030

Our problem then becomes:

24)30 We can now follow the algorithm for dividing a decimal number by a whole number. Note that we have to add two trailing zeros in the dividend to complete the division with a zero remainder. 1.25 24)30.00 24 60 48 120 120 0