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0 . 8215199 ÷ 4 . 113 size 12{0 "." "8215199" div 4 "." "113"} {}

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 4 12 3 can be checked by multiplying 4 and 3, we can check our division by performing the multiplication

4.113 3 decimal places × 0.1997373 7 decimal places

This multiplication produces 3 + 7 = 10 size 12{3+7="10"} {} decimal digits. But our suspected quotient contains only 8 decimal digits. We conclude that the answer is an approximation. Then, rounding to five decimal places, we get 0.19974.

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Practice set c

Find each quotient using a calculator. If the result is an approximation, round to four decimal places.

42 . 49778 ÷ 14 . 261 size 12{"42" "." "49778" div "14" "." "261"} {}

2.98

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0 . 001455 ÷ 0 . 291 size 12{0 "." "001455" div 0 "." "291"} {}

0.005

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7 . 459085 ÷ 2 . 1192 size 12{7 "." "459085" div 2 "." "1192"} {}

3.5197645 is an approximate result. Rounding to four decimal places, we get 3.5198

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Dividing decimals by powers of 10

In problems 4 and 5 of [link] , we found the decimal representations of 8, 162 . 41 ÷ 10 size 12{8,"162" "." "41" div "10"} {} and 8, 162 . 41 ÷ 100 size 12{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.

816.241 10 8162.410 80            ̲ 16          10          ̲ 62        60        ̲ 24      20      ̲ 41    40    ̲ 10  10  ̲

Thus, 8, 162 . 41 ÷ 10 = 816 . 241 size 12{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.

81.6241 100 8162.4100 800            ̲ 162          100          ̲ 62 4       60 0       ̲ 2 41     2 00     ̲ 410   400   ̲ 100 100 ̲ 0

Thus, 8, 162 . 41 ÷ 100 = 81 . 6241 size 12{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.

Sample set d

Find each quotient.

9, 248 . 6 ÷ 100 size 12{9,"248" "." 6 div "100"} {}
Since there are 2 zeros in this power of 10, we move the decimal point 2 places to the left.

9248.6 divided by 100 is equal to 92.480. Notice that the only effect is the movement of a decimal two places to the left of 9248.6.

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3 . 28 ÷ 10 , 000 size 12{3 "." "28" div "10","000"} {}

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.

3.28 divided by 10,000 is equal to 0.000328. Notice that the only effect is the movement of a decimal four places to the left of 0003.28.

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Practice set d

Find the decimal representation of each quotient.

182 . 5 ÷ 10 size 12{"182" "." 5 div "10"} {}

18.25

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182 . 5 ÷ 100 size 12{"182" "." 5 div "100"} {}

1.825

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182 . 5 ÷ 1, 000 size 12{"182" "." 5 div 1,"000"} {}

0.1825

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182 . 5 ÷ 10 , 000 size 12{"182" "." 5 div "10","000"} {}

0.01825

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646 . 18 ÷ 100 size 12{"646" "." "18" div "100"} {}

6.4618

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21 . 926 ÷ 1, 000 size 12{"21" "." "926" div 1,"000"} {}

0.021926

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Exercises

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

4 . 8 ÷ 3 size 12{4 "." 8÷3} {}

1.6

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16 . 8 ÷ 8 size 12{"16" "." 8÷8} {}

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18 . 5 ÷ 5 size 12{"18" "." 5÷5} {}

3.7

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12 . 33 ÷ 3 size 12{"12" "." "33"÷3} {}

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54 . 36 ÷ 9 size 12{"54" "." "36"÷9} {}

6.04

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73 . 56 ÷ 12 size 12{"73" "." "56"÷"12"} {}

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159 . 46 ÷ 17 size 12{"159" "." "46"÷"17"} {}

9.38

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12 . 16 ÷ 64 size 12{"12" "." "16"÷"64"} {}

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37 . 26 ÷ 81 size 12{"37" "." "26"÷"81"} {}

0.46

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439 . 35 ÷ 435 size 12{"439" "." "35"÷"435"} {}

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36 . 98 ÷ 4 . 3 size 12{"36" "." "98"÷4 "." 3} {}

8.6

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46 . 41 ÷ 9 . 1 size 12{"46" "." "41"÷9 "." 1} {}

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3 . 6 ÷ 1 . 5 size 12{3 "." 6÷1 "." 5} {}

2.4

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0 . 68 ÷ 1 . 7 size 12{0 "." "68"÷1 "." 7} {}

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50 . 301 ÷ 8 . 1 size 12{"50" "." "301"÷8 "." 1} {}

6.21

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2 . 832 ÷ 0 . 4 size 12{2 "." "832"÷0 "." 4} {}

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4 . 7524 ÷ 2 . 18 size 12{4 "." "7524"÷2 "." "18"} {}

2.18

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16 . 2409 ÷ 4 . 03 size 12{"16" "." "2409"÷4 "." "03"} {}

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1 . 002001 ÷ 1 . 001 size 12{1 "." "002001"÷1 "." "001"} {}

1.001

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25 . 050025 ÷ 5 . 005 size 12{"25" "." "050025"÷5 "." "005"} {}

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12 . 4 ÷ 3 . 1 size 12{"12" "." 4÷3 "." 1} {}

4

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0 . 48 ÷ 0 . 08 size 12{0 "." "48"÷0 "." "08"} {}

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30 . 24 ÷ 2 . 16 size 12{"30" "." "24"÷2 "." "16"} {}

14

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48 . 87 ÷ 0 . 87 size 12{"48" "." "87"÷0 "." "87"} {}

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12 . 321 ÷ 0 . 111 size 12{"12" "." "321"÷0 "." "111"} {}

111

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64 , 351 . 006 ÷ 10 size 12{"64","351" "." "006"÷"10"} {}

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64 , 351 . 006 ÷ 100 size 12{"64","351" "." "006"÷"100"} {}

643.51006

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64 , 351 . 006 ÷ 1, 000 size 12{"64","351" "." "006"÷1,"000"} {}

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64 , 351 . 006 ÷ 1, 000 , 000 size 12{"64","351" "." "006"÷1,"000","000"} {}

0.064351006

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0 . 43 ÷ 100 size 12{0 "." "43"÷"100"} {}

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

11 . 2944 ÷ 6 . 24 size 12{"11" "." "2944"÷6 "." "24"} {}

Actual Quotient Tenths Hundredths Thousandths
Actual Quotient Tenths Hundredths Thousandths
1.81 1.8 1.81 1.810
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45 . 32931 ÷ 9 . 01 size 12{"45" "." "32931"÷9 "." "01"} {}

Actual Quotient Tenths Hundredths Thousandths
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3 . 18186 ÷ 0 . 66 size 12{3 "." "18186"÷0 "." "66"} {}

Actual Quotient Tenths Hundredths Thousandths
Actual Quotient Tenths Hundredths Thousandths
4.821 4.8 4.82 4.821
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4 . 3636 ÷ 4 size 12{4 "." "3636"÷4} {}

Actual Quotient Tenths Hundredths Thousandths
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0 . 00006318 ÷ 0 . 018 size 12{0 "." "00006318"÷0 "." "018"} {}

Actual Quotient Tenths Hundredths Thousandths
Actual Quotient Tenths Hundredths Thousandths
0.00351 0.0 0.00 0.004
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For the following 9 problems, find each solution.

Divide the product of 7.4 and 4.1 by 2.6.

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Divide the product of 11.01 and 0.003 by 2.56 and round to two decimal places.

0.01

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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.

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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

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Six cans of cola cost $2.58. What is the price of one can?

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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

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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?

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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

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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?

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Calculator problems

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

3 . 8994 ÷ 2 . 01 size 12{3 "." "8994"÷2 "." "01"} {}

1.94

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0 . 067444 ÷ 0 . 052 size 12{0 "." "067444"÷0 "." "052"} {}

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14 , 115 . 628 ÷ 484 . 74 size 12{"14","115" "." "628"÷"484" "." "74"} {}

29.120

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219 , 709 . 36 ÷ 9941 . 6 size 12{"219","709" "." "36"÷"9941" "." 6} {}

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0 . 0852092 ÷ 0 . 49271 size 12{0 "." "0852092"÷0 "." "49271"} {}

0.173

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2 . 4858225 ÷ 1 . 11611 size 12{2 "." "4858225"÷1 "." "11611"} {}

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0 . 123432 ÷ 0 . 1111 size 12{0 "." "123432"÷0 "." "1111"} {}

1.111

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2 . 102838 ÷ 1 . 0305 size 12{2 "." "102838"÷1 "." "0305"} {}

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Exercises for review

( [link] ) Convert 4 7 8 size 12{4 { {7} over {8} } } {} to an improper fraction.

39 8

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( [link] ) 2 7 size 12{ { {2} over {7} } } {} of what number is 4 5 size 12{ { {4} over {5} } } {} ?

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( [link] ) Find the sum. 4 15 + 7 10 + 3 5 size 12{ { {4} over {"15"} } + { {7} over {"10"} } + { {3} over {5} } } {} .

47 30 or 1 17 30

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( [link] ) Round 0.01628 to the nearest ten-thousandths.

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( [link] ) Find the product (2.06)(1.39)

2.8634

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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