{"id":450,"date":"2023-01-05T17:30:04","date_gmt":"2023-01-05T22:30:04","guid":{"rendered":"https:\/\/opentextbc.ca\/alfm4\/?post_type=chapter&p=450"},"modified":"2023-03-14T16:37:33","modified_gmt":"2023-03-14T20:37:33","slug":"dividing-decimals","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/alfm4\/chapter\/dividing-decimals\/","title":{"raw":"Topic A: Dividing Decimals","rendered":"Topic A: Dividing Decimals"},"content":{"raw":"Dividing decimals uses the same method that you learned for dividing whole numbers.\r\n
\r\n\r\nVocabulary Review<\/strong>\r\n\r\n[latex]\\require{enclose}\r\n\\begin{array}{r}\r\n\\text{quotient}\\\\\r\n\\text{divisor}\\enclose{longdiv}{\\text{dividend}} \\\\\r\n\\end{array}[\/latex]\r\n\r\ndividend \u00f7 divisor = quotient\r\n\r\nUsing the above diagram, write the definitions.\r\n\r\nDivisor:\r\n\r\nDividend:\r\n\r\nQuotient:\r\n\r\n<\/div>\r\nRemember to use zeros to hold the places in the [pb_glossary id=\"703\"]quotient[\/pb_glossary] if there is no other [pb_glossary id=\"631\"]digit[\/pb_glossary].\r\n

[latex]\\begin{array}{r}\\downarrow \\phantom{0}\\\\1\\mathbf{0}4\\\\9\\enclose{longdiv}{936} \\\\9\\downarrow \\;\\;\\\\\\hline 03\\phantom{0}\\\\0\\downarrow\\\\\\hline 36\\\\36\\\\ \\hline0\\end{array}[\/latex]<\/p>\r\n\r\n

Where Do You Place the Decimal Point?<\/h1>\r\nDividing decimals follows almost the same steps as dividing whole numbers. Here you will be shown two ways to figure out where to place the decimal point.\r\n

Method One<\/h2>\r\nOne way is to [pb_glossary id=\"647\"]estimate[\/pb_glossary] the [pb_glossary id=\"703\"]quotient[\/pb_glossary] using rounded whole numbers.\r\n
\r\n

Example A<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\n[latex]17.7 \\div3[\/latex]\r\n\r\nEstimate: 17.7 \u00f7 3 \u2248 18 \u00f7 3 = 6\r\n\r\nThis tells us that the correct answer will be around 6 (which is one whole number place).\r\n\r\nWe know that the answer will not be around 0.6 and it will not be around 60.\r\n\r\nIf we take the decimals out and just divide the digits, the answer is 59.\r\n\r\n[latex]\\begin{array}{r}59\\\\3\\enclose{longdiv}{177}\\end{array}[\/latex]\r\n\r\nThe estimate shows that the the decimal point will come after one whole number.\r\n\r\n17.7 \u00f7 3 = 5.9\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example B<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\n[latex]137.88\\div18[\/latex]\r\n\r\nEstimate: 137.88 \u00f7 18 \u2248 140 \u00f7 20 = 7\r\n\r\nThe answer will be around 7. It will not be around 0.7 or 70 or 700.\r\n\r\nIf we take the decimals out and just divide the digits, the answer is 7.66.\r\n\r\n[latex]\\begin{array}{r}7.66\\\\18\\enclose{longdiv}{137.88}\\end{array}[\/latex]\r\n\r\nThe estimate shows that the the decimal point will come after one whole number.\r\n\r\n137.88 \u00f7 18 = 7.66\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n\r\nTo check the accuracy of your division, multiply the [pb_glossary id=\"703\"]quotient[\/pb_glossary] by the [pb_glossary id=\"639\"]divisor[\/pb_glossary].\r\n\r\ndividend \u00f7 divisor<\/span><\/strong> = quotient <\/strong><\/span>\r\n\r\nquotient<\/strong><\/span> \u00d7 divisor<\/span><\/strong>\u00a0= dividend\r\n\r\nThe [pb_glossary id=\"697\"]product[\/pb_glossary] will equal the [pb_glossary id=\"637\"]dividend[\/pb_glossary] if your arithmetic is correct.\r\n\r\n23.72 \u00f7 8 = 2.965\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 [latex]\\begin{array}{rr}&2.965\\\\ \\times&8\\\\ \\hline &23.720\\end{array}[\/latex]\r\n\r\n<\/div>\r\n
\r\n

Exercise 1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nThe division has been done. Your task is to put the decimal point in the quotient by doing a whole number estimate of the question.\r\n
    \r\n \t
  1. [latex]\\begin{array}{r}10.8\\\\2\\enclose{longdiv}{21.6}\\\\ \\end{array}[\/latex] [latex]\\left(\\begin{array}{r}10\\\\2\\enclose{longdiv}{20}\\\\ \\end{array}\\right)[\/latex]<\/li>\r\n \t
  2. [latex]\\begin{array}{r}1.3\\\\8\\enclose{longdiv}{10.4}\\\\ \\end{array}[\/latex] [latex]\\left(\\begin{array}{r}1\\\\8\\enclose{longdiv}{10}\\\\ \\end{array}\\right)[\/latex]<\/li>\r\n \t
  3. [latex]\\begin{array}{r}236\\\\6\\enclose{longdiv}{14.16}\\\\ \\end{array}[\/latex]<\/li>\r\n \t
  4. [latex]\\begin{array}{r}53\\\\7\\enclose{longdiv}{37.4}\\\\ \\end{array}[\/latex]<\/li>\r\n \t
  5. [latex]\\begin{array}{r}434\\\\4\\enclose{longdiv}{173.6}\\\\ \\end{array}[\/latex]<\/li>\r\n \t
  6. [latex]\\begin{array}{r}345\\\\5\\enclose{longdiv}{17.25}\\\\ \\end{array}[\/latex]<\/li>\r\n \t
  7. [latex]\\begin{array}{r}864\\\\7\\enclose{longdiv}{60.48}\\\\ \\end{array}[\/latex]<\/li>\r\n \t
  8. [latex]\\begin{array}{r}182\\\\6\\enclose{longdiv}{10.92}\\\\ \\end{array}[\/latex]<\/li>\r\n \t
  9. [latex]\\begin{array}{r}3369\\\\2\\enclose{longdiv}{6.738}\\\\ \\end{array}[\/latex]<\/li>\r\n \t
  10. [latex]\\begin{array}{r}18\\\\37\\enclose{longdiv}{66.6}\\\\ \\end{array}[\/latex]<\/li>\r\n \t
  11. [latex]\\begin{array}{r}243\\\\18\\enclose{longdiv}{43.74}\\\\ \\end{array}[\/latex]<\/li>\r\n \t
  12. [latex]\\begin{array}{r}43\\\\21\\enclose{longdiv}{90.3}\\\\ \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 1<\/strong>\r\n
      \r\n \t
    1. 10.8<\/li>\r\n \t
    2. 1.3<\/li>\r\n \t
    3. 2.36<\/li>\r\n \t
    4. 5.3<\/li>\r\n \t
    5. 43.4<\/li>\r\n \t
    6. 3.45<\/li>\r\n \t
    7. 8.64<\/li>\r\n \t
    8. 1.82<\/li>\r\n \t
    9. 3.369<\/li>\r\n \t
    10. 1.8<\/li>\r\n \t
    11. 2.43<\/li>\r\n \t
    12. 4.3<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

      Method Two (If the Divisor is a Whole Number)<\/h2>\r\nHave you found the shortcut?\r\n\r\nIf the divisor is a whole number, put the decimal point in the quotient right above the decimal point in the dividend.\r\n\r\nThen just go ahead and divide, ignoring the decimal point all together.\r\n
      \r\n

      Example C<\/p>\r\n\r\n<\/header>\r\n

      \r\n\r\n[latex]18.45\\div9 =[\/latex]\r\n\r\n[latex]\\begin{array}{r}\\mathbf{.}\\phantom{45}\\\\9\\enclose{longdiv}{18\\mathbf{.}45}\\\\ \\end{array}[\/latex]\r\n\r\n\"18.45\r\n\r\n<\/div>\r\n<\/div>\r\n
      \r\n

      Example D<\/p>\r\n\r\n<\/header>\r\n

      \r\n\r\n[latex]420.75\\div25 =[\/latex]\r\n\r\n[latex]\\begin{array}{r}\\mathbf{.}\\phantom{75}\\\\25\\enclose{longdiv}{420\\mathbf{.}75}\\\\ \\end{array}[\/latex]\r\n\r\n\"420.25\r\n\r\n<\/div>\r\n<\/div>\r\n
      \r\n

      Exercise 2<\/p>\r\n\r\n<\/header>\r\n

      \r\n\r\nFind the quotients. Check the answer by multiplying the quotient by the divisor.\r\n\r\nExample<\/strong>\r\n\r\n[latex]\\begin{array}[t]{r}0.24\\\\60\\enclose{longdiv}{14.40}\\\\ 12\\downarrow\\phantom{0}\\\\ \\hline24\\downarrow\\\\ 240\\\\ \\hline0\\end{array}[\/latex]\u00a0\u00a0\u00a0\u00a0 Check:<\/strong> [latex]\\begin{array}[t]{r}&0.24\\\\ \\times &60\\\\ \\hline &14.40\\end{array}[\/latex]\r\n
        \r\n \t
      1. [latex]91\\enclose{longdiv}{263.9}[\/latex]<\/li>\r\n \t
      2. [latex]25\\enclose{longdiv}{207.5}[\/latex]<\/li>\r\n \t
      3. [latex]57\\enclose{longdiv}{96.9}[\/latex]<\/li>\r\n \t
      4. [latex]63\\enclose{longdiv}{176.4}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 2<\/strong>\r\n
          \r\n \t
        1. 2.9<\/li>\r\n \t
        2. 8.3<\/li>\r\n \t
        3. 1.7<\/li>\r\n \t
        4. 2.8<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

          Method 3 (If the Divisor is a Decimal)<\/h2>\r\nThe past section taught us that if the divisor is a whole number, we put the decimal point in the quotient right above the decimal point in the dividend. Then we just go ahead and divide.\r\n\r\nBut what if the divisor has a decimal in it? A decimal divisor must be changed to a whole number before we can divide.\r\n
          \r\n\r\nRemember:<\/strong>\r\n