{"id":428,"date":"2023-01-04T14:09:46","date_gmt":"2023-01-04T19:09:46","guid":{"rendered":"https:\/\/opentextbc.ca\/alfm4\/?post_type=chapter&p=428"},"modified":"2025-09-19T16:55:06","modified_gmt":"2025-09-19T20:55:06","slug":"multiplying-decimals","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/alfm4\/chapter\/multiplying-decimals\/","title":{"raw":"Topic A: Multiplying Decimals","rendered":"Topic A: Multiplying Decimals"},"content":{"raw":"Multiplying decimals uses the same method that you learned for multiplying whole numbers. Review multiplication of whole numbers.\r\n
\r\n\r\nVocabulary Review<\/strong>\r\n\r\nReview the diagram below and try to write in the explanations of the mathematical terms. You may refer to the glossary, if you wish. For right now, it is mostly important to remember what factor means.\r\n\r\n\"3\r\n\r\nProduct:<\/strong>\r\n\r\nFactors:<\/strong>\r\n\r\n<\/div>\r\n

How to Multiply Decimals<\/h1>\r\nMultiplying decimals follows almost the same steps as multiplying whole numbers. On the next few pages, you will be shown two ways to multiply decimals.\r\n

Method One<\/h2>\r\nOne method is to [pb_glossary id=\"647\"]estimate[\/pb_glossary] the [pb_glossary id=\"697\"]product[\/pb_glossary] using whole numbers to determine where the decimal goes.\r\n
\r\n

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

\r\n\r\n[latex]4.3 \u00d7 5.7[\/latex]\r\n\r\nEstimate: 4.3 \u00d7 5.7 \u2248 4 \u00d7 6 = 24\r\n\r\nThis tells us that the correct answer will be around 24 (which is two whole number places). We know that the answer will not be around 2.4 and it will not be around 240.\r\n\r\nIf we take the decimals out and just multiply the [pb_glossary id=\"631\"]digits[\/pb_glossary], the answer is 2451<\/strong>.\r\n\r\n[latex]\\begin{array}{rr}&57\\\\ \\times&43\\\\ \\hline &171\\\\+&2280\\\\ \\hline &2451\\end{array}[\/latex]\r\n\r\nThe estimate shows that the decimal point will come after two whole number places, so 4.3 \u00d7 5.7 = 24.51\r\n\r\n[latex]\\begin{array}{rr}&5.7\\\\ \\times&4.3\\\\ \\hline &171\\\\+&2280\\\\ \\hline &24.51\\end{array}[\/latex]\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]23.24 \u00d7 3.9[\/latex]\r\n\r\nEstimate: 23.24 \u00d7 3.9 \u2248 23 \u00d7 4 = 92\r\n\r\nThe answer will be around 92. It will not be around 9.2 and it will not be around 920.\r\n\r\nIf we take the decimals out and just multiply the digits, the answer is 90636.\r\n\r\nThe estimate shows that the decimal point will come after two whole number places.\r\n\r\nSo [latex]23.24\\times3.9 = 90.636[\/latex]\r\n\r\n[latex]\\begin{array}{rr}&23.24\\\\ \\times &3.9\\\\ \\hline &20916\\\\+&69720\\\\ \\hline &90.636\\end{array}[\/latex]\r\n\r\nIf the whole numbers in the question are large, you can round to the nearest ten or hundred to help you decide where to put the decimal point. This is a quick estimate.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

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

\r\n\r\n[latex]383.298 \u00d7 213.87[\/latex]\r\n\r\n383.298 \u00d7 213.87 \u2248 400 \u00d7 200 = 80000<\/span>\r\n\r\nThe answer will be around 80000<\/span>. \u00a0It will not be around 8000<\/span> or 800000<\/span>.\r\n\r\nIf we take the decimals out and just multiply the digits 383.298 by 213.87, the numerals in the product are 8197<\/span>594<\/span>326<\/span>.\r\n\r\nThe estimate shows that the whole number will go up to the ten-thousands place, which is five whole number places, so\r\n\r\n383.298 \u00d7 213.87 = 81975<\/span>.94326\u00a0 \u00a0W<\/strong>hew!<\/strong>\r\n\r\n[latex]\\begin{array}{rr}&383.298\\\\ \\times &213.87\\\\ \\hline &81975.94326\\end{array}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

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

\r\n\r\nAll the multiplying has been done already. Your task is to put the decimal point in the product by doing a whole number estimate of the question.\r\n\r\nExample<\/strong>\r\n\r\n[latex]\\begin{array}{rr}&2.8\\\\ \\times &4.3\\\\ \\hline &84\\\\+&1120\\\\\\hline&12.04\\end{array}[\/latex]\u00a0\u00a0\u00a0\u00a0 Estimate: [latex]\\begin{array}{rr}&3\\\\ \\times &4\\\\ \\hline &12\\end{array}[\/latex]\r\n
    \r\n \t
  1. [latex]\\begin{array}[t]{rr}&56.9\\\\ \\times&12.3\\\\ \\hline &1707\\\\ &11380\\\\&56900\\\\ \\hline &69987\\end{array}[\/latex]<\/li>\r\n \t
  2. [latex]\\begin{array}[t]{rr}&7.3\\\\ \\times&19.6\\\\ \\hline &14308\\end{array}[\/latex]<\/li>\r\n \t
  3. [latex]\\begin{array}[t]{rrrr}&28.7\\\\ \\times&64.97\\\\ \\hline &1864639\\end{array}[\/latex]<\/li>\r\n \t
  4. [latex]\\begin{array}[t]{rrrr}&73.2\\\\ \\times&1.6\\\\ \\hline &11712\\end{array}[\/latex]<\/li>\r\n \t
  5. [latex]\\begin{array}[t]{rrrr}&1.8\\\\ \\times&1.3\\\\ \\hline &234\\end{array}[\/latex]<\/li>\r\n \t
  6. [latex]\\begin{array}[t]{rrrr}&48.29\\\\ \\times&55.6\\\\ \\hline&2684924\\end{array}[\/latex]<\/li>\r\n \t
  7. [latex]\\begin{array}[t]{rrrr}&91.2\\\\ \\times&1.5\\\\ \\hline&13680 \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 1<\/strong>\r\n
      \r\n \t
    1. 699.87<\/li>\r\n \t
    2. 143.08<\/li>\r\n \t
    3. 1864.639<\/li>\r\n \t
    4. 117.12<\/li>\r\n \t
    5. 2.34<\/li>\r\n \t
    6. 2684.924<\/li>\r\n \t
    7. 136.80<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

      Method Two<\/h2>\r\nAnother way of locating the decimal point in the product is to look at the number of decimal places in the decimals you are multiplying.\r\n
      \r\n

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

      \r\n\r\n[latex]\\begin{array}{lll}&43.23&\\text{2 decimal places}\\\\ \\times&19.6&\\text{1 decimal place}\\\\ \\hline \\end{array}[\/latex]\r\n\r\nThen add the number of decimal places you counted above (2 + 1 = 3)\r\n\r\nThis is the number of decimal places you will have in your answer.\r\n\r\n[latex]\\begin{array}{lrl}&43.23&\\text{2 decimal places}\\\\ \\times&19.6&\\text{1 decimal place}\\\\ \\hline &847.\\mathbf{308}&\\text{3 decimal places}\\end{array}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
      \r\n

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

      \r\n\r\n[latex]\\begin{array}{lrl}&0.35&\\text{2 decimal places}\\\\ \\times&0.47&\\text{2 decimal places}\\\\ \\hline &0.\\mathbf{1645}&\\text{4 decimal places}\\end{array}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
      \r\n

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

      \r\n\r\n[latex]\\begin{array}{lrl}&13.452&\\text{3 decimal places}\\\\ \\times&30&\\text{0 decimal places}\\\\ \\hline &403.\\mathbf{560}&\\text{3 decimal places}\\end{array}[\/latex]\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\nAgain, the multiplying has been done. Use the method of multiplying the understood [pb_glossary id=\"625\"]denominators[\/pb_glossary] to put the decimal point in the product.\r\n
        \r\n \t
      1. [latex]\\begin{array}{lrl}&9.2&\\text{1 decimal place}\\\\ \\times&0.3&\\text{1 decimal place}\\\\ \\hline &2.76&\\text{2 decimal places}\\end{array}[\/latex]<\/li>\r\n \t
      2. [latex]\\begin{array}{lrl}&0.27&\\text{2 decimal places}\\\\ \\times&1.8&\\text{1 decimal place}\\\\ \\hline &0.486&\\text{3 decimal places}\\end{array}[\/latex]<\/li>\r\n \t
      3. [latex]\\begin{array}{rr}&6.8\\\\ \\times&0.4\\\\ \\hline &272\\end{array}[\/latex]<\/li>\r\n \t
      4. [latex]\\begin{array}{rr}&2.4\\\\ \\times &0.13\\\\ \\hline &0312\\end{array}[\/latex]<\/li>\r\n \t
      5. [latex]\\begin{array}{rr}&240\\\\ \\times &0.05\\\\ \\hline &1200\\end{array}[\/latex]<\/li>\r\n \t
      6. [latex]\\begin{array}{rr}&0.91\\\\ \\times&6.2\\\\ \\hline &5642\\end{array}[\/latex]<\/li>\r\n \t
      7. [latex]\\begin{array}{rr}&0.29\\\\ \\times&1.2\\\\ \\hline &0348\\end{array}[\/latex]<\/li>\r\n \t
      8. [latex]\\begin{array}{rr}&0.87\\\\ \\times&0.19\\\\ \\hline &01653\\end{array}[\/latex]<\/li>\r\n \t
      9. [latex]\\begin{array}{rr}&7.3\\\\ \\times &3.8\\\\ \\hline & 2774\\end{array}[\/latex]<\/li>\r\n \t
      10. [latex]\\begin{array}{rr}&4.24\\\\ \\times &0.07\\\\ \\hline &02968\\end{array}[\/latex]<\/li>\r\n \t
      11. [latex]\\begin{array}{rr}&0.042\\\\ \\times &60\\\\ \\hline &2520\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 2<\/strong>\r\n
          \r\n \t
        1. 2.76<\/li>\r\n \t
        2. 0.486<\/li>\r\n \t
        3. 2.72<\/li>\r\n \t
        4. 0.312<\/li>\r\n \t
        5. 12.00<\/li>\r\n \t
        6. 5.642<\/li>\r\n \t
        7. 0.348<\/li>\r\n \t
        8. 0.1653<\/li>\r\n \t
        9. 27.74<\/li>\r\n \t
        10. 0.2968<\/li>\r\n \t
        11. 2.52<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
          \r\n

          Exercise 3<\/p>\r\n\r\n<\/header>\r\n

          \r\n\r\nIf you had trouble with the first two exercises, then get help from your instructor. Here is extra practice if you want or need it.\r\n
            \r\n \t
          1. [latex]\\begin{array}{lrl}&41&\\text{0 decimal places}\\\\ \\times&0.061&\\text{3 decimal places}\\\\ \\hline &2.501&\\text{3 decimal places}\\end{array}[\/latex]<\/li>\r\n \t
          2. [latex]\\begin{array}{lrl}&7.8&\\text{1 decimal place}\\\\ \\times&0.5&\\text{1 decimal place}\\\\ \\hline &3.90&\\text{2 decimal places total}\\end{array}[\/latex]<\/li>\r\n \t
          3. [latex]\\begin{array}{rr}&59.275\\\\ \\times&0.08\\\\ \\hline &474200\\end{array}[\/latex]<\/li>\r\n \t
          4. [latex]\\begin{array}{rr}&7.18\\\\ \\times &23.46\\\\ \\hline &1684428\\end{array}[\/latex]<\/li>\r\n \t
          5. [latex]\\begin{array}{rr}&0.84\\\\ \\times &2.1\\\\ \\hline &1764\\end{array}[\/latex]<\/li>\r\n \t
          6. [latex]\\begin{array}{rr}&18.6\\\\ \\times&0.3\\\\ \\hline &558\\end{array}[\/latex]<\/li>\r\n \t
          7. [latex]\\begin{array}{rr}&100\\\\ \\times&0.35\\\\ \\hline &3500\\end{array}[\/latex]<\/li>\r\n \t
          8. [latex]\\begin{array}{rr}&0.72\\\\ \\times&3.4\\\\ \\hline &2448\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 3<\/strong>\r\n
              \r\n \t
            1. 4.74200<\/li>\r\n \t
            2. 168.4428<\/li>\r\n \t
            3. 1.764<\/li>\r\n \t
            4. 5.58<\/li>\r\n \t
            5. 35.00<\/li>\r\n \t
            6. 2.448<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
              \r\n

              Exercise 4<\/p>\r\n\r\n<\/header>\r\n

              \r\n\r\nMultiply to find the product. Remember to put the decimal point in the correct place; you know two methods!\r\n
                \r\n \t
              1. [latex]\\begin{array}{lr}&13.8\\\\ \\times&3.9\\\\ \\hline &1242\\\\&4140\\\\ \\hline &53.82\\end{array}[\/latex]<\/li>\r\n \t
              2. [latex]\\begin{array}{rrr}&0.076\\\\ \\times&24\\\\ \\hline \\\\ \\end{array}[\/latex]<\/li>\r\n \t
              3. [latex]\\begin{array}{rrr}&2.05\\\\ \\times &1.7\\\\ \\hline \\\\ \\end{array}[\/latex]<\/li>\r\n \t
              4. [latex]\\begin{array}{rrr}&0.429\\\\ \\times &7.5\\\\ \\hline \\\\ \\end{array}[\/latex]<\/li>\r\n \t
              5. [latex]\\begin{array}{rrr}&96\\\\ \\times&0.2\\\\ \\hline \\\\ \\end{array}[\/latex]<\/li>\r\n \t
              6. [latex]\\begin{array}{rrr}&1.79\\\\ \\times&0.33\\\\ \\hline \\\\ \\end{array}[\/latex]<\/li>\r\n \t
              7. [latex]\\begin{array}{rrr}&40.1\\\\ \\times&0.9\\\\ \\hline \\\\ \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 4<\/strong>\r\n
                  \r\n \t
                1. 53.82<\/li>\r\n \t
                2. 1.824<\/li>\r\n \t
                3. 3.485<\/li>\r\n \t
                4. 3.2175<\/li>\r\n \t
                5. 19.2<\/li>\r\n \t
                6. 0.5907<\/li>\r\n \t
                7. 36.09<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

                  Prefixing Zeros<\/h1>\r\nRemember this skill?\r\n