{"id":1148,"date":"2023-01-16T18:23:39","date_gmt":"2023-01-16T23:23:39","guid":{"rendered":"https:\/\/opentextbc.ca\/alfm3\/?post_type=chapter&p=1148"},"modified":"2025-09-18T19:02:13","modified_gmt":"2025-09-18T23:02:13","slug":"dividing-by-two-and-three-digit-divisors","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/alfm3\/chapter\/dividing-by-two-and-three-digit-divisors\/","title":{"raw":"Topic D: Dividing by Two- and Three- Digit Divisors","rendered":"Topic D: Dividing by Two- and Three- Digit Divisors"},"content":{"raw":"

Finding Trial Quotients<\/h1>\r\nWhen dividing by 2-digit numbers, you will need to estimate the quotient. This guess is called a trial quotient.\r\n
\r\n
\r\n
\r\n

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

\r\n\r\n[latex]624 \\div 24[\/latex]\r\n\r\nRewrite as [latex]24\\enclose{longdiv}{624}[\/latex]\r\n
\r\n

Step 1: Divide <\/strong><\/p>\r\n\r\n

    \r\n \t
  1. Think: [latex]2\\enclose{longdiv}{6}[\/latex] is [latex]3[\/latex]. So [latex]24\\enclose{longdiv}{62}[\/latex] is about [latex]3[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Step 2: Multiply and subtract.\u00a0<\/strong><\/p>\r\n\r\n

      \r\n \t
    1. [latex]\\begin{array}{r}3 \\ \\ \\ \\\\ 24\\enclose{longdiv}{624}\\\\72 \\ \\ \\ \\\\ \\hline \\end{array}[\/latex]<\/li>\r\n \t
    2. Since 72 > 62, 3 is too large.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Step 3: Try a smaller number, multiply and subtract.<\/strong><\/p>\r\n\r\n<\/div>\r\n

      \r\n
        \r\n \t
      1. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 24\\enclose{longdiv}{624}\\\\48 \\ \\ \\ \\\\ \\hline 14 \\ \\ \\ \\end{array}[\/latex]<\/li>\r\n \t
      2. Since 14 < 24, 2 is correct.<\/li>\r\n<\/ol>\r\n<\/div>\r\nStep 4: Finish the problem.<\/strong>\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]630 \\div 15[\/latex]\r\n\r\nRewrite as [latex]15\\enclose{longdiv}{630}[\/latex]\r\n
        \r\n

        Step 1: Divide <\/strong><\/p>\r\n\r\n

          \r\n \t
        1. 15 rounds to 20. Think: [latex]2\\enclose{longdiv}{6}[\/latex] is [latex]3[\/latex]. So [latex]15\\enclose{longdiv}{63}[\/latex] is about [latex]3[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n
          \r\n

          Step 2: Multiply and subtract.\u00a0<\/strong><\/p>\r\n\r\n

            \r\n \t
          1. [latex]\\begin{array}{r}3 \\ \\ \\ \\\\ 15\\enclose{longdiv}{630}\\\\45 \\ \\ \\ \\\\ \\hline 18 \\ \\ \\ \\end{array}[\/latex]<\/li>\r\n \t
          2. Since 18 > 15, 3 is too small.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
            \r\n

            Step 3: Try a larger number, multiply and subtract.<\/strong><\/p>\r\n\r\n<\/div>\r\n

            \r\n
              \r\n \t
            1. [latex]\\begin{array}{r}4 \\ \\ \\ \\\\ 24\\enclose{longdiv}{630}\\\\60 \\ \\ \\\\ \\hline 3 \\ \\ \\end{array}[\/latex]<\/li>\r\n \t
            2. Since 3 < 15, 4 is correct.<\/li>\r\n<\/ol>\r\n<\/div>\r\nStep 4: Finish the problem.<\/strong>\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\n\r\nIn each question, the trial quotient is either too large or too small. Multiply. Write too large or too small on the line. Then, write the correct trial quotient beside. Check your work using the answer key at the end of the exercise.\r\n
                \r\n \t
              1. [latex]\\begin{array}{r}2 \\ \\ \\ \\\\ 25\\enclose{longdiv}{475}\\end{array}[\/latex]<\/li>\r\n \t
              2. [latex]\\begin{array}{r}3 \\ \\ \\ \\\\ 15\\enclose{longdiv}{682}\\end{array}[\/latex]<\/li>\r\n \t
              3. [latex]\\begin{array}{r}5 \\ \\ \\ \\\\ 18\\enclose{longdiv}{813}\\end{array}[\/latex]<\/li>\r\n \t
              4. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 25\\enclose{longdiv}{810}\\end{array}[\/latex]<\/li>\r\n \t
              5. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 33\\enclose{longdiv}{891}\\end{array}[\/latex]<\/li>\r\n \t
              6. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 18\\enclose{longdiv}{819}\\end{array}[\/latex]<\/li>\r\n \t
              7. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 27\\enclose{longdiv}{727}\\end{array}[\/latex]<\/li>\r\n \t
              8. [latex]\\begin{array}{r}2 \\ \\ \\ \\\\ 35\\enclose{longdiv}{652}\\end{array}[\/latex]<\/li>\r\n \t
              9. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 25\\enclose{longdiv}{650}\\end{array}[\/latex]<\/li>\r\n \t
              10. [latex]\\begin{array}{r}4\\ \\ \\ \\\\34\\enclose{longdiv}{176}\\end{array}[\/latex]<\/li>\r\n \t
              11. [latex]\\begin{array}{r}4\\ \\ \\ \\\\ 12\\enclose{longdiv}{420}\\end{array}[\/latex]<\/li>\r\n \t
              12. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 43\\enclose{longdiv}{801}\\end{array}[\/latex]<\/li>\r\n \t
              13. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 31\\enclose{longdiv}{899}\\end{array}[\/latex]<\/li>\r\n \t
              14. [latex]\\begin{array}{r}4\\ \\ \\ \\\\ 18\\enclose{longdiv}{648}\\end{array}[\/latex]<\/li>\r\n \t
              15. [latex]\\begin{array}{r}4\\ \\ \\ \\\\ 27\\enclose{longdiv}{946}\\end{array}[\/latex]<\/li>\r\n \t
              16. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 23\\enclose{longdiv}{943}\\end{array}[\/latex]<\/li>\r\n \t
              17. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 24\\enclose{longdiv}{578}\\end{array}[\/latex]<\/li>\r\n \t
              18. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 29\\enclose{longdiv}{406}\\end{array}[\/latex]<\/li>\r\n \t
              19. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 48\\enclose{longdiv}{892}\\end{array}[\/latex]<\/li>\r\n \t
              20. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 28\\enclose{longdiv}{534}\\end{array}[\/latex]<\/li>\r\n \t
              21. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 37\\enclose{longdiv}{939}\\end{array}[\/latex]<\/li>\r\n \t
              22. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 28\\enclose{longdiv}{854}\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 1<\/strong>\r\n
                  \r\n \t
                1. too large, 1<\/li>\r\n \t
                2. too small, 4<\/li>\r\n \t
                3. too large, 4<\/li>\r\n \t
                4. too small, 3<\/li>\r\n \t
                5. too large, 2<\/li>\r\n \t
                6. too small, 4<\/li>\r\n \t
                7. too large, 2<\/li>\r\n \t
                8. too large 1<\/li>\r\n \t
                9. too large, 2<\/li>\r\n \t
                10. too small, 5<\/li>\r\n \t
                11. too large, 3<\/li>\r\n \t
                12. too large, 1<\/li>\r\n \t
                13. too. large 2<\/li>\r\n \t
                14. too large, 3<\/li>\r\n \t
                15. too large, 3<\/li>\r\n \t
                16. too small, 4<\/li>\r\n \t
                17. too large, 2<\/li>\r\n \t
                18. too large, 1<\/li>\r\n \t
                19. too large, 1<\/li>\r\n \t
                20. too large 1<\/li>\r\n \t
                21. too large, 2<\/li>\r\n \t
                22. too small, 3<\/li>\r\n<\/ol>\r\n<\/div>\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]78\\enclose{longdiv}{2706}[\/latex]\r\n
                  \r\n\r\nSince 78 rounds to 80, think [latex]8\\enclose{longdiv}{27}[\/latex]. 8 goes into 27 \u2248 3. 3 would be a good trial quotient.\r\n

                  [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 78\\enclose{longdiv}{2706}\\\\234\\ \\ \\ \\\\ \\hline 36\\ \\ \\ \\end{array}[\/latex]<\/p>\r\nSince 36 < 78, 3 is a good trial quotient.\r\n\r\n<\/div>\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]27\\enclose{longdiv}{2205}[\/latex]\r\n\r\nSince<\/span> 27 rounds to 30, think [latex]3\\enclose{longdiv}{22}[\/latex]<\/span>\u00a0.<\/span> 3 goes into 22 \u2248 7. 7 would be a good trial quotient.<\/span><\/span>\r\n

                  [latex]\\begin{array}{r}7\\ \\ \\ \\\\ 27\\enclose{longdiv}{2205}\\\\189\\ \\ \\ \\\\ \\hline 31\\ \\ \\ \\end{array}[\/latex]<\/p>\r\nSince 31 > 27, so 7 is too small.\u00a0 A better trial quotient would be 8.<\/span><\/span>\u00a0<\/span>\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\n\r\nFind the first digit in the trial quotient.\u00a0 Check your work using the answer key at the end of the exercise.\r\n
                    \r\n \t
                  1. [latex]43\\enclose{longdiv}{1772}[\/latex] [latex]\\begin{array}{r}4\\\\ 4\\enclose{longdiv}{17}\\\\16\\\\ \\hline 1\\end{array}[\/latex]. 1 < 4<\/li>\r\n \t
                  2. [latex]64\\enclose{longdiv}{3276}[\/latex]<\/li>\r\n \t
                  3. [latex]28\\enclose{longdiv}{6008}[\/latex]<\/li>\r\n \t
                  4. [latex]33\\enclose{longdiv}{2731}[\/latex]<\/li>\r\n \t
                  5. [latex]59\\enclose{longdiv}{4164}[\/latex]<\/li>\r\n \t
                  6. [latex]75\\enclose{longdiv}{2420}[\/latex]<\/li>\r\n \t
                  7. [latex]54\\enclose{longdiv}{3316}[\/latex]<\/li>\r\n \t
                  8. [latex]38\\enclose{longdiv}{2759}[\/latex]<\/li>\r\n \t
                  9. [latex]46\\enclose{longdiv}{387}[\/latex]<\/li>\r\n \t
                  10. [latex]35\\enclose{longdiv}{3316}[\/latex]<\/li>\r\n \t
                  11. [latex]83\\enclose{longdiv}{7237}[\/latex]<\/li>\r\n \t
                  12. [latex]77\\enclose{longdiv}{6763}[\/latex]<\/li>\r\n \t
                  13. [latex]93\\enclose{longdiv}{3724}[\/latex]<\/li>\r\n \t
                  14. [latex]52\\enclose{longdiv}{4690}[\/latex]<\/li>\r\n \t
                  15. [latex]86\\enclose{longdiv}{2089}[\/latex]<\/li>\r\n \t
                  16. [latex]26\\enclose{longdiv}{1417}[\/latex]<\/li>\r\n \t
                  17. [latex]72\\enclose{longdiv}{1462}[\/latex]<\/li>\r\n \t
                  18. [latex]27\\enclose{longdiv}{6939}[\/latex]<\/li>\r\n \t
                  19. [latex]32\\enclose{longdiv}{7840}[\/latex]<\/li>\r\n \t
                  20. [latex]24\\enclose{longdiv}{7605}[\/latex]<\/li>\r\n \t
                  21. [latex]16\\enclose{longdiv}{8640}[\/latex]<\/li>\r\n \t
                  22. [latex]45\\enclose{longdiv}{3060}[\/latex]<\/li>\r\n \t
                  23. [latex]38\\enclose{longdiv}{2158}[\/latex]<\/li>\r\n \t
                  24. [latex]42\\enclose{longdiv}{1491}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 2<\/strong>\r\n
                      \r\n \t
                    1. 4<\/li>\r\n \t
                    2. 5<\/li>\r\n \t
                    3. 2<\/li>\r\n \t
                    4. 8<\/li>\r\n \t
                    5. 7<\/li>\r\n \t
                    6. 3<\/li>\r\n \t
                    7. 6<\/li>\r\n \t
                    8. 7<\/li>\r\n \t
                    9. 8<\/li>\r\n \t
                    10. 9<\/li>\r\n \t
                    11. 8<\/li>\r\n \t
                    12. 8<\/li>\r\n \t
                    13. 4<\/li>\r\n \t
                    14. 9<\/li>\r\n \t
                    15. 2<\/li>\r\n \t
                    16. 5<\/li>\r\n \t
                    17. 2<\/li>\r\n \t
                    18. 2<\/li>\r\n \t
                    19. 2<\/li>\r\n \t
                    20. 3<\/li>\r\n \t
                    21. 5<\/li>\r\n \t
                    22. 6<\/li>\r\n \t
                    23. 5<\/li>\r\n \t
                    24. 3<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

                      Two Digit Divisors<\/h1>\r\nDividing by large divisors is a challenge!\r\n\r\nYou must estimate <\/strong>how many times one number will divide into another. Use pencil and have an eraser close by when you do these questions. You will use the same steps that you already know.\r\n
                      \r\n

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

                      \r\n\r\n[latex]964 \\div 75 =[\/latex]\r\n
                      \r\n\r\nStep 1: Divide<\/strong>\r\n
                        \r\n \t
                      1. Does 75 go into 9? \u00a0NO<\/strong><\/li>\r\n \t
                      2. Does 75 go into 96? \u00a0YES<\/strong><\/li>\r\n \t
                      3. Estimate\r\n
                          \r\n \t
                        1. Round 75 to 80 \u2013 think \u201c8\u201d<\/li>\r\n \t
                        2. Round 96 to 100 \u2013 think \u201c10\u201d<\/li>\r\n \t
                        3. How many 8\u2019s in 10?\u00a0 (8 1 = 8, 10\u00a0 8\u00a0 =\u00a0 1)<\/li>\r\n \t
                        4. The estimate for the first digit in the trial quotient is 1.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                        5. Write 1 <\/strong>in the quotient above the 6 tens.<\/span><\/li>\r\n<\/ol>\r\n

                          [latex]\\begin{array}{r}1\\ \\ \\ \\\\ 75\\enclose{longdiv}{964}\\end{array}[\/latex]<\/p>\r\nStep 2:\u00a0 \u00a0Multiply\u00a0<\/strong>\r\n

                          [latex]1 \\times 75 = 75[\/latex]<\/p>\r\n\r\n

                            \r\n \t
                          1. Write 75 under 96.<\/li>\r\n<\/ol>\r\n

                            \u00a0 [latex]\\begin{array}{r}1\\ \\ \\ \\\\ 75\\enclose{longdiv}{964}\\\\75 \\ \\ \\\\ \\hline \\end{array}[\/latex]<\/p>\r\nStep 3:\u00a0 \u00a0Subtract\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>\r\n

                            [latex]96 - 75 = 21[\/latex]<\/p>\r\n\r\n

                              \r\n \t
                            1. Check 21 < 75? YES!<\/strong><\/li>\r\n<\/ol>\r\n

                              \u00a0 [latex]\\begin{array}{r}1\\ \\ \\ \\\\ 75\\enclose{longdiv}{964}\\\\75\\ \\ \\\\ \\hline 21 \\ \\ \\end{array}[\/latex]<\/p>\r\nStep 4: Bring down the next digit <\/strong>in the dividend. 214 is now the number to be divided.<\/span>\r\n\r\n<\/div>\r\n

                              \r\n

                              \u00a0 [latex]\\begin{array}{r}1\\ \\ \\ \\\\ 75\\enclose{longdiv}{964}\\\\75 \\downarrow \\\\ \\hline 214\\end{array}[\/latex]<\/p>\r\nREPEAT<\/strong>\r\n\r\n \r\n\r\nStep 1:\u00a0\u00a0 Divide<\/strong>\r\n

                                \r\n \t
                              1. Does 75 go into 214? \u00a0YES<\/strong><\/li>\r\n \t
                              2. Estimate 75 as 80 \u2013 think \u201c8\u201d\r\n
                                  \r\n \t
                                1. Estimate 214 as 200 \u2013 think \u201c20\u201d<\/li>\r\n \t
                                2. 8 goes into 20 2 times ( 8 2 = 16, so 20\u00a0 8 2)<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                                3. The estimate for the second digit in the trial quotient is 2. Write 2 in the quotient above the 4 in the dividend.<\/li>\r\n<\/ol>\r\n

                                  \u00a0 [latex]\\begin{array}{r}12\\\\ 75\\enclose{longdiv}{964}\\\\75 \\downarrow \\\\ \\hline 214\\end{array}[\/latex]<\/p>\r\nStep 2:\u00a0 \u00a0Multiply<\/strong>\r\n

                                  [latex]2 \\times 75 = 150[\/latex]<\/p>\r\n\r\n

                                    \r\n \t
                                  1. Write 150 under the 214.<\/li>\r\n<\/ol>\r\n

                                    \u00a0 [latex]\\begin{array}{r}12\\\\ 75\\enclose{longdiv}{964}\\\\75 \\downarrow \\\\ \\hline 214 \\\\ 150 \\\\ \\hline\\end{array}[\/latex]<\/p>\r\nStep 3:\u00a0\u00a0 Subtract and check <\/strong>that the remainder is less than the divisor.\r\n

                                    \u00a0 [latex]\\begin{array}{r}12\\\\ 75\\enclose{longdiv}{964}\\\\75 \\downarrow \\\\ \\hline 214 \\\\ 150 \\\\ \\hline 64\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n \r\n\r\nStep 4:\u00a0\u00a0 Bring down <\/strong>\u2013 no more digits in dividend.\r\n\r\nTo check your answer\r\n

                                    [latex]\\begin{array}{rrr}&75\\\\\\times&12\\\\\\hline&150 \\\\ &750 \\\\ \\hline &900 \\\\ + &64* \\\\ \\hline &964\\end{array}[\/latex]<\/p>\r\n*Don\u2019t forget to add the remainder<\/strong> when you check.\r\n\r\n<\/div>\r\n<\/div>\r\n

                                    \r\n

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

                                    \r\n\r\nCarefully divide these questions.\u00a0 Be careful to keep the hundreds in line with the hundreds, the tens with the tens, and so on. You might want to use squared paper for long division. Check your work using the answer key at the end of the exercise.\r\n\r\nIf you are having any <\/strong>difficulty, ask your instructor to watch you doing a few questions to be sure you are using a correct method.\r\n
                                      \r\n \t
                                    1. [latex]10\\enclose{longdiv}{720}[\/latex]<\/li>\r\n \t
                                    2. [latex]12\\enclose{longdiv}{564}[\/latex]<\/li>\r\n \t
                                    3. [latex]21\\enclose{longdiv}{882}[\/latex]<\/li>\r\n \t
                                    4. [latex]22\\enclose{longdiv}{946}[\/latex]<\/li>\r\n \t
                                    5. [latex]32\\enclose{longdiv}{1632}[\/latex]<\/li>\r\n \t
                                    6. [latex]23\\enclose{longdiv}{943}[\/latex]<\/li>\r\n \t
                                    7. [latex]62\\enclose{longdiv}{2528}[\/latex]<\/li>\r\n \t
                                    8. [latex]71\\enclose{longdiv}{2414}[\/latex]<\/li>\r\n \t
                                    9. [latex]24\\enclose{longdiv}{7578}[\/latex]<\/li>\r\n \t
                                    10. [latex]82\\enclose{longdiv}{2958}[\/latex]<\/li>\r\n \t
                                    11. [latex]18\\enclose{longdiv}{6250}[\/latex]<\/li>\r\n \t
                                    12. [latex]25\\enclose{longdiv}{1550}[\/latex]<\/li>\r\n \t
                                    13. [latex]19\\enclose{longdiv}{9595}[\/latex]<\/li>\r\n \t
                                    14. [latex]47\\enclose{longdiv}{3854}[\/latex]<\/li>\r\n \t
                                    15. [latex]58\\enclose{longdiv}{6500}[\/latex]<\/li>\r\n \t
                                    16. [latex]24\\enclose{longdiv}{9648}[\/latex]<\/li>\r\n \t
                                    17. [latex]49\\enclose{longdiv}{1312}[\/latex]<\/li>\r\n \t
                                    18. [latex]67\\enclose{longdiv}{7683}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 3<\/strong>\r\n
                                        \r\n \t
                                      1. 72<\/li>\r\n \t
                                      2. 47<\/li>\r\n \t
                                      3. 42<\/li>\r\n \t
                                      4. 43<\/li>\r\n \t
                                      5. 51<\/li>\r\n \t
                                      6. 41<\/li>\r\n \t
                                      7. 40 R48<\/li>\r\n \t
                                      8. 34<\/li>\r\n \t
                                      9. 315 R18<\/li>\r\n \t
                                      10. 36 R6<\/li>\r\n \t
                                      11. 347 R4<\/li>\r\n \t
                                      12. 62<\/li>\r\n \t
                                      13. 505<\/li>\r\n \t
                                      14. 82<\/li>\r\n \t
                                      15. 112 R4<\/li>\r\n \t
                                      16. 402<\/li>\r\n \t
                                      17. 26 R38<\/li>\r\n \t
                                      18. 114 R45<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nIf the estimate for your trial quotient is too large, the result of the multiplication will be larger than the numbers in the dividend.<\/strong>\r\n\r\nTake, for example, the following division question:\r\n

                                        [latex]23\\enclose{longdiv}{784}[\/latex]<\/p>\r\nWhen you divide<\/strong>, the trial estimate is 4:\r\n

                                        [latex]\\begin{array}{r}4\\ \\ \\ \\\\ 23\\enclose{longdiv}{784}\\end{array}[\/latex]<\/p>\r\nWhen you multiply<\/strong>:\r\n

                                          \r\n \t
                                        1. [latex]4 \\times 23 = 92[\/latex]<\/li>\r\n \t
                                        2. 92 is larger than 78, so 4 is too large an estimate. Erase it. Try 3. [latex]3 \\times 23 = 69[\/latex]<\/li>\r\n \t
                                        3. 3 is the correct estimate and you can complete the division, as seen below:<\/li>\r\n<\/ol>\r\n

                                          [latex]\\begin{array}{l}\\hspace{2.3em}34\\text{ R}2\\\\23 \\enclose{longdiv}{784}\\\\\\hspace{1.5em}69 \\downarrow\\\\\\hline\\hspace{2.3em}94\\\\\\hspace{2.3em}92\\\\\\hline\\hspace{2.8em}2\\end{array}[\/latex]<\/p>\r\nIf the estimate is too small, the result of the subtraction will be larger than the divisor.<\/strong>\r\n\r\nTake this division question as an example:\r\n

                                          [latex]72\\enclose{longdiv}{4487}[\/latex]<\/p>\r\nSay your trial quotient is 5. When you multiply, you get [latex]5 \\times 72 = 360[\/latex]. Since 360 is smaller than 448, you can go on to subtract:\r\n

                                          [latex]448 - 360=88[\/latex]<\/p>\r\nIs 88 less than 72?\u00a0NO<\/strong>, 88 is greater than 72. This means that 5 is too small for the trial quotient. Use the larger number 6.\r\n\r\nMultiply [latex]6 \\times 72 = 432[\/latex]. Subtract [latex]448 - 432 = 16[\/latex]. Since 16 is less than 72, you can continue the division.\r\n\r\nSee the full division process below:\r\n

                                          [latex]\\begin{array}{l}\\hspace{1.1cm}62\\text{ R}23\\\\72 \\enclose{longdiv}{4487}\\\\\\hspace{1.45em}432 \\downarrow\\\\\\hline\\hspace{2.1em}167\\\\\\hspace{2.1em}144\\\\\\hline\\hspace{2.55em}23\\end{array}[\/latex]<\/p>\r\n\r\n

                                          \r\n

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

                                          \r\n\r\nDivide and check your work by multiplying.\u00a0 Check your work using the answer key at the end of the exercise.\r\n
                                            \r\n \t
                                          1. [latex]18\\enclose{longdiv}{648}[\/latex]<\/li>\r\n \t
                                          2. [latex]26\\enclose{longdiv}{6766}[\/latex]<\/li>\r\n \t
                                          3. [latex]52\\enclose{longdiv}{1968}[\/latex]<\/li>\r\n \t
                                          4. [latex]84\\enclose{longdiv}{8640}[\/latex]<\/li>\r\n \t
                                          5. [latex]72\\enclose{longdiv}{2883}[\/latex]<\/li>\r\n \t
                                          6. [latex]94\\enclose{longdiv}{8126}[\/latex]<\/li>\r\n \t
                                          7. [latex]20\\enclose{longdiv}{4060}[\/latex]<\/li>\r\n \t
                                          8. [latex]47\\enclose{longdiv}{1728}[\/latex]<\/li>\r\n \t
                                          9. [latex]33\\enclose{longdiv}{1886}[\/latex]<\/li>\r\n \t
                                          10. [latex]25\\enclose{longdiv}{5750}[\/latex]<\/li>\r\n \t
                                          11. [latex]79\\enclose{longdiv}{2765}[\/latex]<\/li>\r\n \t
                                          12. [latex]42\\enclose{longdiv}{8442}[\/latex]<\/li>\r\n \t
                                          13. [latex]57\\enclose{longdiv}{9144}[\/latex]<\/li>\r\n \t
                                          14. [latex]96\\enclose{longdiv}{20160}[\/latex]<\/li>\r\n \t
                                          15. [latex]75\\enclose{longdiv}{23550}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 4<\/strong>\r\n
                                              \r\n \t
                                            1. 36<\/li>\r\n \t
                                            2. 260 R6<\/li>\r\n \t
                                            3. 37 R44<\/li>\r\n \t
                                            4. 102 R72<\/li>\r\n \t
                                            5. 40 R3<\/li>\r\n \t
                                            6. 86 R42<\/li>\r\n \t
                                            7. 203<\/li>\r\n \t
                                            8. 36 R36<\/li>\r\n \t
                                            9. 57 R5<\/li>\r\n \t
                                            10. 230<\/li>\r\n \t
                                            11. 35<\/li>\r\n \t
                                            12. 201<\/li>\r\n \t
                                            13. 160 R24<\/li>\r\n \t
                                            14. 210<\/li>\r\n \t
                                            15. 314<\/li>\r\n \t
                                            16. <\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

                                              Dividing by 10, 100, 1000...<\/h1>\r\n

                                              What is the pattern?\u00a0 When you divide by 10<\/h2>\r\n
                                                \r\n \t
                                              1. \u00a0<\/strong>\u00a0The ones digit in the dividend becomes the remainder.<\/li>\r\n<\/ol>\r\n

                                                [latex]10\\enclose{longdiv}{324} = 32 R4[\/latex]<\/p>\r\n\r\n

                                                  \r\n \t
                                                1. The other numbers in the dividend stay the same, but each digit is one place value less.<\/li>\r\n<\/ol>\r\n
                                                    \r\n \t
                                                  • \r\n
                                                      \r\n \t
                                                    • the hundreds become tens<\/li>\r\n \t
                                                    • the tens become ones<\/li>\r\n \t
                                                    • the ones become the remainder<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n
                                                      \r\n

                                                      Exercise 5<\/p>\r\n\r\n<\/header>\r\n

                                                      \r\n\r\nFind the quotients. Look for the pattern. Check your work using the answer key at the end of the exercise.\r\n
                                                        \r\n \t
                                                      1. [latex]10\\enclose{longdiv}{46}[\/latex]<\/li>\r\n \t
                                                      2. [latex]10\\enclose{longdiv}{75}[\/latex]<\/li>\r\n \t
                                                      3. [latex]10\\enclose{longdiv}{136}[\/latex]<\/li>\r\n \t
                                                      4. [latex]10\\enclose{longdiv}{832}[\/latex]<\/li>\r\n \t
                                                      5. [latex]10\\enclose{longdiv}{674}[\/latex]<\/li>\r\n \t
                                                      6. [latex]10\\enclose{longdiv}{952}[\/latex]<\/li>\r\n \t
                                                      7. [latex]10\\enclose{longdiv}{2457}[\/latex]<\/li>\r\n \t
                                                      8. [latex]10\\enclose{longdiv}{3685}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 5<\/strong>\r\n
                                                          \r\n \t
                                                        1. 4 R6<\/li>\r\n \t
                                                        2. 7 R5<\/li>\r\n \t
                                                        3. 13 R6<\/li>\r\n \t
                                                        4. 83 R2<\/li>\r\n \t
                                                        5. 67 R4<\/li>\r\n \t
                                                        6. 95 R2<\/li>\r\n \t
                                                        7. 245 R7<\/li>\r\n \t
                                                        8. 368 R5<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

                                                          When you divide by 100<\/h2>\r\n
                                                            \r\n \t
                                                          1. \u00a0The ones and tens digits in the dividend become the remainder.<\/li>\r\n \t
                                                          2. The other digits in the dividend stay the same but each digit is two <\/strong>places less.\r\n
                                                              \r\n \t
                                                            • the thousands become tens<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n
                                                                \r\n \t
                                                              • \r\n
                                                                  \r\n \t
                                                                • the hundreds become ones<\/li>\r\n \t
                                                                • the tens and ones become the remainder<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n
                                                                  \r\n

                                                                  Exercise 6<\/p>\r\n\r\n<\/header>\r\n

                                                                  \r\n\r\nFind these quotients. Look for the pattern when you divide. Check your work using the answer key at the end of the exercise\r\n
                                                                    \r\n \t
                                                                  1. [latex]100\\enclose{longdiv}{386}[\/latex]<\/li>\r\n \t
                                                                  2. [latex]100\\enclose{longdiv}{995}[\/latex]<\/li>\r\n \t
                                                                  3. [latex]100\\enclose{longdiv}{269}[\/latex]<\/li>\r\n \t
                                                                  4. [latex]100\\enclose{longdiv}{175}[\/latex]<\/li>\r\n \t
                                                                  5. [latex]100\\enclose{longdiv}{2948}[\/latex]<\/li>\r\n \t
                                                                  6. [latex]100\\enclose{longdiv}{4671}[\/latex]<\/li>\r\n \t
                                                                  7. [latex]100\\enclose{longdiv}{92045}[\/latex]<\/li>\r\n \t
                                                                  8. [latex]100\\enclose{longdiv}{43821}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 6<\/strong>\r\n
                                                                      \r\n \t
                                                                    1. 3 R86<\/li>\r\n \t
                                                                    2. 9 R95<\/li>\r\n \t
                                                                    3. 2 R69<\/li>\r\n \t
                                                                    4. 1 R75<\/li>\r\n \t
                                                                    5. 29 R48<\/li>\r\n \t
                                                                    6. 46 R71<\/li>\r\n \t
                                                                    7. 920 R45<\/li>\r\n \t
                                                                    8. 438 R21<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

                                                                      When you divide by 1000<\/h2>\r\n
                                                                        \r\n \t
                                                                      1. The ones, tens, and hundreds digits become the remainder.<\/li>\r\n \t
                                                                      2. \u00a0 The other digits stay the same but are three <\/strong>place values less.\r\n
                                                                          \r\n \t
                                                                        • thousands become ones<\/li>\r\n \t
                                                                        • ten thousands become tens<\/li>\r\n \t
                                                                        • hundred thousands become hundreds<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n
                                                                          \r\n

                                                                          Exercise 7<\/p>\r\n\r\n<\/header>\r\n

                                                                          \r\n\r\nTry these. Check your work using the answer key at the end of the exercise.\r\n
                                                                            \r\n \t
                                                                          1. [latex]1000\\enclose{longdiv}{2398}[\/latex]<\/li>\r\n \t
                                                                          2. [latex]1000\\enclose{longdiv}{6475}[\/latex]<\/li>\r\n \t
                                                                          3. [latex]1000\\enclose{longdiv}{4835}[\/latex]<\/li>\r\n \t
                                                                          4. [latex]1000\\enclose{longdiv}{63291}[\/latex]<\/li>\r\n \t
                                                                          5. [latex]1000\\enclose{longdiv}{82405}[\/latex]<\/li>\r\n \t
                                                                          6. [latex]1000\\enclose{longdiv}{293591}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 7<\/strong>\r\n
                                                                              \r\n \t
                                                                            1. 2 R398<\/li>\r\n \t
                                                                            2. 6 R475<\/li>\r\n \t
                                                                            3. 4 R835<\/li>\r\n \t
                                                                            4. 63 R291<\/li>\r\n \t
                                                                            5. 82 R405<\/li>\r\n \t
                                                                            6. 293 R591<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

                                                                              Three-Digit Divisors<\/h1>\r\nIf the divisor has three digits, use the method you know for two-digit divisors, but estimate the divisor to the nearest hundred to find the trial quotient. Be very careful <\/strong>to put the first digit in the quotient in the correct place.\r\n
                                                                              \r\n

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

                                                                              \r\n\r\n17902<\/span> \u00f7 381 =\r\n\r\nStep 1: Divide<\/strong>\r\n
                                                                                \r\n \t
                                                                              1. Does 381 go into 1? NO<\/li>\r\n \t
                                                                              2. Does 381 go into 17?\u00a0 NO<\/li>\r\n \t
                                                                              3. Does 381 go into 179? NO<\/li>\r\n \t
                                                                              4. Does 381 go into 1790<\/span>? YES<\/li>\r\n \t
                                                                              5. Estimate 381 as 400. Think 4.<\/li>\r\n \t
                                                                              6. Estimate 1790<\/span> as 1800<\/span>. Think 18.<\/li>\r\n \t
                                                                              7. 4 goes into 18 \u2248 4 times ([latex]4\\times 4 = 16[\/latex])<\/li>\r\n \t
                                                                              8. Your estimate is 4. Write 4 in the quotient above the 0 in the dividend.<\/li>\r\n<\/ol>\r\n

                                                                                [latex]\\begin{array}{r}4 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\end{array}[\/latex]<\/p>\r\nStep 2: Multiply<\/strong>\r\n

                                                                                  \r\n \t
                                                                                1. 4 \u00d7 381 = 1524<\/span><\/li>\r\n<\/ol>\r\n

                                                                                  [latex]\\begin{array}{r}4 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\\\ 1524 \\ \\ \\ \\\\ \\hline \\end{array}[\/latex]<\/p>\r\n

                                                                                  Step 3: Subtract<\/strong><\/p>\r\n\r\n

                                                                                    \r\n \t
                                                                                  1. 1790<\/span> \u2212 1524<\/span> = 266<\/li>\r\n \t
                                                                                  2. Check 266 < 381? YES!<\/li>\r\n<\/ol>\r\n

                                                                                    [latex]\\begin{array}{r}4 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\\\ 1524 \\ \\ \\ \\\\ \\hline 266 \\ \\ \\ \\end{array}[\/latex]<\/p>\r\nStep 4:\u00a0 Bring down <\/strong>the 2. 2662<\/span> is now the number to be divided\r\n

                                                                                    [latex]\\begin{array}{r}4 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\\\ 1524 \\downarrow \\\\ \\hline 2662 \\end{array}[\/latex]<\/p>\r\nREPEAT<\/strong>\r\n\r\n \r\n\r\nStep 1: Divide [latex]2662 \\div 381 = [\/latex]<\/strong>\r\n

                                                                                      \r\n \t
                                                                                    1. Estimate 381 as 400. Think of 4.<\/li>\r\n \t
                                                                                    2. Estimate 2662<\/span> as 2700<\/span>. Think 27.<\/li>\r\n \t
                                                                                    3. 4 goes into 27 \u2248 6 times ([latex]4 \\times 6 = 24[\/latex]).<\/li>\r\n \t
                                                                                    4. Place this estimate in the quotient above the 2<\/li>\r\n<\/ol>\r\n

                                                                                      [latex]\\begin{array}{r}4 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\\\ 1524 \\downarrow \\\\ \\hline 2662 \\end{array}[\/latex]<\/p>\r\nStep 2: Multiply<\/strong>\r\n

                                                                                        \r\n \t
                                                                                      1. [latex]6\\times381 = 2286[\/latex]<\/li>\r\n<\/ol>\r\n

                                                                                        [latex]\\begin{array}{r}46 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\\\ 1524 \\downarrow \\\\ \\hline 2662 \\\\ 2286 \\\\ \\hline \\end{array}[\/latex]<\/p>\r\nStep 3: Subtract<\/strong>\r\n

                                                                                          \r\n \t
                                                                                        1. 2662<\/span> \u2212 2286<\/span> = 376<\/li>\r\n \t
                                                                                        2. Check 376 < 381? YES!<\/li>\r\n<\/ol>\r\n

                                                                                          [latex]\\begin{array}{r}46 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\\\ 1524 \\downarrow \\\\ \\hline 2662 \\\\ 2286 \\\\ \\hline 376 \\end{array}[\/latex]<\/p>\r\nStep 4: No more digits to bring down.<\/strong>\r\n

                                                                                          17902<\/span> \u00f7 381 = 46 R376<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

                                                                                          \r\n

                                                                                          Exercise 8<\/p>\r\n\r\n<\/header>\r\n

                                                                                          \r\n\r\nDivide and check your answers. These questions are hard work! Check your work using the answer key at the end of the exercise.\r\n
                                                                                            \r\n \t
                                                                                          1. [latex]115\\enclose{longdiv}{8682}[\/latex]<\/li>\r\n \t
                                                                                          2. [latex]205\\enclose{longdiv}{2384}[\/latex]<\/li>\r\n \t
                                                                                          3. [latex]325\\enclose{longdiv}{66321}[\/latex]<\/li>\r\n \t
                                                                                          4. [latex]241\\enclose{longdiv}{13284}[\/latex]<\/li>\r\n \t
                                                                                          5. [latex]860\\enclose{longdiv}{262412}[\/latex]<\/li>\r\n \t
                                                                                          6. [latex]659\\enclose{longdiv}{270190}[\/latex]<\/li>\r\n<\/ol>\r\nAnswers to Exercise 8<\/strong>\r\n
                                                                                              \r\n \t
                                                                                            1. 75 R57<\/li>\r\n \t
                                                                                            2. 11 R129<\/li>\r\n \t
                                                                                            3. 204 R21<\/li>\r\n \t
                                                                                            4. 55 R29<\/li>\r\n \t
                                                                                            5. 305 R112<\/li>\r\n \t
                                                                                            6. 410<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

                                                                                              Topic D: Self-Test<\/h1>\r\nMark\u00a0 \/12\u00a0 \u00a0Aim\u00a0 10\/12<\/strong>\r\n
                                                                                                \r\n \t
                                                                                              1. Divide and check your work for questions b<\/strong> and f<\/strong> using multiplication. (12 marks)\r\n
                                                                                                  \r\n \t
                                                                                                1. [latex]185 \\div 10 = [\/latex]<\/li>\r\n \t
                                                                                                2. [latex]4\\enclose{longdiv}{92}[\/latex]<\/li>\r\n \t
                                                                                                3. [latex]408 \\div 50 =[\/latex]<\/li>\r\n \t
                                                                                                4. [latex]72\\enclose{longdiv}{6768}[\/latex]<\/li>\r\n \t
                                                                                                5. [latex]67\\enclose{longdiv}{5963}[\/latex]<\/li>\r\n \t
                                                                                                6. [latex]53\\enclose{longdiv}{4856}[\/latex]<\/li>\r\n \t
                                                                                                7. [latex]91\\enclose{longdiv}{8736}[\/latex]<\/li>\r\n \t
                                                                                                8. [latex]265\\enclose{longdiv}{133624}[\/latex]<\/li>\r\n \t
                                                                                                9. [latex]606\\enclose{longdiv}{26094}[\/latex]<\/li>\r\n \t
                                                                                                10. [latex]1000\\enclose{longdiv}{83652}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n

                                                                                                  Answers to Topic D Self-Test<\/h2>\r\n
                                                                                                    \r\n \t
                                                                                                  1. Divide and check your work for questions b<\/strong> and f<\/strong> using multiplication.\r\n
                                                                                                      \r\n \t
                                                                                                    1. 18 R5<\/li>\r\n \t
                                                                                                    2. 23<\/li>\r\n \t
                                                                                                    3. 8 R8<\/li>\r\n \t
                                                                                                    4. 94<\/li>\r\n \t
                                                                                                    5. 89<\/li>\r\n \t
                                                                                                    6. 91 R33<\/li>\r\n \t
                                                                                                    7. 96<\/li>\r\n \t
                                                                                                    8. 504 R64<\/li>\r\n \t
                                                                                                    9. 43 R36<\/li>\r\n \t
                                                                                                    10. 83 R652<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>","rendered":"

                                                                                                      Finding Trial Quotients<\/h1>\n

                                                                                                      When dividing by 2-digit numbers, you will need to estimate the quotient. This guess is called a trial quotient.<\/p>\n

                                                                                                      \n
                                                                                                      \n
                                                                                                      \n
                                                                                                      \n

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

                                                                                                      \n

                                                                                                      [latex]624 \\div 24[\/latex]<\/p>\n

                                                                                                      Rewrite as [latex]24\\enclose{longdiv}{624}[\/latex]<\/p>\n

                                                                                                      \n

                                                                                                      Step 1: Divide <\/strong><\/p>\n

                                                                                                        \n
                                                                                                      1. Think: [latex]2\\enclose{longdiv}{6}[\/latex] is [latex]3[\/latex]. So [latex]24\\enclose{longdiv}{62}[\/latex] is about [latex]3[\/latex].<\/li>\n<\/ol>\n<\/div>\n
                                                                                                        \n

                                                                                                        Step 2: Multiply and subtract.\u00a0<\/strong><\/p>\n

                                                                                                          \n
                                                                                                        1. [latex]\\begin{array}{r}3 \\ \\ \\ \\\\ 24\\enclose{longdiv}{624}\\\\72 \\ \\ \\ \\\\ \\hline \\end{array}[\/latex]<\/li>\n
                                                                                                        2. Since 72 > 62, 3 is too large.<\/li>\n<\/ol>\n<\/div>\n
                                                                                                          \n

                                                                                                          Step 3: Try a smaller number, multiply and subtract.<\/strong><\/p>\n<\/div>\n

                                                                                                          \n
                                                                                                            \n
                                                                                                          1. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 24\\enclose{longdiv}{624}\\\\48 \\ \\ \\ \\\\ \\hline 14 \\ \\ \\ \\end{array}[\/latex]<\/li>\n
                                                                                                          2. Since 14 < 24, 2 is correct.<\/li>\n<\/ol>\n<\/div>\n

                                                                                                            Step 4: Finish the problem.<\/strong><\/p>\n<\/div>\n<\/div>\n

                                                                                                            \n
                                                                                                            \n

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

                                                                                                            \n

                                                                                                            [latex]630 \\div 15[\/latex]<\/p>\n

                                                                                                            Rewrite as [latex]15\\enclose{longdiv}{630}[\/latex]<\/p>\n

                                                                                                            \n

                                                                                                            Step 1: Divide <\/strong><\/p>\n

                                                                                                              \n
                                                                                                            1. 15 rounds to 20. Think: [latex]2\\enclose{longdiv}{6}[\/latex] is [latex]3[\/latex]. So [latex]15\\enclose{longdiv}{63}[\/latex] is about [latex]3[\/latex].<\/li>\n<\/ol>\n<\/div>\n
                                                                                                              \n

                                                                                                              Step 2: Multiply and subtract.\u00a0<\/strong><\/p>\n

                                                                                                                \n
                                                                                                              1. [latex]\\begin{array}{r}3 \\ \\ \\ \\\\ 15\\enclose{longdiv}{630}\\\\45 \\ \\ \\ \\\\ \\hline 18 \\ \\ \\ \\end{array}[\/latex]<\/li>\n
                                                                                                              2. Since 18 > 15, 3 is too small.<\/li>\n<\/ol>\n<\/div>\n
                                                                                                                \n

                                                                                                                Step 3: Try a larger number, multiply and subtract.<\/strong><\/p>\n<\/div>\n

                                                                                                                \n
                                                                                                                  \n
                                                                                                                1. [latex]\\begin{array}{r}4 \\ \\ \\ \\\\ 24\\enclose{longdiv}{630}\\\\60 \\ \\ \\\\ \\hline 3 \\ \\ \\end{array}[\/latex]<\/li>\n
                                                                                                                2. Since 3 < 15, 4 is correct.<\/li>\n<\/ol>\n<\/div>\n

                                                                                                                  Step 4: Finish the problem.<\/strong><\/p>\n<\/div>\n<\/div>\n

                                                                                                                  \n
                                                                                                                  \n

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

                                                                                                                  \n
                                                                                                                  \n

                                                                                                                  In each question, the trial quotient is either too large or too small. Multiply. Write too large or too small on the line. Then, write the correct trial quotient beside. Check your work using the answer key at the end of the exercise.<\/p>\n

                                                                                                                    \n
                                                                                                                  1. [latex]\\begin{array}{r}2 \\ \\ \\ \\\\ 25\\enclose{longdiv}{475}\\end{array}[\/latex]<\/li>\n
                                                                                                                  2. [latex]\\begin{array}{r}3 \\ \\ \\ \\\\ 15\\enclose{longdiv}{682}\\end{array}[\/latex]<\/li>\n
                                                                                                                  3. [latex]\\begin{array}{r}5 \\ \\ \\ \\\\ 18\\enclose{longdiv}{813}\\end{array}[\/latex]<\/li>\n
                                                                                                                  4. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 25\\enclose{longdiv}{810}\\end{array}[\/latex]<\/li>\n
                                                                                                                  5. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 33\\enclose{longdiv}{891}\\end{array}[\/latex]<\/li>\n
                                                                                                                  6. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 18\\enclose{longdiv}{819}\\end{array}[\/latex]<\/li>\n
                                                                                                                  7. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 27\\enclose{longdiv}{727}\\end{array}[\/latex]<\/li>\n
                                                                                                                  8. [latex]\\begin{array}{r}2 \\ \\ \\ \\\\ 35\\enclose{longdiv}{652}\\end{array}[\/latex]<\/li>\n
                                                                                                                  9. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 25\\enclose{longdiv}{650}\\end{array}[\/latex]<\/li>\n
                                                                                                                  10. [latex]\\begin{array}{r}4\\ \\ \\ \\\\34\\enclose{longdiv}{176}\\end{array}[\/latex]<\/li>\n
                                                                                                                  11. [latex]\\begin{array}{r}4\\ \\ \\ \\\\ 12\\enclose{longdiv}{420}\\end{array}[\/latex]<\/li>\n
                                                                                                                  12. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 43\\enclose{longdiv}{801}\\end{array}[\/latex]<\/li>\n
                                                                                                                  13. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 31\\enclose{longdiv}{899}\\end{array}[\/latex]<\/li>\n
                                                                                                                  14. [latex]\\begin{array}{r}4\\ \\ \\ \\\\ 18\\enclose{longdiv}{648}\\end{array}[\/latex]<\/li>\n
                                                                                                                  15. [latex]\\begin{array}{r}4\\ \\ \\ \\\\ 27\\enclose{longdiv}{946}\\end{array}[\/latex]<\/li>\n
                                                                                                                  16. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 23\\enclose{longdiv}{943}\\end{array}[\/latex]<\/li>\n
                                                                                                                  17. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 24\\enclose{longdiv}{578}\\end{array}[\/latex]<\/li>\n
                                                                                                                  18. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 29\\enclose{longdiv}{406}\\end{array}[\/latex]<\/li>\n
                                                                                                                  19. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 48\\enclose{longdiv}{892}\\end{array}[\/latex]<\/li>\n
                                                                                                                  20. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 28\\enclose{longdiv}{534}\\end{array}[\/latex]<\/li>\n
                                                                                                                  21. [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 37\\enclose{longdiv}{939}\\end{array}[\/latex]<\/li>\n
                                                                                                                  22. [latex]\\begin{array}{r}2\\ \\ \\ \\\\ 28\\enclose{longdiv}{854}\\end{array}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                    Answers to Exercise 1<\/strong><\/p>\n

                                                                                                                      \n
                                                                                                                    1. too large, 1<\/li>\n
                                                                                                                    2. too small, 4<\/li>\n
                                                                                                                    3. too large, 4<\/li>\n
                                                                                                                    4. too small, 3<\/li>\n
                                                                                                                    5. too large, 2<\/li>\n
                                                                                                                    6. too small, 4<\/li>\n
                                                                                                                    7. too large, 2<\/li>\n
                                                                                                                    8. too large 1<\/li>\n
                                                                                                                    9. too large, 2<\/li>\n
                                                                                                                    10. too small, 5<\/li>\n
                                                                                                                    11. too large, 3<\/li>\n
                                                                                                                    12. too large, 1<\/li>\n
                                                                                                                    13. too. large 2<\/li>\n
                                                                                                                    14. too large, 3<\/li>\n
                                                                                                                    15. too large, 3<\/li>\n
                                                                                                                    16. too small, 4<\/li>\n
                                                                                                                    17. too large, 2<\/li>\n
                                                                                                                    18. too large, 1<\/li>\n
                                                                                                                    19. too large, 1<\/li>\n
                                                                                                                    20. too large 1<\/li>\n
                                                                                                                    21. too large, 2<\/li>\n
                                                                                                                    22. too small, 3<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
                                                                                                                      \n
                                                                                                                      \n

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

                                                                                                                      \n

                                                                                                                      [latex]78\\enclose{longdiv}{2706}[\/latex]<\/p>\n

                                                                                                                      \n

                                                                                                                      Since 78 rounds to 80, think [latex]8\\enclose{longdiv}{27}[\/latex]. 8 goes into 27 \u2248 3. 3 would be a good trial quotient.<\/p>\n

                                                                                                                      [latex]\\begin{array}{r}3\\ \\ \\ \\\\ 78\\enclose{longdiv}{2706}\\\\234\\ \\ \\ \\\\ \\hline 36\\ \\ \\ \\end{array}[\/latex]<\/p>\n

                                                                                                                      Since 36 < 78, 3 is a good trial quotient.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

                                                                                                                      \n
                                                                                                                      \n

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

                                                                                                                      \n

                                                                                                                      [latex]27\\enclose{longdiv}{2205}[\/latex]<\/p>\n

                                                                                                                      Since<\/span> 27 rounds to 30, think [latex]3\\enclose{longdiv}{22}[\/latex]<\/span>\u00a0.<\/span> 3 goes into 22 \u2248 7. 7 would be a good trial quotient.<\/span><\/span><\/p>\n

                                                                                                                      [latex]\\begin{array}{r}7\\ \\ \\ \\\\ 27\\enclose{longdiv}{2205}\\\\189\\ \\ \\ \\\\ \\hline 31\\ \\ \\ \\end{array}[\/latex]<\/p>\n

                                                                                                                      Since 31 > 27, so 7 is too small.\u00a0 A better trial quotient would be 8.<\/span><\/span>\u00a0<\/span><\/p>\n<\/div>\n<\/div>\n

                                                                                                                      \n
                                                                                                                      \n

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

                                                                                                                      \n
                                                                                                                      \n

                                                                                                                      Find the first digit in the trial quotient.\u00a0 Check your work using the answer key at the end of the exercise.<\/p>\n

                                                                                                                        \n
                                                                                                                      1. [latex]43\\enclose{longdiv}{1772}[\/latex] [latex]\\begin{array}{r}4\\\\ 4\\enclose{longdiv}{17}\\\\16\\\\ \\hline 1\\end{array}[\/latex]. 1 < 4<\/li>\n
                                                                                                                      2. [latex]64\\enclose{longdiv}{3276}[\/latex]<\/li>\n
                                                                                                                      3. [latex]28\\enclose{longdiv}{6008}[\/latex]<\/li>\n
                                                                                                                      4. [latex]33\\enclose{longdiv}{2731}[\/latex]<\/li>\n
                                                                                                                      5. [latex]59\\enclose{longdiv}{4164}[\/latex]<\/li>\n
                                                                                                                      6. [latex]75\\enclose{longdiv}{2420}[\/latex]<\/li>\n
                                                                                                                      7. [latex]54\\enclose{longdiv}{3316}[\/latex]<\/li>\n
                                                                                                                      8. [latex]38\\enclose{longdiv}{2759}[\/latex]<\/li>\n
                                                                                                                      9. [latex]46\\enclose{longdiv}{387}[\/latex]<\/li>\n
                                                                                                                      10. [latex]35\\enclose{longdiv}{3316}[\/latex]<\/li>\n
                                                                                                                      11. [latex]83\\enclose{longdiv}{7237}[\/latex]<\/li>\n
                                                                                                                      12. [latex]77\\enclose{longdiv}{6763}[\/latex]<\/li>\n
                                                                                                                      13. [latex]93\\enclose{longdiv}{3724}[\/latex]<\/li>\n
                                                                                                                      14. [latex]52\\enclose{longdiv}{4690}[\/latex]<\/li>\n
                                                                                                                      15. [latex]86\\enclose{longdiv}{2089}[\/latex]<\/li>\n
                                                                                                                      16. [latex]26\\enclose{longdiv}{1417}[\/latex]<\/li>\n
                                                                                                                      17. [latex]72\\enclose{longdiv}{1462}[\/latex]<\/li>\n
                                                                                                                      18. [latex]27\\enclose{longdiv}{6939}[\/latex]<\/li>\n
                                                                                                                      19. [latex]32\\enclose{longdiv}{7840}[\/latex]<\/li>\n
                                                                                                                      20. [latex]24\\enclose{longdiv}{7605}[\/latex]<\/li>\n
                                                                                                                      21. [latex]16\\enclose{longdiv}{8640}[\/latex]<\/li>\n
                                                                                                                      22. [latex]45\\enclose{longdiv}{3060}[\/latex]<\/li>\n
                                                                                                                      23. [latex]38\\enclose{longdiv}{2158}[\/latex]<\/li>\n
                                                                                                                      24. [latex]42\\enclose{longdiv}{1491}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                        Answers to Exercise 2<\/strong><\/p>\n

                                                                                                                          \n
                                                                                                                        1. 4<\/li>\n
                                                                                                                        2. 5<\/li>\n
                                                                                                                        3. 2<\/li>\n
                                                                                                                        4. 8<\/li>\n
                                                                                                                        5. 7<\/li>\n
                                                                                                                        6. 3<\/li>\n
                                                                                                                        7. 6<\/li>\n
                                                                                                                        8. 7<\/li>\n
                                                                                                                        9. 8<\/li>\n
                                                                                                                        10. 9<\/li>\n
                                                                                                                        11. 8<\/li>\n
                                                                                                                        12. 8<\/li>\n
                                                                                                                        13. 4<\/li>\n
                                                                                                                        14. 9<\/li>\n
                                                                                                                        15. 2<\/li>\n
                                                                                                                        16. 5<\/li>\n
                                                                                                                        17. 2<\/li>\n
                                                                                                                        18. 2<\/li>\n
                                                                                                                        19. 2<\/li>\n
                                                                                                                        20. 3<\/li>\n
                                                                                                                        21. 5<\/li>\n
                                                                                                                        22. 6<\/li>\n
                                                                                                                        23. 5<\/li>\n
                                                                                                                        24. 3<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n

                                                                                                                          Two Digit Divisors<\/h1>\n

                                                                                                                          Dividing by large divisors is a challenge!<\/p>\n

                                                                                                                          You must estimate <\/strong>how many times one number will divide into another. Use pencil and have an eraser close by when you do these questions. You will use the same steps that you already know.<\/p>\n

                                                                                                                          \n
                                                                                                                          \n

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

                                                                                                                          \n

                                                                                                                          [latex]964 \\div 75 =[\/latex]<\/p>\n

                                                                                                                          \n

                                                                                                                          Step 1: Divide<\/strong><\/p>\n

                                                                                                                            \n
                                                                                                                          1. Does 75 go into 9? \u00a0NO<\/strong><\/li>\n
                                                                                                                          2. Does 75 go into 96? \u00a0YES<\/strong><\/li>\n
                                                                                                                          3. Estimate\n
                                                                                                                              \n
                                                                                                                            1. Round 75 to 80 \u2013 think \u201c8\u201d<\/li>\n
                                                                                                                            2. Round 96 to 100 \u2013 think \u201c10\u201d<\/li>\n
                                                                                                                            3. How many 8\u2019s in 10?\u00a0 (8 1 = 8, 10\u00a0 8\u00a0 =\u00a0 1)<\/li>\n
                                                                                                                            4. The estimate for the first digit in the trial quotient is 1.<\/li>\n<\/ol>\n<\/li>\n
                                                                                                                            5. Write 1 <\/strong>in the quotient above the 6 tens.<\/span><\/li>\n<\/ol>\n

                                                                                                                              [latex]\\begin{array}{r}1\\ \\ \\ \\\\ 75\\enclose{longdiv}{964}\\end{array}[\/latex]<\/p>\n

                                                                                                                              Step 2:\u00a0 \u00a0Multiply\u00a0<\/strong><\/p>\n

                                                                                                                              [latex]1 \\times 75 = 75[\/latex]<\/p>\n

                                                                                                                                \n
                                                                                                                              1. Write 75 under 96.<\/li>\n<\/ol>\n

                                                                                                                                \u00a0 [latex]\\begin{array}{r}1\\ \\ \\ \\\\ 75\\enclose{longdiv}{964}\\\\75 \\ \\ \\\\ \\hline \\end{array}[\/latex]<\/p>\n

                                                                                                                                Step 3:\u00a0 \u00a0Subtract\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><\/p>\n

                                                                                                                                [latex]96 - 75 = 21[\/latex]<\/p>\n

                                                                                                                                  \n
                                                                                                                                1. Check 21 < 75? YES!<\/strong><\/li>\n<\/ol>\n

                                                                                                                                  \u00a0 [latex]\\begin{array}{r}1\\ \\ \\ \\\\ 75\\enclose{longdiv}{964}\\\\75\\ \\ \\\\ \\hline 21 \\ \\ \\end{array}[\/latex]<\/p>\n

                                                                                                                                  Step 4: Bring down the next digit <\/strong>in the dividend. 214 is now the number to be divided.<\/span><\/p>\n<\/div>\n

                                                                                                                                  \n

                                                                                                                                  \u00a0 [latex]\\begin{array}{r}1\\ \\ \\ \\\\ 75\\enclose{longdiv}{964}\\\\75 \\downarrow \\\\ \\hline 214\\end{array}[\/latex]<\/p>\n

                                                                                                                                  REPEAT<\/strong><\/p>\n

                                                                                                                                   <\/p>\n

                                                                                                                                  Step 1:\u00a0\u00a0 Divide<\/strong><\/p>\n

                                                                                                                                    \n
                                                                                                                                  1. Does 75 go into 214? \u00a0YES<\/strong><\/li>\n
                                                                                                                                  2. Estimate 75 as 80 \u2013 think \u201c8\u201d\n
                                                                                                                                      \n
                                                                                                                                    1. Estimate 214 as 200 \u2013 think \u201c20\u201d<\/li>\n
                                                                                                                                    2. 8 goes into 20 2 times ( 8 2 = 16, so 20\u00a0 8 2)<\/li>\n<\/ol>\n<\/li>\n
                                                                                                                                    3. The estimate for the second digit in the trial quotient is 2. Write 2 in the quotient above the 4 in the dividend.<\/li>\n<\/ol>\n

                                                                                                                                      \u00a0 [latex]\\begin{array}{r}12\\\\ 75\\enclose{longdiv}{964}\\\\75 \\downarrow \\\\ \\hline 214\\end{array}[\/latex]<\/p>\n

                                                                                                                                      Step 2:\u00a0 \u00a0Multiply<\/strong><\/p>\n

                                                                                                                                      [latex]2 \\times 75 = 150[\/latex]<\/p>\n

                                                                                                                                        \n
                                                                                                                                      1. Write 150 under the 214.<\/li>\n<\/ol>\n

                                                                                                                                        \u00a0 [latex]\\begin{array}{r}12\\\\ 75\\enclose{longdiv}{964}\\\\75 \\downarrow \\\\ \\hline 214 \\\\ 150 \\\\ \\hline\\end{array}[\/latex]<\/p>\n

                                                                                                                                        Step 3:\u00a0\u00a0 Subtract and check <\/strong>that the remainder is less than the divisor.<\/p>\n

                                                                                                                                        \u00a0 [latex]\\begin{array}{r}12\\\\ 75\\enclose{longdiv}{964}\\\\75 \\downarrow \\\\ \\hline 214 \\\\ 150 \\\\ \\hline 64\\end{array}[\/latex]<\/p>\n<\/div>\n

                                                                                                                                         <\/p>\n

                                                                                                                                        Step 4:\u00a0\u00a0 Bring down <\/strong>\u2013 no more digits in dividend.<\/p>\n

                                                                                                                                        To check your answer<\/p>\n

                                                                                                                                        [latex]\\begin{array}{rrr}&75\\\\\\times&12\\\\\\hline&150 \\\\ &750 \\\\ \\hline &900 \\\\ + &64* \\\\ \\hline &964\\end{array}[\/latex]<\/p>\n

                                                                                                                                        *Don\u2019t forget to add the remainder<\/strong> when you check.<\/p>\n<\/div>\n<\/div>\n

                                                                                                                                        \n
                                                                                                                                        \n

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

                                                                                                                                        \n

                                                                                                                                        Carefully divide these questions.\u00a0 Be careful to keep the hundreds in line with the hundreds, the tens with the tens, and so on. You might want to use squared paper for long division. Check your work using the answer key at the end of the exercise.<\/p>\n

                                                                                                                                        If you are having any <\/strong>difficulty, ask your instructor to watch you doing a few questions to be sure you are using a correct method.<\/p>\n

                                                                                                                                          \n
                                                                                                                                        1. [latex]10\\enclose{longdiv}{720}[\/latex]<\/li>\n
                                                                                                                                        2. [latex]12\\enclose{longdiv}{564}[\/latex]<\/li>\n
                                                                                                                                        3. [latex]21\\enclose{longdiv}{882}[\/latex]<\/li>\n
                                                                                                                                        4. [latex]22\\enclose{longdiv}{946}[\/latex]<\/li>\n
                                                                                                                                        5. [latex]32\\enclose{longdiv}{1632}[\/latex]<\/li>\n
                                                                                                                                        6. [latex]23\\enclose{longdiv}{943}[\/latex]<\/li>\n
                                                                                                                                        7. [latex]62\\enclose{longdiv}{2528}[\/latex]<\/li>\n
                                                                                                                                        8. [latex]71\\enclose{longdiv}{2414}[\/latex]<\/li>\n
                                                                                                                                        9. [latex]24\\enclose{longdiv}{7578}[\/latex]<\/li>\n
                                                                                                                                        10. [latex]82\\enclose{longdiv}{2958}[\/latex]<\/li>\n
                                                                                                                                        11. [latex]18\\enclose{longdiv}{6250}[\/latex]<\/li>\n
                                                                                                                                        12. [latex]25\\enclose{longdiv}{1550}[\/latex]<\/li>\n
                                                                                                                                        13. [latex]19\\enclose{longdiv}{9595}[\/latex]<\/li>\n
                                                                                                                                        14. [latex]47\\enclose{longdiv}{3854}[\/latex]<\/li>\n
                                                                                                                                        15. [latex]58\\enclose{longdiv}{6500}[\/latex]<\/li>\n
                                                                                                                                        16. [latex]24\\enclose{longdiv}{9648}[\/latex]<\/li>\n
                                                                                                                                        17. [latex]49\\enclose{longdiv}{1312}[\/latex]<\/li>\n
                                                                                                                                        18. [latex]67\\enclose{longdiv}{7683}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                          Answers to Exercise 3<\/strong><\/p>\n

                                                                                                                                            \n
                                                                                                                                          1. 72<\/li>\n
                                                                                                                                          2. 47<\/li>\n
                                                                                                                                          3. 42<\/li>\n
                                                                                                                                          4. 43<\/li>\n
                                                                                                                                          5. 51<\/li>\n
                                                                                                                                          6. 41<\/li>\n
                                                                                                                                          7. 40 R48<\/li>\n
                                                                                                                                          8. 34<\/li>\n
                                                                                                                                          9. 315 R18<\/li>\n
                                                                                                                                          10. 36 R6<\/li>\n
                                                                                                                                          11. 347 R4<\/li>\n
                                                                                                                                          12. 62<\/li>\n
                                                                                                                                          13. 505<\/li>\n
                                                                                                                                          14. 82<\/li>\n
                                                                                                                                          15. 112 R4<\/li>\n
                                                                                                                                          16. 402<\/li>\n
                                                                                                                                          17. 26 R38<\/li>\n
                                                                                                                                          18. 114 R45<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                                                                                                                            If the estimate for your trial quotient is too large, the result of the multiplication will be larger than the numbers in the dividend.<\/strong><\/p>\n

                                                                                                                                            Take, for example, the following division question:<\/p>\n

                                                                                                                                            [latex]23\\enclose{longdiv}{784}[\/latex]<\/p>\n

                                                                                                                                            When you divide<\/strong>, the trial estimate is 4:<\/p>\n

                                                                                                                                            [latex]\\begin{array}{r}4\\ \\ \\ \\\\ 23\\enclose{longdiv}{784}\\end{array}[\/latex]<\/p>\n

                                                                                                                                            When you multiply<\/strong>:<\/p>\n

                                                                                                                                              \n
                                                                                                                                            1. [latex]4 \\times 23 = 92[\/latex]<\/li>\n
                                                                                                                                            2. 92 is larger than 78, so 4 is too large an estimate. Erase it. Try 3. [latex]3 \\times 23 = 69[\/latex]<\/li>\n
                                                                                                                                            3. 3 is the correct estimate and you can complete the division, as seen below:<\/li>\n<\/ol>\n

                                                                                                                                              [latex]\\begin{array}{l}\\hspace{2.3em}34\\text{ R}2\\\\23 \\enclose{longdiv}{784}\\\\\\hspace{1.5em}69 \\downarrow\\\\\\hline\\hspace{2.3em}94\\\\\\hspace{2.3em}92\\\\\\hline\\hspace{2.8em}2\\end{array}[\/latex]<\/p>\n

                                                                                                                                              If the estimate is too small, the result of the subtraction will be larger than the divisor.<\/strong><\/p>\n

                                                                                                                                              Take this division question as an example:<\/p>\n

                                                                                                                                              [latex]72\\enclose{longdiv}{4487}[\/latex]<\/p>\n

                                                                                                                                              Say your trial quotient is 5. When you multiply, you get [latex]5 \\times 72 = 360[\/latex]. Since 360 is smaller than 448, you can go on to subtract:<\/p>\n

                                                                                                                                              [latex]448 - 360=88[\/latex]<\/p>\n

                                                                                                                                              Is 88 less than 72?\u00a0NO<\/strong>, 88 is greater than 72. This means that 5 is too small for the trial quotient. Use the larger number 6.<\/p>\n

                                                                                                                                              Multiply [latex]6 \\times 72 = 432[\/latex]. Subtract [latex]448 - 432 = 16[\/latex]. Since 16 is less than 72, you can continue the division.<\/p>\n

                                                                                                                                              See the full division process below:<\/p>\n

                                                                                                                                              [latex]\\begin{array}{l}\\hspace{1.1cm}62\\text{ R}23\\\\72 \\enclose{longdiv}{4487}\\\\\\hspace{1.45em}432 \\downarrow\\\\\\hline\\hspace{2.1em}167\\\\\\hspace{2.1em}144\\\\\\hline\\hspace{2.55em}23\\end{array}[\/latex]<\/p>\n

                                                                                                                                              \n
                                                                                                                                              \n

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

                                                                                                                                              \n

                                                                                                                                              Divide and check your work by multiplying.\u00a0 Check your work using the answer key at the end of the exercise.<\/p>\n

                                                                                                                                                \n
                                                                                                                                              1. [latex]18\\enclose{longdiv}{648}[\/latex]<\/li>\n
                                                                                                                                              2. [latex]26\\enclose{longdiv}{6766}[\/latex]<\/li>\n
                                                                                                                                              3. [latex]52\\enclose{longdiv}{1968}[\/latex]<\/li>\n
                                                                                                                                              4. [latex]84\\enclose{longdiv}{8640}[\/latex]<\/li>\n
                                                                                                                                              5. [latex]72\\enclose{longdiv}{2883}[\/latex]<\/li>\n
                                                                                                                                              6. [latex]94\\enclose{longdiv}{8126}[\/latex]<\/li>\n
                                                                                                                                              7. [latex]20\\enclose{longdiv}{4060}[\/latex]<\/li>\n
                                                                                                                                              8. [latex]47\\enclose{longdiv}{1728}[\/latex]<\/li>\n
                                                                                                                                              9. [latex]33\\enclose{longdiv}{1886}[\/latex]<\/li>\n
                                                                                                                                              10. [latex]25\\enclose{longdiv}{5750}[\/latex]<\/li>\n
                                                                                                                                              11. [latex]79\\enclose{longdiv}{2765}[\/latex]<\/li>\n
                                                                                                                                              12. [latex]42\\enclose{longdiv}{8442}[\/latex]<\/li>\n
                                                                                                                                              13. [latex]57\\enclose{longdiv}{9144}[\/latex]<\/li>\n
                                                                                                                                              14. [latex]96\\enclose{longdiv}{20160}[\/latex]<\/li>\n
                                                                                                                                              15. [latex]75\\enclose{longdiv}{23550}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                Answers to Exercise 4<\/strong><\/p>\n

                                                                                                                                                  \n
                                                                                                                                                1. 36<\/li>\n
                                                                                                                                                2. 260 R6<\/li>\n
                                                                                                                                                3. 37 R44<\/li>\n
                                                                                                                                                4. 102 R72<\/li>\n
                                                                                                                                                5. 40 R3<\/li>\n
                                                                                                                                                6. 86 R42<\/li>\n
                                                                                                                                                7. 203<\/li>\n
                                                                                                                                                8. 36 R36<\/li>\n
                                                                                                                                                9. 57 R5<\/li>\n
                                                                                                                                                10. 230<\/li>\n
                                                                                                                                                11. 35<\/li>\n
                                                                                                                                                12. 201<\/li>\n
                                                                                                                                                13. 160 R24<\/li>\n
                                                                                                                                                14. 210<\/li>\n
                                                                                                                                                15. 314<\/li>\n
                                                                                                                                                16. <\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                                                                                                                                  Dividing by 10, 100, 1000…<\/h1>\n

                                                                                                                                                  What is the pattern?\u00a0 When you divide by 10<\/h2>\n
                                                                                                                                                    \n
                                                                                                                                                  1. \u00a0<\/strong>\u00a0The ones digit in the dividend becomes the remainder.<\/li>\n<\/ol>\n

                                                                                                                                                    [latex]10\\enclose{longdiv}{324} = 32 R4[\/latex]<\/p>\n

                                                                                                                                                      \n
                                                                                                                                                    1. The other numbers in the dividend stay the same, but each digit is one place value less.<\/li>\n<\/ol>\n
                                                                                                                                                        \n
                                                                                                                                                      • \n
                                                                                                                                                          \n
                                                                                                                                                        • the hundreds become tens<\/li>\n
                                                                                                                                                        • the tens become ones<\/li>\n
                                                                                                                                                        • the ones become the remainder<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
                                                                                                                                                          \n
                                                                                                                                                          \n

                                                                                                                                                          Exercise 5<\/p>\n<\/header>\n

                                                                                                                                                          \n

                                                                                                                                                          Find the quotients. Look for the pattern. Check your work using the answer key at the end of the exercise.<\/p>\n

                                                                                                                                                            \n
                                                                                                                                                          1. [latex]10\\enclose{longdiv}{46}[\/latex]<\/li>\n
                                                                                                                                                          2. [latex]10\\enclose{longdiv}{75}[\/latex]<\/li>\n
                                                                                                                                                          3. [latex]10\\enclose{longdiv}{136}[\/latex]<\/li>\n
                                                                                                                                                          4. [latex]10\\enclose{longdiv}{832}[\/latex]<\/li>\n
                                                                                                                                                          5. [latex]10\\enclose{longdiv}{674}[\/latex]<\/li>\n
                                                                                                                                                          6. [latex]10\\enclose{longdiv}{952}[\/latex]<\/li>\n
                                                                                                                                                          7. [latex]10\\enclose{longdiv}{2457}[\/latex]<\/li>\n
                                                                                                                                                          8. [latex]10\\enclose{longdiv}{3685}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                            Answers to Exercise 5<\/strong><\/p>\n

                                                                                                                                                              \n
                                                                                                                                                            1. 4 R6<\/li>\n
                                                                                                                                                            2. 7 R5<\/li>\n
                                                                                                                                                            3. 13 R6<\/li>\n
                                                                                                                                                            4. 83 R2<\/li>\n
                                                                                                                                                            5. 67 R4<\/li>\n
                                                                                                                                                            6. 95 R2<\/li>\n
                                                                                                                                                            7. 245 R7<\/li>\n
                                                                                                                                                            8. 368 R5<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                                                                                                                                              When you divide by 100<\/h2>\n
                                                                                                                                                                \n
                                                                                                                                                              1. \u00a0The ones and tens digits in the dividend become the remainder.<\/li>\n
                                                                                                                                                              2. The other digits in the dividend stay the same but each digit is two <\/strong>places less.\n
                                                                                                                                                                  \n
                                                                                                                                                                • the thousands become tens<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n
                                                                                                                                                                    \n
                                                                                                                                                                  • \n
                                                                                                                                                                      \n
                                                                                                                                                                    • the hundreds become ones<\/li>\n
                                                                                                                                                                    • the tens and ones become the remainder<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
                                                                                                                                                                      \n
                                                                                                                                                                      \n

                                                                                                                                                                      Exercise 6<\/p>\n<\/header>\n

                                                                                                                                                                      \n

                                                                                                                                                                      Find these quotients. Look for the pattern when you divide. Check your work using the answer key at the end of the exercise<\/p>\n

                                                                                                                                                                        \n
                                                                                                                                                                      1. [latex]100\\enclose{longdiv}{386}[\/latex]<\/li>\n
                                                                                                                                                                      2. [latex]100\\enclose{longdiv}{995}[\/latex]<\/li>\n
                                                                                                                                                                      3. [latex]100\\enclose{longdiv}{269}[\/latex]<\/li>\n
                                                                                                                                                                      4. [latex]100\\enclose{longdiv}{175}[\/latex]<\/li>\n
                                                                                                                                                                      5. [latex]100\\enclose{longdiv}{2948}[\/latex]<\/li>\n
                                                                                                                                                                      6. [latex]100\\enclose{longdiv}{4671}[\/latex]<\/li>\n
                                                                                                                                                                      7. [latex]100\\enclose{longdiv}{92045}[\/latex]<\/li>\n
                                                                                                                                                                      8. [latex]100\\enclose{longdiv}{43821}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                        Answers to Exercise 6<\/strong><\/p>\n

                                                                                                                                                                          \n
                                                                                                                                                                        1. 3 R86<\/li>\n
                                                                                                                                                                        2. 9 R95<\/li>\n
                                                                                                                                                                        3. 2 R69<\/li>\n
                                                                                                                                                                        4. 1 R75<\/li>\n
                                                                                                                                                                        5. 29 R48<\/li>\n
                                                                                                                                                                        6. 46 R71<\/li>\n
                                                                                                                                                                        7. 920 R45<\/li>\n
                                                                                                                                                                        8. 438 R21<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                                                                                                                                                          When you divide by 1000<\/h2>\n
                                                                                                                                                                            \n
                                                                                                                                                                          1. The ones, tens, and hundreds digits become the remainder.<\/li>\n
                                                                                                                                                                          2. \u00a0 The other digits stay the same but are three <\/strong>place values less.\n
                                                                                                                                                                              \n
                                                                                                                                                                            • thousands become ones<\/li>\n
                                                                                                                                                                            • ten thousands become tens<\/li>\n
                                                                                                                                                                            • hundred thousands become hundreds<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n
                                                                                                                                                                              \n
                                                                                                                                                                              \n

                                                                                                                                                                              Exercise 7<\/p>\n<\/header>\n

                                                                                                                                                                              \n

                                                                                                                                                                              Try these. Check your work using the answer key at the end of the exercise.<\/p>\n

                                                                                                                                                                                \n
                                                                                                                                                                              1. [latex]1000\\enclose{longdiv}{2398}[\/latex]<\/li>\n
                                                                                                                                                                              2. [latex]1000\\enclose{longdiv}{6475}[\/latex]<\/li>\n
                                                                                                                                                                              3. [latex]1000\\enclose{longdiv}{4835}[\/latex]<\/li>\n
                                                                                                                                                                              4. [latex]1000\\enclose{longdiv}{63291}[\/latex]<\/li>\n
                                                                                                                                                                              5. [latex]1000\\enclose{longdiv}{82405}[\/latex]<\/li>\n
                                                                                                                                                                              6. [latex]1000\\enclose{longdiv}{293591}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                Answers to Exercise 7<\/strong><\/p>\n

                                                                                                                                                                                  \n
                                                                                                                                                                                1. 2 R398<\/li>\n
                                                                                                                                                                                2. 6 R475<\/li>\n
                                                                                                                                                                                3. 4 R835<\/li>\n
                                                                                                                                                                                4. 63 R291<\/li>\n
                                                                                                                                                                                5. 82 R405<\/li>\n
                                                                                                                                                                                6. 293 R591<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                                                                                                                                                                  Three-Digit Divisors<\/h1>\n

                                                                                                                                                                                  If the divisor has three digits, use the method you know for two-digit divisors, but estimate the divisor to the nearest hundred to find the trial quotient. Be very careful <\/strong>to put the first digit in the quotient in the correct place.<\/p>\n

                                                                                                                                                                                  \n
                                                                                                                                                                                  \n

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

                                                                                                                                                                                  \n

                                                                                                                                                                                  17902<\/span> \u00f7 381 =<\/p>\n

                                                                                                                                                                                  Step 1: Divide<\/strong><\/p>\n

                                                                                                                                                                                    \n
                                                                                                                                                                                  1. Does 381 go into 1? NO<\/li>\n
                                                                                                                                                                                  2. Does 381 go into 17?\u00a0 NO<\/li>\n
                                                                                                                                                                                  3. Does 381 go into 179? NO<\/li>\n
                                                                                                                                                                                  4. Does 381 go into 1790<\/span>? YES<\/li>\n
                                                                                                                                                                                  5. Estimate 381 as 400. Think 4.<\/li>\n
                                                                                                                                                                                  6. Estimate 1790<\/span> as 1800<\/span>. Think 18.<\/li>\n
                                                                                                                                                                                  7. 4 goes into 18 \u2248 4 times ([latex]4\\times 4 = 16[\/latex])<\/li>\n
                                                                                                                                                                                  8. Your estimate is 4. Write 4 in the quotient above the 0 in the dividend.<\/li>\n<\/ol>\n

                                                                                                                                                                                    [latex]\\begin{array}{r}4 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\end{array}[\/latex]<\/p>\n

                                                                                                                                                                                    Step 2: Multiply<\/strong><\/p>\n

                                                                                                                                                                                      \n
                                                                                                                                                                                    1. 4 \u00d7 381 = 1524<\/span><\/li>\n<\/ol>\n

                                                                                                                                                                                      [latex]\\begin{array}{r}4 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\\\ 1524 \\ \\ \\ \\\\ \\hline \\end{array}[\/latex]<\/p>\n

                                                                                                                                                                                      Step 3: Subtract<\/strong><\/p>\n

                                                                                                                                                                                        \n
                                                                                                                                                                                      1. 1790<\/span> \u2212 1524<\/span> = 266<\/li>\n
                                                                                                                                                                                      2. Check 266 < 381? YES!<\/li>\n<\/ol>\n

                                                                                                                                                                                        [latex]\\begin{array}{r}4 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\\\ 1524 \\ \\ \\ \\\\ \\hline 266 \\ \\ \\ \\end{array}[\/latex]<\/p>\n

                                                                                                                                                                                        Step 4:\u00a0 Bring down <\/strong>the 2. 2662<\/span> is now the number to be divided<\/p>\n

                                                                                                                                                                                        [latex]\\begin{array}{r}4 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\\\ 1524 \\downarrow \\\\ \\hline 2662 \\end{array}[\/latex]<\/p>\n

                                                                                                                                                                                        REPEAT<\/strong><\/p>\n

                                                                                                                                                                                         <\/p>\n

                                                                                                                                                                                        Step 1: Divide [latex]2662 \\div 381 =[\/latex]<\/strong><\/p>\n

                                                                                                                                                                                          \n
                                                                                                                                                                                        1. Estimate 381 as 400. Think of 4.<\/li>\n
                                                                                                                                                                                        2. Estimate 2662<\/span> as 2700<\/span>. Think 27.<\/li>\n
                                                                                                                                                                                        3. 4 goes into 27 \u2248 6 times ([latex]4 \\times 6 = 24[\/latex]).<\/li>\n
                                                                                                                                                                                        4. Place this estimate in the quotient above the 2<\/li>\n<\/ol>\n

                                                                                                                                                                                          [latex]\\begin{array}{r}4 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\\\ 1524 \\downarrow \\\\ \\hline 2662 \\end{array}[\/latex]<\/p>\n

                                                                                                                                                                                          Step 2: Multiply<\/strong><\/p>\n

                                                                                                                                                                                            \n
                                                                                                                                                                                          1. [latex]6\\times381 = 2286[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                            [latex]\\begin{array}{r}46 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\\\ 1524 \\downarrow \\\\ \\hline 2662 \\\\ 2286 \\\\ \\hline \\end{array}[\/latex]<\/p>\n

                                                                                                                                                                                            Step 3: Subtract<\/strong><\/p>\n

                                                                                                                                                                                              \n
                                                                                                                                                                                            1. 2662<\/span> \u2212 2286<\/span> = 376<\/li>\n
                                                                                                                                                                                            2. Check 376 < 381? YES!<\/li>\n<\/ol>\n

                                                                                                                                                                                              [latex]\\begin{array}{r}46 \\ \\ \\ \\\\ 381\\enclose{longdiv}{17902} \\\\ 1524 \\downarrow \\\\ \\hline 2662 \\\\ 2286 \\\\ \\hline 376 \\end{array}[\/latex]<\/p>\n

                                                                                                                                                                                              Step 4: No more digits to bring down.<\/strong><\/p>\n

                                                                                                                                                                                              17902<\/span> \u00f7 381 = 46 R376<\/p>\n<\/div>\n<\/div>\n

                                                                                                                                                                                              \n
                                                                                                                                                                                              \n

                                                                                                                                                                                              Exercise 8<\/p>\n<\/header>\n

                                                                                                                                                                                              \n

                                                                                                                                                                                              Divide and check your answers. These questions are hard work! Check your work using the answer key at the end of the exercise.<\/p>\n

                                                                                                                                                                                                \n
                                                                                                                                                                                              1. [latex]115\\enclose{longdiv}{8682}[\/latex]<\/li>\n
                                                                                                                                                                                              2. [latex]205\\enclose{longdiv}{2384}[\/latex]<\/li>\n
                                                                                                                                                                                              3. [latex]325\\enclose{longdiv}{66321}[\/latex]<\/li>\n
                                                                                                                                                                                              4. [latex]241\\enclose{longdiv}{13284}[\/latex]<\/li>\n
                                                                                                                                                                                              5. [latex]860\\enclose{longdiv}{262412}[\/latex]<\/li>\n
                                                                                                                                                                                              6. [latex]659\\enclose{longdiv}{270190}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                                                                Answers to Exercise 8<\/strong><\/p>\n

                                                                                                                                                                                                  \n
                                                                                                                                                                                                1. 75 R57<\/li>\n
                                                                                                                                                                                                2. 11 R129<\/li>\n
                                                                                                                                                                                                3. 204 R21<\/li>\n
                                                                                                                                                                                                4. 55 R29<\/li>\n
                                                                                                                                                                                                5. 305 R112<\/li>\n
                                                                                                                                                                                                6. 410<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                                                                                                                                                                                  Topic D: Self-Test<\/h1>\n

                                                                                                                                                                                                  Mark\u00a0 \/12\u00a0 \u00a0Aim\u00a0 10\/12<\/strong><\/p>\n

                                                                                                                                                                                                    \n
                                                                                                                                                                                                  1. Divide and check your work for questions b<\/strong> and f<\/strong> using multiplication. (12 marks)\n
                                                                                                                                                                                                      \n
                                                                                                                                                                                                    1. [latex]185 \\div 10 =[\/latex]<\/li>\n
                                                                                                                                                                                                    2. [latex]4\\enclose{longdiv}{92}[\/latex]<\/li>\n
                                                                                                                                                                                                    3. [latex]408 \\div 50 =[\/latex]<\/li>\n
                                                                                                                                                                                                    4. [latex]72\\enclose{longdiv}{6768}[\/latex]<\/li>\n
                                                                                                                                                                                                    5. [latex]67\\enclose{longdiv}{5963}[\/latex]<\/li>\n
                                                                                                                                                                                                    6. [latex]53\\enclose{longdiv}{4856}[\/latex]<\/li>\n
                                                                                                                                                                                                    7. [latex]91\\enclose{longdiv}{8736}[\/latex]<\/li>\n
                                                                                                                                                                                                    8. [latex]265\\enclose{longdiv}{133624}[\/latex]<\/li>\n
                                                                                                                                                                                                    9. [latex]606\\enclose{longdiv}{26094}[\/latex]<\/li>\n
                                                                                                                                                                                                    10. [latex]1000\\enclose{longdiv}{83652}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                                                                                                                                                                                      Answers to Topic D Self-Test<\/h2>\n
                                                                                                                                                                                                        \n
                                                                                                                                                                                                      1. Divide and check your work for questions b<\/strong> and f<\/strong> using multiplication.\n
                                                                                                                                                                                                          \n
                                                                                                                                                                                                        1. 18 R5<\/li>\n
                                                                                                                                                                                                        2. 23<\/li>\n
                                                                                                                                                                                                        3. 8 R8<\/li>\n
                                                                                                                                                                                                        4. 94<\/li>\n
                                                                                                                                                                                                        5. 89<\/li>\n
                                                                                                                                                                                                        6. 91 R33<\/li>\n
                                                                                                                                                                                                        7. 96<\/li>\n
                                                                                                                                                                                                        8. 504 R64<\/li>\n
                                                                                                                                                                                                        9. 43 R36<\/li>\n
                                                                                                                                                                                                        10. 83 R652<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n","protected":false},"author":125,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1148","chapter","type-chapter","status-publish","hentry"],"part":1076,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/alfm3\/wp-json\/pressbooks\/v2\/chapters\/1148","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/alfm3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/alfm3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/alfm3\/wp-json\/wp\/v2\/users\/125"}],"version-history":[{"count":26,"href":"https:\/\/opentextbc.ca\/alfm3\/wp-json\/pressbooks\/v2\/chapters\/1148\/revisions"}],"predecessor-version":[{"id":1738,"href":"https:\/\/opentextbc.ca\/alfm3\/wp-json\/pressbooks\/v2\/chapters\/1148\/revisions\/1738"}],"part":[{"href":"https:\/\/opentextbc.ca\/alfm3\/wp-json\/pressbooks\/v2\/parts\/1076"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/alfm3\/wp-json\/pressbooks\/v2\/chapters\/1148\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/alfm3\/wp-json\/wp\/v2\/media?parent=1148"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/alfm3\/wp-json\/pressbooks\/v2\/chapter-type?post=1148"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/alfm3\/wp-json\/wp\/v2\/contributor?post=1148"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/alfm3\/wp-json\/wp\/v2\/license?post=1148"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}