{"id":30,"date":"2023-05-11T23:31:57","date_gmt":"2023-05-12T03:31:57","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/math0635\/chapter\/two-and-three-digit-multipliers\/"},"modified":"2025-02-03T12:56:36","modified_gmt":"2025-02-03T17:56:36","slug":"two-and-three-digit-multipliers","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/math0635\/chapter\/two-and-three-digit-multipliers\/","title":{"raw":"Topic B: Two- and Three-Digit Multipliers","rendered":"Topic B: Two- and Three-Digit Multipliers"},"content":{"raw":"When the multiplier is more than one digit, you use the same process and get partial products. You repeat the steps until you have multiplied by every digit, then add the partial products together.\r\n
Multiplying by Two-Digit Multipliers<\/h6>\r\n
\r\n

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

\r\n\r\n[latex]24\\times23=[\/latex]\r\n\r\nPart 1:<\/strong> Multiply by the ones digit in the multiplier.\r\n

Multiply 3 ones by 24 using the method you already know. The first partial product is 72.<\/p>\r\n

[latex]\\begin{array}{rr}&_1\\hspace{0.5em}\\\\&24\\\\\\times&23\\\\\\hline&72\\end{array}[\/latex]<\/p>\r\nPart 2:<\/strong> Multiply by the tens digit in the multiplier. First, put a 0 to hold the ones place in your partial product. We are multiplying by a ten, so we hold the ones place.\r\n

Step 1:<\/strong> Multiply 2 tens \u00d7 4 ones = 8 tens Write the 8 tens under the tens in your first partial product. It is very important to keep the columns straight \u2013 ones under one, tens under tens.<\/p>\r\n

Step 2:<\/strong> Multiply 2 tens \u00d7 2 tens = 4 hundreds Write the 4 hundreds in your partial product. The second partial product is 480.<\/p>\r\n

[latex]\\begin{array}{rr}&_1\\hspace{0.5em}\\\\&24\\\\\\times&23\\\\\\hline&72\\\\&480\\end{array}[\/latex]<\/p>\r\nPart 3:<\/strong> Add the partial products together, being sure to add ones to ones, tens to tens, hundreds to hundreds. The sum is the final product.\r\n

Draw a line under the partial products. Add. Check your addition.<\/p>\r\n

[latex]\\begin{array}{rr}&_1\\hspace{0.5em}\\\\&24\\\\\\times&23\\\\\\hline&72\\\\+&480\\\\\\hline&552\\end{array}[\/latex]<\/p>\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]36\\times425 =[\/latex]\r\n\r\n \r\n\r\nPart 1:<\/strong> Multiply by the ones digit in the multiplier. 6 \u00d7 425 = 2550<\/span>\r\n

[latex]\\begin{array}{rr}&_1\\hspace{0.15em}_3\\hspace{0.5em}\\\\&425\\\\\\times&36\\\\\\hline&2550\\end{array}[\/latex]<\/p>\r\nPart 2:<\/strong> Multiply by the tens digit in the multiplier. First put a 0 to hold the ones place in the second partial product.\r\n

    \r\n \t
  • Step 1:<\/strong> 3 tens \u00d7 5 tens = 15 tens = 1 hundred and 5 tens\r\nWrite the 5 tens in the second partial product and carry the 1 hundred. Now you can see why it is best to cross out the numbers you carry as soon as you have added them to the product.<\/li>\r\n \t
  • Step 2:<\/strong> 3 tens \u00d7 2 tens = 6 hundreds\r\n6 hundreds + 1 hundred (carried) = 7 hundreds. There is nothing to carry.<\/li>\r\n \t
  • Step 3:<\/strong> 3 tens \u00d7 4 hundreds = 12 thousands<\/li>\r\n<\/ul>\r\n

    [latex]\\begin{array}{rr}&_1\\hspace{0.5em}\\\\&_1\\hspace{0.15em}_3\\hspace{0.5em}\\\\&425\\\\\\times&36\\\\\\hline&2550\\\\&12750\\end{array}[\/latex]<\/p>\r\nPart 3:<\/strong> Add the partial products together.\r\n

    [latex]\\begin{array}{rr}&_1\\hspace{0.5em}\\\\&_1\\hspace{0.15em}_3\\hspace{0.5em}\\\\&425\\\\\\times&36\\\\\\hline&2550\\\\+&12750\\\\\\hline&15300\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    Tip: Keeping the columns straight with ones under ones, tens under tens, hundreds under hundreds is very important. Working on large-squared graphing paper using one digit per square is often helpful.<\/div>\r\n
    tens \u00d7 tens = hundreds\r\ntens \u00d7 hundreds = thousands<\/div>\r\n

    Exercise 1<\/h2>\r\nMultiply, being very careful to keep the columns straight when you write your partial products. Check your work using the answer key at the end of the exercise.\r\n

    [latex]\\begin{array}{rr}&84\\\\\\times&12\\\\\\hline&168\\\\+&840\\\\\\hline&1008\\end{array}[\/latex]<\/p>\r\n\r\n

    <\/div>\r\n
      \r\n \t
    1. [latex]\\begin{array}{rr}&73\\\\\\times&12\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    2. [latex]\\begin{array}{rr}&50\\\\\\times&42\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    3. [latex]\\begin{array}{rr}&62\\\\\\times&31\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    4. [latex]\\begin{array}{rr}&61\\\\\\times&42\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    5. [latex]\\begin{array}{rr}&91\\\\\\times&53\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    6. [latex]\\begin{array}{rr}&92\\\\\\times&31\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    7. [latex]\\begin{array}{rr}&91\\\\\\times&49\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    8. [latex]\\begin{array}{rr}&72\\\\\\times&48\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    9. [latex]\\begin{array}{rr}&53\\\\\\times&30\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    10. [latex]\\begin{array}{rr}&41\\\\\\times&53\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    11. [latex]\\begin{array}{rr}&42\\\\\\times&94\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    12. [latex]\\begin{array}{rr}&80\\\\\\times&86\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    13. [latex]\\begin{array}{rr}&31\\\\\\times&79\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    14. [latex]\\begin{array}{rr}&54\\\\\\times&40\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    15. [latex]\\begin{array}{rr}&61\\\\\\times&48\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    16. [latex]\\begin{array}{rr}&60\\\\\\times&31\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    17. [latex]\\begin{array}{rr}&55\\\\\\times&73\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    18. [latex]\\begin{array}{rr}&84\\\\\\times&56\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
    19. [latex]\\begin{array}{rr}&53\\\\\\times&38\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n

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

        \r\n \t
      1. 876<\/li>\r\n \t
      2. 2100<\/span><\/li>\r\n \t
      3. 1922<\/span><\/li>\r\n \t
      4. 2562<\/span><\/li>\r\n \t
      5. 4823<\/span><\/li>\r\n \t
      6. 2852<\/span><\/li>\r\n \t
      7. 4459<\/span><\/li>\r\n \t
      8. 3456<\/span><\/li>\r\n \t
      9. 1590<\/span><\/li>\r\n \t
      10. 2173<\/span><\/li>\r\n \t
      11. 3948<\/span><\/li>\r\n \t
      12. 6880<\/span><\/li>\r\n \t
      13. 2449<\/span><\/li>\r\n \t
      14. 2160<\/span><\/li>\r\n \t
      15. 2928<\/span><\/li>\r\n \t
      16. 1860<\/span><\/li>\r\n \t
      17. 4015<\/span><\/li>\r\n \t
      18. 4704<\/span><\/li>\r\n \t
      19. 2014<\/span><\/li>\r\n<\/ol>\r\nWhen the multiplier has a zero in the ones place, use this shortcut.\r\n
        \r\n

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

        \r\n\r\n[latex]\\begin{array}{rr}&48\\\\\\times&80\\\\\\hline&3840\\end{array}[\/latex]\r\n\r\nStep 1:<\/strong> 0 ones \u00d7 48 = 0. Place one zero in the product and that will hold the ones place.\r\n\r\nStep 2:<\/strong> Multiply by the tens digit and write the product beside the zero.\r\n\r\n<\/div>\r\n<\/div>\r\n
        \r\n

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

        \r\n\r\n[latex]\\begin{array}{rr}&97\\\\\\times&20\\\\\\hline&1940\\end{array}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n

        Exercise 2<\/h2>\r\nFind the products. Use the shortcut for multipliers with a zero in them. Check your work using the answer key at the end of the exercise.\r\n\r\n[latex]\\begin{array}{rr}&76\\\\\\times&70\\\\\\hline&5320\\end{array}[\/latex]\r\n
          \r\n \t
        1. [latex]\\begin{array}{rr}&52\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        2. [latex]\\begin{array}{rr}&91\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        3. [latex]\\begin{array}{rr}&83\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        4. [latex]\\begin{array}{rr}&49\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        5. [latex]\\begin{array}{rr}&61\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        6. [latex]\\begin{array}{rr}&16\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        7. [latex]\\begin{array}{rr}&36\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        8. [latex]\\begin{array}{rr}&398\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        9. [latex]\\begin{array}{rr}&432\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        10. [latex]\\begin{array}{rr}&863\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        11. [latex]\\begin{array}{rr}&907\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        12. [latex]\\begin{array}{rr}&503\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        13. [latex]\\begin{array}{rr}&452\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        14. [latex]\\begin{array}{rr}&943\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        15. [latex]\\begin{array}{rr}&248\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        16. [latex]\\begin{array}{rr}&6287\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        17. [latex]\\begin{array}{rr}&9025\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        18. [latex]\\begin{array}{rr}&8907\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        19. [latex]\\begin{array}{rr}&300\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        20. [latex]\\begin{array}{rr}&9075\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        21. [latex]\\begin{array}{rr}&3952\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        22. [latex]\\begin{array}{rr}&1528\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
        23. [latex]\\begin{array}{rr}&7106\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n
          <\/div>\r\nAnswers to Exercise 2<\/strong>\r\n
            \r\n \t
          1. 3 640<\/li>\r\n \t
          2. 6 370<\/li>\r\n \t
          3. 5 810<\/li>\r\n \t
          4. 3 430<\/li>\r\n \t
          5. 4 270<\/li>\r\n \t
          6. 1 120<\/li>\r\n \t
          7. 2 520<\/li>\r\n \t
          8. 27 860<\/li>\r\n \t
          9. 30 240<\/li>\r\n \t
          10. 60 410<\/li>\r\n \t
          11. 63 490<\/li>\r\n \t
          12. 35 210<\/li>\r\n \t
          13. 31 640<\/li>\r\n \t
          14. 66 010<\/li>\r\n \t
          15. 17 360<\/li>\r\n \t
          16. 440 090<\/li>\r\n \t
          17. 631 750<\/li>\r\n \t
          18. 623 490<\/li>\r\n \t
          19. 21 000<\/li>\r\n \t
          20. 635 250<\/li>\r\n \t
          21. 276 640<\/li>\r\n \t
          22. 106 960<\/li>\r\n \t
          23. 497 420<\/li>\r\n<\/ol>\r\n

            Multiplying by Three-Digit Multipliers<\/h1>\r\nTo multiply by three digit multipliers, use the same method with one more part.\r\n
            \r\n

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

            \r\n\r\n[latex]417\\times368 =[\/latex]\r\n

            [latex]\\begin{array}{rr}&417\\\\\\times&368\\\\\\hline&3336\\\\&25020\\\\+&125100\\\\\\hline&153456\\end{array}[\/latex]<\/p>\r\nPart 1:<\/strong> Multiply by the 8 ones.\r\n\r\nPart 2:<\/strong> Multiply the 6 tens; hold the ones place with 0.\r\n\r\nPart 3:<\/strong> Multiply by the 3 hundreds. Put 00 to hold the ones and tens places in the third partial product.\r\n

              \r\n \t
            • Step 1:<\/strong> 3 hundreds \u00d7 7 ones = 21 hundreds = 2 thousands and 1 hundred. Write the 1 hundred and carry the 2 thousands.<\/li>\r\n \t
            • Step 2:<\/strong> 3 hundreds \u00d7 1 ten = 3 thousands. 3 thousands + 2 thousands (carried) = 5 thousands.<\/li>\r\n \t
            • Step 3:<\/strong> 3 hundreds \u00d7 4 hundreds = 12 ten thousands.<\/li>\r\n<\/ul>\r\nPart 4: <\/strong>Add the partial products.\r\n\r\n<\/div>\r\n<\/div>\r\n

              Exercise 3<\/h2>\r\nFind the products. Check your work using the answer key at the end of the exercise.\r\n
                \r\n \t
              1. [latex]\\begin{array}{rr}&416\\\\\\times&213\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              2. [latex]\\begin{array}{rr}&375\\\\\\times&291\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              3. [latex]\\begin{array}{rr}&361\\\\\\times&475\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              4. [latex]\\begin{array}{rr}&275\\\\\\times&863\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              5. [latex]\\begin{array}{rr}&984\\\\\\times&469\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              6. [latex]\\begin{array}{rr}&489\\\\\\times&578\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              7. [latex]\\begin{array}{rr}&498\\\\\\times&123\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              8. [latex]\\begin{array}{rr}&267\\\\\\times&854\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              9. [latex]\\begin{array}{rr}&613\\\\\\times&368\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              10. [latex]\\begin{array}{rr}&725\\\\\\times&547\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              11. [latex]\\begin{array}{rr}&269\\\\\\times&912\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              12. [latex]\\begin{array}{rr}&752\\\\\\times&697\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              13. [latex]\\begin{array}{rr}&983\\\\\\times&357\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              14. [latex]\\begin{array}{rr}&835\\\\\\times&148\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
              15. [latex]\\begin{array}{rr}&386\\\\\\times&296\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n
                <\/div>\r\nAnswers to Exercise 3<\/strong>\r\n
                  \r\n \t
                1. 88608<\/span><\/li>\r\n \t
                2. 109125<\/span><\/li>\r\n \t
                3. 171475<\/span><\/li>\r\n \t
                4. 237325<\/span><\/li>\r\n \t
                5. 461496<\/span><\/li>\r\n \t
                6. 282642<\/span><\/li>\r\n \t
                7. 61254<\/span><\/li>\r\n \t
                8. 228018<\/span><\/li>\r\n \t
                9. 225584<\/span><\/li>\r\n \t
                10. 396575<\/span><\/li>\r\n \t
                11. 245328<\/span><\/li>\r\n \t
                12. 524144<\/span><\/li>\r\n \t
                13. 350931<\/span><\/li>\r\n \t
                14. 123580<\/span><\/li>\r\n \t
                15. 114256<\/span><\/li>\r\n<\/ol>\r\n
                  \r\n
                  You know to hold the ones place with a zero if the multiplier has a zero in the ones place. Use the same skill if the multiplier has a zero in the tens place.<\/strong><\/div>\r\n
                  \r\n

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

                  \r\n\r\n[latex]927\\times405 =[\/latex]\r\n

                  [latex]\\begin{array}{rr}&927\\\\\\times&405\\\\\\hline&4635\\\\+&370800\\\\\\hline&375435\\end{array}[\/latex]<\/p>\r\n\r\n

                    \r\n \t
                  • Part 1:<\/strong> Multiply by the 5 ones.<\/li>\r\n \t
                  • Part 2:<\/strong> Multiply by the 0 tens.\r\nHold the ones place with a 0; 0 \u00d7 927 = 0; Place one zero in the tens place in the second partial product.<\/li>\r\n \t
                  • Part 3:<\/strong> Multiply by the 4 hundreds. The ones and tens places are already held by zeros. Start this partial product in the hundreds place on the same line.<\/li>\r\n \t
                  • Part 4:<\/strong> Add the partial products.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n

                    Exercise 4<\/h2>\r\nFind the products. Check your work using the answer key at the end of the exercise.\r\n

                    [latex]\\begin{array}{rr}&_2\\hspace{0.15em}_2\\hspace{0.5em}\\\\&698\\\\\\times&301\\\\\\hline&698\\\\+&209400\\\\\\hline&210098\\end{array}[\/latex]<\/p>\r\n\r\n

                    <\/div>\r\n
                      \r\n \t
                    1. [latex]\\begin{array}{rr}&923\\\\\\times&403\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    2. [latex]\\begin{array}{rr}&830\\\\\\times&108\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    3. [latex]\\begin{array}{rr}&482\\\\\\times&206\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    4. [latex]\\begin{array}{rr}&432\\\\\\times&205\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    5. [latex]\\begin{array}{rr}&625\\\\\\times&405\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    6. [latex]\\begin{array}{rr}&275\\\\\\times&306\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    7. [latex]\\begin{array}{rr}&765\\\\\\times&506\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    8. [latex]\\begin{array}{rr}&1576\\\\\\times&702\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    9. [latex]\\begin{array}{rr}&432\\\\\\times&405\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    10. [latex]\\begin{array}{rr}&625\\\\\\times&409\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    11. [latex]\\begin{array}{rr}&175\\\\\\times&306\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    12. [latex]\\begin{array}{rr}&5874\\\\\\times&309\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    13. [latex]\\begin{array}{rr}&7384\\\\\\times&104\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                    14. [latex]\\begin{array}{rr}&6538\\\\\\times&603\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n
                      Answers to Exercise 4<\/strong><\/div>\r\n
                        \r\n \t
                      1. 371969<\/span><\/li>\r\n \t
                      2. 89640<\/span><\/li>\r\n \t
                      3. 99292<\/span><\/li>\r\n \t
                      4. 88560<\/span><\/li>\r\n \t
                      5. 253 125<\/li>\r\n \t
                      6. 84150<\/span><\/li>\r\n \t
                      7. 387090<\/span><\/li>\r\n \t
                      8. 1106<\/span>352<\/span><\/li>\r\n \t
                      9. 174960<\/span><\/li>\r\n \t
                      10. 255625<\/span><\/li>\r\n \t
                      11. 53550<\/span><\/li>\r\n \t
                      12. 1815<\/span>066<\/span><\/li>\r\n \t
                      13. 767936<\/span><\/li>\r\n \t
                      14. 3942<\/span>414<\/span><\/li>\r\n<\/ol>\r\n

                        Multiplying by 10, 100, and 1 000<\/h1>\r\n

                        Exercise 5<\/h6>\r\nDo the following questions and see if you can find the pattern. Check your work using the answer key at the end of the exercise.\r\n
                          \r\n \t
                        1. [latex]\\begin{array}{rr}&83\\\\\\times&10\\\\\\hline&830\\end{array}[\/latex]<\/li>\r\n \t
                        2. [latex]\\begin{array}{rr}&46\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        3. [latex]\\begin{array}{rr}&97\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        4. [latex]\\begin{array}{rr}&123\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        5. [latex]\\begin{array}{rr}&70\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        6. [latex]\\begin{array}{rr}&129\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        7. [latex]\\begin{array}{rr}&1852\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        8. [latex]\\begin{array}{rr}&29871\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        9. [latex]\\begin{array}{rr}&45\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        10. [latex]\\begin{array}{rr}&26\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        11. [latex]\\begin{array}{rr}&432\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        12. [latex]\\begin{array}{rr}&679\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        13. [latex]\\begin{array}{rr}&2482\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        14. [latex]\\begin{array}{rr}&9037\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        15. [latex]\\begin{array}{rr}&46207\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        16. [latex]\\begin{array}{rr}&97512\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        17. [latex]\\begin{array}{rr}&23\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        18. [latex]\\begin{array}{rr}&452\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        19. [latex]\\begin{array}{rr}&207\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        20. [latex]\\begin{array}{rr}&348\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        21. [latex]\\begin{array}{rr}&2118\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        22. [latex]\\begin{array}{rr}&2431\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        23. [latex]\\begin{array}{rr}&23681\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                        24. [latex]\\begin{array}{rr}&48203\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n
                          Answers to Exercise 5<\/strong><\/div>\r\n
                            \r\n \t
                          1. 830<\/li>\r\n \t
                          2. 460<\/li>\r\n \t
                          3. 970<\/li>\r\n \t
                          4. 1230<\/span><\/li>\r\n \t
                          5. 700<\/li>\r\n \t
                          6. 1290<\/span><\/li>\r\n \t
                          7. 18520<\/span><\/li>\r\n \t
                          8. 298710<\/span><\/li>\r\n \t
                          9. 4500<\/span><\/li>\r\n \t
                          10. 2600<\/span><\/li>\r\n \t
                          11. 43200<\/span><\/li>\r\n \t
                          12. 67900<\/span><\/li>\r\n \t
                          13. 248200<\/span><\/li>\r\n \t
                          14. 903700<\/span><\/li>\r\n \t
                          15. 4620700<\/span><\/span><\/li>\r\n \t
                          16. 9751200<\/span><\/span><\/li>\r\n \t
                          17. 23000<\/span><\/li>\r\n \t
                          18. 452000<\/span><\/li>\r\n \t
                          19. 207000<\/span><\/li>\r\n \t
                          20. 348000<\/span><\/li>\r\n \t
                          21. 2118000<\/span><\/span><\/li>\r\n \t
                          22. 2431000<\/span><\/span><\/li>\r\n \t
                          23. 23681000<\/span><\/span><\/li>\r\n \t
                          24. 48203000<\/span><\/span><\/li>\r\n<\/ol>\r\n
                            \r\n\r\nAnd the pattern is \u2026\r\n\r\nWhen multiplying by 10, 100, 1000, 10000, etc., place as many zeros to the right of the number as there are zeros in the 10, 100, 1000, etc.<\/span><\/span><\/span>\r\n
                              \r\n \t
                            • To multiply by 10 put one zero after the number.<\/li>\r\n \t
                            • To multiply by 100 put two zeros after the number.<\/li>\r\n \t
                            • To multiply by 1000 put three zeros after the number.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n
                              Exercise 6<\/h6>\r\nFind the products using the short method.\u00a0 Do not rewrite the questions. Check your work using the answer key at the end of the exercise.\r\n
                                \r\n \t
                              1. \r\n

                                12 \u00d7 10 = 120<\/p>\r\n<\/li>\r\n \t

                              2. 10 \u00d7 3175<\/span> =<\/li>\r\n \t
                              3. 162 \u00d7 10 =<\/li>\r\n \t
                              4. 10 \u00d7 53821<\/span> =<\/li>\r\n \t
                              5. 10 \u00d7 123 =<\/li>\r\n \t
                              6. 27342<\/span> \u00d7 10 =<\/li>\r\n \t
                              7. 10 \u00d7 98 =<\/li>\r\n \t
                              8. 1134<\/span> \u00d7 10 =<\/li>\r\n \t
                              9. 15 \u00d7 100 =<\/li>\r\n \t
                              10. 100 \u00d7 278 =<\/li>\r\n \t
                              11. 9134<\/span> \u00d7 100 =<\/li>\r\n \t
                              12. 651 \u00d7 100 =<\/li>\r\n \t
                              13. 100 \u00d7 5169<\/span> =<\/li>\r\n \t
                              14. 100 \u00d7 24815<\/span> =<\/li>\r\n \t
                              15. 10 \u00d7 905 =<\/li>\r\n \t
                              16. 45683<\/span> \u00d7 10 =<\/li>\r\n \t
                              17. 1000<\/span> \u00d7 87 =<\/li>\r\n \t
                              18. 521 \u00d7 1000<\/span> =<\/li>\r\n \t
                              19. 1000<\/span> \u00d7 68935<\/span> =<\/li>\r\n \t
                              20. 1000<\/span> \u00d7 8902<\/span> =<\/li>\r\n \t
                              21. 1576<\/span> \u00d7 1000<\/span> =<\/li>\r\n \t
                              22. 31584<\/span> \u00d7 1000<\/span> =<\/li>\r\n \t
                              23. 1000<\/span> \u00d7 426 =<\/li>\r\n \t
                              24. \r\n

                                72 \u00d7 1000<\/span> =<\/p>\r\n<\/li>\r\n<\/ol>\r\n

                                Answers to Exercise 6<\/strong><\/div>\r\n
                                  \r\n \t
                                1. 120<\/li>\r\n \t
                                2. 31750<\/span><\/li>\r\n \t
                                3. 1620<\/span><\/li>\r\n \t
                                4. 538210<\/span><\/li>\r\n \t
                                5. 1230<\/span><\/li>\r\n \t
                                6. 273420<\/span><\/li>\r\n \t
                                7. 980<\/li>\r\n \t
                                8. 11340<\/span><\/li>\r\n \t
                                9. 1500<\/span><\/li>\r\n \t
                                10. 27800<\/span><\/li>\r\n \t
                                11. 913400<\/span><\/li>\r\n \t
                                12. 65100<\/span><\/li>\r\n \t
                                13. 516900<\/span><\/li>\r\n \t
                                14. 2481500<\/span><\/span><\/li>\r\n \t
                                15. 9050<\/span><\/li>\r\n \t
                                16. 456830<\/span><\/li>\r\n \t
                                17. 87000<\/span><\/li>\r\n \t
                                18. 521000<\/span><\/li>\r\n \t
                                19. 68935000<\/span><\/span><\/li>\r\n \t
                                20. 8902000<\/span><\/span><\/li>\r\n \t
                                21. 1576000<\/span><\/span><\/li>\r\n \t
                                22. 31584000<\/span><\/span><\/li>\r\n \t
                                23. 426000<\/span><\/li>\r\n \t
                                24. 72000<\/span><\/li>\r\n<\/ol>\r\n

                                  Topic B: Self-Test<\/h1>\r\nMark\u00a0 \/12 \u00a0\u00a0 Aim\u00a0 10\/12<\/strong>\r\n\r\nMultiply these numbers.\r\n
                                    \r\n \t
                                  1. [latex]\\begin{array}{rr}&47\\\\\\times&39\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                                  2. [latex]\\begin{array}{rr}&58\\\\\\times&93\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                                  3. [latex]\\begin{array}{rr}&48\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                                  4. [latex]\\begin{array}{rr}&982\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                                  5. [latex]\\begin{array}{rr}&678\\\\\\times&39\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                                  6. [latex]\\begin{array}{rr}&4579\\\\\\times&86\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                                  7. [latex]\\begin{array}{rr}&8703\\\\\\times&93\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                                  8. [latex]\\begin{array}{rr}&7390\\\\\\times&85\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                                  9. [latex]\\begin{array}{rr}&8047\\\\\\times&236\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                                  10. [latex]\\begin{array}{rr}&4238\\\\\\times&197\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                                  11. [latex]\\begin{array}{rr}&8200\\\\\\times&444\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n \t
                                  12. [latex]\\begin{array}{rr}&7265\\\\\\times&409\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n
                                    Answers to Topic B Self-Test<\/strong><\/div>\r\nMultiply these numbers.\r\n
                                      \r\n \t
                                    1. \r\n
                                        \r\n \t
                                      1. 1833<\/span><\/li>\r\n \t
                                      2. 5394<\/span><\/li>\r\n \t
                                      3. 4800<\/span><\/li>\r\n \t
                                      4. 982000<\/span><\/li>\r\n \t
                                      5. 26442<\/span><\/li>\r\n \t
                                      6. 393794<\/span><\/li>\r\n \t
                                      7. 809379<\/span><\/li>\r\n \t
                                      8. 628150<\/span><\/li>\r\n \t
                                      9. 1899092<\/span><\/span><\/li>\r\n \t
                                      10. 834886<\/span><\/li>\r\n \t
                                      11. 3640800<\/span><\/span><\/li>\r\n \t
                                      12. 2971385<\/span><\/span><\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"

                                        When the multiplier is more than one digit, you use the same process and get partial products. You repeat the steps until you have multiplied by every digit, then add the partial products together.<\/p>\n

                                        Multiplying by Two-Digit Multipliers<\/h6>\n
                                        \n
                                        \n

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

                                        \n

                                        [latex]24\\times23=[\/latex]<\/p>\n

                                        Part 1:<\/strong> Multiply by the ones digit in the multiplier.<\/p>\n

                                        Multiply 3 ones by 24 using the method you already know. The first partial product is 72.<\/p>\n

                                        [latex]\\begin{array}{rr}&_1\\hspace{0.5em}\\\\&24\\\\\\times&23\\\\\\hline&72\\end{array}[\/latex]<\/p>\n

                                        Part 2:<\/strong> Multiply by the tens digit in the multiplier. First, put a 0 to hold the ones place in your partial product. We are multiplying by a ten, so we hold the ones place.<\/p>\n

                                        Step 1:<\/strong> Multiply 2 tens \u00d7 4 ones = 8 tens Write the 8 tens under the tens in your first partial product. It is very important to keep the columns straight \u2013 ones under one, tens under tens.<\/p>\n

                                        Step 2:<\/strong> Multiply 2 tens \u00d7 2 tens = 4 hundreds Write the 4 hundreds in your partial product. The second partial product is 480.<\/p>\n

                                        [latex]\\begin{array}{rr}&_1\\hspace{0.5em}\\\\&24\\\\\\times&23\\\\\\hline&72\\\\&480\\end{array}[\/latex]<\/p>\n

                                        Part 3:<\/strong> Add the partial products together, being sure to add ones to ones, tens to tens, hundreds to hundreds. The sum is the final product.<\/p>\n

                                        Draw a line under the partial products. Add. Check your addition.<\/p>\n

                                        [latex]\\begin{array}{rr}&_1\\hspace{0.5em}\\\\&24\\\\\\times&23\\\\\\hline&72\\\\+&480\\\\\\hline&552\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                        \n
                                        \n

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

                                        \n

                                        [latex]36\\times425 =[\/latex]<\/p>\n

                                         <\/p>\n

                                        Part 1:<\/strong> Multiply by the ones digit in the multiplier. 6 \u00d7 425 = 2550<\/span><\/p>\n

                                        [latex]\\begin{array}{rr}&_1\\hspace{0.15em}_3\\hspace{0.5em}\\\\&425\\\\\\times&36\\\\\\hline&2550\\end{array}[\/latex]<\/p>\n

                                        Part 2:<\/strong> Multiply by the tens digit in the multiplier. First put a 0 to hold the ones place in the second partial product.<\/p>\n

                                          \n
                                        • Step 1:<\/strong> 3 tens \u00d7 5 tens = 15 tens = 1 hundred and 5 tens
                                          \nWrite the 5 tens in the second partial product and carry the 1 hundred. Now you can see why it is best to cross out the numbers you carry as soon as you have added them to the product.<\/li>\n
                                        • Step 2:<\/strong> 3 tens \u00d7 2 tens = 6 hundreds
                                          \n6 hundreds + 1 hundred (carried) = 7 hundreds. There is nothing to carry.<\/li>\n
                                        • Step 3:<\/strong> 3 tens \u00d7 4 hundreds = 12 thousands<\/li>\n<\/ul>\n

                                          [latex]\\begin{array}{rr}&_1\\hspace{0.5em}\\\\&_1\\hspace{0.15em}_3\\hspace{0.5em}\\\\&425\\\\\\times&36\\\\\\hline&2550\\\\&12750\\end{array}[\/latex]<\/p>\n

                                          Part 3:<\/strong> Add the partial products together.<\/p>\n

                                          [latex]\\begin{array}{rr}&_1\\hspace{0.5em}\\\\&_1\\hspace{0.15em}_3\\hspace{0.5em}\\\\&425\\\\\\times&36\\\\\\hline&2550\\\\+&12750\\\\\\hline&15300\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                          Tip: Keeping the columns straight with ones under ones, tens under tens, hundreds under hundreds is very important. Working on large-squared graphing paper using one digit per square is often helpful.<\/div>\n
                                          tens \u00d7 tens = hundreds
                                          \ntens \u00d7 hundreds = thousands<\/div>\n

                                          Exercise 1<\/h2>\n

                                          Multiply, being very careful to keep the columns straight when you write your partial products. Check your work using the answer key at the end of the exercise.<\/p>\n

                                          [latex]\\begin{array}{rr}&84\\\\\\times&12\\\\\\hline&168\\\\+&840\\\\\\hline&1008\\end{array}[\/latex]<\/p>\n

                                          <\/div>\n
                                            \n
                                          1. [latex]\\begin{array}{rr}&73\\\\\\times&12\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          2. [latex]\\begin{array}{rr}&50\\\\\\times&42\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          3. [latex]\\begin{array}{rr}&62\\\\\\times&31\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          4. [latex]\\begin{array}{rr}&61\\\\\\times&42\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          5. [latex]\\begin{array}{rr}&91\\\\\\times&53\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          6. [latex]\\begin{array}{rr}&92\\\\\\times&31\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          7. [latex]\\begin{array}{rr}&91\\\\\\times&49\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          8. [latex]\\begin{array}{rr}&72\\\\\\times&48\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          9. [latex]\\begin{array}{rr}&53\\\\\\times&30\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          10. [latex]\\begin{array}{rr}&41\\\\\\times&53\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          11. [latex]\\begin{array}{rr}&42\\\\\\times&94\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          12. [latex]\\begin{array}{rr}&80\\\\\\times&86\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          13. [latex]\\begin{array}{rr}&31\\\\\\times&79\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          14. [latex]\\begin{array}{rr}&54\\\\\\times&40\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          15. [latex]\\begin{array}{rr}&61\\\\\\times&48\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          16. [latex]\\begin{array}{rr}&60\\\\\\times&31\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          17. [latex]\\begin{array}{rr}&55\\\\\\times&73\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          18. [latex]\\begin{array}{rr}&84\\\\\\times&56\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                          19. [latex]\\begin{array}{rr}&53\\\\\\times&38\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n<\/ol>\n

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

                                              \n
                                            1. 876<\/li>\n
                                            2. 2100<\/span><\/li>\n
                                            3. 1922<\/span><\/li>\n
                                            4. 2562<\/span><\/li>\n
                                            5. 4823<\/span><\/li>\n
                                            6. 2852<\/span><\/li>\n
                                            7. 4459<\/span><\/li>\n
                                            8. 3456<\/span><\/li>\n
                                            9. 1590<\/span><\/li>\n
                                            10. 2173<\/span><\/li>\n
                                            11. 3948<\/span><\/li>\n
                                            12. 6880<\/span><\/li>\n
                                            13. 2449<\/span><\/li>\n
                                            14. 2160<\/span><\/li>\n
                                            15. 2928<\/span><\/li>\n
                                            16. 1860<\/span><\/li>\n
                                            17. 4015<\/span><\/li>\n
                                            18. 4704<\/span><\/li>\n
                                            19. 2014<\/span><\/li>\n<\/ol>\n

                                              When the multiplier has a zero in the ones place, use this shortcut.<\/p>\n

                                              \n
                                              \n

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

                                              \n

                                              [latex]\\begin{array}{rr}&48\\\\\\times&80\\\\\\hline&3840\\end{array}[\/latex]<\/p>\n

                                              Step 1:<\/strong> 0 ones \u00d7 48 = 0. Place one zero in the product and that will hold the ones place.<\/p>\n

                                              Step 2:<\/strong> Multiply by the tens digit and write the product beside the zero.<\/p>\n<\/div>\n<\/div>\n

                                              \n
                                              \n

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

                                              \n

                                              [latex]\\begin{array}{rr}&97\\\\\\times&20\\\\\\hline&1940\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                              Exercise 2<\/h2>\n

                                              Find the products. Use the shortcut for multipliers with a zero in them. Check your work using the answer key at the end of the exercise.<\/p>\n

                                              [latex]\\begin{array}{rr}&76\\\\\\times&70\\\\\\hline&5320\\end{array}[\/latex]<\/p>\n

                                                \n
                                              1. [latex]\\begin{array}{rr}&52\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              2. [latex]\\begin{array}{rr}&91\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              3. [latex]\\begin{array}{rr}&83\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              4. [latex]\\begin{array}{rr}&49\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              5. [latex]\\begin{array}{rr}&61\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              6. [latex]\\begin{array}{rr}&16\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              7. [latex]\\begin{array}{rr}&36\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              8. [latex]\\begin{array}{rr}&398\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              9. [latex]\\begin{array}{rr}&432\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              10. [latex]\\begin{array}{rr}&863\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              11. [latex]\\begin{array}{rr}&907\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              12. [latex]\\begin{array}{rr}&503\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              13. [latex]\\begin{array}{rr}&452\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              14. [latex]\\begin{array}{rr}&943\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              15. [latex]\\begin{array}{rr}&248\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              16. [latex]\\begin{array}{rr}&6287\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              17. [latex]\\begin{array}{rr}&9025\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              18. [latex]\\begin{array}{rr}&8907\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              19. [latex]\\begin{array}{rr}&300\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              20. [latex]\\begin{array}{rr}&9075\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              21. [latex]\\begin{array}{rr}&3952\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              22. [latex]\\begin{array}{rr}&1528\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                              23. [latex]\\begin{array}{rr}&7106\\\\\\times&70\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n<\/ol>\n
                                                <\/div>\n

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

                                                  \n
                                                1. 3 640<\/li>\n
                                                2. 6 370<\/li>\n
                                                3. 5 810<\/li>\n
                                                4. 3 430<\/li>\n
                                                5. 4 270<\/li>\n
                                                6. 1 120<\/li>\n
                                                7. 2 520<\/li>\n
                                                8. 27 860<\/li>\n
                                                9. 30 240<\/li>\n
                                                10. 60 410<\/li>\n
                                                11. 63 490<\/li>\n
                                                12. 35 210<\/li>\n
                                                13. 31 640<\/li>\n
                                                14. 66 010<\/li>\n
                                                15. 17 360<\/li>\n
                                                16. 440 090<\/li>\n
                                                17. 631 750<\/li>\n
                                                18. 623 490<\/li>\n
                                                19. 21 000<\/li>\n
                                                20. 635 250<\/li>\n
                                                21. 276 640<\/li>\n
                                                22. 106 960<\/li>\n
                                                23. 497 420<\/li>\n<\/ol>\n

                                                  Multiplying by Three-Digit Multipliers<\/h1>\n

                                                  To multiply by three digit multipliers, use the same method with one more part.<\/p>\n

                                                  \n
                                                  \n

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

                                                  \n

                                                  [latex]417\\times368 =[\/latex]<\/p>\n

                                                  [latex]\\begin{array}{rr}&417\\\\\\times&368\\\\\\hline&3336\\\\&25020\\\\+&125100\\\\\\hline&153456\\end{array}[\/latex]<\/p>\n

                                                  Part 1:<\/strong> Multiply by the 8 ones.<\/p>\n

                                                  Part 2:<\/strong> Multiply the 6 tens; hold the ones place with 0.<\/p>\n

                                                  Part 3:<\/strong> Multiply by the 3 hundreds. Put 00 to hold the ones and tens places in the third partial product.<\/p>\n

                                                    \n
                                                  • Step 1:<\/strong> 3 hundreds \u00d7 7 ones = 21 hundreds = 2 thousands and 1 hundred. Write the 1 hundred and carry the 2 thousands.<\/li>\n
                                                  • Step 2:<\/strong> 3 hundreds \u00d7 1 ten = 3 thousands. 3 thousands + 2 thousands (carried) = 5 thousands.<\/li>\n
                                                  • Step 3:<\/strong> 3 hundreds \u00d7 4 hundreds = 12 ten thousands.<\/li>\n<\/ul>\n

                                                    Part 4: <\/strong>Add the partial products.<\/p>\n<\/div>\n<\/div>\n

                                                    Exercise 3<\/h2>\n

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

                                                      \n
                                                    1. [latex]\\begin{array}{rr}&416\\\\\\times&213\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    2. [latex]\\begin{array}{rr}&375\\\\\\times&291\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    3. [latex]\\begin{array}{rr}&361\\\\\\times&475\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    4. [latex]\\begin{array}{rr}&275\\\\\\times&863\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    5. [latex]\\begin{array}{rr}&984\\\\\\times&469\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    6. [latex]\\begin{array}{rr}&489\\\\\\times&578\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    7. [latex]\\begin{array}{rr}&498\\\\\\times&123\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    8. [latex]\\begin{array}{rr}&267\\\\\\times&854\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    9. [latex]\\begin{array}{rr}&613\\\\\\times&368\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    10. [latex]\\begin{array}{rr}&725\\\\\\times&547\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    11. [latex]\\begin{array}{rr}&269\\\\\\times&912\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    12. [latex]\\begin{array}{rr}&752\\\\\\times&697\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    13. [latex]\\begin{array}{rr}&983\\\\\\times&357\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    14. [latex]\\begin{array}{rr}&835\\\\\\times&148\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                    15. [latex]\\begin{array}{rr}&386\\\\\\times&296\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n<\/ol>\n
                                                      <\/div>\n

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

                                                        \n
                                                      1. 88608<\/span><\/li>\n
                                                      2. 109125<\/span><\/li>\n
                                                      3. 171475<\/span><\/li>\n
                                                      4. 237325<\/span><\/li>\n
                                                      5. 461496<\/span><\/li>\n
                                                      6. 282642<\/span><\/li>\n
                                                      7. 61254<\/span><\/li>\n
                                                      8. 228018<\/span><\/li>\n
                                                      9. 225584<\/span><\/li>\n
                                                      10. 396575<\/span><\/li>\n
                                                      11. 245328<\/span><\/li>\n
                                                      12. 524144<\/span><\/li>\n
                                                      13. 350931<\/span><\/li>\n
                                                      14. 123580<\/span><\/li>\n
                                                      15. 114256<\/span><\/li>\n<\/ol>\n
                                                        \n
                                                        You know to hold the ones place with a zero if the multiplier has a zero in the ones place. Use the same skill if the multiplier has a zero in the tens place.<\/strong><\/div>\n
                                                        \n
                                                        \n

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

                                                        \n

                                                        [latex]927\\times405 =[\/latex]<\/p>\n

                                                        [latex]\\begin{array}{rr}&927\\\\\\times&405\\\\\\hline&4635\\\\+&370800\\\\\\hline&375435\\end{array}[\/latex]<\/p>\n

                                                          \n
                                                        • Part 1:<\/strong> Multiply by the 5 ones.<\/li>\n
                                                        • Part 2:<\/strong> Multiply by the 0 tens.
                                                          \nHold the ones place with a 0; 0 \u00d7 927 = 0; Place one zero in the tens place in the second partial product.<\/li>\n
                                                        • Part 3:<\/strong> Multiply by the 4 hundreds. The ones and tens places are already held by zeros. Start this partial product in the hundreds place on the same line.<\/li>\n
                                                        • Part 4:<\/strong> Add the partial products.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n

                                                          Exercise 4<\/h2>\n

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

                                                          [latex]\\begin{array}{rr}&_2\\hspace{0.15em}_2\\hspace{0.5em}\\\\&698\\\\\\times&301\\\\\\hline&698\\\\+&209400\\\\\\hline&210098\\end{array}[\/latex]<\/p>\n

                                                          <\/div>\n
                                                            \n
                                                          1. [latex]\\begin{array}{rr}&923\\\\\\times&403\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          2. [latex]\\begin{array}{rr}&830\\\\\\times&108\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          3. [latex]\\begin{array}{rr}&482\\\\\\times&206\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          4. [latex]\\begin{array}{rr}&432\\\\\\times&205\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          5. [latex]\\begin{array}{rr}&625\\\\\\times&405\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          6. [latex]\\begin{array}{rr}&275\\\\\\times&306\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          7. [latex]\\begin{array}{rr}&765\\\\\\times&506\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          8. [latex]\\begin{array}{rr}&1576\\\\\\times&702\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          9. [latex]\\begin{array}{rr}&432\\\\\\times&405\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          10. [latex]\\begin{array}{rr}&625\\\\\\times&409\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          11. [latex]\\begin{array}{rr}&175\\\\\\times&306\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          12. [latex]\\begin{array}{rr}&5874\\\\\\times&309\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          13. [latex]\\begin{array}{rr}&7384\\\\\\times&104\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                          14. [latex]\\begin{array}{rr}&6538\\\\\\times&603\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n<\/ol>\n
                                                            Answers to Exercise 4<\/strong><\/div>\n
                                                              \n
                                                            1. 371969<\/span><\/li>\n
                                                            2. 89640<\/span><\/li>\n
                                                            3. 99292<\/span><\/li>\n
                                                            4. 88560<\/span><\/li>\n
                                                            5. 253 125<\/li>\n
                                                            6. 84150<\/span><\/li>\n
                                                            7. 387090<\/span><\/li>\n
                                                            8. 1106<\/span>352<\/span><\/li>\n
                                                            9. 174960<\/span><\/li>\n
                                                            10. 255625<\/span><\/li>\n
                                                            11. 53550<\/span><\/li>\n
                                                            12. 1815<\/span>066<\/span><\/li>\n
                                                            13. 767936<\/span><\/li>\n
                                                            14. 3942<\/span>414<\/span><\/li>\n<\/ol>\n

                                                              Multiplying by 10, 100, and 1 000<\/h1>\n

                                                              Exercise 5<\/h6>\n

                                                              Do the following questions and see if you can find the pattern. Check your work using the answer key at the end of the exercise.<\/p>\n

                                                                \n
                                                              1. [latex]\\begin{array}{rr}&83\\\\\\times&10\\\\\\hline&830\\end{array}[\/latex]<\/li>\n
                                                              2. [latex]\\begin{array}{rr}&46\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              3. [latex]\\begin{array}{rr}&97\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              4. [latex]\\begin{array}{rr}&123\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              5. [latex]\\begin{array}{rr}&70\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              6. [latex]\\begin{array}{rr}&129\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              7. [latex]\\begin{array}{rr}&1852\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              8. [latex]\\begin{array}{rr}&29871\\\\\\times&10\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              9. [latex]\\begin{array}{rr}&45\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              10. [latex]\\begin{array}{rr}&26\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              11. [latex]\\begin{array}{rr}&432\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              12. [latex]\\begin{array}{rr}&679\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              13. [latex]\\begin{array}{rr}&2482\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              14. [latex]\\begin{array}{rr}&9037\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              15. [latex]\\begin{array}{rr}&46207\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              16. [latex]\\begin{array}{rr}&97512\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              17. [latex]\\begin{array}{rr}&23\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              18. [latex]\\begin{array}{rr}&452\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              19. [latex]\\begin{array}{rr}&207\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              20. [latex]\\begin{array}{rr}&348\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              21. [latex]\\begin{array}{rr}&2118\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              22. [latex]\\begin{array}{rr}&2431\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              23. [latex]\\begin{array}{rr}&23681\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                              24. [latex]\\begin{array}{rr}&48203\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n<\/ol>\n
                                                                Answers to Exercise 5<\/strong><\/div>\n
                                                                  \n
                                                                1. 830<\/li>\n
                                                                2. 460<\/li>\n
                                                                3. 970<\/li>\n
                                                                4. 1230<\/span><\/li>\n
                                                                5. 700<\/li>\n
                                                                6. 1290<\/span><\/li>\n
                                                                7. 18520<\/span><\/li>\n
                                                                8. 298710<\/span><\/li>\n
                                                                9. 4500<\/span><\/li>\n
                                                                10. 2600<\/span><\/li>\n
                                                                11. 43200<\/span><\/li>\n
                                                                12. 67900<\/span><\/li>\n
                                                                13. 248200<\/span><\/li>\n
                                                                14. 903700<\/span><\/li>\n
                                                                15. 4620700<\/span><\/span><\/li>\n
                                                                16. 9751200<\/span><\/span><\/li>\n
                                                                17. 23000<\/span><\/li>\n
                                                                18. 452000<\/span><\/li>\n
                                                                19. 207000<\/span><\/li>\n
                                                                20. 348000<\/span><\/li>\n
                                                                21. 2118000<\/span><\/span><\/li>\n
                                                                22. 2431000<\/span><\/span><\/li>\n
                                                                23. 23681000<\/span><\/span><\/li>\n
                                                                24. 48203000<\/span><\/span><\/li>\n<\/ol>\n
                                                                  \n

                                                                  And the pattern is \u2026<\/p>\n

                                                                  When multiplying by 10, 100, 1000, 10000, etc., place as many zeros to the right of the number as there are zeros in the 10, 100, 1000, etc.<\/span><\/span><\/span><\/p>\n

                                                                    \n
                                                                  • To multiply by 10 put one zero after the number.<\/li>\n
                                                                  • To multiply by 100 put two zeros after the number.<\/li>\n
                                                                  • To multiply by 1000 put three zeros after the number.<\/span><\/li>\n<\/ul>\n<\/div>\n

                                                                     <\/p>\n

                                                                    Exercise 6<\/h6>\n

                                                                    Find the products using the short method.\u00a0 Do not rewrite the questions. Check your work using the answer key at the end of the exercise.<\/p>\n

                                                                      \n
                                                                    1. \n

                                                                      12 \u00d7 10 = 120<\/p>\n<\/li>\n

                                                                    2. 10 \u00d7 3175<\/span> =<\/li>\n
                                                                    3. 162 \u00d7 10 =<\/li>\n
                                                                    4. 10 \u00d7 53821<\/span> =<\/li>\n
                                                                    5. 10 \u00d7 123 =<\/li>\n
                                                                    6. 27342<\/span> \u00d7 10 =<\/li>\n
                                                                    7. 10 \u00d7 98 =<\/li>\n
                                                                    8. 1134<\/span> \u00d7 10 =<\/li>\n
                                                                    9. 15 \u00d7 100 =<\/li>\n
                                                                    10. 100 \u00d7 278 =<\/li>\n
                                                                    11. 9134<\/span> \u00d7 100 =<\/li>\n
                                                                    12. 651 \u00d7 100 =<\/li>\n
                                                                    13. 100 \u00d7 5169<\/span> =<\/li>\n
                                                                    14. 100 \u00d7 24815<\/span> =<\/li>\n
                                                                    15. 10 \u00d7 905 =<\/li>\n
                                                                    16. 45683<\/span> \u00d7 10 =<\/li>\n
                                                                    17. 1000<\/span> \u00d7 87 =<\/li>\n
                                                                    18. 521 \u00d7 1000<\/span> =<\/li>\n
                                                                    19. 1000<\/span> \u00d7 68935<\/span> =<\/li>\n
                                                                    20. 1000<\/span> \u00d7 8902<\/span> =<\/li>\n
                                                                    21. 1576<\/span> \u00d7 1000<\/span> =<\/li>\n
                                                                    22. 31584<\/span> \u00d7 1000<\/span> =<\/li>\n
                                                                    23. 1000<\/span> \u00d7 426 =<\/li>\n
                                                                    24. \n

                                                                      72 \u00d7 1000<\/span> =<\/p>\n<\/li>\n<\/ol>\n

                                                                      Answers to Exercise 6<\/strong><\/div>\n
                                                                        \n
                                                                      1. 120<\/li>\n
                                                                      2. 31750<\/span><\/li>\n
                                                                      3. 1620<\/span><\/li>\n
                                                                      4. 538210<\/span><\/li>\n
                                                                      5. 1230<\/span><\/li>\n
                                                                      6. 273420<\/span><\/li>\n
                                                                      7. 980<\/li>\n
                                                                      8. 11340<\/span><\/li>\n
                                                                      9. 1500<\/span><\/li>\n
                                                                      10. 27800<\/span><\/li>\n
                                                                      11. 913400<\/span><\/li>\n
                                                                      12. 65100<\/span><\/li>\n
                                                                      13. 516900<\/span><\/li>\n
                                                                      14. 2481500<\/span><\/span><\/li>\n
                                                                      15. 9050<\/span><\/li>\n
                                                                      16. 456830<\/span><\/li>\n
                                                                      17. 87000<\/span><\/li>\n
                                                                      18. 521000<\/span><\/li>\n
                                                                      19. 68935000<\/span><\/span><\/li>\n
                                                                      20. 8902000<\/span><\/span><\/li>\n
                                                                      21. 1576000<\/span><\/span><\/li>\n
                                                                      22. 31584000<\/span><\/span><\/li>\n
                                                                      23. 426000<\/span><\/li>\n
                                                                      24. 72000<\/span><\/li>\n<\/ol>\n

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

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

                                                                        Multiply these numbers.<\/p>\n

                                                                          \n
                                                                        1. [latex]\\begin{array}{rr}&47\\\\\\times&39\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                                        2. [latex]\\begin{array}{rr}&58\\\\\\times&93\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                                        3. [latex]\\begin{array}{rr}&48\\\\\\times&100\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                                        4. [latex]\\begin{array}{rr}&982\\\\\\times&1000\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                                        5. [latex]\\begin{array}{rr}&678\\\\\\times&39\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                                        6. [latex]\\begin{array}{rr}&4579\\\\\\times&86\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                                        7. [latex]\\begin{array}{rr}&8703\\\\\\times&93\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                                        8. [latex]\\begin{array}{rr}&7390\\\\\\times&85\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                                        9. [latex]\\begin{array}{rr}&8047\\\\\\times&236\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                                        10. [latex]\\begin{array}{rr}&4238\\\\\\times&197\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                                        11. [latex]\\begin{array}{rr}&8200\\\\\\times&444\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n
                                                                        12. [latex]\\begin{array}{rr}&7265\\\\\\times&409\\\\\\hline\\\\\\end{array}[\/latex]<\/li>\n<\/ol>\n
                                                                          Answers to Topic B Self-Test<\/strong><\/div>\n

                                                                          Multiply these numbers.<\/p>\n

                                                                            \n
                                                                          1. \n
                                                                              \n
                                                                            1. 1833<\/span><\/li>\n
                                                                            2. 5394<\/span><\/li>\n
                                                                            3. 4800<\/span><\/li>\n
                                                                            4. 982000<\/span><\/li>\n
                                                                            5. 26442<\/span><\/li>\n
                                                                            6. 393794<\/span><\/li>\n
                                                                            7. 809379<\/span><\/li>\n
                                                                            8. 628150<\/span><\/li>\n
                                                                            9. 1899092<\/span><\/span><\/li>\n
                                                                            10. 834886<\/span><\/li>\n
                                                                            11. 3640800<\/span><\/span><\/li>\n
                                                                            12. 2971385<\/span><\/span><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n","protected":false},"author":1935,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-30","chapter","type-chapter","status-publish","hentry"],"part":28,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/math0635\/wp-json\/pressbooks\/v2\/chapters\/30","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/math0635\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/math0635\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/math0635\/wp-json\/wp\/v2\/users\/1935"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/math0635\/wp-json\/pressbooks\/v2\/chapters\/30\/revisions"}],"predecessor-version":[{"id":1060,"href":"https:\/\/pressbooks.bccampus.ca\/math0635\/wp-json\/pressbooks\/v2\/chapters\/30\/revisions\/1060"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/math0635\/wp-json\/pressbooks\/v2\/parts\/28"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/math0635\/wp-json\/pressbooks\/v2\/chapters\/30\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/math0635\/wp-json\/wp\/v2\/media?parent=30"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/math0635\/wp-json\/pressbooks\/v2\/chapter-type?post=30"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/math0635\/wp-json\/wp\/v2\/contributor?post=30"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/math0635\/wp-json\/wp\/v2\/license?post=30"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}