Topic B: Two- and Three-Digit Multipliers

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.

Multiplying by Two-Digit Multipliers

Example A

[latex]24\times23=[/latex]

Part 1: Multiply by the ones digit in the multiplier.

Multiply 3 ones by 24 using the method you already know. The first partial product is 72.

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

Part 2: 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.

Step 1: Multiply 2 tens 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 – ones under one, tens under tens.

Step 2: Multiply 2 tens 2 tens = 4 hundreds Write the 4 hundreds in your partial product. The second partial product is 480.

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

Part 3: Add the partial products together, being sure to add ones to ones, tens to tens, hundreds to hundreds. The sum is the final product.

Draw a line under the partial products. Add. Check your addition.

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

Example B

[latex]36\times425 =[/latex]

 

Part 1: Multiply by the ones digit in the multiplier. 6 × 425 = 2550

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

Part 2: Multiply by the tens digit in the multiplier. First put a 0 to hold the ones place in the second partial product.

  • Step 1: 3 tens × 5 tens = 15 tens = 1 hundred and 5 tens
    Write 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.
  • Step 2: 3 tens × 2 tens = 6 hundreds
    6 hundreds + 1 hundred (carried) = 7 hundreds. There is nothing to carry.
  • Step 3: 3 tens × 4 hundreds = 12 thousands

[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]

Part 3: Add the partial products together.

[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]

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.
tens × tens = hundreds
tens × hundreds = thousands

Exercise 1

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.

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

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

Answers to Exercise 1

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

When the multiplier has a zero in the ones place, use this shortcut.

Example C

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

Step 1: 0 ones × 48 = 0. Place one zero in the product and that will hold the ones place.

Step 2: Multiply by the tens digit and write the product beside the zero.

Example D

[latex]\begin{array}{rr}&97\\\times&20\\\hline&1940\end{array}[/latex]

Exercise 2

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.

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

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

Answers to Exercise 2

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

Multiplying by Three-Digit Multipliers

To multiply by three digit multipliers, use the same method with one more part.

Example E

[latex]417\times368 =[/latex]

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

Part 1: Multiply by the 8 ones.

Part 2: Multiply the 6 tens; hold the ones place with 0.

Part 3: Multiply by the 3 hundreds. Put 00 to hold the ones and tens places in the third partial product.

  • Step 1: 3 hundreds × 7 ones = 21 hundreds = 2 thousands and 1 hundred. Write the 1 hundred and carry the 2 thousands.
  • Step 2: 3 hundreds × 1 ten = 3 thousands. 3 thousands + 2 thousands (carried) = 5 thousands.
  • Step 3: 3 hundreds × 4 hundreds = 12 ten thousands.

Part 4: Add the partial products.

Exercise 3

Find the products. Check your work using the answer key at the end of the exercise.

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

Answers to Exercise 3

  1. 88608
  2. 109125
  3. 171475
  4. 237325
  5. 461496
  6. 282642
  7. 61254
  8. 228018
  9. 225584
  10. 396575
  11. 245328
  12. 524144
  13. 350931
  14. 123580
  15. 114256
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.

Example F

[latex]927\times405 =[/latex]

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

  • Part 1: Multiply by the 5 ones.
  • Part 2: Multiply by the 0 tens.
    Hold the ones place with a 0; 0 × 927 = 0; Place one zero in the tens place in the second partial product.
  • Part 3: 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.
  • Part 4: Add the partial products.

Exercise 4

Find the products. Check your work using the answer key at the end of the exercise.

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

  1. [latex]\begin{array}{rr}&923\\\times&403\\\hline\\\end{array}[/latex]
  2. [latex]\begin{array}{rr}&830\\\times&108\\\hline\\\end{array}[/latex]
  3. [latex]\begin{array}{rr}&482\\\times&206\\\hline\\\end{array}[/latex]
  4. [latex]\begin{array}{rr}&432\\\times&205\\\hline\\\end{array}[/latex]
  5. [latex]\begin{array}{rr}&625\\\times&405\\\hline\\\end{array}[/latex]
  6. [latex]\begin{array}{rr}&275\\\times&306\\\hline\\\end{array}[/latex]
  7. [latex]\begin{array}{rr}&765\\\times&506\\\hline\\\end{array}[/latex]
  8. [latex]\begin{array}{rr}&1576\\\times&702\\\hline\\\end{array}[/latex]
  9. [latex]\begin{array}{rr}&432\\\times&405\\\hline\\\end{array}[/latex]
  10. [latex]\begin{array}{rr}&625\\\times&409\\\hline\\\end{array}[/latex]
  11. [latex]\begin{array}{rr}&175\\\times&306\\\hline\\\end{array}[/latex]
  12. [latex]\begin{array}{rr}&5874\\\times&309\\\hline\\\end{array}[/latex]
  13. [latex]\begin{array}{rr}&7384\\\times&104\\\hline\\\end{array}[/latex]
  14. [latex]\begin{array}{rr}&6538\\\times&603\\\hline\\\end{array}[/latex]
Answers to Exercise 4
  1. 371969
  2. 89640
  3. 99292
  4. 88560
  5. 255625
  6. 84150
  7. 387090
  8. 1106352
  9. 174960
  10. 255625
  11. 53550
  12. 1815066
  13. 767936
  14. 3942414

Multiplying by 10, 100, and 1 000

Exercise 5

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.

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

And the pattern is …

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.

  • To multiply by 10 put one zero after the number.
  • To multiply by 100 put two zeros after the number.
  • To multiply by 1000 put three zeros after the number.

 

Exercise 6

Find the products using the short method.  Do not rewrite the questions. Check your work using the answer key at the end of the exercise.

  1. 12 × 10 = 120

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

  24. 72 × 1000 =

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

Topic B: Self-Test

Mark  /12    Aim  10/12

A. Multiply these numbers.

  1. [latex]\begin{array}{rr}&47\\\times&39\\\hline\\\end{array}[/latex]
  2. [latex]\begin{array}{rr}&58\\\times&93\\\hline\\\end{array}[/latex]
  3. [latex]\begin{array}{rr}&48\\\times&100\\\hline\\\end{array}[/latex]
  4. [latex]\begin{array}{rr}&982\\\times&1000\\\hline\\\end{array}[/latex]
  5. [latex]\begin{array}{rr}&678\\\times&39\\\hline\\\end{array}[/latex]
  6. [latex]\begin{array}{rr}&4579\\\times&86\\\hline\\\end{array}[/latex]
  7. [latex]\begin{array}{rr}&8703\\\times&93\\\hline\\\end{array}[/latex]
  8. [latex]\begin{array}{rr}&7390\\\times&85\\\hline\\\end{array}[/latex]
  9. [latex]\begin{array}{rr}&8047\\\times&236\\\hline\\\end{array}[/latex]
  10. [latex]\begin{array}{rr}&4238\\\times&197\\\hline\\\end{array}[/latex]
  11. [latex]\begin{array}{rr}&8200\\\times&444\\\hline\\\end{array}[/latex]
  12. [latex]\begin{array}{rr}&7265\\\times&409\\\hline\\\end{array}[/latex]
Answers to Topic B Self-Test
  1. Multiply these numbers.
    1. 1833
    2. 5394
    3. 4800
    4. 982000
    5. 26442
    6. 393794
    7. 809379
    8. 628150
    9. 1899092
    10. 834886
    11. 3640800
    12. 3012285

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Fundamentals of Mathematics 3 Copyright © 2023 by Wendy Tagami and Liz Girard is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.