1.7: Two- & Three-Digit Multipliers

https://opentextbc.ca/alfm3/ – From Book 3, Unit 1: Multiplication, include Topics B, C, D. Do not include the Unit 1 Review. From Unit 2: Division, include Topics B, C, D, E, F. Do not include Unit 2 Review.

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]

Step 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]

Step 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.

[latex]2 \text{ tens }\times 4 \text{ ones }= 8 \text{ tens}[/latex]

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.

[latex]2 \text{ tens }\times 2 \text{ tens }= 4 \text{ hundreds}[/latex]

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]

Step 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]

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

[latex]6 × 425 = 2\,550[/latex]

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

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

[latex]3 \text{ tens} \times 5 \text{ tens} = 15 \text{ tens} = 1 \text{ hundred and } 5 \text{ tens}[/latex]

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.

[latex]3 \text{ tens} \times 2 \text{ tens} = 6 \text{ hundreds}[/latex]

[latex]6 \text{ hundreds} + 1 \text{ hundred (carried) } = 7 \text{ hundreds}[/latex]

There is nothing to carry.

[latex]3 \text{ tens} × 4 \text{ hundreds} = 12 \text{ thousands}[/latex]

[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]

Step 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]

Multiplication Tips

Here are some tips to keep in mind while multiplying:

  • Keeping the columns straight with ones under ones, tens under tens, and 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.

Example:     [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]

Exercise 1 Answers

  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 the following shortcut.

Example C

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

Step 1: Place one zero in the product, which will hold the ones place.

[latex]0 \text{ ones} \times 48 = 0[/latex]

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

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

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.

Example:     [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]

Exercise 2 Answers

  1. 520
  2. 3640
  3. 4980
  4. 2450
  5. 1830
  6. 1440
  7. 2880
  8. 3980
  9. 8640
  10. 43150
  11. 27210
  12. 20120
  13. 36160
  14. 66010
  15. 22320
  16. 251480
  17. 541500
  18. 712560
  19. 27000
  20. 181500
  21. 118560
  22. 106960
  23. 71060

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]

Step 1: Multiply by the 8 ones.

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

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

[latex]3 \text{ hundreds} \times 7 \text{ ones} = 21 \text{ hundreds} = 2 \text{ thousands and } 1 \text{ hundred}[/latex]

Write the 1 hundred and carry the 2 thousands.

[latex]3 \text{ hundreds} \times 1 \text{ ten} = 3 \text{ thousands}[/latex]

[latex]3 \text{ thousands} \times 2 \text{ thousands (carried) } = 5 \text{ thousands}[/latex]

[latex]3 \text{ hundreds} \times 4 \text{ hundreds} = 12 ten \text{ thousands}[/latex]

Step 4: Add the partial products.

Exercise 3

Find the products.

  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]

Exercise 3 Answers

  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]

Step 1: Multiply by the 5 ones.

Step 2: Multiply by the 0 tens.

Hold the ones place with a 0.

[latex]0 \times 927 = 0[/latex]

Place one zero in the tens place in the second partial product.

Step 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.

Step 4: Add the partial products.

Exercise 4

Find the products.

Example:     [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]

Exercise 4 Answers

  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, & 1000

Exercise 5

Do the following questions and see if you can find the pattern.

  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]

Exercise 5 Answers

  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

The pattern is:

When multiplying by 10, 100, 1000, 10000, and so on, 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.

  1. [latex]12 \times 10 = 120[/latex]
  2. [latex]10 \times 3175 =[/latex]
  3. [latex]162 \times 10 =[/latex]
  4. [latex]10 \times 53821 =[/latex]
  5. [latex]10 \times 123 =[/latex]
  6. [latex]27342 \times 10 =[/latex]
  7. [latex]10 \times 98 =[/latex]
  8. [latex]1134 \times 10 =[/latex]
  9. [latex]15 \times 100 =[/latex]
  10. [latex]100 \times 278 =[/latex]
  11. [latex]9134 \times 100 =[/latex]
  12. [latex]651 \times 100 =[/latex]
  13. [latex]100 \times 5169 =[/latex]
  14. [latex]100 \times 24815 =[/latex]
  15. [latex]10 \times 905 =[/latex]
  16. [latex]45683 \times 10 =[/latex]
  17. [latex]1000 \times 87 =[/latex]
  18. [latex]521 \times 1000 =[/latex]
  19. [latex]1000 \times 68935 =[/latex]
  20. [latex]1000 \times 8902 =[/latex]
  21. [latex]1576 \times 1000 =[/latex]
  22. [latex]31584 \times 1000 =[/latex]
  23. [latex]1000 \times 426 =[/latex]
  24. [latex]72 \times 1000 =[/latex]

Exercise 6 Answers

  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

1.7: Practice Questions

  1. 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]

1.7: Practice Answers

  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

Attribution

This chapter has been adapted from Topic B: Two- and Three-Digit Multipliers in Adult Literacy Fundamental Mathematics: Book 3 – 2nd Edition (BCcampus) by Wendy Tagami and Liz Girard (2023), which is under a CC BY 4.0 license.

License

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Intermediate PreAlgebra: Building Success Copyright © 2024 by Kim Tamblyn, TRU Open Press is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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