1.12: Dividing by Two & Three Digit Divisors

Finding Trial Quotients

When dividing by 2-digit numbers, you will need to estimate the quotient. This guess is called a trial quotient.

Example A

[latex]624 \div 24[/latex]

Rewrite as [latex]24\enclose{longdiv}{624}[/latex]

Step 1: Divide.

Think: [latex]2\enclose{longdiv}{6}[/latex] is [latex]3[/latex].

So, [latex]24\enclose{longdiv}{62}[/latex] is about [latex]3[/latex].

Step 2: Multiply and subtract.

[latex]\begin{array}{r}3 \ \ \ \\ 24\enclose{longdiv}{624}\\72 \ \ \ \\ \hline \end{array}[/latex]

Since 72 > 62, 3 is too large.

Step 3: Try a smaller number, multiply and subtract.

[latex]\begin{array}{r}2\ \ \ \\ 24\enclose{longdiv}{624}\\48 \ \ \ \\ \hline 14 \ \ \ \end{array}[/latex]

Since 4 < 24, 2 is correct.

Step 4: Finish the problem.

Example B

[latex]630 \div 15[/latex]

Rewrite as [latex]15\enclose{longdiv}{630}[/latex]

Step 1: Divide.

15 rounds to 20.

Think: [latex]2\enclose{longdiv}{6}[/latex] is [latex]3[/latex].

So, [latex]15\enclose{longdiv}{63}[/latex] is about [latex]3[/latex].

Step 2: Multiply and subtract.

[latex]\begin{array}{r}3 \ \ \ \\ 15\enclose{longdiv}{630}\\45 \ \ \ \\ \hline 18 \ \ \ \end{array}[/latex]

Since 18 > 15, 3 is too small.

Step 3: Try a larger number, multiply and subtract.

[latex]\begin{array}{r}4 \ \ \ \\ 24\enclose{longdiv}{630}\\60 \ \ \\ \hline 3 \ \ \end{array}[/latex]

Since 3 < 15, 4 is correct.

Step 4: Finish the problem.

Exercise 1

Each question has a trial quotient that is either too large or too small. Multiply. Write too large or too small on the line. Then, write the correct trial quotient beside. Check your work using the answer key at the end of the exercise.

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

Exercise 1 Answers

  1. too large, 1
  2. too small, 4
  3. too large, 4
  4. too small, 3
  5. too large, 2
  6. too small, 4
  7. too large, 2
  8. too large 1
  9. too large, 2
  10. too small, 5
  11. too large, 3
  12. too large, 1
  13. too. large 2
  14. too large, 3
  15. too large, 3
  16. too small, 4
  17. too large, 2
  18. too large, 1
  19. too large, 1
  20. too large 1
  21. too large, 2
  22. too small, 3

Example C

[latex]78\enclose{longdiv}{2706}[/latex]

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

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

Since 36 < 78, 3 is a good trial quotient.

Example D

[latex]27\enclose{longdiv}{2205}[/latex]

Since 27 rounds to 30, think [latex]3\enclose{longdiv}{22}[/latex]. 3 goes into 22 ≈ 7. 7 would be a good trial quotient.

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

Since 31 > 27, so 7 is too small.  A better trial quotient would be 8. 

Exercise 2

Find the first digit in the trial quotient.

Example:     [latex]43\enclose{longdiv}{1772}[/latex] [latex]\begin{array}{r}4\\ 4\enclose{longdiv}{17}\\16\\ \hline 1\end{array}[/latex]   1 < 4

 

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

Exercise 2 Answers

  1. 5
  2. 2
  3. 8
  4. 7
  5. 3
  6. 6
  7. 7
  8. 8
  9. 4
  10. 8
  11. 8
  12. 4
  13. 9
  14. 2
  15. 5
  16. 2
  17. 2
  18. 2
  19. 3
  20. 5
  21. 6
  22. 5
  23. 3

Two Digit Divisors

Dividing by large divisors is a challenge!

You must estimate how many times one number will divide into another. Do these questions using a pencil and have an eraser close by. You will use the same steps that you already know.

Example E

[latex]964 \div 75 =[/latex]

Step 1: Divide.

  1. Does 75 go into 9?  NO
  2. Does 75 go into 96?  YES
  3. Estimate:
    • Round 75 to 80 – think “8”
    • Round 96 to 100 – think “10”
    • How many 8’s in 10?  (8 1 = 8, 10  8  =  1)
    • The estimate for the first digit in the trial quotient is 1.
  4. Write 1 in the quotient above the 6 tens.

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

Step 2: Multiply.

[latex]1 \times 75 = 75[/latex]

Write 75 under 96.

[latex]\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\\75 \ \ \\ \hline \end{array}[/latex]

Step 3: Subtract.

[latex]96 - 75 = 21[/latex]

Check 21 < 75? YES!

[latex]\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\\75\ \ \\ \hline 21 \ \ \end{array}[/latex]

Step 4: Bring down the next digit in the dividend. 214 is now the number to be divided.

[latex]\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214\end{array}[/latex]

Repeat Steps 1 to 4.

Repeat Step 1: Divide.

  1. Does 75 go into 214?  YES
  2. Estimate 75 as 80 – think “8”:
    • Estimate 214 as 200 – think “20”
    • 8 goes into 20 2 times ( 8 2 = 16, so 20  8 2)
  3. The estimate for the second digit in the trial quotient is 2. Write 2 in the quotient above the 4 in the dividend.

[latex]\begin{array}{r}12\\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214\end{array}[/latex]

Repeat Step 2: Multiply.

[latex]2 \times 75 = 150[/latex]

Write 150 under the 214.

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

Repeat Step 3: Subtract and check that the remainder is less than the divisor.

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

Repeat Step 4: Bring down the next digit.

No more digits in dividend.

Check:

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

Exercise 3

Carefully divide these questions. Be careful to keep the hundreds in line with the hundreds, the tens with the tens, and so on. You might want to use squared paper for long division.

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

Exercise 3 Answers

  1. 72
  2. 47
  3. 42
  4. 43
  5. 51
  6. 41
  7. 40 R48
  8. 34
  9. 24 R2
  10. 36 R6
  11. 347 R4
  12. 62
  13. 505
  14. 82
  15. 112 R4
  16. 402
  17. 26 R38
  18. 114 R45

If the estimate for your trial quotient is too large, the result of the multiplication will be larger than the numbers in the dividend.

  1. Divide: Trial estimate is 4.
  2. Multiply:
    • [latex]4 \times 23 = 92[/latex]
    • 92 is larger than 78, so 4 is too large an estimate. Erase it.  Try 3. [latex]3 \times 23 = 69[/latex]
    • 3 is the correct estimate and you can complete the division.

[latex]23\enclose{longdiv}{784}[/latex]     [latex]\begin{array}{r}4\ \ \ \\ 23\enclose{longdiv}{784}\\92\ \ \ \\ \hline \end{array}[/latex]     [latex]\begin{array}{l}\hspace{2.3em}34\text{ R}2\\23 \enclose{longdiv}{784}\\\hspace{1.5em}69 \downarrow\\\hline\hspace{2.3em}94\\\hspace{2.3em}92\\\hline\hspace{2.8em}2\end{array}[/latex]

If the estimate is too small, the result of the subtraction will be larger than the divisor.

  1. Divide: Trial estimate is 5.
  2. Multiply: [latex]5 \times 72 = 360[/latex]
  3. Subtract:
    • [latex]448 - 72 = 360[/latex]
    • Check 88 < 72?  NO, 88 is greater than 72.
    • So 5 is too small. Erase it and use a larger number. 6 will be a better estimate.
  4. Divide: [latex]448 \div 82 \approx 6[/latex]
  5. Multiply: [latex]6 \times 72 = 432[/latex]
  6. Subtract:
    • [latex]448 - 432 = 16[/latex]
    • Check  16<72? YES!
  7. Bring down the next digit and complete the division.

[latex]72\enclose{longdiv}{4487}[/latex]     [latex]\begin{array}{r}5\ \ \ \\ 72\enclose{longdiv}{4487}\\360\ \ \ \\ \hline88\hspace{0.8em} \end{array}[/latex]     [latex]\begin{array}{l}\hspace{1.1cm}62\text{ R}23\\72 \enclose{longdiv}{4487}\\\hspace{1.45em}432 \downarrow\\\hline\hspace{2.1em}167\\\hspace{2.1em}144\\\hline\hspace{2.55em}23\end{array}[/latex]

Exercise 4

Divide and check your work by multiplying.

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

Exercise 4 Answers

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

Dividing by 10, 100, 1000…What Is The Pattern? 

When You Divide by 10

The ones digit in the dividend becomes the remainder.

[latex]10\enclose{longdiv}{324} = 32 R4[/latex]

The other numbers in the dividend stay the same, but each digit is one place value less.

  • The hundreds become tens.
  • The tens become ones.
  • The ones become the remainder.

Exercise 5

Find the quotients. Look for the pattern.

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

Exercise 5 Answers

  1. 4 R6
  2. 7 R5
  3. 13 R6
  4. 83 R2
  5. 67 R4
  6. 95 R2
  7. 245 R7
  8. 368 R5

When You Divide by 100

The ones and tens digits in the dividend become the remainder.

The other digits in the dividend stay the same but each digit is two places less:

  • The thousands become tens.
  • The hundreds become ones.
  • The tens and ones become the remainder.

Exercise 6

Find the quotients. Look for the pattern when you divide.

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

Exercise 6 Answers

  1. 3 R86
  2. 9 R95
  3. 2 R69
  4. 1 R75
  5. 29 R48
  6. 46 R71
  7. 920 R45
  8. 438 R21

When You Divide by 1000

The ones, tens, and hundreds digits become the remainder.

The other digits stay the same but are three place values less:

  • The thousands become ones.
  • The ten thousands become tens.
  • The hundred thousands become hundreds.

Exercise 7

Find the quotients.

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

Exercise 7 Answers

  1. 2 R398
  2. 6 R475
  3. 4 R835
  4. 63 R291
  5. 82 R405
  6. 293 R591

Three Digit Divisors

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

Example F

[latex]17902 \div 381 =[/latex]

Step 1: Divide.

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

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

Step 2: Multiply.

[latex]4 \times 381 = 1524[/latex]

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

Step 3: Subtract.

[latex]1\,790-1\,524=266[/latex]

Check: 266 < 381? YES!

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

Step 4: Bring down the 2.

2662 is now the number to be divided

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

Repeat Steps 1 to 4.

Repeat Step 1: Divide.

[latex]2\,662 \div 381 =[/latex]

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

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

Repeat Step 2: Multiply.

[latex]6\times381=2\,286[/latex]

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

Repeat Step 3: Subtract.

[latex]2\,662-2\,286=376[/latex]

Check: 376 < 381? YES!

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

Repeat Step 4: No more digits to bring down.

[latex]17\,902\div381=46[/latex] R376

Exercise 8

Divide and check your answers. These questions are hard work!

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

Exercise 8 Answers

  1. 75 R57
  2. 11 R129
  3. 204 R21
  4. 55 R29
  5. 305 R112
  6. 410

1.12: Practice Questions

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

1.12: Practice Answers

  1. Divide and check your work for questions b) and f) using multiplication.
    1. 18 R5
    2. 8 R8
    3. 38
    4. 94
    5. 89
    6. 91 R33
    7. 96
    8. 504 R64
    9. 43 R36
    10. 83 R652

Attribution

This chapter has been adapted from Topic D: Dividing by Two and Three Digit Divisors 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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