1.12: Dividing by Two & Three Digit Divisors

Kim Tamblyn

1.12: Dividing by Two & Three Digit Divisors

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

Example A

$624 \div 24$

Rewrite as $24\enclose{longdiv}{624}$

Step 1: Divide.

Think: $2\enclose{longdiv}{6}$ is $3$ .

So, $24\enclose{longdiv}{62}$ is about $3$ .

Step 2: Multiply and subtract.

$\begin{array}{r}3 \ \ \ \\ 24\enclose{longdiv}{624}\\72 \ \ \ \\ \hline \end{array}$

Since 72 > 62, 3 is too large.

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

$\begin{array}{r}2\ \ \ \\ 24\enclose{longdiv}{624}\\48 \ \ \ \\ \hline 14 \ \ \ \end{array}$

Since 4 < 24, 2 is correct.

Step 4: Finish the problem.

Example B

$630 \div 15$

Rewrite as $15\enclose{longdiv}{630}$

Step 1: Divide.

15 rounds to 20.

Think: $2\enclose{longdiv}{6}$ is $3$ .

So, $15\enclose{longdiv}{63}$ is about $3$ .

Step 2: Multiply and subtract.

$\begin{array}{r}3 \ \ \ \\ 15\enclose{longdiv}{630}\\45 \ \ \ \\ \hline 18 \ \ \ \end{array}$

Since 18 > 15, 3 is too small.

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

$\begin{array}{r}4 \ \ \ \\ 24\enclose{longdiv}{630}\\60 \ \ \\ \hline 3 \ \ \end{array}$

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.

$\begin{array}{r}2 \ \ \ \\ 25\enclose{longdiv}{475}\end{array}$
$\begin{array}{r}3 \ \ \ \\ 15\enclose{longdiv}{682}\end{array}$
$\begin{array}{r}5 \ \ \ \\ 18\enclose{longdiv}{813}\end{array}$
$\begin{array}{r}2\ \ \ \\ 25\enclose{longdiv}{810}\end{array}$
$\begin{array}{r}3\ \ \ \\ 33\enclose{longdiv}{891}\end{array}$
$\begin{array}{r}3\ \ \ \\ 18\enclose{longdiv}{819}\end{array}$
$\begin{array}{r}3\ \ \ \\ 27\enclose{longdiv}{727}\end{array}$
$\begin{array}{r}2 \ \ \ \\ 35\enclose{longdiv}{652}\end{array}$
$\begin{array}{r}3\ \ \ \\ 25\enclose{longdiv}{650}\end{array}$
$\begin{array}{r}4\ \ \ \\34\enclose{longdiv}{176}\end{array}$
$\begin{array}{r}4\ \ \ \\ 12\enclose{longdiv}{420}\end{array}$
$\begin{array}{r}2\ \ \ \\ 43\enclose{longdiv}{801}\end{array}$
$\begin{array}{r}3\ \ \ \\ 31\enclose{longdiv}{899}\end{array}$
$\begin{array}{r}4\ \ \ \\ 18\enclose{longdiv}{648}\end{array}$
$\begin{array}{r}4\ \ \ \\ 27\enclose{longdiv}{946}\end{array}$
$\begin{array}{r}3\ \ \ \\ 23\enclose{longdiv}{943}\end{array}$
$\begin{array}{r}3\ \ \ \\ 24\enclose{longdiv}{578}\end{array}$
$\begin{array}{r}2\ \ \ \\ 29\enclose{longdiv}{406}\end{array}$
$\begin{array}{r}2\ \ \ \\ 48\enclose{longdiv}{892}\end{array}$
$\begin{array}{r}2\ \ \ \\ 28\enclose{longdiv}{534}\end{array}$
$\begin{array}{r}3\ \ \ \\ 37\enclose{longdiv}{939}\end{array}$
$\begin{array}{r}2\ \ \ \\ 28\enclose{longdiv}{854}\end{array}$

Exercise 1 Answers

too large, 1
too small, 4
too large, 4
too small, 3
too large, 2
too small, 4
too large, 2
too large 1
too large, 2
too small, 5
too large, 3
too large, 1
too. large 2
too large, 3
too large, 3
too small, 4
too large, 2
too large, 1
too large, 1
too large 1
too large, 2
too small, 3

Example C

$78\enclose{longdiv}{2706}$

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

$\begin{array}{r}3\ \ \ \\ 78\enclose{longdiv}{2706}\\234\ \ \ \\ \hline 36\ \ \ \end{array}$

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

Example D

$27\enclose{longdiv}{2205}$

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

$\begin{array}{r}7\ \ \ \\ 27\enclose{longdiv}{2205}\\189\ \ \ \\ \hline 31\ \ \ \end{array}$

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: $43\enclose{longdiv}{1772}$ $\begin{array}{r}4\\ 4\enclose{longdiv}{17}\\16\\ \hline 1\end{array}$ 1 < 4

$64\enclose{longdiv}{3276}$
$28\enclose{longdiv}{6008}$
$33\enclose{longdiv}{2731}$
$59\enclose{longdiv}{4164}$
$75\enclose{longdiv}{2420}$
$54\enclose{longdiv}{3316}$
$38\enclose{longdiv}{2759}$
$46\enclose{longdiv}{387}$
$35\enclose{longdiv}{3316}$
$83\enclose{longdiv}{7237}$
$77\enclose{longdiv}{6763}$
$93\enclose{longdiv}{3724}$
$52\enclose{longdiv}{4690}$
$86\enclose{longdiv}{2089}$
$26\enclose{longdiv}{1417}$
$72\enclose{longdiv}{1462}$
$27\enclose{longdiv}{6939}$
$32\enclose{longdiv}{7840}$
$24\enclose{longdiv}{7605}$
$16\enclose{longdiv}{8640}$
$45\enclose{longdiv}{3060}$
$38\enclose{longdiv}{2158}$
$42\enclose{longdiv}{1491}$

Exercise 2 Answers

5
2
8
7
3
6
7
8
4
8
8
4
9
2
5
2
2
2
3
5
6
5
3

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

$964 \div 75 =$

Step 1: Divide.

Does 75 go into 9? NO
Does 75 go into 96? YES
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.
Write 1 in the quotient above the 6 tens.

$\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\end{array}$

Step 2: Multiply.

$1 \times 75 = 75$

Write 75 under 96.

$\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\\75 \ \ \\ \hline \end{array}$

Step 3: Subtract.

$96 - 75 = 21$

Check 21 < 75? YES!

$\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\\75\ \ \\ \hline 21 \ \ \end{array}$

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

$\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214\end{array}$

Repeat Steps 1 to 4.

Repeat Step 1: Divide.

Does 75 go into 214? YES
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)
The estimate for the second digit in the trial quotient is 2. Write 2 in the quotient above the 4 in the dividend.

$\begin{array}{r}12\\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214\end{array}$

Repeat Step 2: Multiply.

$2 \times 75 = 150$

Write 150 under the 214.

$\begin{array}{r}12\\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214 \\ 150 \\ \hline\end{array}$

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

$\begin{array}{r}12\\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214 \\ 150 \\ \hline\end{array}$

Repeat Step 4: Bring down the next digit.

No more digits in dividend.

Check:

$\begin{array}{rrr}&75\\\times&12\\\hline&150 \\ &750 \\ \hline &900 \\ + &64 \\ \hline &964\end{array}$

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.

$10\enclose{longdiv}{720}$
$12\enclose{longdiv}{564}$
$21\enclose{longdiv}{882}$
$22\enclose{longdiv}{946}$
$32\enclose{longdiv}{1632}$
$23\enclose{longdiv}{943}$
$62\enclose{longdiv}{2528}$
$71\enclose{longdiv}{2414}$
$24\enclose{longdiv}{7578}$
$82\enclose{longdiv}{2958}$
$18\enclose{longdiv}{6250}$
$25\enclose{longdiv}{1550}$
$19\enclose{longdiv}{9595}$
$47\enclose{longdiv}{3854}$
$58\enclose{longdiv}{6500}$
$24\enclose{longdiv}{9648}$
$49\enclose{longdiv}{1312}$
$67\enclose{longdiv}{7683}$

Exercise 3 Answers

72
47
42
43
51
41
40 R48
34
24 R2
36 R6
347 R4
62
505
82
112 R4
402
26 R38
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.

Divide: Trial estimate is 4.
Multiply:
- $4 \times 23 = 92$
- 92 is larger than 78, so 4 is too large an estimate. Erase it. Try 3. $3 \times 23 = 69$
- 3 is the correct estimate and you can complete the division.

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

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

Divide: Trial estimate is 5.
Multiply: $5 \times 72 = 360$
Subtract:
- $448 - 72 = 360$
- 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.
Divide: $448 \div 82 \approx 6$
Multiply: $6 \times 72 = 432$
Subtract:
- $448 - 432 = 16$
- Check 16<72? YES!
Bring down the next digit and complete the division.

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

Exercise 4

Divide and check your work by multiplying.

$18\enclose{longdiv}{648}$
$26\enclose{longdiv}{6766}$
$52\enclose{longdiv}{1968}$
$84\enclose{longdiv}{8640}$
$72\enclose{longdiv}{2883}$
$94\enclose{longdiv}{8126}$
$20\enclose{longdiv}{4060}$
$47\enclose{longdiv}{1728}$
$33\enclose{longdiv}{1886}$
$25\enclose{longdiv}{5750}$
$79\enclose{longdiv}{2765}$
$42\enclose{longdiv}{8442}$
$57\enclose{longdiv}{9144}$
$96\enclose{longdiv}{20160}$
$75\enclose{longdiv}{23550}$

Exercise 4 Answers

36
260 R6
37 R44
102 R72
40 R3
86 R42
203
36 R36
57 R5
230
35
201
160 R24
210
314

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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.

1.12: Dividing by Two & Three Digit Divisors

Example A

Example B

Exercise 1

Exercise 1 Answers

Example C

Example D

Exercise 2

Exercise 2 Answers

Example E

Exercise 3

Exercise 3 Answers

Exercise 4

Exercise 4 Answers

When You Divide by 10

Exercise 5

Exercise 5 Answers

When You Divide by 100

Exercise 6

Exercise 6 Answers

When You Divide by 1000

Exercise 7

Exercise 7 Answers

Example F

Exercise 8

Exercise 8 Answers

1.12: Practice Answers

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Finding Trial Quotients

Example A

Example B

Exercise 1

Exercise 1 Answers

Example C

Example D

Exercise 2

Exercise 2 Answers

Two Digit Divisors

Example E

Exercise 3

Exercise 3 Answers

Exercise 4

Exercise 4 Answers

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

When You Divide by 10

Exercise 5

Exercise 5 Answers

When You Divide by 100

Exercise 6

Exercise 6 Answers

When You Divide by 1000

Exercise 7

Exercise 7 Answers

Three Digit Divisors

Example F

Exercise 8

Exercise 8 Answers

1.12: Practice Questions

1.12: Practice Answers

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License

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