Unit 2: Division
Topic C: Dividing Larger Numbers by One-Digit Divisors
Several methods are used to divide larger numbers. This book will only teach one method. If you have learned a different method for dividing, ask your instructor to review it with you. You can use the practice exercises in this workbook using whichever method you prefer.
One-Digit Divisors without Remainders
Division has four steps which are repeated until the dividend is completely divided. Work through the three examples which show these steps.
- Divide.
- Multiply
- Subtract and compare the remainder to the divisor.
- Bring down the next digit in the dividend and repeat.
Example A
294 ÷ 7 =
Rewrite as [latex]7\enclose{longdiv}{294}[/latex]
Step 1: Divide.
- Find a trial quotient using multiplication tables or division facts.
- Look at the dividend one digit at a time.
- The first digit is a 2, which is really 2 hundreds.
- Will 7 go into 2? Can you divide 2 by 7? No.
- Look at the first 2 digits, 29, which is really 29 tens.
- Will 7 go into 29? Yes. 4 × 7 = 28.
- The first number in the quotient is 4. Place the 4 in the quotient directly above the 9 tens. The 4 is 4 tens in the quotient.
[latex]\begin{array}{r}4\hspace{0.8em}\\7 \enclose{longdiv}{294}\end{array}[/latex]
Step 2: Multiply.
- Multiply 4 × 7 = 28.
- Write the 28 under the 29. Draw a line.
[latex]\begin{array}{r}4\hspace{0.8em}\\7 \enclose{longdiv}{294}\\28\hspace{0.8em}\\\hline\end{array}[/latex]
Step 3: Subtract.
- Subtract 29 − 28 = 1 (ten).
- Check 1 < 7.
[latex]\begin{array}{r}4\hspace{0.8em}\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline1\hspace{0.8em}\end{array}[/latex]
Step 4: Bring down the next number in the dividend.
- Bring down the next number in the dividend (4), and put it beside the result of the subtraction you just completed (1).
- Together, they make 14. This 14 is the number that you must now divide.
[latex]\begin{array}{r}4\hspace{0.8em}\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline14\end{array}[/latex]
Repeat Step 1: Divide.
- Divide 14 ÷ 7 = 2
- Put the 2 in the quotient right after the 4 in the ones place.
[latex]\begin{array}{r}42\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline14\end{array}[/latex]
Repeat Step 2: Multiply.
- Multiply 2 × 7 = 14.
- Write the 14 under the 14. Draw a line.
[latex]\begin{array}{r}42\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline14\\14\\\hline\end{array}[/latex]
Repeat Step 3: Subtract.
- Subtract 14 − 14 = 0. There is 0 remainder.
- Check 0 < 7.
[latex]\begin{array}{r}42\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline14\\-14\\\hline0\end{array}[/latex]
There are no more numbers in the dividend to bring down, so you are done.
294 ÷ 7 = 42
Example B
128 ÷ 2 =
[latex]\begin{array}{r}64\\2 \enclose{longdiv}{128}\\-12\hspace{0.8em}\\\hline08\\-8\\\hline0\end{array}[/latex]
Step 1: Divide.
- Can 2 go into 1? No.
- Can 2 go into 12? Yes.
- How many times?
- 2 × 6 = 12; 12 ÷ 2 = 6
- The first number in the trial quotient is 6. Put the 6 in the quotient directly above the 2 tens dividend.
Step 2: Multiply.
- 6 × 2 = 12.
- Write the 12 under the 12. Draw a line.
Step 3: Subtract.
- 12 − 12 = 0
- Check 0 < 12
Step 4: Bring down the next number in the dividend.
- Bring down the next number in the dividend (8), and put it beside the result of the subtraction you just completed (0).
- 8 is now the number to be divided.
Repeat
- Divide: 8 ÷ 2 = 4
- Multiple: 4 × 2 = 8
- Subtract: 8 − 8 = 0
- Check 0 < 2
- Bring down the next digit. There are no more digits in the dividend.
128 ÷ 2 = 64
Exercise 1
Find the quotients (divide, multiply, subtract, compare). Check your work using the answer key at the end of the exercise.
- [latex]4\enclose{longdiv}{364}[/latex]
- [latex]2\enclose{longdiv}{144}[/latex]
- [latex]5\enclose{longdiv}{455}[/latex]
- [latex]7\enclose{longdiv}{651}[/latex]
- [latex]8\enclose{longdiv}{144}[/latex]
- [latex]2\enclose{longdiv}{166}[/latex]
- [latex]7\enclose{longdiv}{588}[/latex]
- [latex]2\enclose{longdiv}{196}[/latex]
- [latex]5\enclose{longdiv}{230}[/latex]
- [latex]8\enclose{longdiv}{584}[/latex]
- [latex]6\enclose{longdiv}{366}[/latex]
- [latex]4\enclose{longdiv}{244}[/latex]
- [latex]5\enclose{longdiv}{375}[/latex]
- [latex]8\enclose{longdiv}{200}[/latex]
- [latex]2\enclose{longdiv}{628}[/latex]
- [latex]7\enclose{longdiv}{357}[/latex]
- [latex]9\enclose{longdiv}{837}[/latex]
- [latex]8\enclose{longdiv}{248}[/latex]
- [latex]8\enclose{longdiv}{312}[/latex]
- [latex]7\enclose{longdiv}{462}[/latex]
- [latex]5\enclose{longdiv}{295}[/latex]
- [latex]6\enclose{longdiv}{384}[/latex]
- [latex]2\enclose{longdiv}{276}[/latex]
- [latex]4\enclose{longdiv}{372}[/latex]
Answers to Exercise 1
- 91
- 72
- 91
- 93
- 18
- 83
- 84
- 98
- 46
- 73
- 61
- 61
- 75
- 25
- 314
- 51
- 93
- 31
- 39
- 66
- 59
- 64
- 138
- 93
Example C
856 ÷ 8 =
[latex]8\enclose{longdiv}{856}[/latex]
Divide
Does 8 go into 8? Yes. 8 ÷ 8 = 1.
[latex]\begin{array}{r}1\hspace{1.2em}\\8 \enclose{longdiv}{856}\end{array}[/latex]
Multiply
1 × 8 = 8
[latex]\begin{array}{r}1\hspace{1.2em}\\8 \enclose{longdiv}{856}\\8\hspace{1.2em}\end{array}[/latex]
Subtract
8 − 8 = 0. Check 0 < 8.
[latex]\begin{array}{r}1\hspace{1.2em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline0\hspace{1.2em}\end{array}[/latex]
Bring down the next digit
5 is now the number to be divided.
[latex]\begin{array}{r}1\hspace{1.2em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\end{array}[/latex]
Now repeat.
Divide
Does 8 go into 5? No.
You must put a zero to hold the place in the quotient. If a digit is brought down, a digit must be placed in the quotient.
Multiply
0 × 8 = 0
[latex]\begin{array}{r}10\hspace{0.8em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\end{array}[/latex]
Subtract
5 − 0 = 5. Check 5 < 8.
[latex]\begin{array}{r}10\hspace{0.8em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline5\hspace{0.8em}\end{array}[/latex]
Bring down the next digit
56 is now the number to be divided.
[latex]\begin{array}{r}10\hspace{0.8em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline56\end{array}[/latex]
Now repeat.
Divide
Does 8 go into 56? Yes. 56 ÷ 8 = 7.
Write 7 in the quotient in the ones place above the 6 in the dividend.
[latex]\begin{array}{r}107\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline56\end{array}[/latex]
Multiply
7 × 8 = 56
[latex]\begin{array}{r}107\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline56\\56\end{array}[/latex]
Subtract
56 − 56 = 0. Check 0 < 8.
[latex]\begin{array}{r}107\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline56\\-56\\\hline0\end{array}[/latex]
Bring down the next digit.
No more digits.
856 ÷ 8 = 107
Exercise 2
Find the quotients (divide, multiply, subtract, compare). Check your work using the answer key at the end of the exercise.
- [latex]7\enclose{longdiv}{721}[/latex]
- [latex]9\enclose{longdiv}{954}[/latex]
- [latex]3\enclose{longdiv}{927}[/latex]
- [latex]3\enclose{longdiv}{621}[/latex]
- [latex]4\enclose{longdiv}{824}[/latex]
- [latex]9\enclose{longdiv}{972}[/latex]
- [latex]7\enclose{longdiv}{714}[/latex]
- [latex]2\enclose{longdiv}{416}[/latex]
- [latex]5\enclose{longdiv}{540}[/latex]
- [latex]6\enclose{longdiv}{654}[/latex]
- [latex]8\enclose{longdiv}{832}[/latex]
- [latex]4\enclose{longdiv}{436}[/latex]
- [latex]5\enclose{longdiv}{515}[/latex]
- [latex]2\enclose{longdiv}{814}[/latex]
- [latex]6\enclose{longdiv}{648}[/latex]
- [latex]8\enclose{longdiv}{856}[/latex]
- [latex]5\enclose{longdiv}{525}[/latex]
- [latex]7\enclose{longdiv}{763}[/latex]
- [latex]9\enclose{longdiv}{945}[/latex]
- [latex]3\enclose{longdiv}{315}[/latex]
- [latex]8\enclose{longdiv}{872}[/latex]
- [latex]4\enclose{longdiv}{416}[/latex]
- [latex]6\enclose{longdiv}{618}[/latex]
- [latex]2\enclose{longdiv}{612}[/latex]
Answers to Exercise 2
- 103
- 106
- 309
- 207
- 206
- 108
- 102
- 208
- 108
- 109
- 104
- 109
- 103
- 407
- 108
- 107
- 105
- 109
- 105
- 105
- 109
- 104
- 103
- 306
One-Digit Divisors with Remainders
Do the division exactly the same way that you have been learning. Often there is a remainder after the last subtraction. Write it with the quotient as you already know how to do.
Example D
259 ÷ 8 =
[latex]\begin{array}{l}\hspace{1.5em}32\text{ R}3\\8 \enclose{longdiv}{259}\\\hspace{0.3em}-24\\\hline\hspace{1.7em}19\\\hspace{1em}-16\\\hline\hspace{2.3em}3\end{array}[/latex]
259 ÷ 8 = 32 R3
Exercise 3
Divide and show any remainders. Check your work using the answer key at the end of the exercise.
- [latex]20\enclose{longdiv}{93}[/latex]
- [latex]3\enclose{longdiv}{52}[/latex]
- [latex]5\enclose{longdiv}{94}[/latex]
- [latex]7\enclose{longdiv}{74}[/latex]
- [latex]4\enclose{longdiv}{95}[/latex]
- [latex]9\enclose{longdiv}{96}[/latex]
- [latex]6\enclose{longdiv}{97}[/latex]
- [latex]8\enclose{longdiv}{99}[/latex]
- [latex]9\enclose{longdiv}{98}[/latex]
- [latex]4\enclose{longdiv}{59}[/latex]
- [latex]6\enclose{longdiv}{76}[/latex]
- [latex]3\enclose{longdiv}{79}[/latex]
- [latex]7\enclose{longdiv}{96}[/latex]
- [latex]5\enclose{longdiv}{57}[/latex]
- [latex]2\enclose{longdiv}{47}[/latex]
- [latex]8\enclose{longdiv}{91}[/latex]
- [latex]7\enclose{longdiv}{89}[/latex]
- [latex]6\enclose{longdiv}{82}[/latex]
- [latex]5\enclose{longdiv}{67}[/latex]
- [latex]2\enclose{longdiv}{85}[/latex]
- [latex]4\enclose{longdiv}{71}[/latex]
- [latex]3\enclose{longdiv}{65}[/latex]
- [latex]9\enclose{longdiv}{92}[/latex]
- [latex]8\enclose{longdiv}{94}[/latex]
Answers to Exercise 3
- 46 R1
- 17 R1
- 18 R4
- 10 R4
- 23 R3
- 10 R6
- 16 R1
- 12 R3
- 10 R8
- 14 R3
- 12 R4
- 26 R1
- 13 R5
- 11 R2
- 23 R1
- 11 R3
- 12 R5
- 13 R4
- 13 R2
- 42 R1
- 17 R3
- 21 R2
- 10 R2
- 11 R6
Check Division by Multiplying
To check your division, do this:
- Multiply the quotient by the divisor.
- Add on any remainder.
The product will equal the dividend if your arithmetic is correct.
Example E
[latex]\begin{array}{l}\hspace{1.5em}52\text{ R}1\\9 \enclose{longdiv}{469}\\\hspace{0.3em}-45\\\hline\hspace{1.7em}19\\\hspace{1em}-18\\\hline\hspace{2.3em}1\end{array}[/latex] [latex]\begin{array}{rr}&52\\\times&9\\\hline&468\end{array}[/latex] [latex]468+1=469[/latex]
Exercise 4
Divide and check your answer by multiplying. Check your work using the answer key at the end of the exercise.
- [latex]7\enclose{longdiv}{709}[/latex]
- [latex]2\enclose{longdiv}{423}[/latex]
- [latex]5\enclose{longdiv}{538}[/latex]
- [latex]4\enclose{longdiv}{609}[/latex]
- [latex]9\enclose{longdiv}{406}[/latex]
- [latex]6\enclose{longdiv}{125}[/latex]
- [latex]3\enclose{longdiv}{605}[/latex]
- [latex]9\enclose{longdiv}{928}[/latex]
- [latex]3\enclose{longdiv}{962}[/latex]
- [latex]4\enclose{longdiv}{805}[/latex]
- [latex]8\enclose{longdiv}{301}[/latex]
- [latex]2\enclose{longdiv}{807}[/latex]
- [latex]6\enclose{longdiv}{725}[/latex]
- [latex]7\enclose{longdiv}{320}[/latex]
- [latex]9\enclose{longdiv}{140}[/latex]
- [latex]8\enclose{longdiv}{483}[/latex]
- [latex]2\enclose{longdiv}{197}[/latex]
- [latex]6\enclose{longdiv}{307}[/latex]
- [latex]5\enclose{longdiv}{504}[/latex]
- [latex]8\enclose{longdiv}{709}[/latex]
- [latex]7\enclose{longdiv}{876}[/latex]
- [latex]4\enclose{longdiv}{101}[/latex]
- [latex]3\enclose{longdiv}{269}[/latex]
- [latex]5\enclose{longdiv}{473}[/latex]
Answers to Exercise 4
- 101 R2
- 211 R1
- 107 R3
- 152 R1
- 45 R1
- 20 R5
- 201 R2
- 103 R1
- 320 R2
- 201 R1
- 37 R5
- 403 R1
- 120 R5
- 45 R5
- 15 R5
- 60 R3
- 98 R1
- 51 R1
- 100 R4
- 88 R5
- 125 R1
- 25 R1
- 89 R2
- 94 R3
Topic C: Self-Test
Mark /24 Aim 19/24
- Find the quotient. (6 marks)
- [latex]6\enclose{longdiv}{96}[/latex]
- [latex]4\enclose{longdiv}{92}[/latex]
- [latex]7\enclose{longdiv}{91}[/latex]
- [latex]2\enclose{longdiv}{93}[/latex]
- [latex]5\enclose{longdiv}{94}[/latex]
- [latex]3\enclose{longdiv}{52}[/latex]
- Divide. (6 marks)
- [latex]7\enclose{longdiv}{182}[/latex]
- [latex]8\enclose{longdiv}{736}[/latex]
- [latex]6\enclose{longdiv}{162}[/latex]
- [latex]5\enclose{longdiv}{295}[/latex]
- [latex]4\enclose{longdiv}{184}[/latex]
- [latex]9\enclose{longdiv}{576}[/latex]
- Divide and show your check for each answer (12 marks – 1 mark for question, 1 mark for check).
- [latex]9\enclose{longdiv}{705}[/latex]
- [latex]4\enclose{longdiv}{257}[/latex]
- [latex]7\enclose{longdiv}{899}[/latex]
- [latex]5\enclose{longdiv}{538}[/latex]
- [latex]8\enclose{longdiv}{876}[/latex]
- [latex]6\enclose{longdiv}{628}[/latex]
Answers to Topic C Self-Test
- Find the quotient.
- 16
- 23
- 13
- 46 R1
- 18 R4
- 17 R1
- Divide.
- 26
- 92
- 27
- 59
- 46
- 64
- Divide and show your check for each answer.
- 78 R3
- 64 R1
- 128 R3
- 107 R3
- 109 R4
- 104 R4