Long Division and Repeated Subtraction
This is a complete lesson with examples and exercises about the repeated subtraction process, as it relates to division. I give several examples of comparing division to bagging fruits and using repeated subtraction in that context. Several exercises follow. Lastly the lesson shows a comparison of this process with the actual long division algorithm. The lesson is meant for fifth grade.
You know how multiplication can be seen as repeated
addition. Division is the opposite of multiplication. So, it
should be no surprise that division can be solved by repeated (or continued) subtraction.
Read through the examples carefully in order to understand that. 
Example 1. Bag 771 apples, placing 3 apples in each bag. How many bags
will you need? You might start by putting 3 apples
into one bag, which leaves you with 768
apples. From then on, for each bag you use, subtract 3 apples. Keep counting the bags you
use until you have no apples!
771 
− 3 
− 3 
− 3 
− 3 
− 3 
− 3 
... 
keep subtracting! 

1 bag 
1 bag 
1 bag 
1 bag 
1 bag 
1 bag 
... 
keep counting bags! 

It just takes quite a long time
to do it this way! Instead, you can take a
shortcut: subtract
300 apples at a time (which means 100 bags), as long as you can,
then 30 apples at a time as long as you can (which means 10 bags), and lastly 3
apples at a time.
771 
− 300 
− 300 
− 30 
− 30 
... 

100 bags 
100 bags 
10 bags 
10 bags 
... 



1. Bag fruits the slow way. Fill in the missing parts.
2. Bag fruits the fast way!
3. If there were 765 mangos instead of 762 in problem 2d above, how would the
result change?
4. a. Margie subtracted 24 from a certain number seven times, and
reached zero.
What was the number she started with?
b. This time, Margie subtracted 9 from a certain
number five times, and reached 2.
What was the number she started with?
Let’s compare continued subtraction
with long division. They are actually the same method, just written out differently!
Below, the numbers in long division are
written out
in full, using black and gray digits. The gray digits are the
ones we do not usually write. Also,
in the first example, the three parts of the quotient (200,
60, and 3) are written above each other for comparison's sake.
Fill in the two last examples.

Continued subtraction
789 ÷ 3 = ?
Dividend
(the apples) 

Quotient
(the bags) 
789 


−600


200 



189 


−180 

60 



9 


− 9 

3 
0 

263 















2 
0 
0 


3 
) 
7 
8 
9 

 
6 
0 
0 


1 
8 
9 
Hundreds. Three goes into
7 two times, or 7 ÷ 3 = 2 R1.
200 “bags” get added to the
quotient.
We subtract 7 − 6 = 1 and
drop down the 8, and it is
the same as the subtraction
789 − 600 = 189. 









6 
0 


2 
0 
0 


3 
) 
7 
8 
9 

 
6 
0 
0 


1 
8 
9 

 
1 
8 
0 



0 
9 
Tens. Three goes into
18
six times, or 18 ÷ 3 = 6.
60 “bags” get added to the
quotient.
We subtract
18 tens (180),
and
there are 9 apples left. 





3 



6 
0 


2 
0 
0 


3 
) 
7 
8 
9 

 
6 
0 
0 


1 
8 
9 

 
1 
8 
0 



0 
9 



 
9 




0 




Ones. 9 ÷ 3 = 3. The final answer is 263. 

637 ÷ 5 = ?
Dividend 

Quotient 
637 


−500 

100 



137 


−100 

20 



37 


−35 

7 
2 

127 


Hundreds 


1 
0 
0 


5 
)

6 
3 
7 

 
5 
0 
0 


1 
3 
7 



Tens 


1 
2 
0 


5 
)

6 
3 
7 

 
5 
0 
0 


1 
3 
7 

 
1 
0 
0 



3 
7 



Ones 


1 
2 
7 


5 
)

6 
3 
7 

 
5 
0 
0 


1 
3 
7 

 
1 
0 
0 



3 
7 


 
3 
5 




2 





5. Bag fruits. Also solve the problems using long
division, and compare the methods.
a. Bag 610 apples, 5 apples in each bag.

b. Bag 853 kiwis, 3 kiwis in each bag.
Kiwis  
Bags 
853   
−
 
200 
  
  
− 240   
  
13   
− 12  

 

1   





3 
)

8 5 3 


c. Bag
445 grapefruits, 3 grapefruits in each bag.

d.
Bag 952 plums, 4 plums in each bag.





4 
)

9 5 2 


e. Bag
2,450 pears, 9 pears in each bag.





9 
)

2 4 5 0 


f. Bag
1,496 oranges, 8 oranges in each bag.





8 
)

1 4 9 6 


See also
Long division worksheets
Create an unlimited supply of worksheets for long division (grades 46), including with 2digit and 3digit divisors. The worksheets can be made in html or PDF format  both are easy to print. You can also customize them using the generator.
www.homeschoolmath.net/worksheets/long_division.php
This lesson is taken from Maria Miller's book Math Mammoth Multiplication & Division 3, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.

