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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 ...
Let's keep count of the bags as we subtract (put into bags) the apples. Study carefully the two calculations on the right.

In method 1, we count the bags 100 bags at a time initially, and then 10 bags at a time.

Study method 1 now.

In method 2, we start out by counting 200 bags and subtracting 600 apples all at once, instead of subtracting 300 apples two separate times.

Similarly, we then count 50 bags and subtract 150 apples all at once (150 is the largest possible multiple of 30 that we can subtract from 171).

In total we need 200 + 50 + 7 = 257 bags for all the apples. 

We can write the division 771 ÷ 3 = 257.

Method 1 - slower

Apples  Bags 
771
−  300 100 bags
471
−  300 100 bags
171
−  30 10 bags
141
−  30 10 bags
111
−  30 10 bags
81
−  30 10 bags
51
−  30 10 bags
21
−  21 7 bags
0 257 bags

Method 2 - quicker 

Apples  Bags 
771
−  600 200 bags
171
−  150 50 bags
21 
−  21

0
7 bags

257 bags


Example 2. It will not matter if you do the subtracting the slow way or the fast way. It works either way, and the answer is the same—the slow way just takes longer!

In method 2, instead of using 100 or 10 bags at a time, we use a multiple of 100 bags and a multiple of 10 bags at a time.

In the second step, method 2 uses 60 bags. It would still work if you used 20 bags or 30 bags—you would just have more subtractions to do till you would reach zero apples.

Method 1 - slower

Dividend Quotient
Apples Bags
       
795  
− 300   100
495
− 300 100
195
− 30 10
165
− 30 10
135
− 30 10
105
− 30 10
75
− 30 10
45
− 30 10
15
− 15 5
0 265

Method 2 - quicker 

Dividend Quotient
Apples Bags
     
795
− 600 200
195
− 180 60
15
−  15 5
0  265

1. Bag fruits the slow way. Fill in the missing parts.

a. Bag 657 apples;
    3 apples in each bag.

Apples   Bags
657
100
357
100
57
10
27
−    27  
0

b. Bag 984 peaches;
    8 in each bag.

Peaches   Bags
984
 800  
184
 80  
104
 80  
3
0

c. Bag 536 pineapples;
    4 in each bag.

Pineapples   Bags
536
100
136
 
96
 
56
 
16
 
0


2. Bag fruits the fast way!

a. Bag 474 apples;
    3 apples in each bag.

Apples   Bags
474
100
174
50
24
 
0

b. Bag 2,032 lemons;
    8 lemons in each bag.

Lemons   Bags
2032
200
432
 
32
 
0

c. Bag 3,655 bananas;
    5 in each bag.

Bananas   Bags
3655
 
155
 
5
 
0

d. Bag 762 mangos;
    6 mangos in each bag.

Mangos   Bags
762
100
20
42
 
0

e. Bag 1,152 papayas;
    3 papayas in each bag.

Papayas   Bags
1152
300
 
 
0

f. Bag 4,770 cherries;
    9 in each bag.

Cherries   Bags
4770
 
 
0

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

) 

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

) 

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

) 

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

) 

6 3 7
5 0 0
1 3 7

 

 
Tens
1 2 0

) 

6 3 7
5 0 0
1 3 7
1 0 0
  3 7

 

 
Ones
1 2 7

) 

6 3 7
5 0 0
1 3 7
1 0 0
  3 7
3 5
2

988  ÷  4 = ?

Dividend Quotient
988
200
188
40
28  
 

 

 
 
   
4 

) 

9 8 8
  0 0
    8
    0
     
   
 

2546  ÷  7 = ?

Dividend Quotient
2546
300
446
 
   
 

 

 
 
     
7 

) 

2 5 4 6
    0 0
      6
      0
       
     
   


5. Bag fruits. Also solve the problems using long division, and compare the methods.

a. Bag 610 apples, 5 apples in each bag.

Apples Bags
610
− 500
110
− 100
10
−  10  
0
 

)

 6 1 0

 

b. Bag 853 kiwis, 3 kiwis in each bag.

Kiwis Bags
853

200

 

−  240

13
−  12  
1
 

)

 8 5 3

 

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

Grapefruits Bags
445
   −
145
25
−   
 
 

)

 4 4 5

 

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

Plums Bags
952
200
 

30
 
−   
0
 

)

 9 5 2

 

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

Pears Bags
2450
 
 
−   
 
 

)

 2 4 5 0

 

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

Oranges Bags
1496
 
 

 
 
−   
0
 

)

 1 4 9 6

 




This lesson is taken from my book Math Mammoth Multiplication & Division 3.

Math Mammoth Multiplication & Division 3

A self-teaching worktext for 5th grade that covers multi-digit multiplication, long division, problem solving, simple equations, ratios, divisibility, and factoring.

Download ($7.40). Also available as a printed copy.

=> Learn more and see the free samples!

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