The concept of “borrowing” when subtracting a large number from a smaller number is perhaps one of the most confusing lessons we learn in school. Given a simple subtraction problem like:
134 – 97 = ??
In our head we say something like:
four minus seven, can not be
borrow one from three. fourteen minus seven is seven…
Mental Subtraction is very easy if there is no “borrowing”. We will discuss three ways to avoid borrowing.
Subtraction by Steps:
Imagine subtraction to be going down from a higher number (minuend) to a lower number (subtrahend). The distance between them is their difference. On the way down, we can make a stopover on an intermediate value. The difference then becomes the sum of the distance from the higher number to the stopover and the distance from the stopover to the lower number.
Example 3.1:
1 3 4
---------- 1 0 0
- 9 7
Along the way from 134 to 97 is 100. Instead of going directly to 97, we can stop at 100. It is very easy to see that 100 is 34 units away from 134
1 3 4
- 1 0 0
3 4
And 97 is 3 units away on the other direction.
1 0 0
- 9 7
3
So with 100 in the middle, 134 – 97 = 34 + 3 = 37.
Note that we first subtracted 100 which is 3 more than the 97 that is asked for. So we have to “return” whatever we have over-subtracted.
We can see that this technique is very useful when the numbers involved are near and on the opposite sides of a nice round number. Let us consider more examples.
Example 3.2:
1 6 3
------ 1 3 0
- 1 2 8
Here with 130 in the middle, 163 – 130 = 33, then 130 – 28 = 2, so 33 + 2 = 35.
Example 3.3:
1 5 4 2
------ 1 0 0 0
- 8 7 8
Subtracting 1542 – 1000 = 542 (surely we all know that!)
Then using our previous math short cut “All from 9 and The Last From 10 ” we can easily see that 878 is 122 below 1000. Hence the difference is 542 + 122 = 664.
Exercise 3.1: Compute the difference using “Subtraction by Steps”
a) 724 – 698 =
b) 1, 256 – 994 =
c) 3, 534 – 1, 985 =
d) 5, 463 – 2, 778 =
e) 25, 647 – 8678 =
Subtraction using Addition:
This next method is very handy when the subtrahend is just below a multiple of a power of 10. We could also adjust, by addition, the figures involved in the subtraction. Using the numbers in the previous examples, we can easily see equivalences, and have:
Example 3.4
1 3 4
- 9 7
? ?
|
( 1 3 4 + 3 )
- ( 9 7 + 3 )
? ?
|
1 3 7
- 1 0 0
3 7
|
By adding 3 to both the minuend and the subtrahend, the subtraction becomes a much simpler problem: 137 – 100 = 37
Example 3.5:
1 6 3
- 1 2 8
3 5
We can easily compute this mentally since only 2 will be added.
Example 3.6
1 5 4 2
- 8 7 8
? ? ?
Again, knowledge of “all from 9 and the last from 10” enables us to mentally compute this as 542 + 122 = 664
Exercise 3.2: Solve the following using “Subtraction Using Addition”
a) 525 – 496 =
b) 847 – 293 =
c) 5,743 – 1,976 =
d) 7,352 – 2,684 =
e) 17, 437 – 9, 689 =
Subtraction by Parts.
In Subtraction by Steps, we over-subtract, here we under-subtract.
Example 3.7:
5 2 4
- 3 2 7
? ? ?
|
( 5 2 4 - 3 2 4)
- ( 3 2 7 - 3 2 4)
? ?
|
2 0 0
- 3
1 9 7
|
The goal here is to create zeroes in the minuend. We first subtract only 324 to yield an even 200. But 324 is 3 less than the actual subtrahend. And so we subtract again, this time 3 from 200.
Use this mental shortcut when the value of the last few digits of the subtrahend is a little bit more than the value of the corresponding digits in the minuend.
Example 3.8
7 6 3
- 4 7 8
Notice that the last two digits of the minuend is 15 units smaller than the last two digits of the subtrahend. (Subtract 63 from 78 to get 15). So we first subtract 463. Then we have 300 – 15 or 285.
Exercise 3.3: Solve using “Subtraction by Parts”
a) 452 – 357 =
b) 843 – 255 =
c) 2, 336 – 1,448 =
d) 1, 234 – 365 =
e) 8, 745 – 6, 849 =
Answers to all exercises are found in the answer key.
Discover the 25 Math Short Cuts ( 25 MSC )!