Left-to-Right Subtraction

Like Addition, you can use the left to right approach for subtraction. For most of us, it is easier to add than to subtract. But if you continue to compute from left to right and to break down problems into simpler components, subtraction can become almost as easy as addition.

When subtracting two-digit numbers, your goal is to simplify the problem so that you subtract (or add) a one-digit number. Let’s begin with a very simple subtraction problem:

     67
     −  25

 ?

After each step, you arrive at a new and easier subtraction problem. Here, we first subtract 20 (67 – 20 = 47) then we subtract 5 to reach the simpler subtraction problem 47 - 5 for your final answer of 42. The problem can be diagrammed this way:

67 – 25       =       47 – 5                   =       42

(first subtract 20)    (then subtract 5)

However, it may feel uncomfortable to keep track of borrowings (a borrowing occurs when you subtract a larger digit from a smaller one, like 16 – 9). But the good news is that “hard” subtraction problems can usually be turned into “easy” addition problems. Let’s see an example of this.

     75
     −  28

 ?

There are two different ways to solve this problem mentally:

• First subtract 20, then subtract 8:

75 – 28  =       55 – 8                   =       47

(first subtract 20)    (then subtract 8)

But for this problem, the following strategy would be preferable:

• First subtract 30, then add back 2:

75 – 28  =       45 + 2         =       47

(first subtract 30)     (then add 2)

Here is the rule for deciding which method to use: If a two digit subtraction problem would require borrowing, then round the second number up (to a multiple of ten). Subtract the rounded number, then add back the difference. This way you don’t have to worry about borrowings.

For example, the problem 63 - 27 would require borrowing (since 7 is greater than 3), so round 27 up to 30, compute 63 – 30 = 33, then add back 3 to get 36 as your final answer:

    63
    −  27

?

63 – 27  =       33 + 3         = 36

(first subtract 30)      (then add 3)

Now try your hand (or head) at 92 - 37. Since 7 is greater than 2, we round 37 up to 40, subtract it from 92 (92 – 40 = 52), then add back the difference of 3 to arrive at the final answer:

92 – 37 = 52 + 3 = 55

With just a little bit of practice, you will become comfortable working subtraction problems both ways. Just use the rule above to decide which method will work best.

Now let’s try a three-digit subtraction problem:

   897
   −  273

 ?

This particular problem does not require you to borrow any numbers (since every digit of the second number is less than the digit above it), so you should not find it too hard. Simply subtract one digit at a time, simplifying as you go.

   897
    −  273

624

Now let’s look at a three-digit subtraction problem that requires you to borrow a number:

  817
   −  289

 ?

At first glance this probably looks like a pretty tough problem, but if you first subtract 817 – 300 = 517, then add back 11, you reach your final answer of 517 + 11 = 528.

Using complements to simplify subtractions even more

There is a way to easily calculate 3 or 4 digits subtractions very quickly in your head. This technique makes use of complements. For example, let’s say that you’re facing the following problem:

  674
   −  358

 ?

674 – 358        =       274 + 42     = 316

(first subtract 400) (then add 42)

Notice that, when adding 42 to 274, you can calculate mentally like this:

 274 + 42         =       314 + 2       = 316

(first add 40)          (then add 2)          

Instead of following the standard left to right approach, we solved this problem by subtracting 400 to 674 and then add 42 back to the result. 42 is the difference from 100 and 58. A good question is: how do you find 42?

Note that there’s a simple pattern for calculating the second number. In particular, the first digits (on the left) add to 9 and the second digits (on the right) add to 10. The only exception is when the number ends with 0 (e.g. 30 + 70 = 100), which is simpler.

We say that 42 is the complement of 58, 53 is the complement of 47. Now you find the complements of these two-digit numbers: 37, 59, 93, 44 and 08.

To find the complement of 37, first figure out what you need to add to 3 in order to get 9. (The answer is 6.) Then figure out what you need to add to 7 to get 10. (The answer is 3.) Hence, 63 is the complement of 37.

The other complements are 41, 7, 56 and 92. Notice that, like everything else, the complements are determined from left to right. Let’s consider the last subtraction problem:

  725
   −  358

 ?

To begin, we subtracted 400 instead of 358 to arrive at 325 (725 –400 = 325). But then, having subtracted too much, we needed to figure out how much to add back. Using complements gives us the answer in a flash. How far is 358 from 400? The same distance as 58 is from 100. If we find the complement of 58 the way we have seen, we will arrive at 42. Add 42 to 325, and we will arrive at 367, our final answer.

725 – 358        =       325 + 42     = 367

(first subtract 400) (then add 42)

You can use complements technique to solve any subtraction very easily as they allow you to convert difficult subtraction problems into straightforward addition problems.