Math Explorer's Club Tips for Mental Computations

Introduction
Lesson 2: Subtraction
Lesson 3: Multiplication
Lesson 4: Division
Lesson 5: Calendar computations
Lesson 6: Guessing a number and other tricks

Start from left to right, not from right to left

We normally start adding numbers from right to left, and sometimes we "carry" some digit. This is all well and good, but when doing mental computations, starting from the left may prove more useful. For instance, consider

45
+  32

If we start from the left, then we are doing 45 + 30 + 2, since the first sum changes only the left-most digit. This sum is easier to do in our heads: 45 + 30 = 75, and 75 + 2 = 77. Try to "see" the numbers, and avoid repeating unnecessary information in your head "45 and 30 makes 75, 75 and 2 makes 77". Try to see "45, 75, 77" instead. Of course the above example was very easy since there is no digit being "carried". Now let's try an example where we have to carry a number:

37
+  58
So we think of 37 + 58 as 37 + 50 + 8 =87 + 8 = 95.

415
+  1932

First notice that it's easier to think of 1932 + 415 (add the smaller number to the bigger one). Now, 1932 + 400 + 10 + 5 = 2332 + 10 + 5 (we added 4 to 19) = 2342 + 5 (we added 1 to 3) = 2347 (we added 5 to 2). With practice, you'll learn to appreciate the advantages of starting from the left, and hopefully will find it more useful to do mental computations.

Combine it with subtraction

If you want to add 9 to a number, you can just add then and then subtract 1. Similarly if you want to add 8, you can add then and then subtract 2. This can be particularly helpful when we are adding numbers that end with a "large" digit, such as 7, 8, or 9.

For instance,
43
+  8
51
and we can think about this sum as 43 + 10 &minus 2 = 53 &minus 2 = 51. Thinking about adding 10 first and then subtracting 2 makes the sum easier to compute mentally.

Similarly, if we want to add 37, we can add 40 and then subtract 3. In general, to add a number ab, we can just add (a + 1)0 = 10 × (a + 1), and then subtract 10 &minus b. For instance,
413
+  28
413 + 30 &minus 2 = 441

Apply the techniques discussed above to do the following sums in your head.
1. 35 + 46.
2. 129 + 89.
3. 395 + 207.
4. 9345 + 3245.
5. 1830 + 855.

There exists a way to avoid some carrying in addition by simply writing the total for each column, and then adding these. For instance, consider
2358
+   8239
+   4275
+   5739
31   (8 + 9 + 5 + 9)
18   (7 + 3 + 7 + 3)
14   (3 + 2 + 2 + 7)
19   (2 + 8 + 4 + 5)
20611

Notice that the space is needed, since the 18 in the second row actually represents 18 × 10, since it was obtained by adding up the decimal units.
Activity: More sums
1. Use the technique just described in the sums below
1. 3253 + 4683 + 2234.
2. 4510 + 2719 + 1289.
3. 6204 + 4812 + 4819.

Here is a nice fact: If we divide an integer n by 9, the remainder obtained is the same as the sum of the digits of n, module 9. For instance, take n = 67. Since 67 = 9 × 7 + 4, the remainder is 4. Now 6 + 7 = 13 &equiv 4 (mod 9). Casting out nines to check addition is simply the process of checking if the remainder module 9 of the numbers in the sum add up to the remainder module 9 of our answer. For instance, consider

8243   (8)
+  3434   (5)
11677  (4)

where the numbers in red represent the remainders mod 9. Notice that 8 + 5 &equiv 4 (mod 9). So we have some evidence that our answer is correct.

Notice that casting out nines is not a proof that our answer is correct. However, it is a method to determine if the answer is wrong: If the sum of remainders doesn't add up to the remainder of our answer, we know for sure we made a mistake somewhere. For instance, consider the sum
413
+  831
+    28

Notice that the sum of the remainders mod 9 is 8 + 3 + 1 &equiv 3 (mod 9). So if we obtained, say, 1322 &equiv 8 (mod 9), we can be certain that we made a mistake.

We remark that casting out nines can be used to check subtraction and multiplication.