## 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

# From left to right

Just as in addition, it may prove helpful to do mental subtraction from left to right. For instance, to do
92
&minus 36

we can subtract 3 from 9 (which is the equivalent of doing 92 &minus 30) and then from 62 (92 &minus 30) we subtract 6 to obtain 56.

Notice that the when subtracting two digits in x &minus y that are in the same position (for instance 9 and 3 in our example) we need to subtract one more if the following digit of x (2 in our example) is smaller than the following digit of y (6 in our example).

Let's do the following example,
274
&minus 152

274 &minus 100 = 174, 174 &minus 50 = 124, and 124 &minus 2 = 122. Thus 274 &minus 152 = 122.

A simple way of subtracting x, specially if x ends with 7, 8, or 9, is to subtract the closest multiple of 10 that is bigger than x and then add the difference. For instance, to subtract 48, subtract 50 and then add 2; since 48 = 50 &minus 2, we obtain the same result.

The same idea can be generalized to do more complicated examples. For instance, since 500 &minus 26 = 474, we can get the answer to 1892 &minus 474 by subtracting 500 and then adding 26. In other words, 1892 &minus 474 is

1892
&minus 500
+   26
1392
+    26
1418

Activity: Mental subtraction
1. 78 &minus 24.
2. 173 &minus 45.
3. 8275 &minus 362.
4. 69 &minus 51.
5. 3872 &minus 3221.

# x &minus y, when (b &minus 1) × 100 &le y &le b × 100 and b × 100 &le x &le (b + 1) × 100

Consider
192
&minus    36

The answer 156 can be obtained very easily by subtracting from left to right. However, we can solve the problem by simply adding two numbers. In our case, the numbers to add are 100 &minus 36 = 64 and 192 &minus 100 = 92. Notice that 64 + 92 = 156, which is indeed 192 &minus 36.

In general, we can obtain x &minus y, when (b &minus 1) × 100 &le y &le b × 100 and b × 100 &le x &le (b + 1) × 100 very easily: the answer is the sum of the difference between y and b × 100 (the closest hundred to y rounding up) and the number obtained by subtracting b × 100 from x. For instance,
1127
&minus    1046

In the above example, b = 11, so the numbers to add are 11 × 100 &minus 1046 = 54 and 1127 &minus 11 × 100 = 27. Indeed, we can easily check that 1127 &minus 1046 = 81 = 27 + 54.

The proof of the validity of this method is incredibly simple, and we leave it as an activity.
Activity: x &minus y, when (b &minus 1) × 100 &le y &le b × 100 and b × 100 &le x &le (b + 1) × 100
1. Argue why the above method produces the correct answer
2. Use the method in the operations below:
1. 367 &minus 234.
2. 1902 &minus 1869.
3. 189 &minus 23.
4. 4823 &minus 4746.

# Subtracting without borrowing

Sometimes one can avoid "borrowing" if we subtract a pair (or more) of digits instead of just one digit. For instance, consider 4261 &minus 3929. If we were to solve this problem in the traditional way, we would have to "borrow" 1 from 6 to do 11 &minus 9. However, notice that if we do 61 &minus 29 = 32 there is no need to borrow. Now we do 42 &minus 39 = 3 to obtain the answer 332.

Let's look at a different example:
7826
&minus    1073

Notice that in either 6 &minus 3 and 26 &minus 73 we would have to borrow. So we do 826 &minus 73 = 753 instead. To get the answer, we concatenate 7 &minus 1 = 6 to 753 to obtain 6753.

Activity: Subtraction without borrowing
1. Argue why the above method produces the correct answer
2. Use the method in the operations below:
1. 3167 &minus 239.
2. 1092 &minus 869.
3. 48636 &minus 4863.
4. 57632 &minus 21419.
Cornell University, Department of Mathematics