Often while working, an arithmetical answer is needed — exact is sometimes required, approximate is often acceptable — maybe the basis of a decision or a path forward is based on a few quick calculations, with more rigorous calculations to follow

**Let’s illustrate with examples from two groups**

**First, we’ll learn to multiply by 11 by just writing down the answer**Question: What is ?

Try it by yourself this time

Solution:

Try it by yourself this time

Remember the binomial theorem or Pascal’s Triangle. Let’s go back a few years and refresh.

**Second, we’ll try a few exponent calculations**Question: What is ?

Solution:

And ?

Solution:

Question: What is ?

Now dropping the last two terms, we have

Solution:

Now dropping the last two terms, we have

The exact answer is

Therefore, by using only three terms we have an error of less than

Therefore, by using only three terms we have an error of less than

Now let’s consider a few squares.

Question: What is ?

Recall that

Solution:

Question: What is ?

Recall that

Solution:

Question: What is ?

Again

Solution:

Now let’s consider common temperature conversions.

**Start with determining the Fahrenheit equivalent of a Centigrade temperature ?**Recall

Or

Easier if approached using

Which is simply less plus

Or

Easier if approached using

Which is simply less plus

Question: What is the Fahrenheit equivalent of ?

Say less plus

Say plus

Say plus

Therefore,

Say plus

Say plus

Therefore,

**Next determine the Centigrade equivalent of a Fahrenheit temperature ?**Recall

Or

Easier if approached using

So say times half plus times a tenth of a half and so forth

Or

Easier if approached using

So say times half plus times a tenth of a half and so forth

Question: What is the Centigrade equivalent of ?

Say

Say half of

Say

Therefore, .

Say half of

Say

Therefore, .

Question: What is the Centigrade equivalent of ?

Say

Say half of with a remainder of

Say

Say half of with a remainder of

Say

The extra is to account for the remainder of 1

Therefore, .

Question: What is the Centigrade equivalent of ?

Say

Say half of

Say

Therefore, .

Say half of

Say

Therefore, .

**And last, Aliquot Parts**How to use the divisors of 1, of 10, and of 100

2, 5

To multiply by 5, we can divide by 2 and vice versa, whichever is easier!

**What is ?**Just say 184 over 2 times 10; therefore, .

**What is ?**Just say 121 times 5; therefore, .

**What is ?**Just say 3243 times 2 over 10; therefore, .

**What is ?**Just say 4585 over 5 times 10; therefore, .

**What is ?**Just say 1236 times 2; therefore, .

2, 50

5, 20

4

To multiply by 25, we can divide by 4 if it’s easier that way

What is ?

Just say 160 over 4 times 100;

therefore, .

therefore, .

What is ?

Just say 6 times 4;

therefore, .

therefore, .

8

3

6

7

9

12

**What is tax on ?**Just say 47 over 12; therefore, ~

Could say 47 over 12 less say ~;

therefore, ~.

The exact value is Could say 47 over 12 less say ~;

therefore, ~.

15

**What is of ?**Just say 630 over 15 times 10; therefore, times ; therefore, .

Click for a couple three quick examples using aliquot parts