# Engineering Math Nov 2023

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 ?
Solution:
Question: What is ?
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 ?
Solution:

Now dropping the last two terms, we have

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.
Recall
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,
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
Question: What is the Centigrade equivalent of ?
Say
Say half of
Say

Therefore, .
Question: What is the Centigrade equivalent 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, .

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, .
To multiply by 2, we can divide by 5 if it’s easier that way

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

Just say 6 times 4;
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

15

What is of ?

Just say 630 over 15 times 10; therefore, times ; therefore, .

Click for a couple three quick examples using aliquot parts