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 \bold{742\cdot11} ?
Solution:
Question: What is \bold{6{,}587{,}236\cdot11} ?
Solution:
= \bold{72{,}459{,}596}
Try it by yourself this time

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

    \[(a+b)^2 = a^2+\bold{2}ab+b^2\]

    \[(a+b)^3 = a^3+\bold{3}a^2b+\bold{3}ab^2+b^3\]

    \[(a+b)^4 = a^4+\bold{4}a^3b+\bold{6}a^2b^2+\bold{4}ab^3+b^4\]

    \[\bold{1}\]

    \[\bold{1}\ \ \ \bold{1}\]

    \[\bold{1}\ \ \ \bold{2}\ \ \ \bold{1}\]

    \[\bold{1}\ \ \ \bold{3}\ \ \ \bold{3}\ \ \ \bold{1}\]

    \[\bold{1}\ \ \ \bold{4}\ \ \ \bold{6}\ \ \ \bold{4}\ \ \ \bold{1}\]

    \[\bold{1}\ \ \ \bold{5}\ \ \ \bold{10}\ \ \bold{10}\ \ \ \bold{5}\ \ \ \bold{1}\]

    \[\bold{1}\ \ \ \bold{6}\ \ \ \bold{15}\ \ \bold{20}\ \ \bold{15}\ \ \ \bold{6}\ \ \ \bold{1}\]

Second, we’ll try a few exponent calculations
Question: What is \bold{1.10^2} ?
Solution:
= 1+2(0.10)+(0.10)^2
= 1+0.20+0.01 = \bold{1.21}
And \bold{1.10^3} ?
Solution:
= 1+3(0.10)+3(0.10)^2+(0.10)^3
= 1+0.300+0.030+0.001
= \bold{1.331}
Question: What is \bold{1.05^4} ?
Solution:
1+4(0.05)+6(0.05)^2+4(0.05)^3+(0.05)^4
Now dropping the last two terms, we have
\approx 1+0.200000+6(0.002500) ...
\approx 1+0.200000+ (0.015000) ...
\approx \bold{1.215}
The exact answer is 1.21550625
Therefore, by using only three terms we have an error of less than 0.05\%
Now let’s consider a few squares.
Question: What is \bold{73^2} ?

Recall that (a+b)^2 = a^2+\bold{2}ab+b^2

Solution: (70)^2+\bold{2}(70)(3)+(3)^2

= 4900+6 \cdot 70+9
= 4900+420+9 = \bold{5329}
Question: What is \bold{88^2} ?

Recall that (a-b)^2 = a^2-\bold{2}ab+b^2

Solution: (90)^2-\bold{2}(90)(2)+(2)^2

= 8100-4 \cdot 90+4
= 8100-360+4
= 8100-400+44 = \bold{7744}
Question: What is \bold{36^2} ?

Again (a-b)^2 = a^2-\bold{2}ab+b^2

Solution: (40)^2-\bold{2}(40)(4)+(4)^2

= 1600-8 \cdot 40+4
= 1600-320+16
= 1600-400+96 = \bold{1296}
Now let’s consider common temperature conversions.
Start with determining the Fahrenheit equivalent of a Centigrade temperature ?
Recall \bold{F} = \frac{9}{5} \cdot \bold{C}+32
Or \bold{F} = [1.80] \cdot \bold{C}+32
Easier if approached using \bold{F} = [0.90 \cdot 2] \cdot \bold{C}+32
Which is simply [2 \cdot \bold{C}] less 10\% plus 32
Question: What is the Fahrenheit equivalent of 350^\circ\text{C} ?
Say [2 \cdot \bold{350}] less 10\% plus 32
Say 700-70 plus 32
Say 630 plus 32
Therefore, \bold{662}^\circ\text{F}
Next determine the Centigrade equivalent of a Fahrenheit temperature ?
Recall \bold{C} = \frac{5}{9} \cdot (\bold{F}-32)
Or \bold{C} = [0.55\bar5] \cdot [\bold{F}-32]
Easier if approached using \bold{C} = [0.50+(0.1) \cdot 0.50+(0.01) \cdot 0.50] \cdot [\bold{F}-32]
So say [\bold{F}-32] times half plus [\bold{F}-32] times a tenth of a half and so forth
Question: What is the Centigrade equivalent of 350^\circ\text{F} ?
Say [350-32] = 318
Say half of 318 = 159
Say 159+15.9 \text{ etc}
= 159+16-0.1+1.6+0.16
= 174.9+1.6+0.16 = 176.66
Therefore, \bold{176.66}^\circ\text{C}\approx \bold{177}^\circ\text{C}.
Question: What is the Centigrade equivalent of 351^\circ\text{F} ?
Say [350-32] = 319 \text{ (odd number)}
Say half of 319 = 159 with a remainder of 1
Say 159+15.9 \text{ etc}
= 159+16-0.1+1.6+0.16+[0.6]
= 176.66+[0.6]
The extra 0.6 is to account for the remainder of 1
Therefore, \bold{177.26}^\circ\text{C}\approx \bold{177}^\circ\text{C}.
Question: What is the Centigrade equivalent of 1650^\circ\text{F} ?
Say [1650-32] = 1618}
Say half of 1618 = 809
Say 159+15.9 \text{ etc}
= 809+80.9+8.1+0.8
= 898.8
Therefore, \bold{898.8}^\circ\text{C}\approx \bold{899}^\circ\text{C}\approx \bold{900}^\circ\text{C}.
And last, Aliquot Parts
How to use the divisors of 1, of 10, and of 100
2, 5
5 \cdot 2=10
2 \cdot 5=10

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

What is 5 \cdot 184 ?

Just say 184 over 2 times 10; therefore, \bold{920}.
What is 5 \cdot 121 ?

Just say 121 times 5; therefore, \bold{605}.
What is 3243 / 5 ?

Just say 3243 times 2 over 10; therefore, \bold{648.6}.
To multiply by 2, we can divide by 5 if it’s easier that way

What is 2 \cdot 4585 ?

Just say 4585 over 5 times 10; therefore, \bold{9170}.
What is 2 \cdot 1236 ?

Just say 1236 times 2; therefore, \bold{2472}.
2, 50
50 \cdot 2=100
2 \cdot 50=100

5, 20
20 \cdot 5=100
5 \cdot 20=100

4
25 \cdot 4=100
4 \cdot 25=100
To multiply by 25, we can divide by 4 if it’s easier that way
What is 25 \cdot 160 ?

Just say 160 over 4 times 100;
therefore, \bold{4000}.
To divide by 25, we can multiply by 4 if it’s easier that way
What is 600 / 25 ?

Just say 6 times 4;
therefore, \bold{24}.
8
12.5 \cdot 8=100
8 \cdot 12.5=100
3
33.33 \cdot 3=100
3 \cdot 33.33=100
6
16.67 \cdot 6=16\frac{2}{3} \cdot 6=100
6 \cdot 16.67=6 \cdot 16\frac{2}{3}=100
7
14.29 \cdot 7=14\frac{2}{7} \cdot 7=100
7 \cdot 14.29=12 \cdot 7\frac{2}{7}=100
9
11.11 \cdot 9=11\frac{1}{9} \cdot 9=100
9 \cdot 11.11=9 \cdot 11\frac{1}{9}=100
12
8.33 \cdot 12=8\frac{1}{3} \cdot 12=100
12 \cdot 8.33=12 \cdot 8\frac{1}{3}=100
What is 8.25\% tax on \$47.00 ?
Just say 47 over 12; therefore, ~\bold{\$4.00}
Could say 47 over 12 less say ~\$0.10;
therefore, ~\bold{\$3.90}.
The exact value is \bold{\$3.88}
15
6.67 \cdot 15=6\frac{2}{3} \cdot 15=100
15 \cdot 6.67=15 \cdot 6\frac{2}{3}=100

\frac{2}{3} \cdot \frac{3}{2} = 1
\frac{2}{3} \cdot 1.5 = 1

What is \frac{2}{3} of 630 ?

Just say 630 over 15 times 10; therefore, 42 times 10; therefore, \bold{420}.
Click for a couple three quick examples using aliquot parts


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