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 the 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 [351-32] = 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=7 \cdot 14\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}.


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