QUANT METHODS BOOT CAMP (QMBC)
Director, Dr. John Sorrentino
Economics Dept. Temple University Ambler
215-283-1567; sorrento@temple.edu
FIRST-DAY WARM UP
Forget the outside world for now. Get with the program & please read ahead.
Motivation to Study Math/Stat in General
The study of math can be FUN if it not attached to fear-of-failure & related
anxiety, yes? Right. You can’t go too far in the world without it.
Enough on motivation. (*_*)
Some FUN-da-mental laws on the operations of arithmetic are given. You may
know them, but you might trace how they are needed in many other math/stat
statements that we use.
Associative Laws of + (addition) and  (multiplication)
(a + b) + c = a + (b + c) = a + b + c
(a  b)  c = a  (b  c) = a  b  c
Examples:
(5+3)+2 = 5+(3+2) = 5+3+2 = 10
(53)2 = 5(32) = 532 = 30.
Commutative Laws of + and 
a+b=b+a
ab=ba
Examples:
5+3 = 3+5 = 8
53 = 35 = 15.
Distributive Laws of + , , and / (division)
a  (b + c) = a  b + a  c
(b + c) / a = b/a + c/a
These laws can be used to make other mathematical statements, derivations &
proofs. Take for example the “squaring” of the sum of two hypothetical numbers:
To show:
We have
(a+b)2 = a2 + 2ab + b2
(a+b)2 = (a+b)  (a+b)
= a  (a+b) + b  (a+b)
= a2 + ab + ba + b2
= a2 + ab + ab + b2
= a2 + 2ab + b2
by definition of square
distributive law
distributive law
commutative law
by adding like terms.
Example.1: (2+3) 2 = 22 + 223 + 32 = 4 + 12 + 9 = 25
Example.2: If there were 10 men & 6 women in a room, then show that if the
number of men plus the number of women is squared, this number will exceed
the sum of the squares of the men & the women separately.
(10 + 6) 2 = 256
102 + 62 = 100 + 36 = 136
The difference is 2ab = 2106 = 120.
Example.3: (5+3)/2 = 5/2 + 3/2 = 2.5 + 1.5 = 4.
Additional useful laws are:
Monotonic Law of Addition
a <b a+c<b+c
Monotonic Law of Multiplication
a <b ac<bc
Example.1: 3 < 4  3+2 < 4+2  5 < 6
Example.2: 3 < 4  32 < 42  6 < 8.
Useful laws involving fractions are:
Multiplication of Fractions
(a/b)  (c/d) = (a  c)/(bd)
Division of Fractions (invert & multiply rule)
(a/b)  (c/d)
= (ad)/(bc)
Example.1: (1/2)  (2/3) = (12)/(23) = 2/6 = 1/3
Example.2: (1/2) / (2/3) = (13)/(22) = 3/4
Additional useful formulae can be gotten by the use of the laws & rules above.
Addition/Subtraction of Fractions (common denominator rule)
To show:
(a/b)  (c/d) = (ad+bc)/(bd)
(a/b)  (c/d) = (ad/bd)  (bc/bd)
= (ad+bc)/(bd)
by multiplying (a/b) by (d/d)=1;
multiplying (c/d) by (b/b)=1;
commutative law
using common denominator
Cross-Multiplying
To show:
(a/b) = (c/d)  ad = bc
(a/b) = (ad/bd)
multiplying (a/b) by (d/d)=1
(c/d) = (bc/bd)
multiplying (c/d) by (b/b)=1
(a/b) = (c/d)  (ad/bd) = (bc/bd)
 ad = bc
multiplying both sides by the
common denominator
Equalized Proportions
To show:
:
(a/b) = (c/d)  (a/(a+b)) = (c/(c+d))
(a/(a+b)) = (c/(c+d))
 a(c+d) = c(a+b)
by cross multiplying rule
ac+ad = ac+cb
by distributive law, commutative
law
ad = cb
(ad/bd) = (cb/bd)
(a/b) = (c/d)
subtracting ac from both sides
dividing both sides by bd
canceling the d’s & b’s
:
(a/b) = (c/d)
 ad = bc
 (a/(a+b)) = (c/(c+d))
by cross multiplying rule
by the reverse of the process
above.
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Most people are willing to trust mathematicians with telling them the
truth about rules, & the rules usually work out fine in practice. One might argue,
however, that the process of thinking logically line-by-line is a good
thinking exercise. Many people do NOT have the patience to do this, though
the mental ability is there. They often use an OPPORTUNITY COST
ARGUMENT from the Dismal Science, economics. It’s just about always better
doing something else! Also, when you do it & then stay off of it for a long time, the
brain pathways get “rusty.” It takes a while to get back in the groove if you have
to. It’s almost like the cliché about never forgetting how to ride a bicycle once you
know. The figure below shows you how much you’ve changed on the first day.
Don’t go away; Day 2 will come sooner than you think…