APPLIED MATHEMATICS

a=a

reflexive property

a=b, b=a

symmetric property

if a=b and b=c, then a=c

transitive property

addition property

if a=b then a+c=b+c

subtraction property

if a=b then a-c=b-c

multiplication property

if a=b then ac=bc

division property

if a=b then a/c=b/c

if a= b then a can be replaced by b

substitution property

if a+c=b+c or a/c=b/c then a=b

cancellation property

commutative property

order of numbers does not matter when adding or multiplying

a+b=b+a, ab=ba

commutative property

associative property

sum or product of three or more numbers does not change no matter how they are grouped

a+(b+c)=(a+b)+c, a(bc)=(ab)c

associative property

distributive property

the multiplier gets distributed and multiplied to the addends

a(b+c) = ab+ac

distributive property

ONE SOLUTION

consistent and independent system

INFINITE SOLUTIONS

consistent and dependent system

NO SOLUTION

inconsistent system

y=log(a)x

if x=a^y

lnx

to represent log(e)x

Ratio

a:b, a/b

a and b in ratio

a=antecedent b=consequent

Proportion

statement of equality between two ratios

a:b=c:d

a/b=c/d

b and c in proportion

means

a and d in proportion

extremes

a/b=c/d then b/a=d/c

inversion

a/b=c/d then a/c=b/d

alteration

composition

a/b=c/d then (a+b)/b=(c+d)/d

division

(a-b)/b=(c-d)/d

composition and division

(a+b)/(a-b)=(c+d)/(c-d)

mean proportional to a and b

a/x=x/b then x=√ab

third proportional to a and b

a/b=b/x then x=b²/a

fourth proportional to a, b and c

a/b=c/x then x=bc/a

variation

relates the value of one variable to the others

direct variation

x is directly proportional to y

k=x/y

direct variation

inverse variation

x is inversely proportional to y

k=xy

inverse variation

joint variation

x is directly proportional to y and z

k=x/yz

joint variation

combined variation

x is directly proportional to y and inversely proportional to z

k=xz/y

combined variation

mixture problems

amt of mixture * percent of pure stuff = amt of pure stuff

A(x%)±B(y%)=A±B(z%)

amt of mixture(percent of pure stuff)

age problems

time elapsed for all people concerned is equal

a=b

a and b = ages

Work problems

work=rate*time

Rate=1/t

if a person can finish a job in t days

(1/t1+1/t2+1/t3…)t=1 complete work

if multiple people can finish a job in t days