Direct and Inverse Proportion

Two types of proportions
Direct proportion;
Y=f(x)=kx  k is a constant
or Y/X=k
Y and X are direct proportion
The more X is, the more Y is
ex: buy one book costs $2, two books costs $4, 3 books costs $6.......
(given k=2, when x=1then y-=2,x=2then y=4 ,x=3 then y=6.............)
the more  x is, the more y is
the more the books, the more  the cos is
x and y are direct proportion (books and cost are direct proportion)

Inverse proportion:
Y=f(x)=k /X             k is a constant
or XY=k
X and Y are inverse proportion
The more X is, The less Y is
ex: D=RT
if D is a constant.
The faster the rate is , the less  the time it will take
EX1: k=xy (D=RT)                                                                      k=xy (D=RT)
 Given 120 miles, rate at 30 mph, it takes 4hours(given k=120   120=30x4)
                           rate at 60 mph, it takes 2 hours(given k=120, 120=60x2)
                          rate ate 120 mph, it takes 1hour(given k=120  120=120x1)
                       
The more x is, the less y is (k=xy)
The more R is, the less T is (D=RT)
x and y are inverse proportion (R and T are inverse proportion)

Ex2: It takes 10 workers and works 100 hours to finish a project(Given k=1000=10x100 )
        It takes 20 workers and works  50 hours to finish a project (k=1000, 1000=20x50)
        It takes 40 workers and work 25 hours to finish a project   (k=1000, 1000=40x25)

Problem Solving Strategy:
Step1: to recognize if it is direct or inverse proportion
Step2: If it is direct proportion: y=f(x)=kx
          If It is inverse proportion: y=f(x)=k/x
Step3: find k=? and put the value in the known variable, and solve the unknown variable.

Q1, For the function f , y=f(x) is inversely proportional to x, when x>0.  If f(3) =6, what is the value of f(2)?

Since it is  inversely proportional- xy=k, y=k/x
 y=f(x)=f(3) =k/3=6, so k=18
f(2)=k/x=18/2=9


Q2, If y is directly proportional to x square and y=1/8 when x=1/2 what is the positive value of x when y=9/2
Since it is directly proportional , y=k(x)(x)
1/8=k(1/4), so k=1/2
9/2=k(x)(x)=1/2(x)(x)
so x=3


Q3, If 5 chickens lay 5 eggs in 5 days. How many eggs will 25 chickens lay in 25 days?
The more chicken, the more eggs, and the less days
Chicken(C) and eggs (E)are direct proportional, and chicken(C) and days(D) are inverse 
proportional ,
Therefore, k=CD/E , k=(5x5)/5=5
k=5=(25x25)/D
D= 125


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