Expert: Sherry Wallin - 5/30/2010

Question
My son is having difficulty applying quadratics. The problem is, a complex has 100 units(D)-when the rent is $300 a week(C), all units are rented. For each extra $20 rent, 5 units will become vacant. Write an equation relating C and D with D as the subject. Thanks for your help-it's too long since I did maths for me to help.

Answer
Hi Sue~
    One way to write a relationship between D and C is to write individual equations in terms of another variable and then solve for either C or D. It helps to know which variable is independent. In the wording of the problem we are told that for every $20 extra in rent that the number of units rented goes down, so that means that the number of units rented depends on the amount charged per unit, or in other words that D depends on C.

We know that when the rent is $300 that all 100 units are rented and  that the number of rentals is 5 less for every $20 increase in the rent. You can represent this information by letting x = the number of $20 increments in the rent so
C = 300 + 20x and D = 100 - 5x gives us the number of units rented in multiples of 5. Solve each of C and D in terms of x:

C = 300 + 20x subtract 300 from both sides of the equation
C - 300 = -300 + 300 + 20x  simplify getting
C - 300 = 20x  now divide both sides by 20 getting
(C - 300)/20 = 20x/20 -> (C - 300)/20 = x

do the same thing for the other equation with D in it, solve for x:

D = 100 - 5x  subtract 100 from both sides
D - 100 = -100 + 100 -5x simplify getting
D - 100 = -5x and now divide bot sides by -5 getting
(D - 100)/-5 = -5x/-5 resulting in
x = (D - 100)/ -5 -> x = D/-5 - 100/-5 -> x = -D/5 + 100/5 -> x = (100 - D)/5

You now have two different representations for x which have to be equal so set them equal to each other:
(C - 300)/20 = (100 - D)/5. Now since we decided that D depends on C this means we want to solve for D in terms of C:
multiply both sides by 5:
5(C - 300)/20 = 5(100 - D)/5 which gives us
(C - 300)/4 = 100 - D   I am going to want D not - D so I multiply both sides by -1:
-1(C - 300)/4 = -100 + D -> (-C + 300)/4 = -100 + D  I want to simplify the left hand side so I multiply through by the -1
(-C + 300)/4 = -100 + D and now I add 100 to both sides
(-C + 300)/4 + 100 = D but I need a common denominator so I add 400/4 instead
(-C + 300)/4 + 400/4 = D and putting the fractions over the same denominator
(-C + 300 + 400)/4 = D ->
D = (700 - C)/4

We know that when we charge $300 we rent 100 units and when we increase the rent by 20 we rent 5 fewer rooms so we have the ordered pair (300, 100) and the ordered pair (320, 95) and the ordered pair (340, 90) and so until (680, 5) and (0,0). Feel free to check out these values and see that they work in our representation of D and C.

Math Prof