Relation between gallons of gas and speed

[bajwa2307]

When a car is driven at 70 miles per hour, it uses 0.06 gallon of gas per mile. When the car is driven at 50 miles per hour, it uses 0.04 gallon of gas per mile. Which of the following relations are possible between the gallons of gas used per mile and the speed at which the car is driven?

I. They are directly proportional.
II. They are indirectly proportional.
III. They are linearly proportional.

(A) only I
(B) only II
(C) only III
(D) I and III
(E) I, II and III

the actual answer is C

can anyone please explain the solution.

[Tani]

Direct proportion: X = kY (where K is a constant). In this problem you have

70 = k (.06) K = 1167

and

50 = k (.04) K = 1250

The two values of K are different so it cannot be direct.

Indirect proportion

XY = K

again, trying our values

70*.06 = 4.2

50*.04 = 2

Two different value of K.

Given your answer choices, III must be correct. If you graphed the values with the mph on one axis and the gallons used on the other you would have 2 points and could use a straight line to connect them, therefore the relationship COULD be linear.

[bajwa2307]

so we get the answer as C by method of elimination. Since we proved it is no directly or linearly proportional, the third option must be true.

Thx so much for the explanation.