   SEARCH HOME Math Central Quandaries & Queries  Question from Julie, a student: Please Help I do not understand this question. X varies directly as the square of S and inversely as T. How does X change when S is doubled ? When both S and T are doubled ? I appreciate your help with this problem. Hi Julie.

"Varies directly" means that as one quantity goes up, the other goes up proportionally. For example, if my gas mileage is 42 mpg, then I can say M = 42 x G, where the quantity M is the total miles and G is the total gallons. M varies directly as G. The constant 42 may be anything else, but the important idea is that if you increase the amount of fuel available (G), then M goes up.

"the square of" just means that the quantity is squared. For example, the distance in meters that a rock falls (D) varies directly as the square of the time (T) in seconds that it is freely falling in the relationship D = 4.9 T2. So as the time goes up, the distance goes up faster and faster.

"Varies inversely" means that as one quantity goes up, the other quantity goes down to compensate. For example, when I run at the speed (S) from one end of a Canadian football field to the other, it takes time (T) in seconds. The higher my speed (S), the lower the time (T). In fact, S = 110 / T (Canadian football fields are 110 yards long), so we say that S varies inversely as T in this equation.

There's always some constant involved in these relationships (in my examples, these were 42, 4.9 and 110) and often we just call the constant k if we don't know it.

X varies directly as the square of S and inversely as T.

So X = k S2 / T.     