Lesson 16 MA 152, Section 2.5

The quotient of two numbers or quantities is called a ratio. Ratios often are used for comparisons. For example, if there are 120 pieces of candy for 13 students, that would be a ratio of 120 . 13

An equation indicating two ratios are equal is called a proportion. In any proportion, cross products are equal. a c  b d  a   c  (bd)   (bd)   b   d  ad  bc

Ex 1: Solve each problem by writing and solving a proportion. At a fraternity gathering, there were 2 males for every 3 females. If there were 24 females, how many were males?

Two variables are said to vary directly or be directly proportional if their ratio is constant. For example: examine the following table where number of hours worked is compared with earnings. # of hours earnings 1 $13 3 $39 8 $104 20 $260 260 104 39 13    These ratios are equal. 20 8 3 1 We say that the earnings vary directly as the number of hours worked. y  k  If y varies directly as x: x where k is some constant. y  kx

For k > 0: In direct variation, as x increases so does y; or, as x decreases, so does y.

1 Definition: Direct Variation: The words 'y varies directly with x' or 'y is directly proportional to x' leads to the equation y  kx , for some constant value k. The number k is called the constant of proportionality or the variation constant.

Two variables are said to vary inversely or be inversely proportional if their product is a constant. Examine this table where time of a trip is compared with bus speed. bus speed time of trip 20 mph 1 hr. 40 mph ½ hr. 1 60 mph hr. 3 80 mph ¼ hr.

(80)(1 )  (60)(1)  (40)( 1 )  (20)(1) The products are equal. 4 3 2 We say that the bus speed varies inversely as the time of the trip.

xy  k  If y varies inversely as x: k where k is some constant. y  x

For k > 0: In inverse variation, as x increases (for x > 0), y decreases; or, as x decreases ( for x > 0), y increases.

Definition: Inverse Variation: The words 'y varies inversely as x' or 'y is inversely proportional to x' k means that y  for some constant k. Again, k is called the constant of proportionality x or variation constant.

Ex 2: Find the constant of proportionality for each. Write the resulting variation equation. a) y is directly proportional to x. If x = 30, then y = 15.

b) R varies inversely as the square of I. If I = 25, then R = 100.

2 Definitions:

Joint Variation: The words 'y varies jointly with w and x' means that y = kwx for some constant k. Combined Variation: The words 'y varies directly as w and inversely as x' means that kw y  for a constant k. x

To solve a variation problem: k 1. Translate the problem into a variation format ( y  kx, y  , y  kwx, etc. ) x 2. Replace the given numbers and solve for the variation constant. 3. Re-write the variation format replacing the value of k. This is the variation equation. 4. Use that equation to solve the problem.

Solve each problem. 7 Ex 3: y is directly proportional to x. If y = 15 when x = 4, find y when x = . 5

Ex 4: q varies jointly with the square of m and the square root of n. If q = 45 when m = 3 and n = 25, find q when m = 4 and n = 36.

Ex 5: The time to drive a certain distance (t) varies inversely to the rate of speed (r). Mary drives 47 miles per hour for 4 hours. How long would it take her to make the same trip at 55 mph?

3 Ex 6: The intensity of illumination on a surface varies inversely as the square of the distance from the light source. A surface is 12 meters from a light source and has an intensity of 2. How far must the surface be from the light source to receive twice as much intensity of illumination?

Ex 7: Hooke's law states that the force required to stretch a spring a distance is directly proportional to the distance. A force of 3 newtons stretches a spring 15 centimeters. A force of 4 newtons would stretch the spring how far?

Ex 8: The power, in watts, dissipated as heat in a resistor varies directly with the square of the voltage and inversely with the resistance. If 20 volts are placed across a 20-ohm resistor, it will dissipate 20 watts. What voltage across a 10-ohm resistor will dissipate 40 watts?