Lesson 16MA 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 .
An equation indicating two ratios are equal is called a proportion. In any proportion, cross products are equal.
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 hoursearnings
1 $13
3 $39
8 $104
20 $260
These ratios are equal.
We say that the earnings vary directly as the number of hours worked.
If y varies directly as x: where k is some constant.
Direct Variation: The words 'y varies directly with x' or 'y is directly proportional to x' leads to the equation , 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 speedtime of trip
20 mph1 hr.
40 mph½ hr.
60 mph hr.
80 mph¼ hr.
The products are equal.
We say that the bus speed varies inversely as the time of the trip.
If y varies inversely as x: where k is some constant.
Inverse Variation: The words 'y varies inversely as x' or 'y is inversely proportional to x' means that for some constant k. Again, k is called the constant of proportionality or variation constant.
Ex 2:Find the constant of proportionality for each. Write the resulting variation equation.
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.
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 for a constant k.
To solve a variation problem:
- Translate the problem into a variation format ()
- Replace the given numbers and solve for the variation constant.
- Re-write the variation format replacing the value of k. This is the variation equation.
- Use that equation to solve the problem.
Solve each problem.
Ex 3:y is directly proportional to x. If y = 15 when x = 4, find y when x = .
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?
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?