Proportions and Rates

When you have finished this lesson and worked your way through the text you should be able to:

Proportionality - Direct and Inverse

In this section we will consider the concept of one thing being proportional to something else. Things can be directly or inversely proportional. For example the number of cookies that you can make is directly proportional to the amount of flour that you use. Conversely, if you are driving somewhere the time to get there is inversely proportional to your speed. In mathematics these concepts are expressed as follows:

Directly Proportional

If n = 1 you have the simple case where y = kx. If you graph such an equation you get a figure such as that shown on the right. Note that the line shown passes through the origin. If y is directly proportional to x when x is zero y is equal to the constant k, times zero. The result will always be zero.

For example in a simple DC electrical circuit of fixed resistance, the voltage is equal to the current times the fixed resistance. Mathematically this is V = kI (where I represents current). If you have a flashlight with 9 volt batteries and you measure the current as 1.5 amperes you have 9 = k(1.5) or k = 9/1.5 = 6. Now your equation is V = 6*I. K is fixed at 6 for this circuit. If the current changes you can determine the new voltage from your equation.

It could be the case that n is some number other than 1. For exmaple if n = 4 you would have a relationship of the form y = x4. In this case if x = 2, y = k24. If y = 9 when x = 3 you have y=k*34 or 9 = k*34 = k*81. Solving for k gives k= 9/81 = 1/9. The resulting equation is y = (1/9)*x4.

Practice Question 1 - Suppose you are designing a rollercoaster. We will consider the first hill. The car is towed to the top of the first hill and stopped. The car is given a very slight push and goes down the hill. As a first approximation the velocity of the car (in ft/sec) at the bottom of the hill would be proportional to the square root of the height of the hill (in ft). (Answers at the end of this webpage)

  1. Write the equation that describes this situation using V for velocity, h for the height of the hill and k for the constant of proportionality.
  2. Given that the velocity at the bottom of the hill is 80 ft/sec when the hill is 100 ft high determine the value of k.
  3. If the velocity is 130 ft/sec how high is the hill?

Inversely Proportional

In the first quadrant the graph of an inversely proportional relationship can look like the figure to the right. The curve does not intersect either the x or the y axis. As x gets close to zero the curve grows without bound. As x gets very large the curve approaches the x axis (y value of zero). An example would be the force of gravity between any two particles.The magnitude of this force would be inversely proportional to the square of the distance between the particles. The equation would be F = k/d2.

Practice Queston 2 - Suppose you are driving to a destination that is some fixed distance from your home. The time to get from your home to your destination can be described as being inversely proportional to your speed (S).

  1. Write an equation that describes this situation using T for time, S for speed and k for the constant of proportionality.
  2. If your speed is 60 mph and the time to get to the destination is 2 hours what is the value of k?
  3. How long would the trip take if your speed was 70mph?

Rates and Rate of Change

Rates - Some information comes to us in the form of a rate. Other information is given as a rate of change. For example crime statistics may be quoted as the number of crimes per 100,000 people in an area. The number given is not the total number of crimes, rather it is the crime rate per 100,000 inhabitants of the area. This type of information can be used to compare areas with very different sized populations.

Rate of change - Information can also come to us as a rate of change. When you look at the speedometer in your car it gives your speed in miles per hour. In essence this tells you how many miles you travel in a given amount of time. This reflects the rate of change of your location, in this case the number of miles traveled in a given amount of time. As another example, when you start your car from a full stop during the first second your speed might go from 0 to 10 ft/sec, during the second second your speed might go from 10 to 25 ft/sec and during the third second from 25 to 30 ft/sec (the speed limit). The rate of change of your speed, called your acceleration could be calculated as follows:
TimeSpeedAcceleration
00----------
110(10-0)/(1-0)= 10
225(25-10)/(2-1)= 15
330(30-25)/((3-2)= 5

Let us next look at data presented as a figure.

The point here is that the rate of change of temperature changes for differnt time intervals. We define rate of change for a particular interval, in this case the intervals were related to time. In other problems they could be related to other variables.


Practice Question 3 - In 2002 Timmy the Turtle won the 50-inch race in the spectacular time of 23.1 seconds. The data shown below are the times recorded during this record-breaking performance. You are to calculate Timmy's speed in inches/second for each of the 10 inch segments of the race. Also determine the segment where Timmy hit his highest speed.
Time (sec)05.129.8714.4218.6323.1
Distance (inches)01020304050


Answers to questions

Question 1

  1. V = k h1/2
  2. k = V/h1/2; k = 80/10 = 8.
  3. 130 = 8 h1/2, h = (130/8)2 = 264 ft.

Question 2

  1. T = k / S
  2. 2 = k / 60, k = 2 (60) = 120
  3. T = 120 / 70 = 1.71 hours (rounded)

Question 3

  1. Segment 1, 0 to 10 inches: Speed = (10 - 0) / (5.12 - 0) = 1.95 in/sec
  2. Segment 2, 10 to 20 inches: Speed = (20 -10) / (9.87 - 5.12) = 2.10 in/sec
  3. Segment 3, 20 to 30 inches: Speed = (30 - 20) / (14.42 - 9.87) = 2.20 in/sec
  4. Segment 4, 30 to 40 inches: Speed = (40 - 30) / (18.63 - 14.42) = 2.37 in/sec
  5. Segment 5, 40 to 50 inches: Speed = (50 - 40) / (23.1 - 18.63) = 2.24 in/sec
  6. Timmy hit his highest speed between 30 and 40 inches. The speed was 2.37 inches/second.