## SOLVING PROBLEMS WITH PROPORTIONS

 On this page, a systematic method for using proportions to solve mathematical problems will be presented. This method, while not the most mathematically elegant, will yield correct results when applied carefully. Let's start with a specific proportion to illustrate the technique. We will say that the quantity A is directly proportional to the quantity B. In symbols, A a B. This means A = k B, where k is a constant of proportionality. If we had initial values for A and B, we might label them with the subscript "1". A new set of values for A and B could be labeled with the subscript "2". We can now form a proportion by setting one equation above the other. In general, we choose to put the second equation above the first. Now we can substitute in the values we have and then go on to solve for the remaining unknown. On the next card, this process will be shown for a specific set of values.

 We know that A and B are related by a direct proportion. When B is equal to 5, A is equal to 7.5. What is the value of A when B is equal to 9? In this case we set A1 = 7.5, B1 = 5 and B2 = 9. This equation can be solved by cross-multiplying then dividing by five to obtain the answer 13.5.

 The volume of paint used (V) is directly proportional to the area (A) that can be covered. A 4-liter can of a certain paint will cover a wall space equal to 60 square meters. You calculate the wall space you need painted to be 150 square meters. How much paint should you purchase?

 The time taken to go from Gunn to Paly (T) is inversely proportional to the average speed (S) that one is able to maintain. On a given day, the average speed was 30 mph and the time taken was 10 minutes. However, on another day, the time was 15 minutes. What was the average speed on that day?

 The amount of data that can be recorded on a computer disk (D) is proportional to the square to the radius (R) of the disk. A 5-cm disk can hold 800 kB of data. How much data can be contained on an 8-cm disk?

 The frequency produced by a vibrating string (F) is proportional to the square root of the tension (T) placed on the wire. (Tension is how tightly the wire is stretched.) If a certain string on a guitar has a frequency of 240 Hz under a tension of 40 newtons, how much tension would be needed to achieve a frequency of 480 Hz?

 As could be seen in the previous examples, the process of setting up a proportion is relatively easy. 1. Write out the proportion 2. Make an equation by putting one proportion over the other. 3. Add subscripts to designate different sets of conditions. Use "1" to represent the first set, and "2" to represent the second. 4. Substitute the data you have into your equation. Keep units with the numbers. 5. Solve with basic algebra rules. 6. Identify the final answer, and be sure to use units. We will use proportional reasoning all year long. Use these web pages whenever you forget details on how to think about proportions. For more practice on using proportions, go to the self-test. Best wishes for a successful year of studying Physics!!

## SELF TEST

 The following section contains questions, with the solution found by clicking on the word "ANSWER". Work out your answer to the questions before checking the solutions. 1) The cost of 12 widgets is \$7.50. The cost is directly proportional to the number of widgets purchased. For a party you are giving, you need 52 widgets. How much will you have to pay for that many? (Ignore sales tax)

 1) The cost for 52 widgets is \$32.50 2) A farmer wishes to increase his income by raising a larger crop. He buys adjacent property, increasing the size of his holdings until each side has become three times larger (3X). How much crop can he raise on this new property, if the amount of crop is proportional to the area?

 2) The farmer can raise nine times as much crop (9X) 3) The amount of water that a pipe can handle depends upon the square of the diameter. A 2-cm diameter pipe can carry at most 40 liters of water per minute. If a new pipe with a diameter equal to 1.5 cm is installed, how much water can it carry?

 3) The new pipe can carry 22.5 liters per minute 4) The time it takes a rock to fall is proportional to the square root of the distance it falls. If a rock takes 2 seconds to fall 20 meters, how far will it fall in 1/2 second?

 4) The new distance is 1.25 meters 5) The mass of a disc cut out of aluminum plate is proportional to the square of the radius of the disc. If a disc with a radius equal to 15 cm has a mass of 1.80 kg, what will the radius be if you wanted a disk with a mass of 0.20 kg?

 5) The new radius is 5 cm 6) For a gas enclosed in a cylinder, the pressure is inversely proportional to the volume. At the beginning of our experiment, the pressure inside a cylinder was 10 psi (pounds per square inch). During the experiment, we reduced the volume from 45 ml to 25 ml. What was the pressure at the end of the experiment?

 6) The new pressure was 18 psi This is the end of the self-check. Hopefully you were successful in answering all of the questions. If you need additional help, see your teacher. This will be a very important skill to have mastered by the time tests and quizes roll around.

Updated August 2001