Three sessions to make and report on the home lighting decision, and one more session to address the corporate lighting decision, including interest and present values.
- Experience with using ratios.
- Formula for the sum of a finite geometric series, or enough experience with polynomials to understand the development of the formula.
- Familiarity with the present value (PV) of an amount (A) to be paid in n years in the future under an assumed interest rate i.
- Formula for the sum of a finite geometric series, or enough experience with polynomials to understand the formula.
Ohio Standards Alignment
- (Optional) Organizer, such as a two-pocket folder
Personal finance, rates of interest, present value, adding finite geometric series
If you have been to the supermarket recently, you may have seen compact fluorescent light bulbs that can replace incandescent bulbs in ordinary household light sockets. The fluorescents are more expensive than incandescents but last longer and use less electricity. What would you need to know in order to decide whether to replace your incandescent light bulbs with the newer fluorescents? If you decide to replace them, should you do it all at once or gradually? Why? Now suppose that you are a financial analyst employed by a very large corporation. Your corporation borrows money as part of its standard operating procedures, and any money that could be saved by switching to the new type of bulbs decreases dollar for dollar the amount that must be borrowed. Examine the effect on the profits of the corporation of switching to the new type of bulbs.
What do you need to find out in order to decide? What is your conclusion? Why?
Should you replace the incandescent light bulbs in your home with the newer fluorescents that last longer but cost more?
In the first class session, pose the overall problem: If you have been to the supermarket recently, you may have seen compact fluorescent light bulbs that can replace incandescent bulbs in ordinary household light sockets. The fluorescents are more expensive than incandescents but last longer and use less electricity.
What would you need to know in order to decide whether to replace your incandescents with the newer fluorescents?
How many incandescent bulbs are in use in your house?
What are their wattages?
What wattage for a fluorescent bulb is needed to provide comparable lighting to a particular wattage of incandescent bulb?
Are there places in your house where fluorescent bulbs should not or cannot be used?
How much does an incandescent bulb cost?
How much does a replacement fluorescent bulb cost?
How long does an incandescent bulb last?
How long does a fluorescent bulb last?
How much electricity does an incandescent bulb use?
How much electricity does a fluorescent bulb use?
How much does electricity cost?
Students may pose other things they would like to know. Let them discuss as a group what needs to be researched.
Where can you find the answers to the questions that you want answered?
Try to have the students focus their attention on practical sources for answers. For example, a readily available source for the average unit cost of electricity might be an electric bill that their family receives, while an impractical source might be the rate department of the local electric utility company.
If the students have identified data for which they cannot think of a readily available source, can the problem be solved without using that data?
Now that required data and data sources have been identified, a decision has to be made whether to proceed by working in groups or individually. After that decision is made and groups, if needed, have been organized, students can do the research and prepare to report at session two.
At the second session, the students will find that they have not all found identical data. They will probably have priced different brands of bulbs, and even for the same brand, prices may vary from store to store. Further, the life of bulbs might depend on their rated voltage or on their wattage. And it is possible that not everyone pays the same rate for electricity.
What data should we use? How can we decide?
The main part of the second session should be devoted to a discussion of how to proceed with the analysis of the cost of using the two types of bulbs, so the students can work out the analysis individually or in groups. Finally, they should be assigned the analysis and asked to prepare a report that includes the costs and their decision on whether to replace the bulbs, and whether to replace them all at once or gradually, and reasons for their decisions.
At the third session, everyone reports and discusses the results.
During these three sessions, no mention has been made of interest and a comparison of expenses paid over a long time span as opposed to all at once. It is in order to put these additional considerations into a realistic setting so they can work on the long-term corporate example that includes taking out loans for corporate expenses.
You will need to decide whether to proceed with the corporate example (or a comparable example). If you proceed, you will have an opportunity to introduce interest and present values, and to compare the value of different streams of payments over periods of time. You will also be able to exploit the arithmetical formula for the sum of a finite geometric series as a tool in calculating the present value of the simple patterns of payments that will occur in the corporate light bulb situation.
At some point, even after the students have carried out the entire study, they may realize that they can determine the ratio of cost of using incandescents / cost of using fluorescents by just using the two ratios (1) price of incandescent bulb / price of replacement fluorescent bulb and (2) life of incandescent / life of fluorescent, and that the cost of electricity is not needed at all.
Comparing expenses paid over a long period of time as opposed to all up-front is the second main point of this problem. It uses the concepts of interest and present value, the same concepts that are used in credit, loan, and mortgage calculations. These are important concepts in all kinds of consumer and corporate finance. The importance of these calculations diminishes as the period of time under consideration shortens. It is quite possible that some students will conclude that these considerations are not worth the trouble in the context of the light bulb decisions. But if they want to reach that conclusion, they need to do the figures anyway to support the decision. It is in not being able to compare the costs that they could find themselves disadvantaged in future financial transactions.
The calculations for even the simplest parts of this problem are an opportunity to discuss the issue of significant digits. Generally, the number of significant digits in an answer cannot exceed the number of significant digits in any of numbers used to calculate the answer. There is no point is carrying out a calculation to many decimal points unless the inputs are that accurate, too.
- These compact fluorescents are a serious development. The Province of Ontario has just passed a law prohibiting the sale of “inefficient” incandescents by 2012. Please see the attached news release from the Ontario Ministry of Energy.
- Calculations of energy savings for both the residential and corporate scenarios can be found here.
Here is a commercial sales site for light bulbs.
This is a government information site on compact fluorescent bulbs.
And this government site addresses the concern about mercury.
From the teaching files of Robert B. Brown.