"Let There Be Light!"

### Preface

About ten years ago [1990] CAESS, the electric power company here in El Salvador (where WTM lives) prepared an informational brochure that explained to their customers how a home owner's light bill was computed each month. This was then distributed throughout its service area. When we received our copy, a light turned on (well, in our head anyway). And a classroom math lesson was born. We hope you like it.

Let There Be Light!
Part I

One of the most important monthly bills that every family must pay is the "light" bill (or electricity bill). And if your family is like most, the amount is often a great surprise. It just seems to go up and up all the time.

Of course, CAESS explains that the reasons for their higher charges are based on many factors: destruction of power lines, cost of operations, expansion of services, etc.

Below is CAESS's own chart showing how they compute your bill. Let's use it, and your calculator, to figure out some problems about monthly bills.

 RATE TABLE For the first 40 KWH each one costs ˘0.166 For the next 160 KWH each one costs ˘0.254 For the next 300 KWH each one costs ˘0.419 In excess of 500 KWH each one costs ˘0.738

### Sample Problem

José and Maria Sanchez have just recently moved into their new apartment. Since they've only been there for two weeks, they haven't had much time to use much electricity. So their bill said they had only consumed 256 KWH (kilowatt-hours). What should they have to pay?

Solution: Since they've used 56 KWH over the first 200 (40 + 160), their bill will be figured this way:

```	a)  40 × 0.166 =  6.64
b) 160 × 0.254 = 40.64
c)  56 × 0.419 = 23.464
Total = 70.744

So, their bill is ˘70.74.
```

Exercise Set I: Compute the amount a person would have to pay for these KWH figures.

```	1)  188 KWH		2)  349 KWH		3)  425 KWH
4)  580 KWH		5)  121 KWH		6)  1024 KWH
```

Let There Be Light!
Part II

On the left side of a receipt there are two numbers. They tell the current and previous readings of the electrical usage in your house, as indicated by the electricity meter that is outside your house.

Below are given some possible readings that might have been done by CAESS's meter reader.

Exercise Set II: First find the number of kilowatt-hours consumed, then compute the bill.

```	     Current   Previous     Total        Amount

1)    2455      2266      ________      ________
2)    4096      3725      ________      ________
3)    8112      7616      ________      ________
4)   11043      9648      ________      ________
5)   13954     13427      ________      ________
```

Let There Be Light!
Part III

The following "story problems" are examples of what you might expect to see in an algebra course. They are not exactly what you would see in "real life", so they could be considered as puzzle problems. But if you think carefully about how you worked the exercises in Sets I and II, you can do these as well.

Exercise Set III:

```  1)  Saul paid his light bill yesterday.  If he wrote a check for
˘101.75, what KWH number was on his bill?

2)  Sandra says that when she paid her electricity bill today, she
paid ˘20.95 more than did Saul.  If this was true, what was the
KWH value on her receipt?

3)  Mario went to the bank to make a deposit in his savings account,
while at the same time pay his light bill.  He says he'll save
gas and time if he does it that way.  He pays the bank teller
the sum of ˘78.29.  How many kilowatt-hours of electricity did
he use?

4)  Marina also decided to "kill two birds with one stone" when she
went to her savings and loan office to pay her mortgage payment
on her house as well as take care of some "lighter" business.
If she paid ˘114.74 to CAESS, how many KWHs did her bill show?

5)  Edgar paid ˘166.27 for his bill this month.  What number would
4853 KWH?

6)  Elena had a ˘191.43 payment to make last month.  What was the
was 9557 KWHJ.

7)  Forgetful Fred always forgets where he keeps his important
papers.  His wife is constantly "getting on his case" about
that (making his life miserable).  He could not locate the
June receipt, but knew that he had paid it because his check
book stub showed an amount of ˘209.88.  He did manage to locate
his May receipt, which gave a current readig of 7642 KWH.  Lucky
for you because you must tell me what the current reading on the
June receipt was.  Can you?

8)  Ask your parents if you may look at a recent light bill receipt
to check out the computation made by CAESS.

9)  Write up your own "light" story problem.  Be as interesting and
creative as you can.  Be sure you can solve it, too.
Then put it on a separate sheet of paper.  (Type it if
possible.)  But one more thing: use a fake name instead of your
own in the upper right corner of your paper.  This is because
some of the best ones will be used to form a quiz for you and

tt(8/7/90)
```

### More from the brochure...

CAESS also included a large table of information in the brochure. This was an attempt, a feeble one in my humble opinion, to show the consuming public how dramatically one's bill increases depending on the various appliances used around the house. This table is presented below; you are invited to study it carefuly. How many errors or examples of illogical reasoning can you find in it? It seems to me to have been prepared by someone who didn't understand mathematics very well. (Of course, I doubt very many individuals who received the pamphlet even bothered to examine it at all. Or they just meekly nodded their heads and thought, "Gee! How expensive things are becoming!")

 Typical Household Consumptionat current price rates Householdappliance Usekwh/mo. Accumulatedkwh Monthlycost Accumulatedcost Unit Cost 6 light bulbs60 w, 4 hr/da 54 54 10.20 . 0.30 per day iron1500 w, 6 hr/wk 36 90 9.14 . 0.23 per hr refrigerator250 w, full time 60 150 15.24 . 0.51 per day television250 w, 6 hr/da 45 195 11.43 . 0.06 per hr radio, fan,blender (mo.) 30 225 11.75 . 0.39 per day stereo150 w, 4 hr/da 18 243 7.54 . 0.06 per hr hair dryer1500 w, 10 hr/mo 15 258 6.29 . 6.29 per mo. Subtotal 1 258 258 71.58 71.58 2.39 per day stove, burners only5000 w 120 378 50.28 . 1.68 per day oven of stove4000 w, 10 hr/mo 32 410 13.41 . 0.56 per hr Subtotal 2 152 410 63.69 135.27 2.12 per day cistern motor1/2 HP 36 446 15.08 . 0.50 per day 2nd TV 30 476 12.57 . 0.07 per hr betamax 20 496 8.38 . 0.28 per day Subtotal 3 86 496 36.03 171.30 1.20 per day 6 additionallight bulbs 36 532 24.55 . 0.82 per day 2 reflectorsof 150 w 27 559 19.93 . 0.66 per day water heater1500 w, 3 hr/da 90 649 66.42 . 2.21 per day 3rd TV 24 673 17.71 . 0.15 per hour Subtotal 4 177 673 128.61 299.92 4.29 per day air conditioner18000 BTU10 hr/da 900 . 664.20 . 2.21 per hr TOTAL . 1573 . 964.12 .

### One year later...

As is so true in modern economies, prices usually go up in such businesses such as public utilities. This was true in this case anyway. Below is the rate table that came out about a year later.

It is certainly interesting to note two things here. (1) The public did not receive a brochure this time explaining the system or that a raise in rates even occurred. (2) Notice that in the first table the costs per KWH are "messy" 3-place decimal values, whereas in the second one they are more normal "money" style values. I have always wondered why this was done. But I never got around to asking that question.

 RATE TABLE For the first 40 KWH each one costs ˘0.20 For the next 160 KWH each one costs ˘0.35 For the next 300 KWH each one costs ˘0.60 In excess of 500 KWH each one costs ˘0.75

A couple of interesting questions naturally arise as one contemplates the two rate tables. (1) What are the percent increases for each of the four levels of the chart? And (2) what is the percent increase on sample bills when the amount to pay is computed using the old rates vs. the new rates?

I wonder what the current rates are now in 1999?