Monday, May 12, 2008

A Cab ride in Chicago

Cab fares in Chicago work like this: when the ride starts, the driver drops the flag on the meter. An immediate charge of $2.25 appears. This charge covers the first 1/9 of a mile. For each additional 1/9 of a mile, the meter adds 20 cents. Hence, for miles after the first, the charge is $1.80 per mile. (What is posted in the taxi is: "$2.25 flag drop, $1.80 each additional mile.")

The distance from your hotel to The Art Institute of Chicago is 1.38 miles. You could walk that far in about 20 minutes, but it is raining and you decide to take a cab. What is the meter charge when you arrive?

Hints: for the first mile, you are charged $2.25 plus 20 cents for each of the remaining eight ninths of a mile. You can then convert 1/9, 2/9 etc. into decimals and find how many ninths you have been charged for when you reach 1.38 miles.

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Answer: IV dollars and XLV cents.

Friday, May 9, 2008

How Many Miles Per Year Do I Drive?

I first drove my car on August 30, 2006. I suspect that I drive less than 10 000 miles per year. I notified my insurance company of this and they reduced the price of my car insurance.

The insurance company advised me to check how many miles I drove from time to time.

On March 8, 2008 the car told me that I had driven 15 000 miles and I needed an oil change. About how many miles do I drive per year?

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Hints: this is an estimation problem. You could, for example, assume that a year has 12 months, each of 30 days, and even assume that the the 15 000 mile notice appeared on April 30, 2008. Then get the miles per month and multiply by 12 to get miles per year.

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One approximate answer: DCCL miles per month times XII.

Monday, April 28, 2008

A Tax Question

The school district I live in just had a vote to increase taxes to pay for better schools.

Before I get to that, consider a simple example. Suppose 20 people vote yes and 30 people vote no.

What were the respective percentages? You need to get the total first! In total, 20 plus 30, or 50 people, voted. The percentages were 40% "yes" and 60% "no."

In real life, 4702 people voted "yes" to increase taxes and 3554 people voted "no" to the tax increase. What were the percentages of yes and no votes?

There was a second question on the ballot, to increase taxes even more to cover improvements in education and to cover inflation. To this, 4092 voted "yes" and 4150 voted "no." What were the percentages of yes and no votes?

Was the vote on the second question close? Did the same total number of people vote on both questions?

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Note: V=5, X=10, and L=50. In Roman numerals, if a single smaller unit precedes a larger unit, you subtract. So XL=40.

Vote totals: First vote, 8 thousand + CCLVI; second vote 8 thousand + CCXLII
Percentages: rounded to nearest whole percent, Yes LVII per centum, No XLIII per centum.
Second vote: Yes L per centum, No L per centum, but No won by LVIII votes out of about 8 thousand cast; the Romans didn't use decimals but the vote lost by about VII tenths of one percent.



Hint: this is a favorite question on standardized tests. The test doesn't tell you to, but you must add the votes to get the total first (!) then calculate the percentages.

posted by Stephen Margolis @ 6:15 AM

Another Taxi Story

In the Twin Cities, taxi fare works like this: as soon as the passenger gets in, the driver drops the flag and $2.50 appears on the meter. Every mile, including the first mile, costs an additional $1.90. Fractions of a mile are 38 cents for every 1/5 of a mile.

My car wouldn't start and I needed to take an urgent trip. Using Mapquest, I found the trip was 15.6 miles long. What was the likely fare, including a 20% tip? Round the fare plus tip to the nearest dime (10 cents).

I found out in advance that the taxi company accepts credit cards. But, if not, could I have paid with $40 in cash?

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Answer: XXXVIII dollars and LX cents. I could have paid with two twenty-dollar bills.

Note: V=5, X=10, and L=50.

Thursday, November 8, 2007

Over-the-counter medicine costs

I take a number of vitamins and over-the-counter medicines.

At the end of the year, I get an accounting of all the prescription medicines I have taken and their cost, but I have to figure out the cost of the over-the-counter medicines myself. I give these figures to my accountant (Cindy, M.B.A.) and she uses them to figure part of my income tax form.

I take one per day of the following medicines:

Aspirin, 81 milligrams
Multivitamin
Vitamin C
Vitamin E
Flaxseed Oil

I take two per day of the following medicines:

Glucosamine-chondroitin complex
Fiber Choice

The prices of these pills are:

Aspirin, 500 tablets for $12.99
Multivitamin, 300 for $13.99
Vitamin C, 400 tablets for $5.99
Vitamin E, 400 tablets for $14.99 (the price is higher because the vitamin is packed in a softgel)
Flaxseed oil, 100 for 5.99

Glucosamine-chondroitin complex, 240 pills for $31.05 (remember: 120 day supply)
Fiber Choice, 90 tablets for 9.89 (remember, 45 day supply)

All of these are subject to 7.15% Minneapolis sales tax.

What is the total cost of these medicines for a 365 day year? For example, Flaxseed oil tablets are about 6 cents each, but I need 365 of them.

Tuesday, November 6, 2007

What the heck is a kWh?, continued.

United States money is in dollars and cents. But electric bills use decimal fractions of a cent.

When you see a gasoline price, it might be 2.789 . That is two dollars, 7 dimes (tens of cents) and 8 cents. But what is that little nine? It is nine mils, each mil is a tenth of a cent. On a gasoline price, it is there to fool people, the real price is rounded to the nearest cent, and is really $2.79.

But, electricity prices might be $0.067550 for one kWh (kilowatt hour). That is 6 cents, 7 mills, 5 hundredths of a cent and 5 thousandths of a cent. The price is worked out before the Public Utilities Commission (PUC) of each state. The utility, such as Xcel Energy, requests a rate and the PUC either accepts it or suggests a lower rate. Eventually, a decision is made.

In Minnesota, the summer rate (about 6 cents and 8 mils) is higher than the winter rate (about 5 cents and 8 mils). The reason: supply and demand. In Minnesota, winter demand is only 3/4 of summer demand.

If you multiply summer use by summer rate and winter use by winter rate and round each to the nearest cent, you get $11.46 and $20.69; add these to get $32.15 .

Even if I used no electricity at all, Xcel would need to keep up the wires and equipment to supply me. They charge me $6 per month as my share of this upkeep; it is called "Basic Service Charge."

I used 526 kWh during the month. They charge me $0.020798 (about 2 cents and 1 mil) extra for this, calling it a "Fuel Cost Charge." Presumably, Xcel negotiated this charge with the PUC to cover increases in the cost of fuel. It adds $10.94 to my bill.

On the same 526 kWh, they charge $0.0021 for an "Environment Improvement Rider." My bill does not explain this charge, which is about 2 mils per kWh. It adds $1.11 to my bill. If I find out more about it, I will post an explanation to this blog.

Finally, there is a "Resource Adjustment" of $1.11 . Believe it or not, this is a charge to cover telling me to conserve energy by using fluorescent lamps, etc. The Legislature requires Xcel to tell me this, Xcel does so, then charges me for telling me. Every year, the Legislature approves this procedure.

The sum of all this is $51.30. The state of Minnesota then charges 6% sales tax and when the tax is added, that is my total bill. There are seven (7) separate charges on my bill.


About a comet and magnitudes

The story of my electric bill will be continued in the next posting.

This is about a comet. The math has to do with exponents and powers. If you haven't yet learned about exponents and powers (you will), ask a teacher or post a comment. Part of a news story follows:

In terms of actual measured brightness, 17P has moved from a magnitude of 17, which is visible only through a large telescope, to somewhere just above magnitude 2. That's an increase of about 1 million times, Herbst said, and it makes the comet visible to the naked eye even in urban settings where the night sky is obscured by man-made light.

If B is brightness and M is magnitude, B = (base)^-M ; B is (base) to the minus M power. The number (base) is defined by (base)^5 = 100. The number (base) is about 2.511. Since the magnitude changed from 17 to about 2, a change of about 15, and 15/5 = 3, the brightness increased by 100 cubed, or 100 x 100 x 100, or about 1 million. All the "abouts" in the last sentence are because different observers have made different estimates of the brightness now. They think it is about 2nd magnitude; they are comparing it to other stars. Stars of first magnitude are bright, stars of 2nd magnitude less bright, stars of 3rd magnitude still less bright, and so on.

The comet was first seen on November 6 1892. It comes near Earth every 6.88 years. When did it last come near Earth (and nobody payed attention since it was 17th magnitude)? When will it next come near Earth?

This is the whole news story:

dailypress.com/news/dp-news_cometholmes_1105nov05,0,5662681.story?coll=dp-wid

dailypress.com

Stargazers aglow over comet's stunning visit

A usually dull passer-by named Holmes has local astronomers staying up late to get a celestial eyeful.

By MARK ST. JOHN ERICKSON | 247-4783

November 5, 2007

For more than a century, the comet known as 17P/Holmes has been too shy to shine. It would slip past Earth every seven years with a light so dim, the naked eye couldn't see it. Even astronomers largely ignored its fainter-than-faint, reliably uneventful visits.

But when an observer in the Canary Islands looked up into the northeast sky early on the morning of Oct. 24, an object where the dark comet should have been beamed so brightly that he couldn't explain it. Minutes later, the same strange object was spotted from Barcelona, Spain, by another equally bewildered stargazer.

In less than 24 hours, Comet 17P/Holmes had intensified so much that its once dark mass resembled a star, prompting a rash of excited sightings across the planet. Then — over the following few days — its brightness multiplied by a million times, making it not only visible to the unaided eye but also one of the most prominent objects in the night sky.

"This is the first naked-eye comet since Hale-Bopp in 1997 — so we're really pretty excited," said astronomy curator Kelly Ann Herbst of the Virginia Living Museum.

"Even without binoculars, you can see that this is a pretty big thing. It's immediately visible. And with a telescope, you can see not only the nucleus but also the details of the coma — this beautiful halo that ripples and changes over time."

Often referred to by astronomers as "17P" or "Comet Holmes," the celestial object's sudden outburst during the past couple of weeks echoes almost exactly its discovery on Nov. 6, 1892. Scanning the night sky, amateur English astronomer Edwin Holmes was so startled by its unexpected appearance that he scared his wife, who heard his exclamation from the next room.

After fading somewhat in the weeks that followed, the comet erupted again on Jan. 16. Not long afterward, it fell off the radar screen almost completely until flaring up again.

"Just like now, everybody was amazed — and they hoped to see it again," Herbst said. "But nobody could find it when they thought it should be coming around. This is only the second time the comet has had a period of outburst like this."

In terms of actual measured brightness, 17P has moved from a magnitude of 17, which is visible only through a large telescope, to somewhere just above magnitude 2. That's an increase of about 1 million times, Herbst said, and it makes the comet visible to the naked eye even in urban settings where the night sky is obscured by man-made light.

The comet also appears much bigger than it normally does, vaulting from imperceptible at first to the same size as Jupiter. In recent days, it's surpassed the giant planet, too, giving Herbst and many other night observers in the local area something totally unexpected to see.

Lawrence Taylor, a NASA Langley atmospheric scientist who's a member of the Virginia Peninsula Astronomy/Stargazers group, could barely wait for the trick-or-treaters to leave his Hampton home Halloween night so he could take a quick peek into the heavens.

So impressive was the sight that he ended up staying outside until 1 a.m. — all the while shuttling larger and better pieces of equipment out into his yard.

"I was just going to take a quick look with the binoculars, but then I saw this huge thing — a great big fuzzy ball," he said. "The fact that you could see it with the naked eye — even with a quarter moon — is pretty amazing. It's just awesome — and it could be up there a couple of weeks."

Just how long 17P's surprising outburst will last remains to be determined. But right now, its unexpected behavior can be seen not long after dusk — and at an unusually favorable time of the lunar cycle.

"The moon is waning now, so we'll have some really dark nights to see the sun reflecting off its dust," Herbst says. "It's really just perfect."

Both the Living Museum and the Peninsula Stargazers plan to stage numerous public observing sessions over the next few weeks, including such relatively dark and undisturbed viewing locations as York River State Park in James City County and Grundland Park in the Grandview section of Hampton.

But even in Newport News, where Herbst has been following Comet Holmes with the museum's big 16-inch-diameter telescope, the sight of the large golden ball rising in the northeast sky has been compelling.

"You can't see the tail because it's extended out behind it. But it's still really quite spectacular," Herbst said. "It has this huge envelope of dust and gas surrounding the nucleus — and it's so big that it almost fills the field of vision on our largest telescope."