Wikipedia:Reference desk/Archives/Mathematics/2008 September 15

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September 15

having problems with conversion of measurements

somethings i can do but this is not one of them. here are some examples.15 meters to millimters?3.5 tons to pounds?6800 seconds to hours?could someone please help me understand how you change these? —Preceding unsigned comment added by 71.185.146.144 (talk) 09:14, 15 September 2008 (UTC)

I put them in google and it does them for me...

http://www.google.co.uk/search?hl=en&q=15+meters+in+millimeters&meta= http://www.google.co.uk/search?hl=en&q=3.5+tonnes+in+pounds&meta= http://www.google.co.uk/search?hl=en&q=6800+seconds+in+hours&meta= Simple as that. Of course without google things get a bit harder... 194.221.133.226 (talk) 10:35, 15 September 2008 (UTC)

To change between units you need to know how many of the one unit are in the other unit. Then you multiply by how many of the first unit you have. Say that there are 5 "apples" in a "bag" and you have 2.4 bags, then you have 2.4 * 5 = 12 apples. To go the other way, there is 1/5 = 0.2 bags per apple, so 9 apples is 9 * 0.2 = 1.8 bags. So all you need to know is how many millimetres in one metre, how many pounds in one ton, seconds in an hour etc. If you don't know these from memory then the relevant wikipedia articles will help you. -- SGBailey (talk) 11:28, 15 September 2008 (UTC)

Annual Percentage Rate

So I am in a confusion about what “APR” is. In particular, in the simplest case (ignoring everything but the principal) whether it is a nominal or effective rate.

The article annual percentage rate claims it is an effective rate. Through some online searching I’ve found conflicting sites that seem to say opposite things. Eventually I dug out my old theory of interest book (Theory of interest by Kellison) which said it was a nominal rate. Asking a friend, she pulled out hers (Mathematics of interest rate and finance by Guthrie and Lemon), which also says it is a nominal rate.

So I just wanted to check with anyone that might know more about this than me to see if the article is actually wrong, before changing anything (or telling the students in 101 my class something incorrect next week!) GromXXVII (talk) 12:53, 15 September 2008 (UTC)

I'm not going by the article, I have a Ba in finance, and graduated with high honors from the best college in the nation. The APR is not an effective rate. Here's two examples to clear things up.

A $100 bond pays a $3 coupon every 6 months.

The APR is 6.000%

The effective interest rate on the bond is 6.090%

A $100 bond pays a $4 coupon, every 12 months

The APR is 4.000%

The effective interest rate on the bond is 4.000%

I'll answer any further questions, and will provide relevant sources, at your request. The following example may be confusing, but here it goes anyway.

A credit card has a 29.9% APR

If I charged $200, and pay off $100 of it, then I'll have a $100 balance remaining. When my new bill comes in, I'll owe $102.492

The effective interest rate is 34.358%, and if you figure out how I computed this, you'll have came away with full understanding, as every assumption is reflected. Sentriclecub (talk) 13:11, 15 September 2008 (UTC)

Surely a $100 bond that pays $4 each 12 months is not a 6% rate? The credit card example is also flawed, in that many cards compute interest based on an average daily balance, thus not knowing when the partial payment was made makes it impossible to computer the next month's finance charge. Offering sources "upon request" is redundant, the request is implicit in the use of the reference desk. --LarryMac | Talk 13:24, 15 September 2008 (UTC)

Thanks fixed it, ask me how to derive a physics formula, no problem. Trying to multiply 4x1, that's a challenge. The credit card example isn't flawed, if you reread the stipulation that I started the month with a $100 remainder balance and made no new charges. The reason I don't want to offer sources, is to avoid the conundrum that its a real possibility that some sources may calculuate an EAR and APR in different ways. If I gave a source, I would be forced to say that the wikipedia article is wrong, which I don't believe. I would have given the source right away, but I want to compare my author's accuracy to the wikipedia article's accuracy. Secondly, I will point out that you are correct about when the payment is applied, however the example implies in my favor. Since I'm providing clearification, I don't want to give more information than necessary, but I also agree on your point that there is risk resulting with not specificating every detail such as was it februrary on or off leap year, or was it a month with 31 days, etc... Upon further inspection, the credit card example is not flawed. The rate per period is APR/n. LarryMac, may I respectfully ask if you have studied this stuff in school? Something is intriguing me, a subtle mistake you've made in your post. I take it you are a math expert in another area? I don't want to come across bad, I'm representing my school now, and I hope my behavior today is helpful and respectful. I'm 24, and one of the worst feelings I ever had on wikipedia was upsetting someone at the ref desk. So please be patient with me, and my only intention here today is to help answer this guy's question, while Primum non nocere to the other helpers. Sentriclecub (talk) 14:02, 15 September 2008 (UTC)

I understand the effective interest rate in your credit card example being 34.358%=(1+0.299/12)^12. But I’m still a little confused in that APR then appears to be a nominal interest rate (in that I could divide by 12 to get the monthly interest charged of 2.49%), but you said the Wikipedia article isn’t wrong? The first line is “Annual percentage rate (APR) is an expression of the effective interest rate” which seems to say the opposite. (as far as I know it’s not possible to be both an effective and nominal rate unless if the frequency of compounding is the period). GromXXVII (talk) 23:02, 15 September 2008 (UTC)

Yeah, I discussed that earlier over at a discussion page somewhere. Here is the quote--The first sentence is already prone to having people miseducated. Annual percentage rate (APR) is an expression of the effective interest rate the borrower will pay on a loan which is accurate, but completely misleading, because the APR is not an effective interest rate. It is an expression for one, but not one itself. -/quote I spoke to some of the editors in finance articles, and they're okay with me saying that the line is wrong. I just didn't want to offend anyone over there. APR is a nominal rate, (and is an effective rate if and only if n=1). I am going to work with Wikipedia:FINANCE to help fix inaccurate statements in existing prominent articles. The finance articles on wikipedia are in poor shape (maybe 5 or so major ones are rife with errors) so I'm going to volunteer some time over there, even though I'm going to med-school and leaving finance behind me. It's where I'm needed the most, so I'll try and make myself compare articles to my textbooks and rule in favor of my book. At least that way, APR and the articles which link to APR will all be consistent. If someone else wants to use a source that APR is an effective interest rate, they can come behind me--I dont care. But as it stands now nominal interest rate, effective interest rate, real interest rate and annual percentage rate contradict, so I already have the permission from the wikiproject over there, to do a mass clean-thru of the pages. My 5 finance professors and all of their textbooks (5+5=10) are all consistent with each other, and I'll source all my texts, so in a few days, the articles will be clear and those negative tags will come off the tops of all 6 pages that I plan to work on. Sentriclecub (talk) 00:04, 16 September 2008 (UTC)

Ahhh. I see. I was erroneously taking “is an expression of” to mean “is”. You have my thanks in advance for any clarifications you’re able to make on those pages. GromXXVII (talk) 00:17, 16 September 2008 (UTC)

addendum 1

APR = (Per-period rate) X (Periods per year)

EAR = $e^{APR}-1\,$

APR = $ln(1+EPR)\,$

I'm going to look more into all this for you, and will have a full report back in an hour--it looks as though neither the APR nor EAR are independent of the number of compounding periods. The best way to interpret interest rates is with calculus. It appears that the APR and EAR are dependent on the number of periods per year, which bothers me. I'll re-read everything, and will also consult my HP-10Bii user manual, 2 feet away from me at all times. While you are waiting, check out Khan academy on youtube, this guy is a bona fide expert. The finance videos he has produced are wonderful. Sentriclecub (talk) 13:35, 15 September 2008 (UTC)

To clarify, the EAR is the Effective Annual Rate and in the formula above, the APR is the force of interest or the continuously compounded rate of interest. Zain Ebrahim (talk) 13:43, 15 September 2008 (UTC)

It's both nominal and effective. The opposite of nominal is real. See nominal interest rate, real interest rate and effective interest rate. --Tango (talk) 12:59, 15 September 2008 (UTC)

From reading the above definitions I would say it is always effective but only nominal if the interest period is annual (once a year). -- Q Chris (talk) 13:09, 15 September 2008 (UTC)

For this purpose, you need to look at the second definition at nominal interest rate, in which case a rate can't be nominal and effective. When I studied financial maths, the lecturers intentionally avoided the term APR because it changes from person to person. Zain Ebrahim (talk) 13:12, 15 September 2008 (UTC)

Actually, it depends where you are in the world - see annual percentage rate#Region-specific details. In the US, APR is defined as periodic interest rate times the number of compounding periods in a year, so it is a nominal interest rate. In the UK, however, APR is the effective interest rate, which factors in compounding and is not the same as the nominal rate unless the interest is only compounded once a year (in the US the effective rate is called the Annual Percentage Yield or APY - see this Investopedia article). So a loan with monthly interest payments of 0.5% per month would be described in the US as having an APR of 6%, but in the UK its APR would be 6.17% (this example is used in the nominal interest rate article). As in many instances of financial terminology, US and UK just follow different rules. Gandalf61 (talk) 13:27, 15 September 2008 (UTC)

Here are some sentences straight out of my book. Returns on assets with regular cash flows, such as mortgages and bonds, usually are quoated as an APR. The APR can be translated to an EAR by remembering that APR = (Per-period rate)x(Periods per year). Therefore to obtain the EAR if there are n compounding periods in the year, we solve the equation $1+EAR=(1+{\frac {APR}{n}})^{n}$. Sentriclecub (talk) 14:09, 15 September 2008 (UTC)

So for n large, 1+EAR is approximately e^APR. Algebraist 14:23, 15 September 2008 (UTC)

Here are some sentences straight out of my book. Returns on assets with regular cash flows, such as mortgages and bonds, usually are quoated as an APR. The APR can be translated to an EAR by remembering that APR = (Per-period rate)x(Periods per year). Therefore to obtain the EAR if there are n compounding periods in the year, we solve the equation $1+EAR=(1+{\frac {APR}{n}})^{n}$.

Additionally, I'm glad to see the point made that different sources say different things. I personally think any interest rate which is dependent on the number of periods per year, is a flawed measure. So I think with this information, we can call it a closed case. Finance has a way of trying to simplify formulas which are inherently exponential into algebraic. This is why the APR is probably best described as a nominal rate, not as an effective rate. Since the APR is usually divided into 12 periods per year (afterall, people make mortgage payments, and credit card payments once per month, not once per 30 days) so its good for consumers ultimately. Afterall, what if a homeowner took out a 8% home equity loan to pay a 9% credit card, and one of the APR's was calculuated assuming 12 periods per year, and the other APR was confounded and ended up losing EV on the deal. So with this reasoning, (and in my education, I studied the interest rate schemes which lead to federal standards--probably in the APR article) therefore I would argue that the APR is a nominal interest rate, since the reason the government makes Rent-a-Center publish an APR is so that consumers know they can compare it apples to apples like a mortgage payment, which they are familiar with.

Okay, even further into the philosophy of all this, I have finally concluded that the APR is a number to help consumers from getting duped. Say someone goes to rent-a-center and buys $800 rims, for their car and finances it. That guy can ask virtually anyone if 18% APR is a good deal, or not. He's not actually interested in his monthly payment, or in computing the principle to interest ratio of each minimum monthly payment. He just wants to know if the interest rate, if he's getting ripped off on the financing part of the purchase. In other words, I conjecture that an APR is a way for consumers to know if they are being taken advantage of, under the financing terms. As an anecdote, I heard with my own ears, a Ron Legrand audio lecture, on how to purchase a stream of cash flows from a landlord. Suppose the mark goes up to the landlord (who is a landlord, uses seller-financed a mortgage to the buyer) and if the principle on the mortgage is worth $160,000, make the landowner an offer to buy all the cash flows from now until when the principle on the mortgage is $120,000. You can play around with some of these numbers, but the mark ends up "buying" the $40,000 difference at a negotiated price. These are similar to the Carlton Sheets real estate courses. You'd be amazed at how much profit this comes out to be for the mark. So, I believe that APR is a nominal measure, which most consumers in america know that the default rate on a credit card is extremely undesirable. How do they know that a 30% APR is undesirable? Because APR is a nominal number which is widespread (all mortgages and credit cards I believe are federally required to report APR) so that consumers don't have to take a finance course to understand that banks borrow money at a very low rate, around 4%, and that people with good credit can borrow at around 8%, and that you should only finance at 18% APR, rims which cost $800 if you wish to take on a undesirable interest rate. Therefore, you should only buy them if you make enough utility on the $800 price, to justify the exorbitant interest rate. This would be the case if the rims were of some special case of tremendous value, like maybe a clearance sale. My closing point, which I hope I've made, is that you should ask your self where APR's came from? What is the history of APR's. They are designed to help consumers understand (through the easy to understand simple interest method) a complex formula. Most consumers aren't good at calculating exponents and natural logarithms, but almost anyone that can get a loan, understands how to divide by 12 and multiply by 12. Thus APR's are understood by the majority of consumers. Consumers don't understand EAR in the same proportion as those that would understand simple interest. Therefore the APR which is most helpful, is the one most everyone can understand. APR is nominal. To calculuate the interest for one month, you divide the APR by 12. This is within the abilities of the largest proportion of consumers. Then again, I also question if credit card lobbyists may have been behind this whole enigma. Afterall, it lets them quote an APR of 29.900, but if this value were computed using exponents and non-simple-interest, they would have to report 34.358. But I think its that the greatest proportion consumers understand simple interest. The government (that regulates how consumers are told the interest rate) will guide corporations to quote an interest rate which people will understand. Sentriclecub (talk) 14:42, 15 September 2008 (UTC) Also, I went back and read the APR article and the third sentence is APR is intended to make it easier to compare lenders and loan options. which summarizes my point.

combinatorics question

A bridge club has ten members. Four get together to play everyday. Show that in two years one group of four people gets together at least four times.

The only thing that I can think of is that in the first 211 days one group definitely gets together twice. Any other ideas?--Shahab (talk) 19:03, 15 September 2008 (UTC)

Your argument uses the basic version of the pigeonhole principle, but there's also a generalized version, pointing out that since 3×210 = 630 days is the longest possible period of time without any group playing more than three times, in 631 days some group must play four times. -- Jao (talk) 19:36, 15 September 2008 (UTC)

Thank you. Here's another problem which is troubling me. I can't figure out how to apply the pigeonhole principle here: Prove if the numbers 1, 2, 3, . . . 12 are randomly placed around a circle, there must be three consecutive numbers whose sum is at least 19.--Shahab (talk) 07:00, 16 September 2008 (UTC)

Going round the circle, there are 12 sums of three consecutive numbers. You can work out the total of these 12 sums (each number on the circle is counted three times, in three different sums), so you can work out the average value of these 12 sums. The 12 sums cannot all be less than their average, so at least one sum must be greater than or equal to their average. Gandalf61 (talk) 09:23, 16 September 2008 (UTC)

As far as I understand this the total of the 12 sums must be 234, which when divided by 12 would give 19.5 proving more then asked for. Clearly this is an elegant solution. Thanks. But I want to explicitly involve the the pigeonhole principle in the solution coz the problem occurs in that section in my book. (Another way that I thought of was that if each of the sums is less then or equal to 18 then their total can be at most 12 times 18 i.e. 216, contradicting the fact that the total is 234).--Shahab (talk) 09:46, 16 September 2008 (UTC)

Divide the circle into groups of three. If each number occurs once there will be 4 such groups. By the generalized pigeonhole principle if none of these groups sum to more than 18 then the whole circle can not sum to more than 18*4=72, but 1+2+...+12=78.Taemyr (talk) 14:21, 16 September 2008 (UTC)

Another approach, less combinatoric but more direct: try to create an arrangement in which each sum of 3 consecutive numbers is 18 or less. Note that no two out of 9, 10, 11 and 12 can be in the same sum, so no two of these four can be neighbours or have just one number between them. So your arrangement must be a cyclic permutation of abbabbabbabb where the as are 9, 10, 11 and 12. Now show that there is nowhere left to put 8 without creating a sum of 19 or greater. Gandalf61 (talk) 09:33, 16 September 2008 (UTC)

Population and area question

Population and area question

How many times greater is the population of China than Australia? Secondly, how many times greater is the land area of China than Australia? —Preceding unsigned comment added by 81.151.147.129 (talk) 19:17, 15 September 2008 (UTC)

You can certainly find the relevant figures in our articles on Australia and People's Republic of China. -- Coneslayer (talk) 19:29, 15 September 2008 (UTC)

Word Problem- how to setting up the equations

Two pipes feed into a tank. The large pipe can fill the tank in 2 hours, the smaller pipe can fill the tank in 6 hours. If both pipes are used together, how long would it take to fill the tank.

Thank you for your help. —Preceding unsigned comment added by Evansranch (talk • contribs) 20:20, 15 September 2008 (UTC)

A good way to start is by answering the question: "It would take x hours." Then think about it: how many tanks will the large pipe fill in x hours? How many tanks will the small pipe fill? How many will they fill together? -- Jao (talk) 20:29, 15 September 2008 (UTC)

Per hour, the large pipe fills half the tank, the small one one sixth of it. So what fraction together? How many hours to make this fraction into one, i.e. a full tank?—86.148.186.156 (talk) 21:42, 16 September 2008 (UTC)

In one hour, the 'faster' pipe can fill 1/2 of the tank and the 'slower' pipe can fill 1/6 of the tank. (1/2 + 1/6) = 4/6 = 2/3 of the tank. Suppose it takes 'x' hours to fill the tank. Then 2x/3 = 1 which implies that x = 3/2 hours. Therefore the tank is filled in 1 hour and 30 minutes.

Topology Expert (talk) 13:15, 17 September 2008 (UTC)

We usually avoid actually give the answer to questions like this. It's much better to guide the OP towards getting the answer themselves than just giving it. --Tango (talk) 17:29, 17 September 2008 (UTC)

Combinations

A question I am working on reads:

A bag contains 20 chocolates, 15 toffees and 12 peppermints. If three sweets are chosen at random what is the probability that they are:

a) All different,

b) All chocolates,

c) All the same,

d) All not chocolates?

I have calculated the answer to a, to be 0.222 (which is correct), using the following steps:

47C3 = 16215

Chocolates * Toffees * Peppermints = 3600

3600/16215 = 0.222

Firstly, while I know this works, I do not fully understand why, and I am completely stuck on all the other sections. Can anyone point me in the right direction? Thanks. --AFairfax (talk) 20:39, 15 September 2008 (UTC)

What you've done is calculate the number of possible choices of three sweets, that's the 47C3, and the number of choices satisfying the condition that they all be different, the 3600. The reason you can just multiply the number of each type together is because you know there must be one chocolate, one toffee and one peppermint, so the only choice is in which of the chocolates, which of the toffees and which of the peppermints your choose. You then divide one number by the other to get the probability. Does that make any sense? The other questions are answered the same way - you need to calculate how many different ways you can satisfy each condition. So for (b), you need to work out how many ways you can choose 3 chocolates from a bag containing 20 chocolates (you can ignore the other sweets since they aren't allowed) - that's just 20C3. I'll let you try (c) and (d). Good luck! --Tango (talk) 21:11, 15 September 2008 (UTC)

The annoying thing is I already knew all that really, just didn't put it into practice in the correct way. Thanks for the pointer. --AFairfax (talk) 22:17, 15 September 2008 (UTC)

There are definitely 16215 ways of drawing three sweets and 3600 ways of drawing three sweets of different kinds. But when you divide the one by the other to get a probability you're introducing a new assumption: that all 16215 draws are equally likely. It's a standard assumption in word problems like this, and it's probably meant to be implied by the phrase "chosen at random", but it would be tricky to achieve in reality with a heterogeneous collection like this one. Shaking the bag and then letting one candy fall out would probably favor some types over others by a significant margin. So always think carefully before you treat a ratio of integers as a probability. -- BenRG (talk) 23:55, 15 September 2008 (UTC)

(edit conflict)

$20\cdot 15\cdot 12/{\tbinom {47}{3}}=20\cdot 15\cdot 12\cdot {\frac {3!44!}{47!}}=20\cdot 15\cdot 12\cdot {\frac {3!}{45\cdot 46\cdot 47}}={\frac {20}{47}}\cdot {\frac {15}{46}}\cdot {\frac {12}{45}}\cdot 3!$

Explanation of the derived formula: First we choose among 20 chocolates out of 47 sweets, second we choose among 15 toffees out of 46 remaining sweets, third we choose among 12 peppermints out of 45 remaining sweets. 3! is the number of possible orders of selections (chocolates, toffees, peppermints; or chocolates, peppermints, toffees; or toffees, peppermints, chocolates etc.). According to mathematics of outs in Texas Hold'em, b) is determined by formula ${\frac {20}{47}}\cdot {\frac {19}{46}}\cdot {\frac {18}{45}}=0.070$.

Similarly, c) ${\frac {20}{47}}\cdot {\frac {19}{46}}\cdot {\frac {18}{45}}+{\frac {15}{47}}\cdot {\frac {14}{46}}\cdot {\frac {13}{45}}+{\frac {12}{47}}\cdot {\frac {11}{46}}\cdot {\frac {10}{45}}=0.112$

And d) ${\frac {47-20}{47}}\cdot {\frac {46-20}{46}}\cdot {\frac {45-20}{45}}=0.180$

--Admiral Norton ^(talk) 21:19, 15 September 2008 (UTC)

This is a Hypergeometric distribution.

Topology Expert (talk) 13:10, 17 September 2008 (UTC)

ms^-3

how would you describe the concept of m/s/s/s? 84.13.30.238 (talk) 20:41, 15 September 2008 (UTC)

• Jerk. --PaulTaylor (talk) 21:00, 15 September 2008 (UTC)

  Nitpick: The units of jerk. Zain Ebrahim (talk) 22:02, 15 September 2008 (UTC)

Thanx Paul and Zain! I was just curious... thnx again ATMarsden^{Talk · {Semi-Retired}} 15:50, 16 September 2008 (UTC)

m/s/s means the change in m/s per second; or the change in velocity per second or acceleration. Therefore, m/s/s/s means the change in m/s/s per second; or the change in acceleration per second. This has no particular name but the third derivative of a function represents this. —Preceding unsigned comment added by Topology Expert (talk • contribs) 13:09, 17 September 2008 (UTC)

As already noted, it has a name: jerk. -- Jao (talk) 19:08, 17 September 2008 (UTC)

Gauss and triangular numbers

from Carl Friedrich Gauss:

"...every positive integer is representable as a sum of at most three triangular numbers..."

A list of triangular numbers taken from triangular number:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

I know I'm missing something here. Integers are in the set {..., -2, -1, 0, 1, 2, ...}, and 1, 3, 6, and 10 are all triangular numbers. 1 + 3 + 6 + 10 = 20 and 20 is an integer. Therefore, the sum of four triangular numbers is equal to an element in the set of integers. What am I missing? Ζρς ι'β' ^¡hábleme! 22:27, 15 September 2008 (UTC)

20 = 10 + 10? Confusing Manifestation(Say hi!) 22:48, 15 September 2008 (UTC)

And just to clarify, what Gauss found was that given any positive integer, you can break it down into a sum of three or fewer triangular numbers. Not that adding more than three triangular numbers together suddenly throws you out of the set of integers entirely. Confusing Manifestation(Say hi!) 22:49, 15 September 2008 (UTC)

Ok, thanks. That's what I was missing. 67.54.224.199 (talk) 23:15, 15 September 2008 (UTC)

Or since 0 is a triangular number then 20=10+10+0 with no change to the statement. Dmcq (talk) 08:19, 16 September 2008 (UTC)

The statement says "at most three", so there is no need to pad it out with zeros. --Tango (talk) 10:44, 16 September 2008 (UTC)