## Math Riddle

For the discussion of math. Duh.

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### Math Riddle

Usually I am able to mentally deduce math riddles immediately upon seeing them. However, even after calculator work, I could not figure out the following (with the *correct* answer):

"Kevin's electricity costs him 22 cents per kilowatt every hour. He's decided to install solar cells on his roof for \$40,000, which will save 5% on his power bill.
How many days will it take to pay for itself if Kevin uses 454 kilowatts every hour of every day?"

Before I give the "correct" answer, I want to see how you do on this.

(This was found on "The Brain Game" 2009 Calendar, and as far as I've found, a Google search yields nothing.)

Include work.

Nyktos
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### Re: Math Riddle

Spoiler:
He pays \$0.22 × 454 × 24 = \$2 397.12 per day to start. After installing the solar panel, he saves \$2 397.12 × 0.05 = \$119.86 (approximately) per day. Thus it will take 40 000 ÷ 119.86 = 333.723 days for it to pay for itself, which rounds up to 334.

I'm guessing I'm wrong, since that seems pretty obvious.

Birk
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### Re: Math Riddle

I did it a little differently.

Spoiler:
I did 40,000 is 5% of 800,000 so when he spends 800,000 dollars he has saved 40,000. Then I figured the same cost/day you did of 2391.12. 800,000/2397.12 = 333.73 days. So the same answer. Yay. I guess I just like big numbers so I went the 800k route.

I mean is there some trick that makes the obvious answer wrong?

valarauca
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### Re: Math Riddle

Does his power bill decrease after the installation or does he just consume more power to equal the previous bill?
Under the assumption that it doesn't (because then the solar panel would never pay for it self):

Spoiler:
454 kilowatts per hour per day =
10896 kilowatts per day=
\$2397.12 every day (in NRG)
so if the solar panel causes a 5% reducation that means
\$2397.12 becomes
\$2277.264
which is a saving of 119.85 per day
which means 333.75 days?
So like 334 days

Are we all wrong?
Industrial Software Specialist.

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### Re: Math Riddle

You are all getting the same answer that I did!

Spoiler:
However, the answer given on this calendar is 8009 days. How did they get that?

Ieatsoap6
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### Re: Math Riddle

Spoiler:
However, the answer given on this calendar is 8009 days. How did they get that?

It could be that they meant to say that he uses 454 kilowatt hours per day and his rate is \$0.22/kWh. \$2,000 seems a bit high for a per day for electricity. So that would make the time it take for the panels to pay for themselves 24 times longer, since you're dividing the energy use by 24. Multiplying the answer everyone's getting by 24 yields the correct answer.

I think the question may have simply been poorly phrased or a typo or something.

Buttons
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### Re: Math Riddle

Since Kevin is using approximately five thousand times as much electricity as the average person (read: me), I imagine the Feds will figure out what that power is for and bust him before the solar panels can pay for themselves.

Leztat
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### Re: Math Riddle

hm... my answer to that is 16.686690695501268188492858096382 days...

Oh me yarm how much electricity does he use?

Leztat
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### Re: Math Riddle

Buttons wrote:Since Kevin is using approximately five thousand times as much electricity as the average person (read: me), I imagine the Feds will figure out what that power is for and bust him before the solar panels can pay for themselves.

hahahahah xD Oh me yarm there's something wrong with you!!

Nyktos
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### Re: Math Riddle

Ieatsoap6 wrote:It could be that they meant to say that he uses 454 kilowatt hours per day and his rate is \$0.22/kWh. \$2,000 seems a bit high for a per day for electricity. So that would make the time it take for the panels to pay for themselves 24 times longer, since you're dividing the energy use by 24. Multiplying the answer everyone's getting by 24 yields the correct answer.

I think the question may have simply been poorly phrased or a typo or something.

*Smacks head* Duh. I guess I really am a mathematician inside, since I saw absolutely no problem with the idea of a \$60 000 per month electricity bill.

Qaanol
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### Re: Math Riddle

Can we assume he deposits the monthly savings into an interest-bearing account that compounds daily?
wee free kings

valarauca
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### Re: Math Riddle

Nyktos wrote:*Smacks head* Duh. I guess I really am a mathematician inside, since I saw absolutely no problem with the idea of a \$60 000 per month electricity bill.

I think he could cover the 60 grand by selling the pot that he growing in his basement.
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Nlelith
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### Re: Math Riddle

I got 8009.611534 days. Would you really only save 5% on electricity with a \$40,000 solar panel setup?

Edit: Oh, right. Show your work.

Spoiler:
5% savings gives 20.9 cents/kw*hr. With the setup, it costs him 20.9 [cents/kw*hr]*454 [kw*hr/day]*1/100 [dollar/cents]=94.886 dollars/day. So he saves 99.8-94.886=4.994 dollars/day. Multiply it by x number of days and set it equal to \$40,000, solve for x, and get 8009.611534 days.

Qaanol
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### Re: Math Riddle

Can we further assume he would have otherwise invested the 40 grand into the same interest-bearing account?

If so, at what interest rate would the solar panels switch from "Paying themselves off in a finite amount of time" to "Never paying themselves off"? What if the interest is compounded continuously?
wee free kings

Buttons
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### Re: Math Riddle

And if Kevin's electricity consumption were remotely normal (let's be generous and say \$300/month, which is what you might expect using space heaters or air conditioning at all times), then it'll take more than a century for the solar panels to pay for themselves anyway.

jestingrabbit
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### Re: Math Riddle

Its normal to always have something on altering the temp? Really? Because I go more than six months of the year with no such object.

And seriously, \$300 a month? When I have to pay that for 3 months I'm pissed off. Or is this for a family or something?
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Buttons
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### Re: Math Riddle

jestingrabbit wrote:Its normal to always have something on altering the temp? Really? Because I go more than six months of the year with no such object.

And seriously, \$300 a month? When I have to pay that for 3 months I'm pissed off. Or is this for a family or something?

I dunno, I was trying to shoot for as big a number as possible. Our bill is usually \$25 or so, but our neighbors use space heaters and were paying \$250 during the coldest months.

Cosmologicon
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### Re: Math Riddle

Qaanol wrote:Can we further assume he would have otherwise invested the 40 grand into the same interest-bearing account?

If so, at what interest rate would the solar panels switch from "Paying themselves off in a finite amount of time" to "Never paying themselves off"? What if the interest is compounded continuously?

I'd say that assumption is more or less cancelled out by the assumption that electricity remains at a constant price for the next couple decades.

Averazul
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### Re: Math Riddle

Electricity bills usually aren't linear either. My dad has solar panels on his roof and saves more money (% of normal bill) than power he produces (% of power consumed) because the solar panels take off the top kWh, which are the most expensive.

mr-mitch
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### Re: Math Riddle

The answer of 8009 is 100% correct.
Where you're probably going wrong, is with the units.

(question copied here for my convenience :p)

"Kevin's electricity costs him 22 cents per kilowatt every hour. He's decided to install solar cells on his roof for \$40,000, which will save 5% on his power bill.
How many days will it take to pay for itself if Kevin uses 454 kilowatts every hour of every day?"

Spoiler:
Bill is 0.22 dollars / kWh
Savings is 5% so you are saving 0.05 * .22 dollars / kWh = 0.011 dollars / kWh
He uses 454 kWh/day (where'd I get this? he uses 454kW * 24 hour = 1 day, 1 day = 24 hour, giving 454kWh/day)
Multiplying the two together, you get
4.994 dollars / day
so we obtain the equation
4.994x = 40000, where x is the number of days
giving x = 8009.6115 days.

Problems like these can be dramatically simplified when considering the units.

gmalivuk
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### Re: Math Riddle

mr-mitch wrote:(where'd I get this? he uses 454kW * 24 hour = 1 day, 1 day = 24 hour, giving 454kWh/day)

No, that's wrong. You didn't simplify the question, you just made it wrong.

454kW * 24hr = 10896 kWh, and that's in one day. 10896 kWh/day, in other words. I'm sure you got what the question writer intended, but that's not what the question actually says.
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lilpele22
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### Re: Math Riddle

nevermind.. once I actually read it I felt stupid that I had missed that I forgot to convert hours to days .

mr-mitch
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### Re: Math Riddle

No, that's wrong. You didn't simplify the question, you just made it wrong.

454kW * 24hr = 10896 kWh, and that's in one day. 10896 kWh/day, in other words. I'm sure you got what the question writer intended, but that's not what the question actually says.

Actually, I did realise that this morning when I woke up, having posted that at a very early hour.

To be honest, I have no idea why or I confused myself, probably upon simplifying the explanation. Anywho, I just shouldn't have simplified to letters.

Spoiler:
Bill is 0.22 dollars / (kW every hour)
Savings is 5% so you are saving 0.05 * .22 dollars / kW every hour = 0.011 dollars / kW every hour
He uses 454 (kW every hour)/ day
Multiplying the two together, you get
4.994 dollars / day
so we obtain the equation
4.994x = 40000, where x is the number of days
giving x = 8009.6115 days.

Blatm
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### Re: Math Riddle

mr-mitch wrote:Bill is 0.22 dollars / (kW every hour)
Savings is 5% so you are saving 0.05 * .22 dollars / kW every hour = 0.011 dollars / kW every hour
He uses 454 (kW every hour)/ day

The correct calculation would be 454 (kW every hour) * 24 hours/day, making the answer 334 days.

As gmalivuk said, though the author of the question probably meant to say 454 kwh every day, he said every hour. However the answer to the question as written is 334 days.

{delta}
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### Re: Math Riddle

I am reporting this to the bureau of poorly stated word problems.

mr-mitch
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### Re: Math Riddle

The correct calculation would be 454 (kW every hour) * 24 hours/day, making the answer 334 days.

As gmalivuk said, though the author of the question probably meant to say 454 kwh every day, he said every hour. However the answer to the question as written is 334 days.

No, treat kW every hour as a unit, and it all makes sense. He probably meant kWh by kW every hour. There is no conversion required - the units are already 454 (kW every hour)/day and the cost is \$/kW every hour.
So multiplying cancels out kW every hour giving you \$ and days allowing you to solve the problem.
I like to apply dimensional reasoning to questions which just don't make sense/badly worded.

gmalivuk
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### Re: Math Riddle

mr-mitch wrote:He probably meant kWh by kW every hour.

Yes, but that would be wrong.

Because if I go 10 meters every second, it's 10 m/s, not 10 m*s.

The problem as stated is worded incorrectly if the answer they want is 8009 days. "Fixing" that mistake by then interpreting something in a way that would be incorrect for any properly worded question isn't really helping anyone.
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mr-mitch
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### Re: Math Riddle

Yes it would be wrong, however, the question is still attemptable - by treating "kW every hour" as a unit, as I have done.
Actually, even as kW every hour as kW/h, that's perfectly fine.

Cost = \$0.22/(kW/h)
Savings = \$0.011../(kW/h)
Usage = 454 kW/hdays
multiplying savings * usage (cost * usage = price, etc)
yields 4.994/d *x days = \$40000
solving for x gives 8009.

There is actually nothing wrong with the problem, we've just all assumed things and followed along with them. By sticking with the facts, as you should do with any problem, the question remains valid and you are claim is not.
If you consider reality, it is obvious the author meant kWh instead of or as kW every hour, however, given this mistake, it doesn't matter. As I have shown, the question is still valid.

gmalivuk
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### Re: Math Riddle

mr-mitch wrote:Actually, even as kW every hour as kW/h, that's perfectly fine.

Sure, but then you get a completely different answer. Because you have power per time, instead of power * time (which is energy, which is proportional to cost)

"Kevin's electricity costs him 22 cents per kilowatt every hour." In other words, cost is (22 cents / kW) / hour, which we can simplify to 22 cents / (kWh). There is no reason to treat that sentence as actually meaning 22 cents per (1 kilowatt every hour). Especially as electricity is charged by the kWh, not by the kW/h.

So yes, there most definitely *is* still something wrong with the question as stated. It is poorly worded, because the most natural way to read any part of it leads to an answer different from the one they wanted people to get.
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ILMTitan
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### Re: Math Riddle

I read it that he was using 454kWh/day after he installed the solar panels, so...
Spoiler:
his savings rate would be
[math]454kWh/day *(.05/.95).[/math]
making his payoff time

[math]\$40,000 /(454kWh/h * 24h/day * (.05/.95) * \$0.22/kWh) = 317 days[/math]

mr-mitch
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### Re: Math Riddle

"Kevin's electricity costs him 22 cents per kilowatt every hour." In other words, cost is (22 cents / kW) / hour, which we can simplify to 22 cents / (kWh). There is no reason to treat that sentence as actually meaning 22 cents per (1 kilowatt every hour). Especially as electricity is charged by the kWh, not by the kW/h.
.

No, you still get the same answer, and as I said, where you're going wrong is the units. He gets charged (and hence saves) by his "kW every hour" as stated in the question, so no matter how you want to represent this, it doesn't matter, because he uses some amount of kW every hour, every day. So as I have said, you must multiply these two rates together to get his savings every day, which then allows you to find the amount of days.

Even if, as you say, there is no reason to treat the sentence meaning as such, as you have stated (correctly) electricity is charged by kWh 's, it still works perfectly fine. Since electricity is charged by kWh, you'd say the rate is \$/kWhday, and multiplying the two together still works as expected. Then yes, there is a problem with the wording. But it doesn't matter, for any question with the same circumstances, "kW every hour," or a similar such phrase, can always be treated as a unit, as per dimensional reasoning.

I still don't know why you are ignoring this fact.

famousp
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### Re: Math Riddle

Ok guys, your problem is the unit kWh. You've clearly never looked at an electric bill before.

A standard "unit" of power is the kilowatt * hour. This means the amount of energy consumed at a 1 kW rate for 1 hour.

For example:
A 1000 W toaster running for 1 hour = 1 kWh
A 100 W light bulb running for 10 hours = 1 kWh
A 10 W night light running for 100 hours = 1 kWh

They all used the exact same amount of energy, just at different rates.

So when subby says 454 kWh, he means 454 kWh used per day, not 454 kWh * 24 hours per day. The time is already taken into consideration in the unit.

BTW, 22 cents per kWh seems a little high for the U.S., at least based on this link.
EDIT: Got all semantic-y on myself.

Nitrodon
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### Re: Math Riddle

famousp: I'm sure that everyone here already knows that kWh is the standard unit of energy (not power) used in an electricity bill. Let's look at the references to power/energy in the actual problem statement though.

Kevin's electricity costs him 22 cents per kilowatt every hour.

This states that electricity costs 22 cents per kWh. It's an unusual way to phrase it, but that's the only reasonable interpretation. Nobody is disputing this part, as far as I know.

Kevin uses 454 kilowatts every hour of every day

I fail to see any reasonable interpretation of this clause other than "Kevin uses 454 kilowatts of power continuously, 24 hours a day, 7 days a week, 365.2422 days a year." I certainly don't see any way to interpret it as "Kevin uses 454 kWh per day" while remaining consistent with the rules of English grammar. If the latter interpretation is intended, then the problem is stated incorrectly.

gmalivuk
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### Re: Math Riddle

Nitrodon wrote:
Kevin's electricity costs him 22 cents per kilowatt every hour.

This states that electricity costs 22 cents per kWh. It's an unusual way to phrase it, but that's the only reasonable interpretation. Nobody is disputing this part, as far as I know.

Actually, I think mr-mitch is, when claiming that we should take (kilowatt every hour) as its own unit in this case. Which is stupid, because that would be like taking (second every second) as a unit in "His speed was changing by a meter per second every second". So it's true, as you say, that the other interpretation is the only reasonable one.

But then when we come to the second part, as you also say, there's no indication whatsoever that now "kilowatts every hour" means kWh, and the "of every day" part just tells us his usage is constant.
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mr-mitch
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### Re: Math Riddle

Which is stupid, because that would be like taking (second every second) as a unit in "His speed was changing by a meter per second every second".

It's not like taking second every second as a unit in "his speed was changing by a meter per second every second." First of all, who would say such a thing, and second of all, they're totally different scenarios. Dimensional reasoning works and validates the question, whether you like it or not.

It would, however, work if it was a similar situation. If, for example, there was some rate against second every second, it would work. This is pointless, because there probably doesn't exist such a rate, unless you invent one.

ian
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### Re: Math Riddle

You're still arguing that 1 kW every hour of every day is some how just 1kWh a day, which is clearly wrong.

You can use 1 kW for an hour, or one kWh an hour, but 1000 joules a second every hour makes no sense.

Macbi
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### Re: Math Riddle

ian wrote:You're still arguing that 1 kW every hour of every day is some how just 1kWh a day, which is clearly wrong.

You can use 1 kW for an hour, or one kWh an hour, but 1000 joules a second every hour makes no sense.

It does as a measure of the rate of change of power. Maybe in the OP they're charging him for changing how much power he uses, rather than how much he uses. Interpreting it literally, that's what it says.
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mr-mitch
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### Re: Math Riddle

ian wrote:You're still arguing that 1 kW every hour of every day is some how just 1kWh a day, which is clearly wrong.

You can use 1 kW for an hour, or one kWh an hour, but 1000 joules a second every hour makes no sense.

Actually, I'm not.

Cosmologicon
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### Re: Math Riddle

mr-mitch wrote:It's not like taking second every second as a unit in "his speed was changing by a meter per second every second." First of all, who would say such a thing,

Sounds pretty reasonable to me. What's wrong with it?

mr-mitch
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### Re: Math Riddle

That's not what I was referring to, however, I'd like for the mod to stop posting statements without proving them. In maths, as you should all know, proofs require us to step from one truth to another. Do not state things without backing them up.