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simultaneous equations
following on from the previous answer to micina's question, i've been asked to try to explain the other explaination - the one given in terms of simulateous equations. here is an attempt, which may well need itterating..... one day people will start to realise that maths isn't some arcane magic divorced from reality, but just another, albeit extremely simple [that being the very reason it is usefulness], way of describing that reality. when that day comes there shall be much rejoicing from this corner of the world.


first off, what is an equation?
an equation is just a ratio - a way of saying that if you have a certain number of :a:s i'll give you another number of :b:s.. like an exchange rate at the bank: the bank tells you that if you bring them a hundred dollars they'll give you sixty five quid, or in maths-speak:
100d=65q
a very simple ratio. now say instead the bank told you that if you gave them a hundred dollars and ten euros they'd give you seventy quid. another very simple ratio that goes into that very simple language called maths like so:
100d+10e=70q

here we have two ratios.. another way of saying this is that we have two equations... or even that we have "simultaneous equations": ratios that all apply at the same time... yes everytime you look at the currency board at a bank, or a stock ticker, or a restaurant price list, or a receipt, or... you are looking at simultaneous equations. useful buggers. ^_^.......
suppose we wanted to know how much the bank would give us if we just gave them the ten euros... well we know how much they'd give us for the hundred dollars [sixty-five quid] so obviously that extra five quid was down to the ten euros. or in maths speak:
10e=5q

now i just made it slightly simpler by having the number of dollars the same both times.. say instead you gave the bank two hundred dollars and a thousand yen, and they decided to give you a hundred and thirty-five quid.. or in maths-speak:
200d+1000y=135q

and i want to know how much money they'd give me for my yen on its own. well i already know that a hundred dollars gets me sixty-five quid, so two hundred dollars should get me a hundred and thirty quid. that leaves five quid to cover the yen. or...
1000y=5q


now for another example...
A group of old ladies meet for an afternoon tea party. They bring all their cats. In all, there are 22 heads and 72 feet. How many old ladies and how many cats are in the room?
so lets see what we know:
1. we know that both cats and grannies tend to have one head each.
2. we know that cats tend to have four feet each and that grannies tend to have two each.
3. we know that the total number of feet is seventy-two and the total number of heads is twenty-two.

ok, so what does that give us in maths-speak?
1. h=c+g [translation: the number of heads can be found by counting the cats and counting the grannies and adding those numbers together]
2. f=4c+2g [translation: the number of feet can be found by counting the cats, multiplying that count by four [or alternatively just counting all the cat feet], and adding it to the number of grannies multiplied by two [or counting the granny feet]]
3. h=22 and f=72 [translation: we have twenty-two heads and seventy-two feet]

so we can change the first ratio a bit to see how many cats we get per granny:
h=c+g -> c=h-g -> c=22-g [so this bank is going to give us cats in number equal to the number grannies we give subtracted from the total number of heads, we we've been told is twenty-two].
now the second ratio/equation talks about four times the number of cats [actually just the number of cats' feet], so lets see how much four cats would fetch us via the head count:
4c=4x(22-g)=88-4g

now we're getting somewhere: we know that if we take some grannies to the bank they promise that eighty-eight minus four times the number of grannies will be four times the number of cats they'll give in exchange*... lets take them to the bank/equation (2)..
f=4c+2g [same as (2)] or f=(88-4g)+2g or f=88-2g

so if we take eighty-eight bodies minus two times the number of grannies, the bank will give use lots of feet.. well seventy-two feet, as if you remember that's the deal they made earlier. or in maths-speak:
72=88-2g or 2g=16 or even g=8
so we have eight grannies [you've actually already answered the question, as only one of the answers you were given has eight grannies, but let's keep going]...

now remember we found out that if we took some grannies to the bank they promised to give us twenty-two minus that number of grannies in cats?
or c=22-g from above. so if we hand over our eight grannies, they'll give us fourteen juice cats to fry:
c=22-8 -> c=14

so now we know if you take our twenty-two heads and seventy-two feet to this very ghoulish bank, they'll hand over eight grannies and fourteen cats. problem solved.

*this might give you an idea of why mathematicians normally use maths to communicate rather english! it means the same thing, but it's sounds a hell of a lot more complicated in english, because english is a vastly more complex and difficult language.



micina: hope that helped, if not please tell me where it isn't clear.... put as an entry rather than as a reply, because i thought it might be more generally useful/amusing.


and here's a slightly more complicated question to see whether you *really* understood ;-)
you have a room full of kangeroos [two feet honest], dogs and people. there are 18 heads, 56 feet and 15 tails. how many of each animal are there?