this article is part of a section on explaining mathematical terminology.

these documents are highly colour coded for greater clarity. the key is as follows:

this explanation start as an amusing example of the difference between someone who can do maths, and someone who is nummerate!... micina asked her readers for help in solving the following..

not often that i get examples of differences in response between a mathematician and a nummerate.. the mathematician [unless nummerate of course] will tend to do it the formula way every time no matter how long it takes, and is highly unlikely to see lower tech shortcuts.. for someone else's comments on this sort of thing see here.

update: it occured to me in the shower that my method is better even if there are no answers...

• make a stupid guess. for example: 1 granny and 21 cats - only thinking about making the number of heads right.
• work out the number of feet. in this case 86 feet.
• work out how many out the number of feet is, and in what direction. in this case 14 too many.
• using the feet reduction worked out above, work out how many cats to swap to grannies or vice versa. in this case swap 7 cats into grannies.
• voila you have an answer. in this case: 8 grannies and 14 cats.

still much quicker and easier than simultaneous equations.

following on from that answer, i was 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, way of describing that reality - that simplicity being the root cause of maths' great usefulness. when that day comes there shall be much rejoicing from this corner of the world.

to the explanation..... let's start from the very beginning and ask a fundamental question:

there are a number of ways to describe what we now have:

• we have two ratios.
• we have two equations.
• we have "simultaneous equations"

"simultaneous equations" just means "ratios that all apply at the same time".

in other words, 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.

so what do we do with these equations? it entirely depends what you want to know. for example, suppose we wanted to know how much will the bank give us if we just gave them the ten euros?...

well, we know how much they'd give us for the hundred dollars from the first equation/contract in the box above: sixty-five quid. so obviously that extra five quid they are offering in the second equation/contract is down to the ten euros.

or in maths speak:

10e = 5q
 

now, i just made that example slightly simpler by having the number of dollars the same in both contracts... let's see what happens if 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 will they 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 let's return to the original example...

the first stage is to translate into maths speak - as said early this is just to simplify the language, making it easier to see what is going on.

h = c + g (equation 1)
 

this simply means that the number of heads can be found by counting the cats and counting the grannies and adding those numbers together.

f = 4c + 2g (equation 2)
 

this one means that 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 the total from counting the granny feet]

h = 22 and f = 72 (equation 3)
 

the final one means that we have twenty-two heads and seventy-two feet in the room.

ok, let's change the first ratio a bit to see how many cats we get per granny:

h = c + g c = h - g c = 22 - g
 

in other words, 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 let's see how much four times the number of cats would fetch us via the head count we just worked out:

4c = 4 x (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 exchange¹...

let's take them to the bank/equation 2..

f = 4c + 2g [same as 2] f=(88 - 4g) + 2g f=88 - 2g
 

and back into english: 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 back into maths-speak:

72 = 88 - 2g 2g = 16 g = 8
 

so finally, we have eight grannies. we can actually already stop here, as only one of the answers we 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?

c=22-g
 

so if we hand over our eight grannies, they'll give us fourteen juicy cats to fry [strangely i can never find the recipe when i need it]:

c=22-8 c=14
 

so now we know that 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. phewww.

and here's a slightly more complicated question to see whether you *really* understood ;-)

you have a room full of kangeroos [they have two feet honest], dogs and people. there are 18 heads, 56 feet and 15 tails. how many of each animal are there?

well... i actually made that problem while half asleep; some kind soul pointed out that the answer is in fact incredibly simple. just think of the tails... so i made another, silly more/less sensible problem:

martian have three heads, three legs and three tails. venusians have two heads, one leg and five tails. earthians i think you know about.... in a room, containing only the above types, there are ten heads, eleven legs and thirteen tails. how many from each planet?

1610 words.

© 30/11/2003 the auroran sunset

original diary entries 22/nov/2003 and 23/nov/2003.

footnotes

footnote 1: 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.