Transposing Formulae

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Equations and Transposing Formula

Equations are a set of symbols representing quantities of things that are equal and like a set of balanced scales can be rearranged provided that whatever is subtracted or added to one side of the scales the exact same quantity is added or subtracted from the other side in order to balance the scales out.

How are equations made?

Sometimes it is more convenient to represent quantities of things by letters or symbols. A simple example of this is to represent an apple by the letter (a) and a number of apples are represented as follows:-

An equation comes about when someone notices equality between two or more different things. For instance one day a greengrocer named Fred Lawton observes that a banana is usually twice as heavy as an apple, so he says:

“In order to balance my scales I need two apples for every one banana”

he uses this relationship without realising he has invented an equation describing the weight relationship between apples and bananas, all he needs to do is write it down in shorthand form with letters or symbols:-

What greengrocer Fred is mentally picturing is:-

Mathematically he could have jotted this down on paper:-

b = 2a

Although the observation was an approximation due to apples and bananas being slightly different weights, in general he found this relationship to be a good one.

One day Fred the greengrocer is asked by a customer:

“How many apples are there in a Kilogramme?”

He knows that two apples are the same weight as one banana, but that’s of no use to him now, he asks himself:

“What’s a kilogramme?”

The customer tells him that he has observed that when he asks other greengrocers for a kilogramme of bananas he gets six more often than not.

Lots of things were going on in the Fred’s mind all of them ve:Then in a flash of inspiration:-

Fred remembers that one banana is equivalent in weight to two apples, and the customer had told him that there are 6 bananas in one kilogramme he quickly writes all this down lest he should forget:-

b = 2a

K = 6b

Just by looking at what was written down he empirically denotes that there are twelve apples to a kilogramme, not realising he has solved two simultaneous equations with two unknowns.

Other equations that Fred found were:-

One kilogramme equals 2.2 pounds; or as Fred wrote it down:-

K = 2.2P; where P denoted pounds or (Ibs)

Fred pondered on this for some time and thought::-

“This is very useful for my customers, because I know that they don’t like to buy their produce in kilogrammes, they like pounds, ounces, yards, bushels and pecks! “

Fred was loathe to change the old ways of measuring and weighing things, so this useful little equation would be of great benefit to his older customers, who steadfastly refused to deal in the new imposed metric system.

Fred, Old Money and Mrs Dangle

One day an elderly lady enters Fred’s greengrocer shop and asks if he’s got change for half a crown:

“Oh no I’ll have to convert this into new money and back again for Mrs Dangle” said Fred

He racked his brains to try and remember how many old pennies were in a half crown.

He remembered at last that, in the old money that Mrs Dangle was used to contained 240 pennies to one pound Stirling, and also a half crown had 30 pennies, and there were eight half crowns to each pound.

Fred very nearly fainted trying to work all this out in his head.

He scratched the following with his pen:-

240p = 100n

Where, p represented old pennies and n represented new pennies.

Again purely by observing this little relationship between old and new money he had scratched down on his piece of paper, he deduced that one new penny was equal to 2.4 old pennies. Fred had found an equation albeit a simple one to covert Mrs Dangles Old Money into the new decimalised money.

Hence he noted: n =2.4p

“I’m getting good at this” said a smug looking Fred to Mrs Dangle.

“I don’t know about that, I’m in a hurry Mr Lawton! The gas has nearly run out and Mr Dangle is not very well and he does feel the cold” said a worried looking Mrs Dangle

“Well, what would you like it changed into Mrs Dangle – five pound notes?”

Mrs Dangle laughed at Mr Lawton’s little joke, he was always jovial with his customers.

“Well my gas meter takes old pennies you see Mr Lawton and it won’t take half-crowns, and I was wondering, how long will the gas last on half a crowns worth of old money, if it’s five new pence a therm, and I use three therms an hour when Mr Dangle’s got the gas fire on”

“Well half a crown’s only going to last you 50 minutes Mrs Dangle”

How did Mr Lawton the greengrocer work that one out?

Paul Dirac

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