# The Maths They Never Taught Us - Part Three

## It does work both ways

**Let's Mix Politics and Sport**

Now the next idea I would like to draw your attention to, as touched upon in the previous section on opposite operations, back in The Maths They Never Taught Us - Part Two , is that if numbers work out one way, then they should work out another.

So, just as the order of the terms in some operations do not matter, then there is more than one way to solve a mathematical problem, and by doing it more than one way, we can even confirm that our first answer was correct.

To illustrate this, let us check out a fictitious future Netball Test between the Australian Diamonds and the New Zealand Silver Ferns.

## Australia vs. Silver Ferns, Brisbane, Saturday, 20th July, 2019

1st Quarter 2nd Quarter 3rd Quarter 4th Quarter Aust 12 Aust 16 Aust 13 Aust 17 NZ 10 NZ 21 NZ 13 NZ 16

## FINAL SCORE : Australia 58 New Zealand 60

For this fictitious test, yet one that could be real and therefore very relevant, another way to check the margin of victory, as opposed to looking just at the final score, is to add all the quarters together, and compare them. So, let us take it one quarter at a time

## Brisbane Test Analysis by Quarter

1st : Aust + 2 2nd : NZ + 5 3rd : Even 4th : Aust + 1

If we look at it from **Australia’s** point of view, quarter by quarter, we see that after the **1**^{st}, they were up by two. In the **2 ^{nd} Quarter**,

**New Zealand**came back to win it by five, meaning that at Half Time,

**Australia**was now

*down by three*. In the third quarter, honours were even. Thusfore,

**Australia**remained

*down*

*by three*, but in the final Quarter, the

**Australians**scored

*one net more*, leaving them a total of

*two points short*, as the final score confirms. Now, this leads us to another interesting mathematical idea, which has relevance to a number of things, from sport to politics.

## Golden Oldies Division Tournament, Australian Open Final, Centre Court, Rod Laver Arena, Flinders Park, Melbourne, January, 2056

Roger Federer ( SUI ) 6 0 5 7 10 Rafael Nadal ( ESP ) 4 6 7 6 8

We see here that once again the **Swiss Maestro **has triumphed in a mammoth **five setter**, by **three** sets to **two**, but if we look at the total number of games won by each player, we can determine that although he lost the match, the **Spaniard** actually *won***31** games to **Federer’s** **28**, which works out if you also calculate it set by set. This is the nature of tennis. It is ** not** won by games, but by sets, and each set is treated individually.

And yet if we were to investigate the details of this contest further, ** even though it is not evident on our scorecard**, we would find that in this particular match at least, most of the time

**Mr. Federer**won a game, he did so by going to

**forty love**, then winning the next point, while

**Mr.**

**Nadal**was more likely to win by taking the game to

**deuce**, then getting the next two points in a row. It works out then, that although

**Nadal**win more games,

*did***Federer**actually won more points. And this kind of thing doesn’t only occur in Tennis.

## Was our HART really in it ?

In 1981, the South African Rugby Union team, the Springboks, toured New Zealand, amid huge protest. Sporting contact with such a country always caused considerable controversy, due to the policy of Apartheid then practiced by South Africa. In 1981 people witnessed the greatest civil unrest seen here in New Zealand, as those opposed to the tour went out to do battle with police riot squads outside each game’s venue, in an effort to stop each match.

Now with all this happening, the All Blacks won the First Test at an unjaded Lancaster Park, in Christchurch, by **14 - 9**. After this, they lost the Second Test in Wellington by **24 - 12**. The Third Test went to Eden Park in Auckland, to give the All Blacks a victory by **25 - 22**, and a hard earned test series win, beating South Africa by two tests to one.

Yet again, though, we see that if we add up all the points scored by both teams in the test series, New Zealand in three tests scored **51** points to South Africa’s** 55**. This gave South Africa **four **more points overall. You can calculate this margin either by adding the teams' points for each match together, then comparing, or even adding the margins of victory for each match, and working it out from there.

Now, the fact that South Africa *did* score more points in all three tests combined, is actually of no real relevance, since each test is played separate from any of the others, and it’s what happens on the day that decides defeat or victory. But this analysis of the previous serves to prove the point that you can work out totals of any kind of thing in more than one way, and if both answers *don’t* match,* then *you know that you have a bit of a problem. Doing calculations in more than one way is in fact a common practice for scientists and mathematicians to enable them to check their answers.

Actually, recalling recent ( November 2013 ) events, there are times that points in an individual game do count, and that is in World Cup Soccer qualifying. Two countries will be paired up to play off against each other, in these cases at least, the *aggregate number of goals* count, especially away goals. There is a lot of relevance for mathematics in looking at how this works, which may be done at a future time.

## Last past any Post

Now, this idea of separate games counting, as opposed to total overall points in a series, may be the same sort of story in the way some elections are carried out. Historically, up to and including 1993, General Elections in New Zealand were carried out using a method known as First Past the Post. Since 1996, New Zealand changed to employ a form of Proportional Representation known as MMP.

In the old days there were a certain number of electorates, which only changed as population increased, and some seats were added and boundaries redrawn. These electorates were geographical areas determined so that each contained about the same number of voters. In every electorate, which represented a seat in Parliament, a number of candidates would stand to be chosen as that electorate’s Member of Parliament. This meant that each voter, wherever they were in the country, would be choosing a local candidate, rather than all the votes throughout New Zealand being pooled together, added up, and the party that got the most votes winning.

This was where First Past the Post resembles the set by set play of tennis, as well as the on the day rugby score, regardless of total games won in a tennis match, or points scored in a test series. In fact, on at least two occasions, in 1978 and 1981, the ruling National Party under Robert Muldoon won with *less* votes than their major rival, the New Zealand *Labour* Party, lead then by Bill Rowling. How could this outrage occur ? Well, remember the tennis match, as I have just said ? Basically, a first past the post election can be simplified to resemble one such match :

## Example of First Past the Post Election

## Analysis of First Past the Post

Now we can see that even though the Happy Party was given a total of** 71 **votes *more* than the Noddy Party, the Happy Party loses the election by** four** seats to **three**. This kind of thing is also why the *loser** *of the 2000 United States Presidential Election, Mr. Al Gore, could get *more* votes than the winner, who went on to become two term President George Bush, Jr., so that although America narrowly missed being gored, they may by now be well and truly bushed. It’s known in Florida as the Butterfly Effect. This is because, even though Americans do not employ a first past the post system exactly like New Zealand once had, their electoral system is still engineered so that votes are not used effectively in a similar way.

We can see from the table that the reason for these kinds of occurrences, is that when the winner of the election wins a seat, they do so by a narrow margin, and yet when they lose one, they get well and truly beaten, so that all the extra votes the losing party gets in a given electorate are sort of wasted.

We look now at another clever aspect of mathematics, which forms the basis of how we figure out equations in many different forms, and how we can go from expressing a truth in a certain way, to ways in which the same thing can be put from a different point of view

## For every Action there is an equal and opposite Reaction - not just in Physics

Having been looking at how things can work out in more than one way, we come now to the subject of opposite operations. Look at this equation : **16** ×** 9** =** 12 **× **?**

So how do we work out the mystery number which will make the equation true ? Let us imagine we changed the **16 **on the left into the **12 **on the right. To balance out the equation, we would then have to change the **9 **on the left into another number which corresponds to it, just as **16** here corresponds to **12**.

Now, in order to know what to change the **9** into, we need to find out what we have done to the **16 **to turn it into the **12**, and then we go and do the *opposite *thing to the **9**, which will balance out both sides of the equation.

What is **16 **to **12 **? Well, in order to turn **16 **into **12**, we actually had to multiply by **¾**.

That is : **16** ×** ¾ **=** 12**, or **12 **is **¾ **of **16**.

But what do we do to **9 **? The opposite - we *divide* it by **¾**. **9 **÷** ¾ **=** 12**

And so we end up with : **16** ×** 9 **=** 12 **×** 12 **=** 144**

An even simpler example may also help to explain : **3 **×** 2 **=** 6 ** ; ** 6 **×** 1** =** 6**.

This time, in the equation at left we d*oubled ***3 **to get **6.** If we want to instead use **6** in place of **3 **on the left hand side of the multiplication symbol, then in order to balance the equation, we have to *halve *the **2 **that is there to get **1**. So we could say :

**3 **×** 2 **=** 6 **; ** ( 3 **×** 2 )** ×** ( 2 **÷** 2 ) **also = **6**.

Because in the second part above, the **2’s **cancel out.

In more general form : **(** **a **×** b ) **×** ( c **÷** b ) **=** a **×** c**

## Since, as we can see :

## There's a Fraction too much Fraction

Speaking of fractions, because there are some interesting ways in which these entities behave, they do tend to catch people unawares, so when confronted with them, we need to be very careful.

**20 **÷** 10** =** 2 **÷** 1 **=** 2**, and so ( **20 **÷ **10** ) ÷** ****2** = **1**, but **20 **÷ (**10 **÷** 2**) =** 20 **÷** 5 **=** 4**.

We got a larger answer the second time, because we divided our denominator, the number we were originally dividing by, by **two**, which means we ended up the second time dividing by a lesser amount, and when we do that, our answer ends up greater.

For example : **20 **÷** 20 **=** 1 **; **20 **÷** 10** =** 2 **; **20 **÷** 5 **=** 4 **In the cases you have just seen, as the denominator (the value on the bottom of the fraction), decreases, the value of the fraction as a number increases. Correspondingly, though, if we decrease the value of the numerator, (the top part of the fraction), then the value of the fraction also decreases.

Incidentally, decreasing the value of the denominator has the same effect as increasing that of the numerator, and vice versa, so that if I, instead of *dividing* the denominator by **two** as I did before, *multiplied* the numerator by **two** : (** 20 **×** 2 **) ÷** 10 **=** 40** ÷** 10 **=** 4**, which is the same answer I had before.

You need to be very much aware of what you can get away with when using fractions, so it is always best, when playing around with them, to check out your answers once you’ve finished.

For example : **20 **÷ (** 10 **÷** 2 **) is *not* the same as **20 **÷** 10** ÷** 2**, which equals **1**. This is because the parentheses in the first equation allow **10 **÷** 2 **to be considered as *one number*. This is from the principles of **BODMAS**, which I shall go into more detail about later. Suffice to say, that the numbers in the parentheses ( )** **are worked out first, but where there are no parentheses, as in our 2nd example : **20 **÷** 10 **÷** 2**, we end up dividing** **twice, first we divide **20 **by **10**, and then divide the answer to this by **2**.

In effect, actually, what we have above without the parentheses is the same as saying **20 **÷ (** 10 **×** 2 **), or **20 **÷** 20**, which itself is not to be confused with the expression **20 **÷** 10** ×** 2**, which, without the parentheses, ends up equalling **4**.

Now, you may also be aware how to multiply and divide fractions. We see that if are to multiply two fractions, we times the top numbers ( numerators ) together, and then the bottom ones, ( denominators ), together, as well.

For instance : (** 3** ÷** 5** ) × (** 4 **÷** 7 **) = (** 3 **×** 4 **) ÷ (** 5 **×** 7 **) =** 12** ÷** 35**,

But if we wanted to *divide *(** 3 **÷** 5** ) ÷ (** 4 **÷** 7** ), we flip the second fraction, then multiply :

(** 3 **÷** 5 **) ÷ (** 4 **÷** 7 **) = (** 3** ÷** 5 **) × (** 7** ÷** 4 **) = (** 3 **×** 7 **) ÷ (** 5 **×** 4 **) = (** 21 **÷** 20 **)

Now the reason you flip the *second* fraction is because *that* is the one being divided by the other. If you went and did it the other way, and inverted the first fraction, you would then end up with the answer (** 20 **÷** 21** ), which is not really the same.

## Do unto others ? Let's reciprocate

As a matter of fact, this second number is the *reciprocal* of the first. We obtain the *reciprocal* of any number by dividing **1 **by it, ( thus, **½** is the *reciprocal* of **2**, since it is the same as saying **one ***divided by*** two **). We can also find the *reciprocal* if it is a fraction, by *inverting* it, such that (** 4 **÷** 3 **)** **is the *reciprocal* of the fraction ( **3 **÷** 4 **), and is quite simply also the same as saying **1 **÷ (** 3 **÷** 4** ). Another thing to be aware of with these fractions, is that, for *example*, ( **3 **÷** 5 **÷** 4** ) ÷** 7 **= (** 3 **÷** 5** ) ÷ (** 4** ÷** 7 **), but *not* **3 **÷** 5** ÷** 4 ÷ 7**. **3 **÷** 5** ÷** 4 **÷** 7**, though, does equal (** 3 **÷** 5 **) ÷ (** 4 **×** 7 **), which also equals **3**** ** ÷( **5** ×** 4 **×** 7 **).

Along with this, there are a number of other facts with fractions which are very useful to know :

If **a **÷** b **=** x **÷** y **( **a **is to **b** what **x **is to **y** ) **ay** =** xb**

Example : **3** ÷** 6** =** 12 **÷** 24**, so that **3 **×** 24** =** 6** ×** 12**

Because : if, as we said, **a** ÷** b **=** x **÷** y**, this also means that **a **=** b **× (** x **÷** y **), which equals **bx **÷** y**, so that since **bx** ÷** y = a**, then we see that **bx ***does equal ***ay**. All this is simple manipulation of formulae by changing the subject. As long as we know what we’re doing.

## Not so dirty Double Cross

So, what if you want to add fractions with different denominators ? Sure, you could cross multiply, where you multiply the numerator of each fraction by the denominator of the other, then multiply both denominators together, but there is another way.

I have come up with what I call the Fractional Difference Formula, which allows us to add or subtract fractions with different denominators and numerators.

Imagine you have a two fractions, ( **y** ÷** x **), and the other, (** y **+** b** ) ÷ (** x** +** a **), where **y **+** b **represents whatever **y **is equal to at the time, *plus* the value of **b **to make a new number, and the same with **x **and **a**.

Say, for example, we wanted to add **¾** to **⅔**. If we say that **three quarters** is represented by the symbols (** y **÷** x **), then **two thirds **would be equal to (** y** +** b **) ÷ (** x **+** a** ), where **2** is **y **+** b**, and **3 **is **x **+** a**.

In this case, since we decided that **¾ **= (** y **÷** x )**, then **y **=** 3**, and **x** =** 4**, so to make **y **+** b** equal to **2**, **b **must =** - 1**, and then for **x **+** a **to be equal to **3**, **a** also equals **- 1**, ( but not necessarily always so. ) It is also true then that **a**, **b**, **x **and **y **can be either *positive* or *negative*.

Now our formula to *add* these two is :

(** y **÷** x **) + [ (** y **+** b **) ÷ (** x** +** a** ) ] = { [** x**(** 2y** +** b **) ] +** ay **} ÷ (** x² **+** ax **)

{ [** 4**×(** 6 **+** -1 **) ] +** -3 **} ÷ (** 16 **+** -4 **) =** 17 **÷** 12**, which is indeed correct.

( Here we see **x²**, which, as I mentioned before, is just a shorthand way of saying **x **times** x**. )

Another formula exists if we wish to *subtract* any fractions, so let’s try **¾ **-** ⅔**. The equation for this is : **( y **÷** x ) **- [ (** y** +** b **) ÷ (** x **+** a** ) ], = (** ay **-** xb **) ÷ (** x² **+** ax** )

and so my formula for working this out will be :** **

(** ay **-** xb** ) ÷ (** x² **+** ax **) = (** -3 **-** - 4 **) ÷ ( **16 **+** -4 **) = (** 1 **÷** 12 **), which is also certainly the correct answer.

## By the Formula

Now, whether or not you find cross multiplying easier, this is still the kind of formula that can be very useful, especially if you have one of those programmable scientific calculators which allows you to plug formulae in, and can automatically give you an answer once you give it the value of each variable. These two equations were developed over a period of time as I experimented with various formulae before realising that I could whittle them down to two basic ones to handle all eventualities. The formulae themselves can be shown to work by applying some basic algebra, using the same method one would need to employ to cross multiply the normal way :

( **y** ÷ **x** ) + [ ( **y** +** b** ) ÷ ( **x** + **a** ) ]

= [ **y**(** x** +** a** ) + **x**( **y** + **b** ) ] ÷** x**( **x** + **a** )

= ( **yx** + **ay** + **xy** + **xb** ) ÷ ( **x²** + **ax** )

= ( **2xy** + **ay** +** xb** ) ÷ ( **x²** + **ax** )

= [ **x**( **2y** + **b** ) + **ay** ] ÷ ( **x²** + **ax** ), which is a simplified form for ease of calculation.

The second formula, for *subtracting* one fraction from another, is therefore also achieved the same way, by cross multiplication, even though at the start, I had actually worked these equations out a different way, using a table of values, only for the cross multiplication to confirm them.

( **y** ÷ **x** ) + [ ( **y** +** b** ) ÷ ( **x** + **a** ) ]

= [** y**(** x **+** a** ) -** x**(** y **+** b **) ] ÷** x**(** x **+** a **)

= [ (** xy **+** ay** ) - (** xy **+** xb **) ] ÷ (** x²** +** ax** )

= (** ay **-** xb** ) ÷ (** x² **+** ax** )

So these two interesting formulae will work for the subtraction and addition of any pair of fractions you can think of. All you need be aware of, is that if the numerator or denominator on one fraction differs from the other, your **b **and **a** values are the *differences* between the two numbers, and do not represent numbers in a fraction themselves.

## Don't be so Mean

Next, let us look at what in mathematics is known as the Geometric Mean, where, if we know then that **a** ÷** b **=** b **÷** y**, then **b² **=** ay**, so that **b** =** √**(**ay**)

If you are not already familiar with the small numbers raised up to the right of an ordinary number as we see here with **b²**, we shall be studying more of these later. All you need to know is that **b²**, which is pronounced **b squared**, is the same as saying **b** times** b**, so that if **b **equalled **9**, then **b² **would be the same as **9²**, or **9** ×** 9**, which is **81**. Now, back to our Geometric Mean, an example of this using numbers represented by the letters, is :

**4 **÷** 10 **=** 10 **÷** 25 **; **10² **( =** 10 **×** 10 **=** 100** ) =** 4** ×** 25**,

and the reason for this working is explained in identical fashion to the one before, by manipulating the fractions to change the subjects.

See, if **b **÷** a **=** y **÷** b**, then **ay **=** b² **For instance, **3 **÷** 9** =** 1 **÷** 3**, and **1 **×** 9 **=** 3²**

This again works when you manipulate the fractions and change the subject, so let’s look next at adding fractions with different denominators, and see what happens. Here’s a formula from Physics to do with how light is emitted. **1 **÷** f **=** 1 **÷** v** +** 1 **÷** u**

If we see this, we might want to find out what **f **equals, so how do we do that ? Well, **1 **÷** v** +** 1 **÷** u** obviously equals the *reciprocal* of **f**, (**1 **÷** f**), so all we do is cross multiply, then flip the fraction, thus : ** 1 **÷** v** +** 1 **÷** u** = (** v **+** u** ) ÷** uv**, so **f **itself =** uv **÷ (** v **+** u **).

The same cross multiplication will work to manipulate any formula in order to change its subject. When we cross multiply, all we do is multiply each denominator of one fraction by the numerator of the other, add or subtract those totals within the numerator, and divide by the product of both denominators

## Moving On Even Further

We continue this journey in the next Hub, The Very Next Step - Squares and the Power of Two and if You are curious, take a look at the other Hubs, The Maths They Never Taught Us - Part One, The Maths They Never Taught Us - Part Two , And then there were Three - a Study on Cubes, Moving on to Higher Powers - a First look at Exponents, The Power of Many More - more on the Use of Exponents, Mathematics - the Science of Patterns , More on the Patterns of Maths , Mathematics of Cricket , The Shape of Things to Come , Trigonometry to begin with, Pythagorean Theorem and Triplets, Things to do with Shapes, Pyramids - How to find their Height and Volume, How to find the Area of Regular Polygons, The Wonder and Amusement of Triangles - Part One, The Wonder and Amusement of Triangles - Part Two, the Law of Missing Lengths, The Wonder and Amusement of Triangles – Part Three : the Sine Rule, and The Wonder and Amusement of Triangles - Part Four : the Cosine Rule.

Also, feel free to check out my non Maths Hubs :

Bartholomew Webb , They Came and The Great New Zealand Flag

Just take a good look then see if you can come up with anything along the same lines. By all means Your comments are very welcome, as well as feedback and suggestions which would be given due credit, or indeed have a go and publishing Your ideas Yourselves, but firstly, by all means, add Your comments - it's a free Country.

## Disclaimer

Any reference to any Copyright or Registered Trademark is credited as such. Some discoveries are my own, but may also have been found independently by others as Mathematics is a living language. Some information has been referenced in a number of publications, most in the public domain, as well as on Wikipedia ( copyright 2013 Wikimedia Foundation ).

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