Math Is Fun Forum

hi mishin05

The correct 'formula' for (iii) is

If you put U = x and V = 1 then

That looks ok to me.

Bob

hi bobbym and Tonia,

Very impressed by this, bobbym. That's got to be the best method so far.  :):)

You are very welcome.  I have enjoyed trying the problem and haven't given up on finding the 'grade 8' solution.

Watch this space!

Bob

hi Otsdarva,

Start at http://www.mathsisfun.com/geometry/index.html

Then post to the help page when you get stuck.

Welcome to the forum!

Bob

hi Mandy,

Still here.  Have you tried the factorising exercises ? (post 22)

Bob

hi Tonia,

If I may use the analogy that we are at war with this problem, then I have a racial characteristic that " We may lose the first battle, but we go on to win the war!".  If that was good enough for Winston Churchill, it's good enough for me.  In terms of helping you, that means I'm not ready to give up yet.

If I've understood correctly what 'grade 8' means then I'm very surprised.  It is far from a 'ridiculously simple problem'.  Bobbym's method uses Pythagoras as you first specified, but requires a lot of advanced algebra to get to an answer.  The trick of substituting y = x^2 to solve a quartic (highest power x^4) is taught to mathematics students in England at age 17/18 (=Advanced level), not earlier.  No simultaneous equations with more than 2 variables would occur before this level too.  Maybe we, all three, are missing a direct and simple method, but I doubt it.  And no one else on the forum has jumped in with an answer either, so I think the person writing this book may have set a problem that is just too hard for grade 8.  Nor does it lend itself to an easy scale drawing type solution.  I have a program called 'Sketchpad' that allows an accurate construction, but even with that, it took me three attempts before I got one that worked.

Now to the methods.

bobbym wrote:

You can make x the subject of D and substitute this into C. ***

Then eliminate w and z using A and B to leave an equation in y.

This is a quartic but can be converted to a quadratic that can be solved using the formula.  (at grade 8 ?)

(bobbym, now I look more closely at this I'm stuck also as to how the step *** works as there are some terms with w and z rather than w^2 and z^2. This approach is tough for those of us who don't have Mathematica.)  These are for you:

So I thought I'd try another approach.  The formula

is one I learnt in 1968 when I was doing A level maths.  This is the first time I've actually used it since!

It is derived from the cosine formula and double angle formulas, so again is an Advanced level topic.  's' is called the semi-perimeter and is introduced only to make the final version of the formula neat to remember.  Otherwise it is

I used it to get the area of triangles, AEB, BEC and AEC.  Then added them together and set this equal to the area of half the square.  Advantage: only one unknown; I used x as half the length of a side because this kept lots of ½s out of the equation.  The result was again an equation that wouldn't yield to analytic solutions so I resorted to Excel.  I constructed formulas for the left hand side and the right hand side of the equation and then trialled to make them equal.  Excel has a macro that does trials for you where you specify the target value and what you want to vary and the procedure 'homes in' on the solution.

I will post the proof for the formula and the spreadsheet if you want.

Bob

hi Aartt

This is the equation.

These are the numbers:

so

so

so

Does that explain it?

Bob

OK I'm impressed.

Are you going to reveal your method? 

Bob

hi Aartt

This looks like the equation

with

so rearrange to give

Does that make sense?

My advice is the usual  "Break the problem down into short, easier steps."  In particular:

(i) Identify which you know out of v, u, a, d, t and which you want to find.
(ii) Write down the correct equation.
(iii) If all the units are correct, m/s etc, then leave them out during the calculation
(iv) Substitute the numbers.
(v) Work out the missing letter.
(vi)  Put in the correct units.
(vii)  Is this answer sensible for the problem?

Bob

hi mandy jane,

Let's have a look at your answers.

Let's learn some Latex.  This is the computer language that allow you do write this as

Then I put p^2 for each 'p squared'

math] p^2 + p^2 + p^2 [/math]

Although it will take a while to learn Latex, it'll be worth it as your answers will look like 'proper maths'.

Now, is your answer correct?

These are all similar and I think we need to go back to the beginning with what you are being asked to do for these.  I'll cover that at the end.

Now let's look at factorising more slowly.

Work out 3 x 4 + 3 x 7

= 12 + 21 = 33

But you can also add together 4 + 7 and then times the answer by 3

3 x (4 + 7) = 3 x 11 = 33.

You get the same answer.

So 3 x 4 + 3 x 7 = 3 x (4 + 7) = 3 x 11

This is what factorising is based on.  3 is the factor.

Let's stick with numbers only for the moment and try some more.

Copy and complete these by replacing the ? marks.

Q7  3 x 6 + 3 x 8 = 3 x ( 6 + ? ) = 3 x ?  = ??

Q8  7 x 2 + 7 x 5 = 7 x ( ? + 5 ) = 7 x ? = ??

Q9  5 x ? + 5 x 9 = 5 x ( 3 + 9 ) = ? x 12 = ??

Q10  9 x 7 - 9 x 5 = 9 x (?????) = ??

Q11  42 + 56 = 7 x ? + 7 x ? = 7 x ? = ?  (42 and 56 are both in the what? times table)

Q12  542 x 98 + 542 x 2  = 542 x (???) = ???

Once you have a clear idea what is happening with just numbers, the algebra will be easier.

I've tried several ways of printing the text.  Please say which is easiest for you.

I'm going to make a post that will start algebra at the beginning.  We are so close to Christmas; so I'll try to do it for early next week.

Bob

hi mandy jane,

Let's try this way.

Here are some algebra questions from an old foundation exam paper.

If you think you can do a question, post back your answer.

Or say if you are stuck with that question.

Q1  Simplify

Q2  Simplify

Q3  Factorise

Q4  Solve

Q5  Expand

Q6  Factorise

Bob

hi mandy jane,

Go ahead.  What help would you like?

Bob

Hi Mariner,

I looked at these yesterday and hoped someone else would pick up your problem.  But alas, no one has.

The trouble is: there's no obvious pattern so no wonder you are stuck.  If you write down any five numbers at random, it is possible to fit a formula to them but that's not what you want I suspect.

These go up and down so that counts out 'plus x', 'subtract x' , 'multiply by x' and 'divide by x'.  So I thought maybe there are two interlocked sequences, but I haven't got enough numbers for that, so I have to admit defeat at the moment.  Sorry

But this post will  refresh the thread with New Post so maybe someone else will spot something.

Bob