Fair Shares?
Congratulations to the many people who sent in excellent solutions.
Jeni and Lucy , St James Middle School, Bury St Edmunds and Jackie
of Madras College, St Andrews used trial and improvement. All the
following used algebra: Timothy of Munsang College, Hong Kong; Kate
and Katherine of Emmbrook School, Wokingham; Lyndsey, Christiane
and Joanne of The Mount School, York; Fiona and Elizabeth of
Stamford High School, Lincs.
Here is the solution from Prav, Sheli, Meg, Ruoyi andLiz The North
London Collegiate School Puzzle Club.
Dear Cambridge,
We have a solution to the Rollerball question published in the May
problems. We made a formula to find out the prize money which we
called $\phi$.
$$\begin{eqnarray} 100 + \frac{1}{10} (x - 100) &=& 200 +
\frac{1}{10} [ ( x - 300 ) - \frac{1}{10} (x - 100) ] \\ 100 +
\frac{x}{10} - 10 &=& 200 + \frac{1}{10} (x - 300 -
\frac{x}{10} + 10)\\ \frac{x}{10} + 90 &=& 200 +
\frac{1}{10} ( -290 + \frac{9x}{10})\\ \frac{x}{10}+ 90 &=&
200 - 29 + \frac{9x}{100}\\ \frac{x}{10} + 90 &=& 171 +
\frac{9x}{100}\\ \frac{x}{10} &=& 81 + \frac{9x}{100} \\
\frac{x}{100} &=& 81 \\ x &=& 8100 \end{eqnarray}$$
The first child gets:
$$ 100 + \frac{1}{10} (8100 - 100) = 900. $$
Therefore, if every child gets the same amount, so $$
\frac{£8100}{£900} = 9 $$
there are nine children.
Love from, Prav, Sheli, Meg, Ruoyi and Liz