Miss

Scheme 1: amount paid = p pence for the first k mile travelled [= pk] plus q pence for each mile travelled beyond the first k miles [= q(200-k) ].

So the amount paid = pk + q(200 - k)

Scheme 2: amount paid = 200r

3. The 'break-even point' is the distance for which the two schemes pay the same amount. The manager believes that there will always be a 'break-even point' for the two schemes.

(a) Set up expressions involving p, q, r and k which will work out the amount paid in travel expenses under each of schemes 1 and 2.

Where m is the number of miles travelled:

Scheme 1: (simply replace 200 in the above equation by m) amount paid = pk + q(m - k) = pk + mq - kq

Scheme 2: amount paid = mr

(b) Hence set up an equation which will determine the 'break-even point' for various choices of p, q, r and k.

The break even point occurs when the amount paid by the two schemes is equal, ie when

pk + mq - qk = mr

At this point you may like to select various values p, q, r and k and determine the break-even point. A graph might be handy (a graph of cost against number of miles travelled for the two schemes?).

4. Obtain a range of solutions for this equation, confirming or otherwise, with justification explanation or proof, whether or not there is always a 'break-even point'.

A good answer to this part of the question is needed to obtain the top marks (over 21 out of 24).

One possible way of approaching