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 this question might be to determine when there will be a