show that the manager can increase his profit by reducing his mark-up below 20% provided that 20k>F?

Question

The manager of a supermarket usually adds a mark-up of 20% to the wholesale prices of all the goods he sells. He reckons that he has a loyal core of F customers and that, if he lowers his mark-up to x % he will attract an extra k(20-k) customers from his rivals. Each week the average shopper buys goods whose wholesale value is $A. Show that with a mark up of x% the supermarket will have an anticipated weekly profit of
$ 1/100 Ax((F+20k)-kx)

Who can help me with this maths problem?

Answer 1

If he reduces his markup to x%
No of Customers = F + k(20 - k)

And so gross profit = x% of $A * (F + k(20 - k))

=$1/100 * Ax(F + k(20 - k))

= $1/100 * Ax(F + 20k - k²)

≠$1/100 * Ax(F + 20k - kx) (unless k = x)