Divisional performance

Please, help with following:
Business Solutions is a firm of management consultants, which experienced considerable business
growth during the last decade. Recently the firm’s senior managers had begun to experience difficulties
in managing the business, so at the end of last year the firm was reorganised and a regional divisional
structure was introduced with individual profit targets being set for each of the semi-autonomous profit
centres. Although North division has its own customer base that is distinct from that of its sister
division South, it does occasionally call upon the services of a South consultant to assist with its
projects. North has to pay a cross charge to South per consulting day. HQ determines the amount of
the charge. North is free to choose whether it employs a South consultant or subcontracts the project to
an external consultant. The manager of North division believes that the quality of the external
consultant and the one from South division are identical and on this basis will always employ the one
who is prepared to work for the lower fee.
The following information is also available:
? North division is very busy and it charges its clients $1,200 per consulting day;
? North division pays its external consultant $500 per consulting day;
? The variable cost per internal consulting day is $100.
Required:
Determine a possible optimal daily cross charge that should be paid by North for the services of a
consultant from South for each of the following scenarios:
(a) South division has spare consulting capacity;
(b) South division is fully occupied earning fees of $400 per consulting day;
(c) South division is fully occupied earning fees of $700 per consulting day.

In solution for (b) it is explained:If North uses the external consultant, the total contribution for one day’s consultancy in both
North and South divisions would be:
Income Variable cost
North 1,200 500
South 400 100
––––– –––––
1,600 – 600 = $1,000
––––––
If the South consultant goes to North, then the total contribution would be:
$1200 – $100 = $1,100
Therefore it is best if North employs the South consultant – by setting the cross charge above
$400 and below $500, say $450, both parties will benefit and agree to the transaction.

I didn’t understand this explanation.Maybe I don’t understand meaning of cross-charge but in my opinion, the above explanation is contradicting:if we use South, then cross charge is 100.Why then this 400 -500 appeared?I myself solved it as 400-500 but using simpler logic:it’s not advantageous for South to provide service below 400 then it is minimum price.Then it is not acceptable for North to pay more than 500 then it’s maximum price.
But I suppose in above explanation theré is smth important I miss.Could you please explain, what they mean.
Thank you.