Capital Allowance on cars

I found a question where I do not agree with the answer. Could you please help me to decide who has wrong?

Having a haulier company with the following information.
01/04/2014 TWDV b/f £18,000 (main pool)
10/04/2014 equipment purchase £30,200
01/09/2014 overhead crane purchase £510,000
05/02/2015 motor car 1 purchase (CO2 187g/km) £13,600
31/03/2015 motor car 2 purchase (CO2 117g/km) £ 9,400

Overhead crane is planned to be used for a min. 30 years
motor car 2 is used by the sales manager and 20% of the mileage is private use.

Question is how much writing down allowance they are entitled to.

My answer would be: the equipment and the crane qualifies to AIA but because it can be max. 500,000, the excess amount goes to the pools (30,200 to the main pool, 10,000 to the special rate pool). Motor car 1 is a high CO2 car, goes to the special rate pool. Motor car 2 – because it has private usage – goes to a separate category, medium CO2 car with PU.
Adding these figures and calculate the 18% and 8% WDA, I got 7,380 allowance in the main pool, 1,888 allowance in the special rate pool and 1,692 allowance regarding to the car. BUT, because there is a 20% private use of the car, I would calculate the 80% of the 1,692 which is 1354 and with this the total allowance would be 510,622.

The book answer is: the motor car 2 has been put in the main pool and got 510,960 as allowance at the end.