I am trying to do this fixed income futures question by calculating the QF for the underlying bond, then comparing it with the QF of the futures contract.

Quoted futures price
Quoted bond price

Conversion factor
Accrued interest since last coupon payment

Time remaining to contract expiration
Three months
Accrued interest at futures contract expiration

Accrued interest over life of futures contract

Risk-free rate is 0.30%

a) QF of the underling bond:

F=[(112+0.08)](1.003)3/12 - 0.2 = 111.9639656

QF= 111.9639656/0.9 = 124.40441

b) Difference between the QFs of the futures contract and underlying bond: 125-124.40441 = 0.595559
c) Discounting the above back to the present: (.59559)/(1.003)3/12=0.59514

The answer is 0.5356.

Apparently my step c) is wrong and it should be (.59559 *.9)/(1.003)3/12. Can someone explain why after finding the QF difference  in step b), I need to re-convert the QF difference back to the underlying bond price before discounting back to the present?

Thanks in advance!