Predicted future rate and Expected future price ( Future contract)

Dear tutors,

I am confused that

i) Question 4 – Specimen Sept 2018, using Futures contract, calculating the future rate in four month’s time = spot rate + ((future rate – spot rate)*4/6) with future rate in the six months’ time.

ii) whereas for the interest rate using the future contract:
In the case of interest rate increasing by 1.1% to 5.3%
Why we calculate the future price on 31 Jan 2018 = Spot price + basis * 4/6 ( Now: 01 Oct 2017? Instead, we must calculate based on the 5.3% and minus unexpired basis?

Thanks your for your explaination.