182-Day T-Bill vs. 91-Day T-Bill

Question

I am trying to understand how T-Bills work and it would be great if someone could explain me using the following question

At $t=0$ Smith buys a 182-Day T-Bill with a simple annual discount rate of $10\%$. At $t=91$ Smith sells his T-Bill. At that moment, the "prevailing quoted annual discount rate of a 91-Day T-Bill is also $10%$." Find the actual rate of return (91-Day interest rate) that Smith earned during the time he held the T-Bill.

I am thinking that if $d=10\%$ then $i=11.111 \dots \%$ .

So at $t=91$ the interest earned is approximately

$$\frac{1}{4}(11.111 \dots \%)$$

So, I want to say that Smith earned that much, but the answer seems to be about $2.66\%$.

How do we get this?

Answer 1

I have no idea whether this is correct but $$\frac{\left(100-10\times\frac{91}{365}\right)-\left(100-10\times\frac{182}{365}\right)}{\left(100-10\times\frac{182}{365}\right)}\times 100\approx 2.623991$$ while if you replaced $365$ by $360$ it would give about $2.662376$.

This looks like a stupid way to work out interest rates, but who said bankers were clever?