I dont understand the answer key with this math question (GRE exam):
The toal amount that Bob paid for a lock was equal to the price of the lock plus a tax that was 4% of the price of the lock. Bob paid for the lock with 10 dollar bill and received the correct change, which was less than 3 dollar. Which of the following statements must be true?
a) The price of the lock was less than 9.5 dollar
b) The price of the lock was greater than 6.9 dollar
c) The tax was less than 0.45 dollar
My attempt:
let $P$=price of the lock. $T$=the tax dollar
$0<10-1.04P<3$
$6.73<P<9.615$, and $0.269<T<0.38$
Therefore, I think both A and B are correct. But the answer key says the opposite, it says only C is correct. Answer key says A and B are not correct because that a price could be as high as 9.61, which is out of the range of a) and still be valid according to our calculation. However, I could also argue that a tax dollar of 0.39 (which is less than 0.45 as indicate in c) )doesn't nesscarry fall under my calculation range of T.
I am really having trouble to under the logic behind these arguments provided by the answer key.