Base 5 help! determine if the solution is correct or incorrect

Question

Okay so the question is : Below is a solution for rewriting/representing 321(five) in a different way. Determine if the solution is correct or incorrect and why (by relating it to the original number). 1(five) flat, 12(five) rods, 101(five) small squares

This is really confusing to me, base-5 really messes with my head. I have been working on this problem for 4 hours. Every time I come up with an explanation, I read it and it does not make any sense. Please help! :(

Answer 1

To make this easier, first look at the number $325$ in standard base $10$. One can clearly see that $$325=100\cdot 3+10\cdot 2 +1\cdot 5$$ Or in other words, three $10$ "flats", two $10$ "rods", and five "small squares". However in base $5$, the "flats" are $25=5^2$, the "rods" are $5=5^1$, and the "small squares" are $1=5^0$. Thus, in base $5$, \begin{align*} 1\space\text{base five"flats"}+&12\space\text{base five "rods"}+101\space\text{base five "small squares"} \\ & =1\cdot (25)+12\cdot(5)+101\cdot (1) \\ & =25+60+101=186\neq 325 \end{align*}

Which shows that this representation does not add up to $325$, and thus the solution is incorrect.