A$.$ Intercontinental Bank

First $6$ months: $$15\mathrm{.}00/$month $($balance above $$5,000)$

Last $6$ months: $$25\mathrm{.}00/$month $($balance below $$5,000)$

Total Annual Cost:

$(6\backslash$times $15)+(6\backslash$times $25)=90+150=240$

B$.$ City National Bank

Monthly Fee: $$4\mathrm{.}50$

Checks: $30$ checks $-10$ free $=20$ checks charged at $$0\mathrm{.}50$ each

Debit Transactions: $3$ transactions at $$1\mathrm{.}00$ each

Monthly Cost:

$4\mathrm{.}50+(20\backslash$times $0\mathrm{.}50)+(3\backslash$times $1\mathrm{.}00)=4\mathrm{.}50+10+3=17\mathrm{.}50$

Total Annual Cost:

$17\mathrm{.}50\backslash$times $12=210$

C$.$ Bank of America

Monthly Fee: $$6\mathrm{.}00$

Checks: $30$ checks at $$0\mathrm{.}25$ each

Debit Transactions: $3$ transactions at $$2\mathrm{.}00$ each

Monthly Cost:

$6+(30\backslash$times $0\mathrm{.}25)+(3\backslash$times $2\mathrm{.}00)=6+7\mathrm{.}50+6=19\mathrm{.}50$

Total Annual Cost:

$19\mathrm{.}50\backslash$times $12=234$

D$.$ First Union Bank

Monthly Fee: $$9\mathrm{.}00$

Checks: $30$ checks at $$0\mathrm{.}15$ each

Debit Transactions: $3$ transactions at $$1\mathrm{.}50$ each

Monthly Cost:

$9+(30\backslash$times $0\mathrm{.}15)+(3\backslash$times $1\mathrm{.}50)=9+4\mathrm{.}50+4\mathrm{.}50=18$

Total Annual Cost:

$18\backslash$times $12=216$

Conclusion

Based on the calculations, City National Bank offers the lowest annual cost at $$210.$ Therefore, Sienna should choose City National Bank for her checking account.