Sheila wants to deposit $2,145 into savings accounts at three different banks: Bank of the US, 7/4 Bank, and Catch Bank. She will deposit five times as much into Catch Bank as 7/4 Bank because they offer a higher interest rate. She also expects the Bank of the US deposit to be 30% of the total of the other two deposits. How much will Sheila deposit into the Bank of the US savings account?

Let x = deposit for Bank of US; y = deposit for 7/4 Bank and z = deposit for Catch Bank. Since we have 3 unknowns, we have to formulate 3 independent equations from the problem statement.

She will deposit five times as much into Catch Bank as 7/4 Bank = > z = 5y

Bank of the US deposit to be 30% of the total of the other two deposits = >

x = 0.3 (y+z)

Total deposit = > $2145 = x + y + z

There, we have formulated three. Let's assign these equations:

Eq. 1: z = 5y

Eq. 2: x = 0.3 (y+z)

Eq. 3: $2145 = x + y + z

Substitute Eq 1 to Eq 2, we get an equation for x in terms of y. This would be Eq. 4. Then, we substitute Eq 4 to Eq 3, to solve for y.

Eq. 1 to Eq. 2:

x = 0.3 (y + 5y) = 0.3 (6y)

x = 1.8y = > Eq. 4

Eq 4 to Eq 3:

$2145 = 1.8y + y + 5y = 7.8y

y = $2145/7.8 = $275

Substitute y to Eq. 4 to solve for x:

x = 1.8y = 1.8 ($275) = $495

Substitute y to Eq 1 to solve for z:

z = 5y = 5 ($275) = $1375

Let's counter-check if we got the right answers

$2145 = x + y + z = $495 + $275 + $1375

$2145 = $2145 (Correct)

Thus, these are the deposits for each bank:

Bank of US: $495

7/4 Bank: $275

Catch Bank: $1375