Two equal amounts are deposited in two banks at the same interest rate of 12%…

Simple Interest is SI = (P × R × T) / 100, where P is the principal, R is the annual rate, and T is the time in years. When the SAME principal earns simple interest over two different tenures at the SAME rate, the difference between the two interests depends only on the difference in the tenures: SI1 − SI2 = (P × R × (T1 − T2)) / 100.

1. Let the principal deposited in each bank be P (the two amounts are equal).

2. Interest earned over 7 years: SI1 = (P × 12 × 7)/100 = 84P/100.

3. Interest earned over 4.5 years: SI2 = (P × 12 × 4.5)/100 = 54P/100.

4. The difference between the two interests is given as Rs. 585: SI1 − SI2 = 585.

5. Substitute the expressions: 84P/100 − 54P/100 = 585, so 30P/100 = 585.

6. Solve for P: P = 585 × 100/30 = 1950.

Cross-check: for P = Rs. 1950, SI1 = 1950 × 12 × 7/100 = Rs. 1638 and SI2 = 1950 × 12 × 4.5/100 = Rs. 1053; the difference 1638 − 1053 = Rs. 585, matching the given condition.

Therefore, each amount deposited was Rs. 1950.