DS: Interest & Growth – Level 3

Solution (E): Interest in 2nd year = (Amount after Year 1) \(\times r\). Amount after Year 1 = \( 1000(1+r) \). 2nd Year Interest = \( 1000(1+r) \times r = 1000r + 1000r^2 \). We need to know if \( 1000r + 1000r^2 > 50 \). (1) \(P=1000\). No rate. (2) \(r > 0.04\). No Principal. Combined: Let \(r = 0.041\) (4.1%). \( 1000(0.041) + \text{small} \approx 41 \). (No). Let \(r = 0.10\) (10%). \( 1000(0.1) + \dots = 100+10 = 110 \). (Yes). Result depends on the specific rate. Not sufficient.

Solution (A): (1) If rates are equal, more frequent compounding (Quarterly) ALWAYS yields more than Annual compounding for any time \(t > 0\) (except exactly 0). Since investment implies \(t>0\), Bank B yields more. Sufficient. (2) No rates given. Not sufficient.

Solution (D): (1) 100 to 400 is multiplication by 4. \(4 = 2^2\). This is 2 doubling periods. 2 periods = 10 hours \(\implies\) 1 period = 5 hours. Sufficient. (2) 400 to 800 is multiplication by 2. This is exactly 1 doubling period. 1 period = 5 hours. Sufficient.

Solution (B): \( SI = P(2r) \). \( CI = P((1+r)^2 – 1) = P(1 + 2r + r^2 – 1) = P(2r + r^2) \). Ratio \( \frac{SI}{CI} = \frac{P(2r)}{P(2r+r^2)} = \frac{2r}{2r+r^2} = \frac{2}{2+r} \). The ratio depends ONLY on the rate \(r\), not the Principal \(P\). (1) Gives P. Not sufficient. (2) Gives rate. Sufficient.

Solution (C): (1) Total growth is 20%. Without time \(t\), we can’t find annual rate. (e.g., if t=1, rate=20%. If t=100, rate < 1%). Not sufficient. (2) Time = 3. No growth info. Not sufficient. Combined: \( 1.2X = X(1+r)^3 \). \( (1+r)^3 = 1.2 \). Check 5%: \( (1.05)^3 \approx 1.157 \). Since \( 1.2 > 1.157 \), the rate \(r\) must be greater than 5%. Sufficient.