Comparing simple-interest arrangements

Learning outcome

Compare simple-interest arrangements on clearly stated bases, including unequal terms and changing rates.

Concepts and assumptions

For simple interest, I = P × r × t ÷ 100. A comparison must identify which quantities are equal. With the same principal and time, interest is proportional to the annual rate. With the same principal but different terms, compare r × t, not rates alone.

If principals differ, compare P × r × t to find the actual interest amounts. More rupee interest does not necessarily mean a higher rate: more money or more time may explain the difference. Equal interest also does not imply equal final amounts when principals differ.

For two arrangements with the same positive principal, equal simple interest requires r1 × t1 = r2 × t2. This shortcut cancels a common principal; it is invalid when the principals differ.

If a simple-interest rate changes, calculate each segment on the unchanged original principal and add the interest. An equivalent annual simple rate is the time-weighted mean of segment rates. Equal durations allow ordinary averaging; otherwise use time weights.

Assume no additional deposits, withdrawals, interim payments or fees. Rates are annual simple rates; interest never becomes principal. Use 12 months per year and a 365-day year for day-based questions unless another convention is stated. Keep intermediate values exact; round final money to ₹0.01 only when needed, rounding half a paise upward. Compare numerical results, not unstated features of an arrangement.

Worked examples

Example 1 — Equal principal and time. Compare ₹11,500 for 2 years at 8% and 9.5%. First interest = 11500 × 8 × 2 ÷ 100 = ₹1,840. Second interest = 11500 × 9.5 × 2 ÷ 100 = ₹2,185. Difference = 2185 - 1840 = ₹345.

Example 2 — Different durations. Compare ₹9,600 at 10% for 9 months with the same principal at 6% for 15 months. Interests are 9600 × 10 × (9/12) ÷ 100 = ₹720 and 9600 × 6 × (15/12) ÷ 100 = ₹720. Equal interest is earned over different durations; the annual rates are not equal.

Example 3 — Changing simple rates. ₹14,000 earns 6% for the first 6 months and 8% for the next 18 months. Interest = 14000 × 6 × 0.5/100 + 14000 × 8 × 1.5/100 = 420 + 1680 = ₹2,100. A constant 7.5% for 2 years also gives 14000 × 7.5 × 2/100 = ₹2,100. The equivalent rate is (6 × 0.5 + 8 × 1.5) ÷ 2 = 7.5%.

Common mistakes

Do not compare rates without checking principal and time. Do not compound when the question specifies simple interest. Segment interest uses its own duration, not the entire term.

Practice questions

1. On ₹16,800 for 15 months, find the interest difference between 7% and 8.5% simple interest.
2. Compare interest on ₹5,000 at 12% for 2 years and ₹12,000 at 8% for 2 years.
3. The same ₹9,100 earns equal interest at 8% for 15 months and at 10% for an unknown time. Find that time.
4. Compare ₹18,000 at 6% for 4 months followed by 9% for 8 months with 8% throughout one year.

Worked answers

1. Time = 15/12 = 1.25 years. Interests are ₹1,470 and ₹1,785, from 16800 × 7 × 1.25/100 and 16800 × 8.5 × 1.25/100. Difference = ₹315.
2. Interests are 5000 × 12 × 2/100 = ₹1,200 and 12000 × 8 × 2/100 = ₹1,920. The second is ₹720 greater despite its lower rate.
3. Cancel the common principal: 8 × (15/12) = 10 × t. Thus t = 1 year, or 12 months. Each interest is ₹910.
4. Segment interests are 18000 × 6 × (4/12)/100 = ₹360 and 18000 × 9 × (8/12)/100 = ₹1,080. Their sum is ₹1,440, equal to 18000 × 8 × 1/100.