Chapter 5.4 — Exercise 5.3 — Simple and Compound Interest

Simple Interest

Sudhakar borrows ₹15,000 at 9% per annum simple interest over 8 years: A = 15000 × (100 + 8×9)/100 = 15000 × 172/100 = ₹25,800. Spread over 96 equal monthly instalments (8 years × 12), each payment is 25800/96 = ₹268.75.

Compound Interest

₹8,000 at 5% per annum for 2 years, compounded annually: A = 8000 × (105/100)² = ₹8,820, so the compound interest is 8820 − 8000 = ₹820. Unlike simple interest, this amount can't be found with one multiplication using the total time — each year's interest is calculated on the previous year's amount, not on the original principal.

When the Rate Changes Each Year

₹6,500 compounded annually at 5% in the first year and 6% in the second: after year one, A = 6500 × 105/100 = ₹6,825, and this becomes the principal for year two: A = 6825 × 106/100 = ₹7,234.50, for a compound interest of ₹734.50. Whenever the rate changes between periods, each period has to be computed separately in sequence — there's no single combined formula that skips straight to the final amount.

Compounding More Often Than Once a Year

When interest compounds half-yearly or quarterly rather than annually, both the rate and the number of periods need adjusting: halve the annual rate and double the number of periods for half-yearly compounding; quarter the rate and quadruple the periods for quarterly compounding.

• ₹80,000 at 10% for 1½ years: compounded annually (using 1 full year at 10%, then 6 months as simple interest on the result) gives ₹92,400. Compounded half-yearly instead (rate 5%, n = 3 periods) gives 80000 × (105/100)³ = ₹92,610 — ₹210 more, purely from compounding more frequently on the same nominal rate.
• ₹1,000 at 10% for 1 year, compounded quarterly: rate per quarter = 2.5%, n = 4 periods: A = 1000 × (102.5/100)⁴ ≈ ₹1,103.81, giving a compound interest of about ₹103.81 — noticeably more than the ₹100 a straightforward 10% simple interest would give.

Interest Across a Fractional Time Period

₹10,000 at 8½% per annum for 1 year and 3 months: the first full year compounds normally, A = 10000 × 108.5/100 = ₹10,850. For the remaining 3 months (¼ year), this amount becomes the new principal, and the same rate is applied proportionally: A = 10850 × (100 + ¼×8.5)/100 ≈ ₹11,080.56, giving a compound interest of about ₹1,080.56 over the full 1 year 3 months. The general pattern — compound over whole periods, then apply simple interest proportionally over the remaining fraction of a period — handles any mixed time span like this one.

Comparing Simple and Compound Interest Directly

Borrowing ₹12,000 at 6% for 2 years: simple interest gives A = 12000 × 112/100 = ₹13,440, while compound interest gives A = 12000 × (106/100)² = ₹13,483.20 — an extra ₹43.20 purely from compounding. A larger comparison, with different rates for each method: Bharathi borrows ₹12,500 at 12% simple interest for 3 years (interest = 12500×3×12/100 = ₹4,500), while Madhuri borrows the same amount at 10% compound interest for the same period (A = 12500×(110/100)³ = ₹16,637.50, interest = ₹4,137.50). Even with a lower rate, compounding over multiple years can land close to — or, given enough time, exceed — simple interest at a higher rate; here Bharathi still pays ₹362.50 more.

The Same Formula Describes Growth and Decline

Population growth, bacteria multiplying, and machinery depreciating are all the compound interest formula in disguise — growth uses (100 + R)/100, decline uses (100 − R)/100, applied repeatedly period by period:

• A population of 68,09,000 growing at 4.7% per year reaches roughly 81,82,199 after 4 years.
• A bacteria count of 5,06,000 growing at 2.5% per hour reaches roughly 5,31,216 after 2 hours.
• Machinery worth ₹10,000 depreciating by 5% is worth 10000 × 95/100 = ₹9,500 after 1 year.

Recognising "increases/decreases by a fixed percentage, repeated over several periods" is the cue to reach for the compound interest formula, even when the situation has nothing to do with borrowed money at all.

Why Compound Interest Eventually Overtakes Simple Interest

Simple interest adds the same fixed amount every period, since it's always calculated on the unchanging original principal. Compound interest adds a growing amount every period, since each period's interest is calculated on a principal that already includes all the previous periods' interest. Over a short time — a year or two, at ordinary rates — the difference between the two is small, often just a few rupees, exactly as seen in the ₹43.20 gap on a two-year, 6% loan above. But because compound interest's base keeps growing while simple interest's does not, the gap between them widens every additional period, eventually becoming large even at modest rates over enough years — which is exactly why compound interest is the standard for long-term savings and investment growth, while simple interest is more common for short-term loans.

Where Interest Calculations Quietly Go Wrong

• Using simple interest's single-step formula for compound interest. Compound interest needs the amount raised to a power (or period-by-period recalculation), not one multiplication using the total time.
• Forgetting to adjust rate and period count for half-yearly or quarterly compounding. Both the rate and the number of periods change together — halving one without doubling the other gives a wrong answer.
• Applying compound growth to a fractional final period incorrectly. Whole compounding periods use the compound formula; a leftover fraction of a period (like 3 extra months) is usually handled with simple interest on the amount accumulated so far.

Reading "n" Correctly in the Compound Formula

The exponent n in A = P(1 + R/100)ⁿ is always the number of times the interest is actually calculated and added, not simply "the number of years." For annual compounding over 2 years, n = 2. For the same 2 years compounded half-yearly, n = 4 (twice a year, for 2 years), and the rate used is also halved to match each shorter period. Miscounting n — using years directly even when compounding happens more often than yearly — is one of the most common sources of a wrong final amount in this exercise, even when every other part of the calculation is done correctly — always ask "how many times does the interest actually get added?" before writing down a value for n, since that single number, more than any other part of the setup, determines whether the final answer is right.

One Formula, Money or Otherwise

The compound-growth reasoning from this exercise — the same formula describing money, populations, and depreciating machinery — is exactly the pattern Direct and Inverse Proportions builds on when quantities grow or shrink together. For the percentage formulas this exercise assumes, revisit the Introduction to Ratios and Proportion and Exercise 5.2.