CI − SI Difference: The 2-Year and 3-Year Formulas, Derived

Simple interest (SI) is charged on the original principal every year. Compound interest (CI), with annual compounding, is charged on the principal plus all interest added so far. For the first year the two are identical. From the second year on, CI also earns interest on interest, and the gap between CI and SI is exactly that interest on interest.

Questions on the CI − SI difference can be solved by computing both values in full, but the two short formulas below are faster and, once you see where they come from, easy to rebuild.

Let the principal be P and the rate r% a year, and write R = r/100. Over 2 years, SI = 2PR. The compound amount is P(1 + R)² = P(1 + 2R + R²), so CI = 2PR + PR². Subtracting:

In words: the extra amount is the first year's interest (PR) earning interest for one more year (× R).

The long way agrees. The compound amount is 10,000 × 1.21 = 12,100, so CI = 2,100, while SI = 2,000; the difference is 100.

Over 3 years, SI = 3PR and P(1 + R)³ = P(1 + 3R + 3R² + R³), so CI = 3PR + 3PR² + PR³. Subtracting:

The 3R² part comes from three pairs of years in which interest earns interest; the R³ part is interest on interest on interest. That is why the 3-year difference is a little more than three times the 2-year difference.

Check in full: the amount is 8,000 × 1.157625 = 9,261, so CI = 1,261; SI = 3 × 400 = 1,200, and the difference is 61.

Rearranging the 2-year formula gives P = difference ÷ (r/100)² and r = 100 × √(difference ÷ P).

Dividing the two formulas gives a useful ratio: the 3-year difference ÷ the 2-year difference = 3 + r/100. If you know one difference and the rate, you get the other without finding P.

Banks and exam questions often compound interest every six months. The formulas still work if you count in periods instead of years. Halve the annual rate to get the rate per period, and double the number of years to get the number of periods. Simple interest is unaffected, because it never compounds.

The common question is CI − SI for one year with half-yearly compounding. One year is two half-year periods, so this is the two-period difference at half the annual rate: P × (r/200)².

In full: the amount is 10,000 × 1.1² = 12,100, so CI is 2,100, while SI for one year at 20% is 2,000. The difference is 100, as the formula says.

Quarterly compounding works the same way, with a quarter of the rate and four times the number of periods. Beyond three periods, compute the amount directly rather than extending the difference formulas.

• ✘ Using the 2-year formula for 3 years. The 3-year difference needs the extra factor (3 + r/100).
• ✘ Forgetting to square the rate as a fraction: at 8%, (0.08)² is 0.0064, not 0.64.
• ✘ Applying the formulas to half-yearly compounding without converting to periods.
• ✘ Mixing up which value the question asks for: CI, SI, the amount, or the difference.

The same multiplier thinking appears in successive discounts. A broader formula list is in the Quant cheat code.

Why are CI and SI equal for the first year?

With annual compounding, interest is first added to the principal at the end of year one. During that first year both methods charge interest on the original principal only, so the interest for year one is identical and the difference starts in year two.

Do these formulas work for half-yearly compounding?

Yes, if you work in periods instead of years. Halve the annual rate and double the number of periods, then apply the two-period or three-period formula with the per-period rate.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

• Quant Cheat Code: 20 Must-Know Shortcuts & Formulas