Aptitude · Compound Interest · Model 3

A different rate each year, and no average to find

Q: Give the exact two-year growth for rates of a% and b%.

a + b + ab/100 . For 10% and 20% that is 10 + 20 + 2 = 32%, matching 1.1 × 1.2 = 1.32. Note the plus sign on the product term — successive discounts have the same structure with a minus, because there the changes are downward.

When the rate changes from year to year, students look for an average rate to use instead. There isn’t one, and there doesn’t need to be — each year simply gets its own multiplier and you multiply them together.

Set a different rate for each year and multiply them out →

A = P(1 + a/100)(1 + b/100)(1 + c/100). One bracket per year, and the order never matters.

01 The idea

One multiplier per year

A sum earns 10% in the first year, 20% in the second and 25% in the third. The instinct is to average the rates — that would be about 18.3% — and use the standard formula. It gives the wrong answer, and the reason is worth understanding.

Compounding is repeated multiplication, and the average of a set of multipliers is not the multiplier of their product. Averaging 1.1, 1.2 and 1.25 gives 1.1833, but their product is 1.65 while 1.1833 cubed is about 1.657. Close, but not equal, and not what the question asked.

The correct method is simpler than the wrong one. Give each year its own bracket and multiply: P × 1.1 × 1.2 × 1.25. No average, no root, and no new formula — the fixed-rate formula was only ever the special case where all the brackets happened to be identical.

Two useful consequences. First, the order of the rates is irrelevant, because multiplication commutes — a question that presents the rates in a different order has the same answer. Second, if a question does ask for a single equivalent rate, it is the geometric mean of the multipliers, not the arithmetic mean of the rates.

There is no average rate to find. Give each year its own multiplier and multiply them together — that is the whole method.

Per-year multiplier(1 + rate/100) for that particular year. The fixed-rate formula is just the case where every year has the same one.

Overall growthThe product of all the multipliers, minus one. For 10%, 20% and 25% it is 1.1 × 1.2 × 1.25 = 1.65, so 65% overall.

Equivalent annual rateThe geometric mean: the n-th root of the product. Not the arithmetic average of the rates, which is a different and larger-looking number.

02 Worked example

₹10,000 at 10%, then 20%, then 25%

Three years, three different rates. ₹10,000 is invested at compound interest, earning 10% in the first year, 20% in the second and 25% in the third. Find the amount and the total interest.

Year one at 10%Ten per cent of ₹10,000, added on. Nothing unusual yet.10,000 × 1.10 = ₹11,000

Year two at 20%Twenty per cent of the ₹11,000 now standing — not of the original ₹10,000.11,000 × 1.20 = ₹13,200

Year three at 25%Twenty-five per cent of ₹13,200. Each year compounds on the last, exactly as with a fixed rate.13,200 × 1.25 = ₹16,500

Read off the interestAs always, the interest is the amount less the principal.CI = 16,500 − 10,000 = ₹6,500

The one-line versionAll three brackets at once. Twenty-five per cent is 5/4, which cancels neatly.10,000 × 1.1 × 1.2 × 1.25 = 10,000 × 1.65 = ₹16,500

Now test the averaging instinct. The arithmetic mean of 10, 20 and 25 is 18.33%, and 10,000 × 1.1833³ is about ₹16,570 — roughly ₹70 too high. The true equivalent annual rate is the cube root of 1.65, about 18.17%. Averaging the rates always overstates the growth, for the same reason that averaging speeds overstates the average speed of a journey.

03 The method

The method, and the average that is actually correct

One rule for the calculation, and one for the rare question that genuinely wants a single rate.

A = P(1 + a/100)(1 + b/100)(1 + c/100)… — one bracket per year, in any order. The overall growth is the product minus one.

If a question asks for a single equivalent annual rate, take the geometric mean. For n years, it is (product)^1/n − 1. For 10%, 20% and 25% that is 1.65^1/3 − 1 ≈ 18.17%, not the 18.33% arithmetic mean. For two years there is a neat exact form for the overall growth: a + b + ab/100 — note the plus, which is where successive-discount problems have a minus.

Rates	Product	Overall growth	Arithmetic mean (wrong)
10%, 20%	`1.32`	`32%`	`15% → 32.25%`
10%, 20%, 25%	`1.65`	`65%`	`18.33% → 65.7%`
5%, 10%	`1.155`	`15.5%`	`7.5% → 15.56%`
20%, 20%	`1.44`	`44%`	`20% → 44%`
10%, −10%	`0.99`	`−1%`	`0% → 0%`
Two years, exact	`a + b + ab/100`	—	—

05 Cheat sheet

Model 3 on one page

One method, one correct average, and the two things that are never right.

Case	Route	On 10%, 20%, 25%
Amount	`P ∏(1 + rᵢ/100)`	10,000 × 1.65 = 16,500
Overall growth	`∏(1 + rᵢ/100) − 1`	65%
Two years, exact growth	`a + b + ab/100`	10, 20 → 32%
Equivalent annual rate	`(product)^(1/n) − 1`	1.65^(1/3) → 18.17%
Order of the rates	`irrelevant`	same answer either way
Averaging the rates	`wrong`	18.33% overstates it
Adding the rates	`wrong`	55% ≠ 65%

There is no average rate to findEach year gets its own bracket. The fixed-rate formula is the special case where all the brackets are equal, not a rule you have to force the question into.

The order never mattersMultiplication commutes, so rates of 10% then 20% give exactly the same amount as 20% then 10%. Questions that vary the order are testing whether you know that.

A correct single rate is the geometric meanTake the n-th root of the product, not the average of the rates. The arithmetic mean always comes out too high.

06 Where & why

Where Model 3 shows up

Varying rates are how a paper checks whether you understood compounding as multiplication rather than as a formula to be filled in.

Bank PO · SSC CGL

Two or three stated yearly rates

The standard form. Straightforward once you stop looking for an average, and the trap answer is what averaging produces.

Population and growth questions

“grew 8% then fell 5%”

The same multiplication with one bracket below 1. A rise and an equal fall do not cancel, which is the point of these.

Successive percentage change

Shared machinery with discounts

Two successive changes of a% and b% give a + b + ab/100, with the sign of the product term following the signs of the changes.

Depreciation with changing rates

Machinery valued down each year

Identical arithmetic with multipliers under 1. The order still does not matter, and averaging still overstates.

This model is really a check on whether you see compound interest as repeated multiplication. If you do, a changing rate is not a complication at all — it is the general case, and the fixed rate was the special one.

07 Interview questions

What gets asked

Nine, and the second is the misconception the model is built around.

₹10,000 earns 10%, then 20%, then 25% in successive years. Find the amount.

₹16,500. Multiply one bracket per year: 10,000 × 1.1 × 1.2 × 1.25 = 10,000 × 1.65. The interest is ₹6,500 and the overall growth is 65%.

Can you average the rates and use the standard formula?

No. Averaging 10, 20 and 25 gives 18.33%, and 10,000 × 1.1833³ is about ₹16,570 — roughly ₹70 too high. The average of several multipliers is not the multiplier of their product, so averaging always overstates the growth.

Does the order of the rates matter?

No. The amount is the principal times the product of the brackets, and multiplication commutes. So 10% then 20% gives exactly the same result as 20% then 10%. Questions that present the rates in an unusual order are checking whether you know this.

If a question does want a single equivalent annual rate, how do you find it?

Take the geometric mean: the n-th root of the product of the multipliers, minus one. For 10%, 20% and 25% the product is 1.65, so the equivalent rate is 1.65 to the power one-third minus one, about 18.17% — below the 18.33% arithmetic mean.

Give the exact two-year growth for rates of a% and b%.

a + b + ab/100. For 10% and 20% that is 10 + 20 + 2 = 32%, matching 1.1 × 1.2 = 1.32. Note the plus sign on the product term — successive discounts have the same structure with a minus, because there the changes are downward.

A sum grows 10% in one year and falls 10% in the next. Where does it end up?

Down 1%. The multipliers are 1.1 and 0.9, and their product is 0.99. A rise and an equal fall never cancel, because the fall is applied to a larger base than the rise was. This is the same structural fact as a markup and an equal discount.

How does this relate to the fixed-rate formula?

The fixed-rate formula is the special case where every bracket is identical, so the product becomes a power. Varying rates are the general case. Seeing it that way means you do not need a separate method — you need one method with fewer assumptions.

₹8,000 earns 5% then 10%. Find the compound interest.

Hard

(a) For two years at rates a% and b%, show algebraically that the overall growth is a + b + ab/100. (b) Show that compounding at the arithmetic mean (a+b)/2 for two years gives an overall growth of a + b + (a+b)²/400. (c) Hence prove that the averaging method always overstates the growth unless a = b, and identify the exact size of the overstatement in terms of a and b.

Follow-up

This is the model’s claim made rigorous. The overstatement works out to (a − b)²/400, which is zero exactly when the rates are equal and positive otherwise — so the error is not a rounding artefact but a structural consequence of the rates differing.

Show the hint

Subtract your part (a) result from your part (b) result and factorise what is left.