Model 5: Times-Time and Repeated Growth

Compound Interest · 25 min

Aptitude · Compound Interest · Model 5

Doubles in 5 years, quadruples in 10 — not triples

Under compound interest the multiples multiply rather than add. Money that doubles in five years is four times over in ten and eight times in fifteen. Under simple interest the same start would only be three times over in ten, and confusing the two is the most punished error here.

Set a multiple and a span, then stack it →
CI: the multiples multiply — doubles in T, so 4× in 2T, 8× in 3T. SI: the interest is proportional to time, so doubles in T means 3× in 2T.

01 The idea

Stacking the same span

A sum doubles itself in 5 years at compound interest. What does it do in 10 years? The answer is that it becomes four times, not three — and the reason is one line.

Doubling in five years means the five-year multiplier is 2. Ten years is that span twice over, so the multiplier is 2 × 2 = 4. Fifteen years is three spans, giving 8. The multiples multiply because compounding multiplies, and there is no need to find the rate at any point.

Compare with simple interest, where the interest is proportional to time rather than the multiple. If a sum doubles in five years under simple interest, the interest earned equals the principal, so in ten years it earns twice that and the sum becomes three times, not four. The same sentence gives different answers under the two rules, which is precisely why examiners set it.

Because no rate is ever needed, these questions look much harder than they are. If a sum becomes 8 times in 12 years, notice 8 = 2³, so three equal spans of 4 years each doubled it — and it doubles in 4 years. Recognising the multiple as a power is the whole technique.

If the multiple is x over a span of T, then over kT it is xk. Read the multiple as a power and the answer needs no rate at all.
Span multiplierThe factor the money grows by over a stated period. Doubling in 5 years means the 5-year multiplier is 2, whatever the rate happens to be.
Multiples multiplyOver k copies of the same span, the multiple is the k-th power. This is the compound rule and it is what distinguishes it from simple interest.
Recognising powers4 = 2², 8 = 2³, 9 = 3², 27 = 3³, 16 = 2⁴. Exam multiples are almost always powers of a small number, which is the intended route in.

02 Worked example

Doubles in 5 years — what about 10, 15 and 20?

The model in its purest form. A sum of money doubles itself in 5 years at compound interest. In how many years will it become 4 times, 8 times and 16 times itself at the same rate?

1
Translate the doublingFive years multiplies the money by 2. That fixes the five-year multiplier without telling you the rate.(1 + R/100)⁵ = 2
2
Stack two spans for 4 timesTen years is two lots of five years, so the multiplier is applied twice.(1 + R/100)¹⁰ = 2 × 2 = 4  ⇒  4 times in 10 years
3
Three spans for 8 timesFifteen years is three spans. The multiples keep multiplying.2³ = 8  ⇒  8 times in 15 years
4
Four spans for 16 timesTwenty years, and the pattern is now clear: the multiple is 2 raised to the number of spans.2⁴ = 16  ⇒  16 times in 20 years
5
Contrast with simple interestUnder simple interest the interest is proportional to time, so doubling in 5 years means 1P of interest per 5 years.SI: 4 times needs 3P interest ⇒ 15 years, not 10

That last line is the whole examinable point. Compound interest gives 4 times in 10 years; simple interest needs 15 years for the same multiple. A question that says “doubles in 5 years” and asks about 4 times has two entirely different answers depending on which rule applies, and both will be in the options. Always check which interest the question named.

03 The method

The rule, its reverse, and the rate if you really need it

One rule forwards, one backwards, and a note on when a rate is genuinely required.

If the multiple is x over T years, it is xk over kT years. Equivalently, to reach a multiple M, the time is T × log M / log x — but exam numbers always make this an integer you can spot.
Read the multiple as a power. Becomes 8 times in 12 years? 8 = 2³, so three spans of 4 years, and it doubles in 4. Becomes 9 times in 6 years? 9 = 3², so two spans of 3 years, and it triples in 3. You only need the actual rate when the question asks for it, and then it is R = (x1/T − 1) × 100.
GivenCompound interestSimple interest
Doubles in 5 yr4× in 10, 8× in 153× in 10, 4× in 15
Triples in 4 yr9× in 8, 27× in 125× in 8
4 times in 6 yrdoubles in 3 yr—
8 times in 12 yrdoubles in 4 yr—
9 times in 6 yrtriples in 3 yr—
16 times in 8 yrdoubles in 2 yr—
Adding the multiplesthat is the SI rule—

05 Cheat sheet

Model 5 on one page

The forward rule, the backward reading, and the simple-interest contrast that the whole model is set to test.

CaseRouteDoubles in 5 years
Multiple over k spansx^k4× in 10, 8× in 15
Span from a known multipleread it as a power8× in 15 → 2³
Rate, only if asked(x^(1/T) − 1) × 100≈ 14.87%
Simple interest insteadinterest ∝ time3× in 10 yr
Doubling condition, SIR × T = 100—
Adding multiples under CIwrongnot 2+2 = 4× in 10
Powers to recognise4, 8, 9, 16, 27, 322², 2³, 3²…
Multiples multiply under compoundingDoubling twice is quadrupling, not tripling. This is the single most tested point in the model and the trap answer is always the simple-interest one.
No rate is neededEvery forward and backward question here is answerable from the multiple and the span alone. Reaching for the rate first is what makes these look hard.
Check which interest the question names“Doubles in 5 years” gives 4× in 10 under CI and 3× in 10 under SI. Both answers will be in the options.

06 Where & why

Where Model 5 shows up

A short, sharp model that appears in almost every bank paper because the wrong answer is so easy to reach.

Bank PO · RRB · LIC
“Doubles in n years, when 4 times?”

Set constantly. Ten seconds if you know the multiples multiply, and the simple-interest answer is always among the options.

SSC CGL
The reverse reading

“Becomes 8 times in 12 years — when does it double?” Recognise 8 as 2 cubed and the answer is 4 years.

Comparison questions
SI against CI on the same phrasing

“A sum doubles in 5 years. Compare the time to quadruple under simple and compound interest.” Fifteen years against ten.

Population and depreciation
The same arithmetic, different words

“A population doubles every 30 years” behaves identically, and so does halving for depreciation with the multiple below 1.

The instinct to build: when a question gives a multiple and a time, look for the power before you look for the rate. Almost none of these questions need a rate, and the ones that do say so explicitly.

07 Interview questions

What gets asked

Nine, and the first three cover the entire model.

A sum doubles in 5 years at compound interest. When does it become 4 times?
In 10 years. Doubling in 5 years means the 5-year multiplier is 2, so 10 years applies it twice and gives 4. The multiples multiply, so the answer is not 15 years.
Why isn't the answer 15 years?
Because 15 years would be the simple-interest answer. Under simple interest the interest is proportional to time: doubling in 5 years means earning 1P of interest per 5 years, so 4 times needs 3P of interest and therefore 15 years. Under compounding it is the multiple, not the interest, that stacks.
A sum becomes 8 times in 12 years. When does it double?
In 4 years. Recognise 8 as 2 cubed, so 12 years is three equal spans and each one doubles the money. No rate is needed anywhere — reading the multiple as a power is the whole method.
A sum triples in 4 years. When does it become 9 times?
In 8 years, since 9 = 3² is two spans of 4 years. And 27 times would take 12 years. The same pattern applies for any base multiple, not just doubling.
Do you ever need the actual rate in this model?
Only if the question asks for it. Then it is R = (x^(1/T) − 1) × 100 — for doubling in 5 years, about 14.87%. That is an irrational number, which is exactly why questions are phrased in multiples rather than rates.
Compare the two rules on the same sentence.
“A sum doubles in 5 years.” Under compound interest it is 4 times in 10 years and 8 times in 15. Under simple interest it is 3 times in 10 and 4 times in 15. Same premise, entirely different answers, which is why the question always specifies which interest applies.
What multiples should you learn to recognise?
The small powers: 4 = 2², 8 = 2³, 16 = 2⁴, 32 = 2⁵, 9 = 3², 27 = 3³. Exam multiples are essentially always one of these, so spotting the base and the exponent is the intended route.
A sum becomes 16 times in 8 years. When does it become 4 times?
In 4 years. Sixteen is 2⁴, so 8 years is four doubling spans of 2 years each. Four times is two doublings, which is 4 years. Working through the doubling time is usually the cleanest route for these.
How does this connect to the simple interest n-times model?
They are mirror images and worth learning together. Under simple interest the useful translation is that becoming n times means interest of (n−1)P, and time is proportional to that. Under compounding the multiple itself is what compounds. Knowing both, and checking which the question wants, is what stops the two from blurring.

08 Practice problems

Six on stacking multiples

None of the first five needs a rate. Read every multiple as a power before doing anything else.

Forward

Easy
A sum of money doubles itself in 7 years at compound interest. In how many years will it become 8 times itself at the same rate?
Follow-up
Eight is a power of two, so this is a matter of counting spans rather than calculating. Check your answer is not the simple-interest one.
Show the hint
How many doublings make eight?

Backward

Easy
A sum becomes 4 times itself in 6 years at compound interest. In how many years does it double?
Follow-up
The reverse reading, and it should take one line. Four times is two doublings, so the six years covers two equal spans.
Show the hint
Split the 6 years into the number of doubling spans.

A different base

Medium
A sum triples itself in 5 years at compound interest. In how many years will it become 27 times itself?
Follow-up
The base here is 3 rather than 2, and 27 is a power of it. Nothing about the method changes, which is the point of varying the base.
Show the hint
27 is 3 cubed.

Compare the two rules

Medium
A sum doubles itself in 8 years. Find the time it takes to become 4 times itself (a) if the interest is compound, and (b) if the interest is simple. Explain the difference in one sentence.
Follow-up
The same premise gives two different answers, and being able to say why in one sentence is worth more than either number. Both will appear as options in real questions.
Show the hint
Under compounding the multiples stack; under simple interest the interest earned stacks.

Not a whole number of spans

Medium
A sum becomes 9 times itself in 6 years at compound interest. (a) In how many years does it triple? (b) In how many years does it become 81 times? (c) Can you find, without logarithms, the time for it to become 27 times?
Follow-up
Part (c) is the interesting one: 27 is an odd power of 3, so it needs an odd number of tripling spans — which is fine, and gives a whole number of years. Not every multiple needs an even count of spans.
Show the hint
Find the tripling time first, then count how many triplings each target needs.

Where the two rules cross

Hard
A sum doubles itself in 10 years. (a) Under compound interest, find the multiple after 30 years. (b) Under simple interest, find the multiple after 30 years. (c) Find the compound rate and the simple rate implied by the original doubling, and explain why the simple rate is the larger of the two even though compound interest produces the bigger multiple over 30 years.
Follow-up
Part (c) is the resolution of an apparent paradox: to double in the same 10 years, simple interest needs a higher rate (10% against about 7.18%), because it never earns interest on interest. Over longer periods the compound rule wins anyway, and seeing both facts at once is what makes the two models finally separate.
Show the hint
For (c), the simple rate comes from R × T = 100 and the compound rate from the tenth root of 2.