Aptitude · Simple Interest · Model 2

When the question never mentions a rupee

Q: What condition makes a sum double under simple interest?

R × T = 100 . Doubling means the interest equals the principal, so the interest must be 100% of it — and the interest is (R × T)% of the principal. So 8 years needs 12.5%, and 10% needs 10 years. Tripling is the same reasoning with 200.

“A sum becomes five times itself.” “The interest is ₹4500 less than the principal.” “The interest is 3/8 of the amount.” None of these give you a principal, and none of them need one — because the principal always cancels.

Set the multiple and the years, and watch the principal cancel →

If the amount becomes n times the principal, the interest is (n − 1) times the principal. Almost every question in this model opens with that one translation.

01 The idea

The principal cancels, so stop looking for it

Model 1 handed you numbers. Model 2 hands you a relationship: the sum doubles, the interest falls short of the principal by ₹4500, the principal and amount are in the ratio 5 : 6. There is often no principal anywhere in the question, and students freeze looking for one.

You do not need it. Write the principal as P, and it divides out of both sides of every equation in this model. That is not a trick — it is the direct consequence of interest being a fixed percentage of the principal, so any statement comparing interest to principal is a statement about R and T alone.

The single most useful translation is the one in the box above. If a sum becomes 5 times itself, the amount is 5P, so the interest is 5P − P = 4P. Not 5P. Students who write 5P here get every question in the model wrong by the same factor, so it is worth over-learning.

Once the interest is expressed as a multiple of the principal, set the principal to 100. Then “interest is 4P” becomes “interest is 400%”, and dividing by the years gives the rate immediately. Choosing your own principal is legitimate precisely because the answer cannot depend on it.

Amount = nP means interest = (n − 1)P. Set P = 100, read the interest as a percentage, divide by the years, and you have the rate.

n timesThe amount, not the interest. “Becomes 5 times” means A = 5P, so SI = 4P. Doubling means the interest equals the principal exactly.

The 100-boxSetting P = 100 by choice. Legitimate because the principal cancels, and it turns every multiple and ratio in the question straight into a percentage.

Linear growthUnder simple interest the same amount is earned every year, so total interest is proportional to time. That is why time needed is proportional to the interest multiple at a fixed rate.

02 Worked example

10 times in 12 years — so when is it 13 times?

One question, two halves, and the second half is where the model earns its keep. A sum becomes 10 times itself in 12 years at simple interest. In how many years will it become 13 times itself at the same rate?

Turn 10 times into interestThe amount is 10P, so the interest is the amount less the principal. This is the step that decides whether the rest is right.A = 10P ⇒ SI = 10P − P = 9P

Read it as a percentageSet the principal to 100. Then the interest of 9P is 900, which is 900% of the principal — earned over the full 12 years.P = 100, A = 1000, SI = 900 ⇒ SI = 900% of P

Get the rate, if you want itNine hundred percent spread evenly over 12 years. The rate is large because the growth is large; nothing is wrong.R = 900% / 12 = 75% per annum

Now the second half, without the rateBecoming 13 times needs interest of 12P. The rate is unchanged, so time is proportional to the interest: 9P took 12 years, so scale.9P → 12 yr 1P → 12/9 yr 12P → 12 × 12/9

Finish the proportionTwelve times twelve over nine. The answer is a whole number of years, as exam numbers usually are.12 × 12 / 9 = 144/9 = 16 years

The rate was computed in step 3 and then never used. That is deliberate — going through 75% works, but it invites a decimal and an extra chance to slip. At a fixed rate, time is proportional to the interest multiple, so 9P → 12P is just multiplying the time by 12/9. Reach for the proportion, keep the rate as a check.

03 The method

Every phrasing, and what it translates to

This model is really a translation exercise. Once the sentence becomes an equation in P, the arithmetic is trivial. The table is the phrasebook.

A = P(1 + R·T/100), and the translation that matters: A = nP ⇒ SI = (n − 1)P. With P = 100 that reads as SI = (n − 1) × 100%, so R = (n − 1) × 100 / T.

At a fixed rate, time is proportional to the interest. If a sum reaches n times in T years, it reaches m times in T × (m−1)/(n−1) years. This one line answers the most common Model 2 question with no rate and no formula.

The question says	It means	As a percentage of P
Becomes double	SI = P	100%
Becomes 5 times	SI = 4P	400%
Becomes 10 times	SI = 9P	900%
P : A = 5 : 6	SI is 1 part on 5	20%
P : A = 5 : 7	SI is 2 parts on 5	40%
Interest is ₹4500 less than P	P − SI = 4500	SI% below 100%
Interest is ₹950 more than P	SI − P = 950	SI% above 100%
Interest is 11/10 of P	SI = 1.1P	110%
Interest is 3/8 of the amount	8SI = 3P + 3SI, so SI = 0.6P	60%

05 Cheat sheet

Model 2 on one page

Rows one to three are the translations. Rows four and five are the two routes to an answer. Row six is the check that catches a wrong translation immediately.

Situation	Use this	Worked on 10 times in 12 years
Amount is n times P	`SI = (n − 1)P`	SI = 9P
P : A = a : b	`SI = (b − a)/a`	—
SI is a fraction of A	`SI/P = f/(1 − f)`	—
Rate, from a multiple	`R = (n−1)×100/T`	900/12 = 75%
Time, for a new multiple	`T₂ = T(m−1)/(n−1)`	12 × 12/9 = 16 yr
Doubling	`R × T = 100`	Interest equals principal exactly
Tripling	`R × T = 200`	Interest is twice the principal

n times is the amount, not the interestFive times means SI = 4P. Writing SI = 5P makes every answer in the model wrong by the same ratio, and it is the single most common error here.

Rates over 100% are fine900% earned over 12 years is 75% a year. Large multiples force large rates; there is nothing to correct.

Prefer the proportion to the rateTime scales with the interest multiple, so a second multiple needs no rate at all. Fewer decimals, fewer slips.

06 Where & why

Where Model 2 shows up

This is where simple interest stops being arithmetic and starts being comprehension, which is exactly why exams like it.

Bank PO · RRB · LIC

“Becomes n times” and ratio phrasings

The staple. Usually a one-line question with the multiple and the time given and the rate wanted — ten seconds if the translation is automatic.

TCS Digital · Infosys

The two-multiple question

“Becomes 10 times in 12 years, when 13 times?” The proportional route makes this instant; substituting into the formula twice makes it a minute.

SSC CGL · CHSL

Interest compared to principal in rupees

“The interest was ₹4500 less than the sum.” Convert both sides to percentages of the principal and the difference gives the answer in one division.

Compound interest, later

The same phrasings, a different rule

CI asks “becomes n times” too, but there the answer comes from powers rather than proportion. Knowing the simple case cold is what makes the contrast obvious.

If a question about interest contains no rupee figure at all, it is Model 2 and your first line should be a translation into a multiple of P — not a search for the principal.

07 Interview questions

The questions this model attracts

Ten, escalating from the core translation to the honest limits of the technique.

A sum becomes 5 times itself. What is the interest?

Four times the principal. The amount is 5P and the interest is the amount less the principal, so 5P − P = 4P. The commonest error in this model is calling it 5P, which scales every subsequent answer wrongly by 5/4.

Why are you allowed to just decide the principal is 100?

Because the principal cancels out of every equation in this model. The question only ever compares interest to principal, and that comparison is a statement about the rate and time. Setting P = 100 is choosing a convenient unit, not making an assumption — the answer is identical for any P.

A sum becomes 5 times itself in 25 years. Find the rate.

The interest is 4P, which with P = 100 is 400%. That 400% was earned over 25 years, so the rate is 400/25 = 16% per annum. You can check it forward: 16% for 25 years is 400% interest, giving an amount of 500% — five times the principal.

A sum becomes 10 times in 12 years. When does it become 13 times?

Sixteen years. Ten times means interest of 9P over 12 years, and 13 times needs 12P. At the same rate the time is proportional to the interest, so it is 12 × 12/9 = 16 years. Doing it through the rate of 75% gives the same answer with more arithmetic.

What condition makes a sum double under simple interest?

R × T = 100. Doubling means the interest equals the principal, so the interest must be 100% of it — and the interest is (R × T)% of the principal. So 8 years needs 12.5%, and 10% needs 10 years. Tripling is the same reasoning with 200.

The interest is ₹4500 less than the principal, at 5.5% for 10 years. Find the principal.

At 5.5% for 10 years the interest is 55% of the principal. So the shortfall against the principal is 100% − 55% = 45%, and that 45% is the ₹4500 you were given. Therefore 100% is ₹10,000. Converting both quantities to percentages of the principal is the whole method.

The interest for 20 years is 11/10 of the principal. Find the rate.

11/10 of the principal is 110% of it, earned over 20 years, so the rate is 110/20 = 5.5% per annum. Fractions of the principal go straight to percentages, which is why this type is easier than it looks.

The interest for 5 years is 3/8 of the amount. Find the rate.

Careful — it is a fraction of the amount, not the principal. Read it in parts: the amount is 8 parts and the interest is 3, so the principal is 8 − 3 = 5 parts. Then the interest is 3/5 of the principal, which is 60% over 5 years, giving 12% per annum.

P : A is 5 : 6 now, and 5 : 7 after another 4 years. Find the rate.

Hard

At a fixed rate of simple interest, the ratio of a principal to its amount is 4 : 5 after a certain period. After a further 5 years the ratio is 4 : 6. (a) Find the rate per annum. (b) Find the original period. (c) A student reasons: “the ratio went from 4 : 5 to 4 : 6, so the amount rose by one part in 5 years, and one part on an amount of 5 parts is 20%, so the rate is 20/5 = 4% per annum.” Their answer disagrees with yours. Say exactly which quantity they divided by, which one they should have divided by, and why the distinction is not a technicality.

Follow-up

Part (c) is the real question. Every percentage in this chapter is a percentage of the principal, and the student used the amount as the base. It is worth doing because this is the error that quietly wrecks ratio questions — the working looks orderly and the answer is close enough to a listed option to feel safe.

Show the hint

Both ratios are written on the same base of 4 parts. Express each interest as a share of that 4, and compare what you get with the student's 20%.