Aptitude · Simple Interest · Model 1

Three numbers given, one missing, and always the same two percentages

Q: State the simple interest formula and say what each letter is.

SI = P × R × T / 100 , where P is the principal, R is the rate percent per annum and T is the time in years. The 100 is only there because R is written as a percentage. The amount is A = P + SI .

Q: How do you find the principal when you are given the interest?

Rearrange to P = 100 × SI / (R × T) , or think in percentages: the interest is (R·T)% of the principal, so scale it back to 100%. If the interest is ₹1920 at 6% for 5 years, that is 30% → 1920, so 100% → ₹6400.

Q: A rate is given as 6⅔%. What do you do with it?

Convert it to the fraction 20/3 % and keep it as a fraction. Rates like 6⅔%, 8⅓% and 16⅔% are set precisely because their fractional forms cancel cleanly against the principal — 20/3% of ₹4875 is exactly ₹325. Turning it into 6.67 introduces rounding for no reason.

Model 1 is every direct question in the chapter: find the interest, the amount, the principal, the rate or the time. They look like six different questions. They are one question asked from six directions, and this lesson is about spotting which direction you are facing.

Choose which quantity is missing and watch the route change →

Before you write anything, answer one question: which of the five quantities is missing? Everything after that is the same two percentages.

01 The idea

One scenario, six questions, no new formula

Here is a complete simple interest situation. A principal of ₹6400 is lent at 6% per annum for 5 years. It earns ₹1920 in interest, so the amount repaid is ₹8320.

An examiner can hide any one of those five numbers and hand you the rest. Hide the interest and you get “find the SI”. Hide the principal and give the amount, and you get one of the most-set questions in placement papers. Hide the rate and it becomes “at what rate percent…”. Same scenario every time.

So the useful habit is not memorising six rearranged formulas. It is reading the question once and naming the missing letter before you touch the arithmetic. Students lose marks here not because the algebra is hard but because they solved for the interest when the amount was asked for.

The route through all six runs via two percentages. Rate times time gives you the interest as a percentage of the principal — here 6 × 5 = 30%. Add a hundred and you have the amount as a percentage of the principal — 130%. Every question in this model is one multiplication or one division by one of those two numbers.

Interest is (R × T)% of the principal. The amount is (100 + R × T)% of it. Find those two percentages first and every Model 1 question becomes a single step.

Total interest percentageR × T, the interest expressed as a percentage of the principal for the whole period. At 6% for 5 years it is 30%, whatever the principal happens to be.

Growth percentage100 + R × T, the amount as a percentage of the principal. 130% here. This is the number the amount-given questions turn on.

Unitary stepGoing from a part to the whole: if x% corresponds to a value, then 100% is that value × 100 / x. It is how every reverse question in this model is finished.

02 Worked example

The hard direction first: ₹8320 back to ₹6400

The forward case is easy and you have already met it. So the worked example takes the direction students actually get wrong — the amount is given and the principal is wanted. A sum amounts to ₹8320 in 5 years at 6% per annum. Find the principal, and the interest earned.

Name what is missingYou have the amount, the rate and the time. The principal is missing, and the interest is missing with it.A = 8320 R = 6% p.a. T = 5 years P = ?

Get the total interest percentageRate times time. Nothing here depends on the principal, which is exactly why this step can be done before you know it.6% × 5 = 30% of the principal

Turn it into the growth percentageThe ₹8320 is not the interest — it is principal and interest together. So it stands for 100% plus the 30%.A = 100% + 30% = 130% of the principal

Do the unitary step130% corresponds to ₹8320, so divide down to 100%.130% → 8320 100% → 8320 × 100 / 130 = 6400

Subtract for the interest, and checkThe principal is ₹6400, so the interest is the difference — and it should come to 30% of ₹6400.SI = 8320 − 6400 = 1920 and 30% of 6400 = 1920 ✓

The trap in this question is dividing ₹8320 by 1.30 or by 0.30 without deciding which. Dividing by 30% gives ₹27,733 — larger than the amount, which is impossible for a principal and should be spotted instantly. Whenever a reverse answer comes out bigger than the amount you started with, you divided by the wrong percentage.

03 The method

All six rearrangements, and the one you should actually use

The table is here for completeness and for revision the night before. In the exam you will be faster and less error-prone going through the percentages instead.

SI = P × R × T / 100. Make any letter the subject and you get the row you need: P = 100·SI/(R·T), R = 100·SI/(P·T), T = 100·SI/(P·R).

The habit worth building: never divide by R × T when the question gave you an amount. Interest given means divide by (R·T)%; amount given means divide by (100 + R·T)%. Getting that one choice right is most of this model.

The question says	You want	Fast route on our numbers
P, R, T given	SI	30% of 6400 = 1920
P, R, T given	Amount	130% of 6400 = 8320
SI, R, T given	P	30% → 1920, so 100% → 6400
Amount, R, T given	P	130% → 8320, so 100% → 6400
P, SI, T given	R	1920/6400 = 30% over 5 yr, so 6% a year
P, SI, R given	T	6% of 6400 = 384 a year; 1920/384 = 5 yr

05 Cheat sheet

Model 1 on one page

The first two rows are the two numbers to find before anything else. The rest is which of them to divide by, which is the only real decision in this model.

Case	Route	On ₹6400 at 6% for 5 years
Interest percentage	`R × T`	30%
Amount percentage	`100 + R × T`	130%
Find SI from P	`SI = (R·T)% of P`	30% of 6400 = 1920
Find A from P	`A = (100+R·T)% of P`	130% of 6400 = 8320
Find P from SI	`P = SI × 100/(R·T)`	1920 × 100/30 = 6400
Find P from A	`P = A × 100/(100+R·T)`	8320 × 100/130 = 6400
Find R	`R = (SI/P × 100) / T`	30% / 5 = 6%
Find T	`T = SI / (one year’s interest)`	1920 / 384 = 5 years

Amount given? Divide by the bigger percentageAn amount carries the interest inside it, so it is (100 + R·T)%, never (R·T)%. This one choice accounts for most wrong answers in Model 1.

Convert the time before the percentage3 years 9 months is 3.75 years, so at 8% the interest percentage is 30% — not 8 × 3 with the months forgotten.

A principal can never exceed its amountIf a reverse calculation returns something larger than the amount you were given, you divided by the wrong percentage. Catch it before you pick an option.

06 Where & why

Where Model 1 shows up

This is the most-set simple interest model in Indian competitive exams, and it is set as a speed question rather than a thinking question.

TCS NQT · Wipro · Capgemini

One direct question, generous options

Nearly always P, R, T given with the interest or amount wanted. Options are spaced far apart, so the percentage route gets it without long multiplication.

Bank PO · RRB · SSC

The amount-given reverse question

The paper's favourite variant, because dividing by 30% instead of 130% produces a wrong answer the setter has already put in the options.

Time-in-months variants

3 yr 9 mo, 2 yr 8 mo, 219 days

The arithmetic is Model 1 but the marks are lost in the conversion. Exam numbers are chosen so the fraction is exact — treat an untidy fraction as a signal you converted wrongly.

Later chapters

Compound interest, instalments, partnership

All three need Model 1 fluent, because each one applies it repeatedly to a principal that changes between applications.

If you can look at “a sum amounts to X in T years at R%” and write down the growth percentage without pausing, this model is finished and you can spend your time on Models 2 and 3, where the marks actually are.

07 Interview questions

The questions this model attracts

Ten in escalating order — the formula, then the rearrangements, then the reading errors that examiners deliberately set traps for.

State the simple interest formula and say what each letter is.

SI = P × R × T / 100, where P is the principal, R is the rate percent per annum and T is the time in years. The 100 is only there because R is written as a percentage. The amount is A = P + SI.

How do you find the principal when you are given the interest?

Rearrange to P = 100 × SI / (R × T), or think in percentages: the interest is (R·T)% of the principal, so scale it back to 100%. If the interest is ₹1920 at 6% for 5 years, that is 30% → 1920, so 100% → ₹6400.

And when you are given the amount instead of the interest?

The amount is (100 + R·T)% of the principal, not (R·T)%, because it already includes the interest. At 6% for 5 years it is 130%, so ₹8320 × 100 / 130 = ₹6400. Using 30% here is the most common mistake in the whole model.

A question gives 3 years 9 months. What goes into the formula?

3.75 years. Nine months is 9/12 = 0.75 of a year, so it is 3.75, not 3.9. At 8% that makes the interest percentage 8 × 3.75 = 30%. Exam numbers are almost always chosen so this comes out exact, so an untidy result usually means the conversion went wrong.

How do you handle a time given in days?

Divide by 365 unless the question names a different convention. 219 days is 219/365 = 0.6 of a year and 292 days is 0.8. Those specific numbers recur precisely because they divide exactly, so recognising them saves time.

Find the rate if ₹6400 becomes ₹8320 in 5 years.

Take the interest first: 8320 − 6400 = ₹1920. As a fraction of the principal that is 1920/6400 = 30%, and that 30% covers all five years, so the annual rate is 30/5 = 6%. Finding the total percentage and dividing by the years beats substituting into the formula.

Find the time if ₹9600 earns ₹4200 at 12.5% per annum.

One year's interest is 12.5% of 9600 = ₹1200. The total needed is ₹4200, so the time is 4200/1200 = 3.5 years. Dividing the total interest by one year's interest is the cleanest route to a time, and it works whether or not the answer is a whole number.

What is the fastest way to get an amount without finding the interest first?

Multiply the principal by (100 + R·T)% in one go. At 4.8% for 2.5 years the growth is 12%, so the amount is 112% of the principal — on ₹12,500 that is ₹14,000 directly. Finding the interest and adding it takes two steps to reach the same place.

A rate is given as 6⅔%. What do you do with it?

Hard

A sum of ₹4500 becomes ₹5400 in 5 years at simple interest. A second sum of ₹7500 is lent at the same rate. (a) Find the rate. (b) In how many years will the second sum amount to ₹9300? (c) Without computing a new rate, explain why doubling both the principal and the interest in part (b) would leave your answer to (b) unchanged.

Follow-up

Part (c) is the real question. It asks you to see that time depends only on the ratio of interest to principal and not on their sizes — which is the idea the whole of Model 2 is built on, so getting it here makes the next lesson much easier.

Show the hint

Find one year's interest on ₹7500 at the rate from part (a), then see how many of those make up the interest part (b) needs.