A Sum Doubles in Eight Years at One Particular Rate
Learn to calculate simple interest and the amount from the formula, rearrange it to find the principal, rate or time, and handle sums lent at different rates or for part of a year.
In what time does a sum double itself at simple interest?
Whenever the interest earned equals the principal — and the time depends only on the rate, never on how much money you started with.
For a sum to double, the interest must come to the whole of the principal. Putting into the formula, the cancels from both sides and leaves
So at per annum any sum doubles in years — whether it is or lakh.
That cancellation is the most useful feature of the simple interest formula, and this page covers the first part of the ICSE Class 8 Mathematics chapter on interest.
For a sum to double, the interest must come to the whole of the principal. Putting into the formula, the cancels from both sides and leaves
So at per annum any sum doubles in years — whether it is or lakh.
That cancellation is the most useful feature of the simple interest formula, and this page covers the first part of the ICSE Class 8 Mathematics chapter on interest.
Formula
How do you calculate simple interest and the amount?
where is the principal — the money borrowed or invested, is the rate per cent per annum, and is the time in years.
Simple interest is calculated on the original principal every year. The interest earned in one year does not itself earn interest, which is what makes it simple and what distinguishes it from the compound interest of the next part.
Worked example 1. Find the simple interest and the amount on at per annum for years.
A check worth making. The interest for one year is , and three years at each gives . Under simple interest the yearly interest is always the same amount, so multiplying the first year's interest by the number of years must give the total.
Two conditions the formula assumes. The rate must be per annum and the time must be in years. If the question gives a rate per month or a time in months, one of them has to be converted before the formula is used — which is the subject of the last section.
A boundary case. If then , whatever the principal and rate. Interest is a payment for the use of money over time, so with no time elapsed there is nothing to pay — which is why a loan repaid the same day costs nothing in interest.
How do you find the principal, rate or time?
Rearrange the same formula. Since , multiplying both sides by gives , and then any one of the three can be made the subject:
Each is the same statement read a different way, so there is only one formula to remember.
Worked example 1 — the principal. A sum lent at per annum for years earns in interest. Find the sum.
Worked example 2 — the rate. lent for years earns . Find the rate.
Worked example 3 — the time. at per annum earns . Find the time.
All three examples are the same situation approached from different directions, which is why the answers are consistent — and running the forward calculation as a check takes one line.
Worked example 4 — when the amount is given instead of the interest. A sum of becomes in years at simple interest. Find the rate.
First find the interest, which is the amount minus the principal:
The step students skip. A question giving the amount is not giving the interest. Substituting into the formula in place of produces a rate of , which is plainly too large — and the size of the error is itself the warning.
Worked example 5 — the doubling question. In what time will a sum double itself at per annum?
Doubling means the interest equals the principal, so :
For trebling, the interest must be twice the principal, so and — which at gives years.
Each is the same statement read a different way, so there is only one formula to remember.
Worked example 1 — the principal. A sum lent at per annum for years earns in interest. Find the sum.
Worked example 2 — the rate. lent for years earns . Find the rate.
Worked example 3 — the time. at per annum earns . Find the time.
All three examples are the same situation approached from different directions, which is why the answers are consistent — and running the forward calculation as a check takes one line.
Worked example 4 — when the amount is given instead of the interest. A sum of becomes in years at simple interest. Find the rate.
First find the interest, which is the amount minus the principal:
The step students skip. A question giving the amount is not giving the interest. Substituting into the formula in place of produces a rate of , which is plainly too large — and the size of the error is itself the warning.
Worked example 5 — the doubling question. In what time will a sum double itself at per annum?
Doubling means the interest equals the principal, so :
For trebling, the interest must be twice the principal, so and — which at gives years.
How do you handle part of a year, or two different rates?
Convert the time into years as a fraction, and for two rates set up an equation with the split as the unknown.
Worked example 1 — months. Find the simple interest on at per annum for months.
Eight months is of a year, so
A check: one year's interest would be , and two-thirds of is . Correct.
Worked example 2 — days. Find the simple interest on at per annum for days.
Taking a year as days, days is of a year:
The conversion that must not be forgotten. Putting for eight months treats them as eight years and multiplies the answer twelvefold. Always write the fraction explicitly — or — as a line of working.
Worked example 3 — a sum split between two rates. A man lends in two parts, one at per annum and the other at per annum, and receives as total interest in one year. Find each part.
Let the part lent at be , so the other part is . For one year:
Multiplying through by :
So was lent at and at .
Checking: and , and . Correct.
A sanity check on that answer without any algebra. If the whole had been lent at the interest would be ; at it would be . The actual lies between them, and nearer the figure — so more than half the money must have been at , which agrees with . Any answer putting the larger part at would be wrong on that ground alone.
Worked example 1 — months. Find the simple interest on at per annum for months.
Eight months is of a year, so
A check: one year's interest would be , and two-thirds of is . Correct.
Worked example 2 — days. Find the simple interest on at per annum for days.
Taking a year as days, days is of a year:
The conversion that must not be forgotten. Putting for eight months treats them as eight years and multiplies the answer twelvefold. Always write the fraction explicitly — or — as a line of working.
Worked example 3 — a sum split between two rates. A man lends in two parts, one at per annum and the other at per annum, and receives as total interest in one year. Find each part.
Let the part lent at be , so the other part is . For one year:
Multiplying through by :
So was lent at and at .
Checking: and , and . Correct.
A sanity check on that answer without any algebra. If the whole had been lent at the interest would be ; at it would be . The actual lies between them, and nearer the figure — so more than half the money must have been at , which agrees with . Any answer putting the larger part at would be wrong on that ground alone.
Exam tip
Exam tip: subtract the principal before using the amount
If a question gives the amount, find the interest first as . Substituting the amount for the interest is the commonest error in this chapter and always gives an absurdly large rate or time.
Write the time as a fraction of a year whenever it is given in months or days — or — and show that line.
Remember the rate is per annum unless stated otherwise, and that both the rate and the time must match before the formula is used.
For , or , rearrange the one formula rather than memorising three. Multiply by first and then divide by the other two.
For a doubling question, set and note the principal cancels, giving . For trebling, and .
For a split sum, let one part be and the other , and write one equation for the total interest.
Carry the symbol on every money line and state per annum with every rate.
And check by computing one year's interest and multiplying by the number of years — under simple interest the yearly figure never changes, so this always works.
Write the time as a fraction of a year whenever it is given in months or days — or — and show that line.
Remember the rate is per annum unless stated otherwise, and that both the rate and the time must match before the formula is used.
For , or , rearrange the one formula rather than memorising three. Multiply by first and then divide by the other two.
For a doubling question, set and note the principal cancels, giving . For trebling, and .
For a split sum, let one part be and the other , and write one equation for the total interest.
Carry the symbol on every money line and state per annum with every rate.
And check by computing one year's interest and multiplying by the number of years — under simple interest the yearly figure never changes, so this always works.
Did you know
Why does the doubling time not depend on the amount?
It seems natural that a larger sum should take longer to double. It does not — and lakh both double in exactly the same time at the same rate.
The reason is that both sides of the comparison scale together. A ten-thousandfold larger principal earns ten thousand times as much interest each year, so it needs the same number of years for the interest to catch up with the principal. The two quantities grow in step, and the ratio between them is all that matters.
Algebraically the principal simply cancels: setting in leaves , with no anywhere in it.
That is worth noticing because it is why rate rather than amount is the number quoted on every loan and deposit. A rate is a statement about the proportion gained per year, and being independent of the sum involved is exactly what makes it a fair basis for comparing one offer with another.
The reason is that both sides of the comparison scale together. A ten-thousandfold larger principal earns ten thousand times as much interest each year, so it needs the same number of years for the interest to catch up with the principal. The two quantities grow in step, and the ratio between them is all that matters.
Algebraically the principal simply cancels: setting in leaves , with no anywhere in it.
That is worth noticing because it is why rate rather than amount is the number quoted on every loan and deposit. A rate is a statement about the proportion gained per year, and being independent of the sum involved is exactly what makes it a fair basis for comparing one offer with another.
Key takeaways
Simple interest: quick revision
- and , with per annum and in years.
- Simple interest is calculated on the original principal every year, so the yearly interest never changes.
- at for years gives and an amount of — and one year's times three confirms it.
- Rearranged: , , — one formula, three readings.
- at for years gives ; the same data gives and years.
- Given the amount, find the interest first: becoming in years means , so .
- Doubling: , the principal cancels, and — so years at . Trebling: .
- Part of a year: months is , giving on at ; and days is , giving on at .
- Two rates: with at and at giving , the equation gives , so at and at — and checks it.
- Sanity-check a split by seeing which single rate the actual interest lies closer to.
Work a set where the amount rather than the interest is given — remembering to subtract the principal first is what separates a correct answer from a wildly wrong one.
- Simple interest is calculated on the original principal every year, so the yearly interest never changes.
- at for years gives and an amount of — and one year's times three confirms it.
- Rearranged: , , — one formula, three readings.
- at for years gives ; the same data gives and years.
- Given the amount, find the interest first: becoming in years means , so .
- Doubling: , the principal cancels, and — so years at . Trebling: .
- Part of a year: months is , giving on at ; and days is , giving on at .
- Two rates: with at and at giving , the equation gives , so at and at — and checks it.
- Sanity-check a split by seeing which single rate the actual interest lies closer to.
Work a set where the amount rather than the interest is given — remembering to subtract the principal first is what separates a correct answer from a wildly wrong one.