COLOUR THEME
Simple and compound interest, the CI minus SI shortcut, instalments and true versus banker's discount.
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Interest is the price paid for using money. The syllabus pairs "interest" with "discounting", so this section covers both simple and compound interest and the true and banker's discount.
In brief
Covers Principal (P), Amount (A).
Principal (P): the sum borrowed or lent. Rate (R): interest per 100 per year unless stated otherwise. Time (T or n): in years.
Amount (A) = principal + interest.
Convert time to years: months divided by 12, days divided by 365.
In brief
Concept Formula Simple interest SI = P x R x T/100 Amount under SI A = P + SI = P(1 + RT/100) Principal from SI P = SI x 100/(R x T) Rate from SI R = SI x 100/(P x T) Time…
| Concept | Formula |
|---|---|
| Simple interest | SI = P x R x T/100 |
| Amount under SI | A = P + SI = P(1 + RT/100) |
| Principal from SI | P = SI x 100/(R x T) |
| Rate from SI | R = SI x 100/(P x T) |
| Time from SI | T = SI x 100/(P x R) |
| Amount under CI (annual compounding) | A = P(1 + R/100)^n |
| Compound interest | CI = A - P |
| Half-yearly compounding | Rate R/2, periods 2n |
| Quarterly compounding | Rate R/4, periods 4n |
| Different rates R1, R2, R3 in successive years | A = P(1 + R1/100)(1 + R2/100)(1 + R3/100) |
| Fractional time (for example 2 and a half years) | A = P(1 + R/100)^2 x (1 + (R/2)/100) |
| CI minus SI for 2 years | P(R/100) squared |
| CI minus SI for 3 years | P(R/100) squared x (3 + R/100) |
| Time for a sum to become n times (SI) | (n - 1) x 100/R years |
| Doubling time under SI | 100/R years |
| Doubling time under CI (rule of 72, approximate) | 72/R years |
In brief
Covers Effective rates for CI.
Simple interest is the same every year, so it grows in a straight line.
Compound interest is paid on the principal and on the accumulated interest, so it grows faster, and the gap widens over time.
For the first year with annual compounding, SI and CI are equal.
Effective rates for CI: 2 years at R% is equivalent to a single rise of 2R + R squared/100 percent (10% for 2 years gives 21%); 3 years at 10% gives 33.1%.
In brief
If a loan is repaid in equal annual instalments of x at R% compound interest, the loan equals the sum of the present values of the instalments: P = x/(1 + R/100) + x/(1 +…
If a loan is repaid in equal annual instalments of x at R% compound interest, the loan equals the sum of the present values of the instalments: P = x/(1 + R/100) + x/(1 + R/100)^2 + ...
In brief
Covers Face value (FV), True discount (TD), Banker's discount (BD).
Face value (FV) is the amount due at the end of the time; present worth (PW) is the sum that, put at interest, grows to FV in that time.
True discount (TD) = FV - PW, which equals the simple interest on PW.
Banker's discount (BD) = simple interest on the face value for the unexpired time.
| Concept | Formula |
|---|---|
| Present worth | PW = FV x 100/(100 + RT) |
| True discount | TD = FV x RT/(100 + RT), or PW x RT/100 |
| Banker's discount | BD = FV x RT/100 |
| Banker's gain | BG = BD - TD = TD squared/PW |
Example. A bill of 1100 is due in 1 year and the rate is 10%.
In brief
Example 1.
Example 1. Find the SI and amount on 5000 at 8% for 3 years.
Example 2. Find the CI on 10,000 at 10% for 2 years, and its difference from SI.
Example 3. Find the amount on 8000 at 10% per annum for 1 year, compounded half-yearly.
Example 4. At 10% simple interest, in how many years will a sum triple?
Example 5. In what time does 1000 become 1331 at 10% compound interest?
Example 6. A sum becomes 1.5 times in some years at 5% SI. Find the time.
Example 7. A loan is repaid in two equal annual instalments of 1210 at 10% compound interest. Find the loan.
Example 8. 10,000 is invested for two years, with a 10% rate in the first year and 20% in the second, compounded annually.
Example 9. Approximate doubling time at 8% compound interest.
In brief
Covers CI as a chain of percentages, Use the difference formula, Spot the compounding period, Year-by-year CI.
CI as a chain of percentages: treat each year as a percentage rise and use the multiplier method (CI at 10% for 3 years is 1.1^3 = 1.331, a 33.1% rise).
Use the difference formula when the question gives CI minus SI for 2 or 3 years, to find P or R directly.
Spot the compounding period: half-yearly, quarterly and monthly compounding change both the rate and the number of periods, in the same proportion.
Year-by-year CI: the CI earned in the nth year alone equals one year's simple interest on the amount at the end of year n-1.
In brief
Forgetting to subtract the principal to find CI. Using R and n directly when compounding is half-yearly or quarterly. Using months in the formula without converting to…
Forgetting to subtract the principal to find CI.
Using R and n directly when compounding is half-yearly or quarterly.
Using months in the formula without converting to years.
Mixing up true discount (on present worth) with banker's discount (on face value).
Practise with free MCQs and answer keys, then return here to fix what you got wrong.
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