Simple interest
Interest is charged only on the original principal.
SI = P × R × T / 100, and the amount A = P + SI.
Example. ₹5,000 at 8% for 3 years: SI = 5000 × 8 × 3 / 100 = ₹1,200.
Because SI grows by the same amount each year, two amounts for different years reveal the yearly interest. If a sum becomes ₹7,200 in 2 years and ₹8,400 in 4 years, then 2 years of interest is ₹1,200, so the principal is 7,200 − 1,200 = ₹6,000 and the rate is 600/6000 = 10%.
Doubling with SI. A sum doubles when SI = P, so R × T = 100. It triples when R × T = 200.
Compound interest
Interest is added to the principal at the end of each period, so later interest is earned on earlier interest.
A = P(1 + R/100)ⁿ and CI = A − P.
Example. ₹10,000 at 10% for 2 years: A = 10000 × 1.21 = 12,100, so CI = ₹2,100.
Half-yearly compounding: halve the rate and double the number of periods. ₹8,000 at 10% a year, compounded half-yearly for 1 year: A = 8000 × 1.05² = 8,820, CI = ₹820.
CI minus SI: the shortcut
Time | CI − SI |
|---|---|
2 years | P (R/100)² |
3 years | P (R/100)² (3 + R/100) |
Example. ₹5,000 at 10% for 2 years: 5000 × 0.01 = ₹50.
Growth multiples with CI
If a sum doubles in n years at CI, it becomes 4 times in 2n years and 8 times in 3n years, because each n years multiplies it by 2 again.
Finding the principal
Divide the amount by the growth factor. ₹8,820 after 2 years at 5% CI: P = 8820 ÷ 1.05² = 8820 ÷ 1.1025 = ₹8,000.
Useful growth factors
Rate | 2 years | 3 years |
|---|---|---|
5% | 1.1025 | 1.157625 |
10% | 1.21 | 1.331 |
20% | 1.44 | 1.728 |
Shortcuts and tricks
The standard way is to apply SI = PRT/100 or A = P(1 + R/100)ⁿ. For the usual 2- and 3-year questions, these shortcuts avoid the powers.
Total percentage for two years of compound interest
Example: CI on ₹20,000 at 5% for 2 years.
Standard way: A = 20,000 × 1.05² = 20,000 × 1.1025 = 22,050, so CI = ₹2,050.
Shortcut: two years at r% is r + r + r²/100 = 5 + 5 + 0.25 = 10.25% in all. 10.25% of 20,000 = ₹2,050.
CI − SI without finding either
Example: the difference between CI and SI on ₹40,000 at 5% for 2 years.
Standard way: CI = 40,000 × 0.1025 = 4,100 and SI = 40,000 × 0.10 = 4,000, so the difference is ₹100.
Shortcut: for 2 years, CI − SI = P(R/100)² = 40,000 × 1/400 = ₹100. For 3 years, use P(R/100)²(3 + R/100).
Multiples at simple interest
Example: at what rate does a sum become 4 times itself in 15 years at SI?
Standard way: the interest is 3P, so 3P = P × R × 15/100 and R = 20%.
Shortcut: rate = (multiple − 1) × 100 ÷ years = 3 × 100 ÷ 15 = 20%.
Multiples at compound interest
Example: a sum doubles in 4 years at CI. When does it become 16 times?
Standard way: if (1 + r)⁴ = 2, then (1 + r)¹⁶ = 2⁴ = 16, so 16 years.
Shortcut: 16 = 2⁴, so four doublings of 4 years each: 16 years.
Exam tips
Read carefully whether the question asks for interest or amount.
For CI over more than three periods, multiply step by step rather than expanding powers.
With SI, the yearly interest never changes; with CI, it grows every year.