Interest (SI & CI)
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Simple Interest
Simple interest
SI = \frac{PRT}{100}
Amount
A = P + SI = P\left(1 + \frac{RT}{100}\right)
Recovering inputs
P = \frac{100\,SI}{RT},\quad R = \frac{100\,SI}{PT},\quad T = \frac{100\,SI}{PR}
n-times in T years
R = \frac{100(n-1)}{T}
Equal yearly interest
SI_{\text{per year}} = \frac{SI_{\text{total}}}{T}
Compound Interest
Compound amount
A = P\left(1 + \frac{R}{100}\right)^T
Compound interest
CI = P\left[\left(1 + \frac{R}{100}\right)^T - 1\right]
Year-wise multipliers
A = P \times \frac{100 + r_1}{100} \times \frac{100 + r_2}{100} \times \cdots
Half-yearly compounding
A = P\left(1 + \frac{R}{200}\right)^{2T}
rate halves, periods double
Quarterly compounding
A = P\left(1 + \frac{R}{400}\right)^{4T}
CI vs SI: differences & doubling
2-year difference
CI - SI = P\left(\frac{R}{100}\right)^2
3-year difference
CI - SI = P\left(\frac{R}{100}\right)^2\left(3 + \frac{R}{100}\right)
CI doubling chain
2\times \text{ in } T \Rightarrow 2^k\times \text{ in } kT
SI multiple pace
n\times \text{ in } T \Rightarrow n'\times \text{ in } T',\ (n'-1) = (n-1)\frac{T'}{T}
SI and CI both given
P\left(\frac{R}{100}\right)^2 = CI_2 - SI_2
Equal annual instalments
Present value (CI)
P = \sum_{k=1}^{n} \frac{x}{\left(1 + \frac{r}{100}\right)^k}
Simple interest instalment
P = nx - \frac{x\,r}{100} \cdot \frac{n(n-1)}{2}
Tabular step
\text{debt}_{k+1} = \text{debt}_k\left(1 + \frac{r}{100}\right) - x
Finding P, R, T from amount data
Principal from amount
P = \frac{A}{\left(1 + \frac{r}{100}\right)^T}
Rate from consecutive amounts
1 + \frac{r}{100} = \frac{A_{t+1}}{A_t} \quad (\text{CI})
Yearly SI from amounts
SI_{\text{year}} = A_{t+1} - A_t \quad (\text{SI})
Two-amount system
\frac{A_2}{A_1} = 1 + \frac{r}{100} \Rightarrow P = \frac{A_1}{1 + r/100}