Financial Mathematics

(c) Rs. 4000

True discount is the simple interest on the present worth: $TD = PW \times r \times t$. $200 = PW \times 0.12 \times \frac{5}{12} = 0.05\,PW \Rightarrow PW = \text{Rs. } 4000$

(a) 2.5 months

With equal drawings at the end of each month for 6 months, the money stays out for $5, 4, 3, 2, 1, 0$ months until the end of the period. The average period is $\dfrac{5+4+3+2+1+0}{6} = \dfrac{15}{6} = 2.5$ months, which equals $\dfrac{n-1}{2}$.

Let the annual profit be Rs. $P$.

Salaries: A gets $0.20P$ and B gets $0.10P$. Residue $= P - 0.30P = 0.70P$.

Capitals are in the ratio $50000 : 35000 : 25000 = 10 : 7 : 5$ (total 22 parts).

A's share of residue $= \dfrac{10}{22}(0.7P)$, B's share $= \dfrac{7}{22}(0.7P)$, C's share $= \dfrac{5}{22}(0.7P)$.

A receives Rs. 1,200 more than B: $\left(0.2P + \tfrac{10}{22}(0.7P)\right) - \left(0.1P + \tfrac{7}{22}(0.7P)\right) = 1200$ $0.1P + \frac{3}{22}(0.7P) = 1200 \Rightarrow \frac{4.3}{22}P = 1200 \Rightarrow P = \frac{26400}{4.3} \approx \text{Rs. } 6139.53$

Hence

• A $= 0.2P + \tfrac{10}{22}(0.7P) \approx$ Rs. 3,181.40
• B $= 0.1P + \tfrac{7}{22}(0.7P) \approx$ Rs. 1,981.40
• C $= \tfrac{5}{22}(0.7P) \approx$ Rs. 976.74

Check: A $-$ B $= 3181.40 - 1981.40 = 1200$, and the three shares add up to about Rs. 6,139.53.

Here $t = \dfrac{3}{12} = \dfrac14$ year and $r = 4\%$, so $r\times t = 0.01$.

The banker's discount is the interest on the face value and the true discount is the interest on the present worth, so $BD - TD = \text{interest on } TD = TD \times r \times t$ $5 = TD \times 0.01 \Rightarrow TD = \text{Rs. } 500$ $BD = TD + 5 = \text{Rs. } 505$

Amount of the bill: $BD = A \times r \times t \Rightarrow 505 = A \times 0.01 \Rightarrow A = \text{Rs. } 50{,}500$.

Check: $PW = 50500 - 500 = 50000$, and $TD = 50000 \times 0.01 = 500$. This agrees.

So true discount is Rs. 500, banker's discount is Rs. 505 and the amount of the bill is Rs. 50,500.

(b) Rs. 9,600

Simple interest $= \frac{8000 \times 10 \times 2}{100} = \text{Rs. } 1,600$, so total amount $= 8000 + 1600 = \text{Rs. } 9,600$.

Given: Principal, $P = \text{Rs } 50{,}000$ Annual rate, $R = 12\%$ per annum Time, $T = 2$ years Compounded half-yearly, so number of conversion periods $n = 2 \times 2 = 4$ and rate per half-year $r = \dfrac{12\%}{2} = 6\%$.

Formula: $A = P\left(1 + \frac{R}{200}\right)^{2T}$ where $\dfrac{R}{200}$ is the half-yearly interest rate in decimal.

Substitution: $A = 50{,}000\left(1 + \frac{12}{200}\right)^{4}$ $A = 50{,}000(1 + 0.06)^4$ $A = 50{,}000(1.06)^4$

Now, $(1.06)^4 = 1.26247696$ $A = 50{,}000 \times 1.26247696 = \text{Rs } 63{,}123.85 \text{ (approx.)}$

Compound interest: $CI = A - P$ $CI = 63{,}123.85 - 50{,}000 = \text{Rs } 13{,}123.85$

Therefore, the compound interest is approximately Rs 13,123.85.

Given: Annual payment, $R = \text{Rs. } 5{,}000$ Time, $n = 4$ years Rate of interest, $i = 8\% = 0.08$ per annum compounded annually

To find: Present value of the ordinary annuity, $PV$

Formula: $PV = R \times \frac{1-(1+i)^{-n}}{i}$

Substituting the values: $PV = 5000 \times \frac{1-(1+0.08)^{-4}}{0.08}$

$PV = 5000 \times \frac{1-(1.08)^{-4}}{0.08}$

$PV = 5000 \times \frac{1-0.7350298528}{0.08}$

$PV = 5000 \times \frac{0.2649701472}{0.08}$

$PV = 5000 \times 3.31212684$

$PV = 16{,}560.63$

Result: The present value of the ordinary annuity is approximately Rs. 16,560.63.

Principal $P=$ Rs 2,00,000; monthly rate $i=\dfrac{12\%}{12}=0.01$; number of months $n=3\times12=36$.

(a) Monthly EMI

$EMI=\frac{P\,i\,(1+i)^n}{(1+i)^n-1}$

$(1.01)^{36}\approx1.430769$, so $EMI=\frac{200000\times0.01\times1.430769}{0.430769}\approx\text{Rs }6{,}642.86$

(b) Total interest paid

Total payment $=6{,}642.86\times36\approx$ Rs 2,39,142.96

$\text{Interest}=2{,}39{,}142.96-2{,}00{,}000\approx\text{Rs }39{,}142.96$

(c) EMI at 10% p.a. compounded monthly

$i=\dfrac{10\%}{12}=0.008333$, $n=36$, and $(1+i)^{36}\approx1.348182$: $EMI=\frac{200000\times0.008333\times1.348182}{0.348182}\approx\text{Rs }6{,}453.44$

So the EMI falls by about Rs 189.42 per month.