Arithmetic

Population grows like compound interest, so $P_T = P_O\left(1+\frac{Q}{100}\right)^T.$

VAT $=13\%$ of Rs 3000 $=\dfrac{13}{100}\times 3000=$ Rs 390.

Price after VAT $=3000+390=$ Rs 3,390.

VAT $=13\%$ of Rs 2000 $=\dfrac{13}{100}\times 2000=$ Rs 260.

Price after VAT $=2000+260=$ Rs 2,260.

VAT $= 13\%$ of Rs $1000 = \dfrac{13}{100}\times 1000 =$ Rs $130$.

Price after VAT $= 1000 + 130 =$ Rs $1130$.

VAT $=13\%$ of Rs 1800 $=\dfrac{13}{100}\times1800=$ Rs 234.

Price after VAT $=1800+234=$ Rs 2,034.

VAT $=13\%$ of Rs 1500 $=\dfrac{13}{100}\times1500=$ Rs 195.

Price after VAT $=1500+195=$ Rs 1,695.

Taking Rs 1300 as the price before VAT:

VAT $= 13\%$ of $1300 = \dfrac{13}{100}\times 1300 =$ Rs 169

Price after VAT $= 1300 + 169 =$ Rs 1,469

Price with VAT $= 1400 + 13\% \text{ of } 1400 = 1400 + 182 = \text{Rs } 1582$.

First year (annual compounding):

Second year (semi-annual compounding): rate per half-year $= 5\%$, number of half-years $= 2$.

Difference $= 6765 - 6000 =$ Rs 765.

The interest of the second year is $12.75\%$ more than that of the first year. The reason is that after the policy change the interest is added twice a year (so interest is earned on interest sooner), and the second year also starts from a larger principal of Rs 66,000.

First year (annual compounding): $P = \text{Rs. } 50000$, $R = 10\%$, $T = 1$.

$CI_1 = 50000 \times \frac{10}{100} = \text{Rs. } 5000, \qquad A_1 = \text{Rs. } 55000$

Second year (semi-annual compounding): $P = \text{Rs. } 55000$, rate per half-year $= 5\%$, number of half-years $= 2$.

$A_2 = 55000 \left(1 + \frac{5}{100}\right)^2 = 55000 \times 1.1025 = \text{Rs. } 60637.50$

$CI_2 = 60637.50 - 55000 = \text{Rs. } 5637.50$

Difference $= 5637.50 - 5000 = \text{Rs. } 637.50$.

$\text{Percentage difference} = \frac{637.50}{5000} \times 100\% = 12.75\%$

The second year's compound interest is $12.75\%$ more than the first year's. The reason is that, in the second year, interest is added every six months, so the interest earned in the first half also earns interest in the second half.

First year (annual compounding)

$CI_1 = 1{,}00{,}000\times\frac{10}{100} = \text{Rs } 10{,}000$

Amount after one year $=$ Rs $1{,}10{,}000$.

Second year (semi-annual compounding)

Rate per half-year $= 5\%$, number of half-years $= 2$.

$A = 1{,}10{,}000\left(1 + \frac{5}{100}\right)^2 = 1{,}10{,}000\times 1.1025 = \text{Rs } 1{,}21{,}275$

$CI_2 = 1{,}21{,}275 - 1{,}10{,}000 = \text{Rs } 11{,}275$

Difference

$CI_2 - CI_1 = 11{,}275 - 10{,}000 = \text{Rs } 1{,}275$

$\text{Percentage difference} = \frac{1275}{10000}\times 100\% = 12.75\%$

The compound interest of the second year is $12.75\%$ more than that of the first year, because the interest in the second year is added (compounded) twice and is calculated on the larger amount of Rs $1{,}10{,}000$.

First year (compounded annually).

$I_1=\frac{55000\times10\times1}{100}=\text{Rs }5{,}500,\qquad A_1=55000+5500=\text{Rs }60{,}500.$

Second year (compounded semi-annually). Rate per half-year $=\dfrac{10}{2}=5\%$, number of half-years $=2$, principal $=$ Rs 60,500.

$A_2=60500\left(1+\frac{5}{100}\right)^2=60500\times1.1025=\text{Rs }66{,}701.25.$

$I_2=66701.25-60500=\text{Rs }6{,}201.25.$

Difference between the two years' interest.

$I_2-I_1=6201.25-5500=\text{Rs }701.25.$

Percentage difference (compared with the first year's interest):

$\frac{701.25}{5500}\times100\%=12.75\%.$

Reason. In the second year the interest is added twice (every six months), so the interest earned in the first half also earns interest in the second half. Hence the compound interest of the second year is 12.75% more than that of the first year.

First year (yearly compounding):

$A_1 = 40000 \times \left(1+\dfrac{10}{100}\right) =$ Rs 44,000

$CI_1 = 44000 - 40000 =$ Rs 4,000

Second year (semi-annual compounding): principal $=$ Rs 44,000, rate per half-year $= 5\%$, number of half-years $= 2$.

$A_2 = 44000 \times \left(1+\dfrac{5}{100}\right)^2 = 44000 \times 1.1025 =$ Rs 48,510

$CI_2 = 48510 - 44000 =$ Rs 4,510

Difference: $4510 - 4000 =$ Rs 510.

$\text{Percentage difference} = \frac{510}{4000}\times 100\% = \textbf{12.75\%}$

The interest of the second year is $12.75\%$ more than that of the first year. The reason is that the principal for the second year is larger (Rs 44,000) and the interest is added twice in the year (semi-annually), so interest is earned on interest within the year.

First year (annual compounding): $P = 80000$, $R = 10\%$, $T = 1$.

$A_1 = 80000 \times 1.10 = 88000, \qquad CI_1 = 88000 - 80000 = \text{Rs } 8000$

Second year (semi-annual compounding): principal $= 88000$, rate per half-year $= 5\%$, number of half-years $= 2$.

$A_2 = 88000 \left(1 + \frac{5}{100}\right)^2 = 88000 \times 1.1025 = 97020$ $CI_2 = 97020 - 88000 = \text{Rs } 9020$

Difference $= 9020 - 8000 = \text{Rs } 1020$.

$\text{Percentage difference} = \frac{1020}{8000} \times 100\% = 12.75\%$

The compound interest of the second year is $12.75\%$ more than that of the first year. The reason is twofold: the second year starts with a larger principal (Rs 88,000), and interest is added every six months, so interest is earned on interest within the year.

Dollars bought $= \dfrac{5{,}50{,}000}{110} =$ USD $5000$.

When the Nepali currency is revalued by $10\%$, it becomes stronger, so the dollar becomes cheaper:

$\text{New rate} = 110 - 10\% \text{ of } 110 = 110 - 11 = \text{NRs } 99 \text{ per USD}$

Amount received $= 5000\times 99 =$ Rs $4{,}95{,}000$.

$\text{Loss} = 5{,}50{,}000 - 4{,}95{,}000 = \text{Rs } 55{,}000$

He has a loss of Rs $55{,}000$.

Step 1: dollars bought.

$\text{USD}=\frac{660000}{110}=6000.$

Step 2: new rate after revaluation. When Nepali currency is revalued by $10\%$, it becomes stronger, so one dollar is worth $10\%$ fewer rupees:

$\text{New rate}=110-10\%\text{ of }110=110-11=\text{Rs }99\text{ per USD}.$

Step 3: rupees obtained back.

$6000\times99=\text{Rs }5{,}94{,}000.$

Step 4: gain or loss.

$\text{Loss}=6{,}60{,}000-5{,}94{,}000=\textbf{Rs 66,000}.$

So the businessman suffers a loss of Rs 66,000.

Dollars obtained: $\frac{1100000}{110} = 10000 \text{ USD}$

When the Nepali rupee is revalued (its value rises) by $5\%$, a dollar buys fewer rupees:

$\text{New rate} = 110 - 5\% \text{ of } 110 = 110 - 5.5 = \text{Rs } 104.5 \text{ per USD}$

Rupees received on re-exchanging: $10000 \times 104.5 = \text{Rs } 10{,}45{,}000$

$\text{Loss} = 11{,}00{,}000 - 10{,}45{,}000 = \text{Rs } 55{,}000$

The businessman has a loss of Rs 55,000.

(a) $V_T = V_0\left(1-\frac{R}{100}\right)^T$

(b) Selling price after 2 years $= 80000(0.8)^2 = 80000 \times 0.64 = \text{Rs }51,200$

Total receipt $= 51200 + 30000 = \text{Rs }81,200$

Profit $= 81200 - 80000 =$ Rs 1,200 (profit)

(c) Selling price after 3 years $= 80000(0.8)^3 = 80000 \times 0.512 = \text{Rs }40,960$

Purchase price $-$ selling price $= 80000 - 40960 =$ Rs 39,040. The selling price is Rs 39,040 less than the purchase price.

Here $P=20{,}00{,}000$, $A=14{,}58{,}000$, $R=10\%$. Let the time be $T$ years.

$A=P\left(1-\frac{R}{100}\right)^T \Rightarrow 1458000=2000000\left(\frac{9}{10}\right)^T$

$\left(\frac{9}{10}\right)^T=\frac{1458000}{2000000}=0.729=\left(\frac{9}{10}\right)^3$

So $T=3$. The price will be Rs 14,58,000 after 3 years.

(a) $P_T = P\left(1+\frac{R}{100}\right)^T$

(b) $P = 20000$, $R = 2\%$, $P_T = 20808$.

$20808 = 20000(1.02)^T \Rightarrow (1.02)^T = 1.0404 = (1.02)^2 \Rightarrow T = \textbf{2 years}$

(c) $R = 3\%$, $T = 2$: $P_2 = 20000(1.03)^2 = 20000 \times 1.0609 = 21218$

Increase $= 21218 - 20000 =$ 1,218 people

(a) Australian dollars $= \frac{129090}{86.06} =$ AUSD 1,500

(b) Revaluation by 2% lowers the rate: $86.06 \times 0.98 = 84.3388$

Rupees received $= 1500 \times 84.3388 =$ Rs 1,26,508.2

(c) Loss $= 129090 - 1,26,508.2 = \text{Rs }2,581.8$

$\text{Loss \%} = \frac{2,581.8}{129090} \times 100 = \textbf{2\%}$

(a) $P_T = P\left(1+\frac{R}{100}\right)^T$

(b) $10404 = 10000(1.02)^T \Rightarrow (1.02)^T = 1.0404 = (1.02)^2 \Rightarrow T =$ 2 years

(c) $R = 4\%$, $T = 2$: $P_2 = 10000(1.04)^2 = 10816$

Increase $= 10816 - 10000 =$ 816

(a) $V_T = V_0\left(1-\frac{R}{100}\right)^T$

(b) $V_3 = 2000000(0.8)^3 = 2000000 \times 0.512 =$ Rs 10,24,000

(c) Total receipt $= 10,24,000 + 900000 = \text{Rs }19,24,000$

Loss $= 2000000 - 19,24,000 = \text{Rs }76,000$

$\text{Loss \%} = \frac{76,000}{2000000} \times 100 = \textbf{3.8\%}$

Pounds bought $= \dfrac{840000}{140} = £6000$.

When the Nepali currency is revalued by $5\%$, it becomes stronger, so one pound buys fewer rupees:

$\text{New rate} = 140 - 5\% \text{ of } 140 = 140 - 7 = \text{Rs. } 133 \text{ per } £1.$

Rupees received $= 6000 \times 133 = \text{Rs. } 798000$.

$\text{Loss} = 840000 - 798000 = \text{Rs. } 42000$

The businessman has a loss of Rs. 42,000.

(a) $V_T=V_o\left(1-\frac{R}{100}\right)^T$

(b) $V_o=\text{Rs } 40{,}00{,}000$, $R=15\%$, $T=3$:

$V_T=4000000\times(0.85)^3=4000000\times0.614125=\text{Rs } 24{,}56{,}500$

The selling price after three years is Rs 24,56,500.

(c) Total money received $=24{,}56{,}500+15{,}00{,}000=\text{Rs } 39{,}56{,}500$.

Cost price $=\text{Rs } 40{,}00{,}000$, so there is a loss of $40{,}00{,}000-39{,}56{,}500=\text{Rs } 43{,}500$.

$\text{Loss \%}=\frac{43500}{4000000}\times100\%=1.0875\%\approx \textbf{1.09\% loss}$

(a)

$\text{Australian dollars}=\frac{356000}{80}=AUSD 4{,}450.$

(b) Revaluation makes the Nepali rupee stronger by 2%, so one Australian dollar buys 2% fewer rupees:

$\text{New rate}=80\times\left(1-\frac{2}{100}\right)=80\times0.98=\text{Rs }78.40.$

$\text{Rupees received}=4450\times78.40=\text{Rs }3{,}48{,}880.$

(c) Loss $=3{,}56{,}000-3{,}48{,}880=$ Rs 7,120.

Australian dollars bought $= \dfrac{150000}{77.02} \approx 1947.53$.

The Nepali currency is devalued by $5\%$, so one Australian dollar is now worth $77.02 \times \dfrac{105}{100} =$ Rs 80.871.

Amount received $= 1947.53 \times 80.871 \approx$ Rs 1,57,500 (exactly $150000 \times 1.05$).

Profit $= 157500 - 150000 =$ Rs 7,500.

Using $V = P\left(1 - \dfrac{R}{100}\right)^T$ with $P = 40,00,000$, $R = 10\%$, $V = 29,16,000$:

$2916000 = 4000000\times(0.9)^T \Rightarrow (0.9)^T = 0.729 = (0.9)^3$

$T = \textbf{3 years}$

Check: $4000000 \to 3600000 \to 3240000 \to 2916000$.

Dollars bought $= \dfrac{330000}{110} = 3000$ dollars.

When the Nepali currency is devalued by $10\%$, one dollar becomes more expensive in rupees:

Amount received on exchanging back $= 3000 \times 121 =$ Rs 3,63,000.

Profit $= 363000 - 330000 =$ Rs 33,000.

Answer: profit of Rs 33,000

Pounds bought $= \dfrac{700000}{140} = 5000$ pounds.

When the Nepali currency is devalued by $5\%$, one pound becomes more expensive in rupees:

Amount received on exchanging back $= 5000 \times 147 =$ Rs 7,35,000.

Gain $= 735000 - 700000 =$ Rs 35,000.

Answer: gain of Rs 35,000

Using $P_T = P_0\left(1 - \dfrac{R}{100}\right)^T$ with $P_0 = 35000$, $R = 10$, $P_T = 25515$:

$25515 = 35000 \times (0.9)^T \Rightarrow (0.9)^T = \frac{25515}{35000} = 0.729 = (0.9)^3$

So $T = 3$.

The price becomes Rs 25,515 after $3$ years.

Compare the compound amounts after 2 years for $P =$ Rs 12,500.

Account M (5% p.a., compounded semi-annually): $R = \dfrac{5}{2} = 2.5\%$ per half-year, $T = 2\times 2 = 4$ half-years.

$A_M = 12500\left(1+\frac{2.5}{100}\right)^4 = 12500\times(1.025)^4 = \text{Rs } 13{,}797.66$

Compound interest in M $= 13797.66 - 12500 =$ Rs 1,297.66.

Account N (6% p.a., compounded annually): $T = 2$ years.

$A_N = 12500\left(1+\frac{6}{100}\right)^2 = 12500\times(1.06)^2 = \text{Rs } 14{,}045$

Compound interest in N $= 14045 - 12500 =$ Rs 1,545.

Since Rs 1,545 $>$ Rs 1,297.66, I will deposit in account N, because it gives Rs 247.34 more interest in 2 years.

Compare the amounts after 2 years on a deposit of Rs 25,000.

Account A (10% p.a., compounded semi-annually): $R=\dfrac{10}{2}=5\%$ per half-year, $T=2\times2=4$ half-years.

$A_A=25000\left(1+\frac{5}{100}\right)^4=25000\times(1.05)^4=\text{Rs } 30{,}387.66$

Interest in A $=30387.66-25000=$ Rs 5,387.66.

Account B (12% p.a., compounded annually): $T=2$ years.

$A_B=25000\left(1+\frac{12}{100}\right)^2=25000\times1.2544=\text{Rs } 31{,}360$

Interest in B $=31360-25000=$ Rs 6,360.

Since Rs 6,360 $>$ Rs 5,387.66, I will deposit in account B, because it earns Rs 972.34 more interest in 2 years.

(a) Compound interest is the interest calculated on the principal together with the interest already earned (accumulated) in the earlier periods. So the interest of each period is calculated on the amount of the previous period.

(b) Given: $P = \text{Rs }5,50,000$, $R = 10\%$ p.a. Year 1 is compounded yearly; in year 2 the interest is compounded semi-annually.

Amount after 1 year (yearly): $A_1 = 550000\left(1+\frac{10}{100}\right) = \text{Rs }6,05,000$

For year 2, principal $= \text{Rs }6,05,000$, $T = 1$ year $= 2$ half-years, rate per half-year $= \frac{10}{2} = 5\%$.

$A_2 = 605000\left(1+\frac{5}{100}\right)^2 = 605000 \times 1.1025 = \text{Rs }6,67,012.5$

So the amount at the end of 2 years is Rs 6,67,012.5.

(c) Interest of first year $= 605000 - 550000 = \text{Rs }55,000$

Interest of second year $= 6,67,012.5 - 605000 = \text{Rs }62,012.5$

The interest of the second year is more than that of the first year by $62012.5 - 55000 = \text{Rs }7,012.5$. This is because in the second year the interest is also earned on the first year's interest and it is compounded semi-annually.

(a) $P_T = P\left(1+\frac{R}{100}\right)^T$, where $P$ is the initial population, $R$ the annual growth rate and $T$ the time in years.

(b) Given: present population $P_T = 126075$, $R = 2.5\%$, $T = 2$ years. Population before 2 years $= P$.

$126075 = P\left(1+\frac{2.5}{100}\right)^2 = P \times (1.025)^2$

$P = \frac{126075}{1.050625} = 1,20,000$

The population before 2 years was 1,20,000.

(c) Population after 1 year (growth) $= 126075 \times 1.025 = 1,29,226.875 \approx 1,29,227$

After 1075 people migrate out: $\approx 1,29,227 - 1075 =$ 1,28,152.

(a) To exchange Nepali rupees into American dollars the bank sells dollars, so the selling rate (NRs 142.24) is used.

(b) Dollars $= \frac{796544}{142.24} = USD 5,600$

(c) After 0.3% devaluation of the Nepali rupee, the dollar becomes costlier, so the buying rate rises:

New buying rate $= 141.64 \times \left(1+\frac{0.3}{100}\right) = 141.64 \times 1.003 = 142.06492$

Rupees received on exchanging USD 5,600 back $= 5600 \times 142.06492 = \text{Rs }7,95,563.55$

Rupees paid at first $= \text{Rs }7,96,544$

Loss $= 796544 - 7,95,563.55 =$ Rs 980.45 (a loss, because the bank buys at a lower rate than it sells).

(a) Annual compound interest: $CI = P\left[\left(1+\frac{R}{100}\right)^T - 1\right]$

(b) Given: $P = \text{Rs }70,000$, $R = 6\%$, $T = 2$ years.

$CI = 70000\left[\left(1+\frac{6}{100}\right)^2 - 1\right] = 70000 \times (1.1236 - 1) = 70000 \times 0.1236 = \text{Rs }8,652$

(c) Simple interest $= \frac{PTR}{100} = \frac{70000 \times 2 \times 6}{100} = \text{Rs }8,400$

Compound interest $= \text{Rs }8,652$

Difference $= \text{Rs }8,652 - \text{Rs }8,400 = \text{Rs }252$. Compound interest is more than simple interest by Rs 252, because in compound interest the interest of the first year also earns interest in the second year.

(a) Compound depreciation is the decrease in the value of an article in which each year's depreciation is calculated on the reduced (depreciated) value of the previous year, not on the original value.

(b) Given: $V_0 = \text{Rs }1,60,000$, $R = 20\%$, $T = 2$ years.

$V_2 = 160000\left(1-\frac{20}{100}\right)^2 = 160000 \times 0.64 =$ Rs 1,02,400

(c) Total depreciation $= 160000 - 1,02,400 =$ Rs 57,600

(a) To get dollars, the bank sells them at the selling rate NRs 133.17.

Rupees needed $= 500 \times 133.17 =$ Rs 66,585

(b) On devaluation of the Nepali currency by 0.5%, the dollar rate rises by 0.5%.

New buying rate $= 132.57 \times 1.005 =$ NRs 133.23285

(c) New selling rate $= 133.17 \times 1.005 = 133.83585$

Rupees needed after devaluation $= 500 \times 133.83585 = \text{Rs }66,917.93$

Difference $= 66,917.925 - 66,585 =$ Rs 332.93 more is required after devaluation.

(a) Let the sum be $P$ and $R = 10\%$.

Amount in 1 year $= P(1.1) = 1.1P$; amount in 2 years $= P(1.1)^2 = 1.21P$.

Difference: $1.21P - 1.1P = 0.11P = 47250$

$P = \frac{47250}{0.11} = \text{Rs 4,29,545.45}$

The sum is Rs 4,29,545.45 (approximately Rs 4,29,545).

(b) Semi-annual (10% p.a., 2 half-years at 5%): $CI = P\left[(1.05)^2 - 1\right] = 4,29,545.45 \times 0.1025 = \text{Rs }44,028.41$

Quarterly (4 quarters at 2.5%): $CI = P\left[(1.025)^4 - 1\right] = 4,29,545.45 \times 0.1038129 = \text{Rs }44,592.36$

Quarterly CI is more than semi-annual CI by $44,592.36 - 44,028.41 =$ Rs 563.95 (approximately).

(a) Compound depreciation is the decrease in the value of an article in which each year's depreciation is calculated on the reduced (depreciated) value of the previous year, not on the original value.

(b) Depreciation in the first year $= 20\%$ of $50000 = \frac{20}{100} \times 50000 =$ Rs 10,000

(c) Price after 3 years: $V_3 = 50000\left(1-\frac{20}{100}\right)^3 = 50000 \times 0.512 = \text{Rs }25,600$

Half of the present price $= \frac{50000}{2} = \text{Rs }25,000$

Since the price after 3 years (Rs 25,600) is still more than Rs 25,000, the machine's price will not fall to half in 3 years. So it will not be possible to buy the machine at half of its present price after 3 years.

(a) While exchanging US dollars into Nepalese rupees the bank buys the dollars, so the buying rate is used.

(b) Rupees $= 3000 \times 142.48 =$ Rs 4,27,440

(c) After 0.5% devaluation both rates rise by 0.5%.

New buying rate $= 142.48 \times 1.005 =$ NRs 143.1924

New selling rate $= 143.08 \times 1.005 =$ NRs 143.7954

The selling rate is still higher than the buying rate (difference rises from Rs 0.60 to Rs 0.603).

(d) Rupees after devaluation $= 3000 \times 143.1924 = \text{Rs }4,29,577.2$

Gain $= 4,29,577.2 - 4,27,440 =$ Rs 2,137.2

(a) Annual compound interest: $CI = P\left[\left(1+\frac{R}{100}\right)^T - 1\right]$

(b) Interest of the first year $= \frac{150000 \times 10 \times 1}{100} = \text{Rs }15,000$

Out of Rs 85,000 paid, Rs 15,000 is interest, so the loan (principal) returned $= 85000 - 15000 =$ Rs 70,000

(c) Remaining principal $= 150000 - 70000 = \text{Rs }80,000$

Interest of the second year $= \frac{80000 \times 10 \times 1}{100} = \text{Rs }8,000$

Amount paid at the end of the second year $= 80000 + 8000 = \text{Rs }88,000$

Total money paid $= 85000 + 88000 =$ Rs 1,73,000

(a) Compound depreciation is the decrease in the value of an article in which each year's depreciation is calculated on the reduced (depreciated) value of the previous year, not on the original value.

(b) $V_3 = 2000000\left(1-\frac{8}{100}\right)^3 = 2000000 \times (0.92)^3 = 2000000 \times 0.778688 =$ Rs 15,57,376

(c) Total receipt $=$ selling price $+$ earning $= 15,57,376 + 743124 = \text{Rs }23,00,500$

Cost price $= \text{Rs }20,00,000$

Profit $= 23,00,500 - 20,00,000 = \text{Rs }3,00,500$

$\text{Profit \%} = \frac{3,00,500}{2000000} \times 100 = \textbf{15.025\%}$

The owner makes a profit of 15.025%.

(a) Rupees are exchanged into dollars at the selling rate (NRs 143.75).

Dollars $= \frac{1300000}{143.75} =$ USD 9,043.48 (approximately)

(b) Revaluation of 0.5% lowers the dollar rate by 0.5%. Dollars are now converted into rupees at the buying rate.

New buying rate $= 143.15 \times (1 - 0.005) = 143.15 \times 0.995 = 142.43425$

Rupees received $= 9,043.478 \times 142.43425 =$ Rs 12,88,101.04

(c) The student gets Rs 11,898.96 less than the amount paid. The loss is due to (i) the bank buying dollars at a lower rate (143.15) than it sells them (143.75), and (ii) the revaluation (strengthening) of the Nepali currency, which makes each dollar worth fewer rupees.

(a) Amount in 2 years $A_2 = 14580$, amount in 3 years $A_3 = 15746.40$. The 3rd year's interest is on the 2-year amount:

$A_3 = A_2\left(1+\frac{R}{100}\right) \Rightarrow 1+\frac{R}{100} = \frac{15746.40}{14580} = 1.08$

$\frac{R}{100} = 0.08 \Rightarrow R = \textbf{8\%}$

(b) $A_2 = P(1.08)^2 \Rightarrow P = \frac{14580}{1.1664} =$ Rs 12,500

(c) Simple interest for 1 year $= \frac{12500 \times 8 \times 1}{100} = \text{Rs }1,000$

Quarterly CI for 1 year (rate per quarter $= 2\%$, 4 quarters):

$CI = 12500\left[(1.02)^4 - 1\right] = 12500 \times 0.08243216 = \text{Rs }1,030.4$

Quarterly compound interest is more than simple interest by Rs 30.4.

(a) $V_T = V_0\left(1-\frac{R}{100}\right)^T$, where $V_0$ is the original price, $R$ the rate of depreciation and $T$ the time in years.

(b) $V_3 = 4000000\left(1-\frac{15}{100}\right)^3 = 4000000 \times 0.614125 =$ Rs 24,56,500

(c) Total receipt $= 24,56,500 + 907000 = \text{Rs }33,63,500$

Cost price $= \text{Rs }40,00,000$. Since receipt is less than cost, there is a loss.

Loss $= 4000000 - 33,63,500 = \text{Rs }6,36,500$

$\text{Loss \%} = \frac{6,36,500}{4000000} \times 100 = \textbf{15.9125\%}$

(a) Currency exchange rate is the value (price) of one country's currency expressed in terms of another country's currency, e.g. how many Nepali rupees are equal to 1 US dollar.

(b) Rupees are changed into dollars at the selling rate (NRs 143.44).

Dollars $= \frac{215160}{143.44} =$ USD 1,500

(c) Dollars are changed back into rupees at the buying rate (NRs 142.84).

Rupees received $= 1500 \times 142.84 =$ Rs 2,14,260

(He gets Rs 900 less than he paid, a loss caused by the difference of the buying and selling rates.)

(a) Annual compound interest: $CI = P\left[\left(1+\frac{R}{100}\right)^T - 1\right]$

(b) Ram pays Shyam annual CI: $P = 50000$, $R = 10\%$, $T = 2$.

$CI = 50000\left[(1.1)^2 - 1\right] = 50000 \times 0.21 =$ Rs 10,500

(c) Ghanshyam pays Ram semi-annual CI (rate per half-year $5\%$, 4 half-years):

$CI = 50000\left[(1.05)^4 - 1\right] = 50000 \times 0.21550625 = \text{Rs }10,775.31$

Gain to Ram $= 10,775.31 - 10,500 =$ Rs 275.31 (gain).

(a) $P = 4,00,000$, $P_T = 5,32,400$, $T = 3$ years (2075 to 2078).

$532400 = 400000\left(1+\frac{R}{100}\right)^3 \Rightarrow \left(1+\frac{R}{100}\right)^3 = 1.331 = (1.1)^3$

$1+\frac{R}{100} = 1.1 \Rightarrow R = \textbf{10\%}$

(b) Using $P_T = P\left(1+\frac{R}{100}\right)^T$ with $P_T = 644204$, $P = 400000$, $R = 10$:

$644204 = 400000(1.1)^T \Rightarrow (1.1)^T = 1.61051 = (1.1)^5 \Rightarrow T = 5$

Counting from the beginning of 2075 B.S., the population was 6,44,204 at the beginning of 2080 B.S. (after 5 years).

(a) Currency exchange rate is the value (price) of one country's currency expressed in terms of another country's currency, e.g. how many Nepali rupees are equal to 1 US dollar.

(b) Rupees are changed into pounds at the selling rate (NRs 197.12).

Pounds $= \frac{49280}{197.12} =$ £250

(c) After 2% devaluation the rates rise by 2%. New buying rate $= 196.32 \times 1.02 = 200.2464$.

Rupees for £500 (pounds sold to the bank at the buying rate) $= 500 \times 200.2464 =$ Rs 1,00,123.2

(d) New buying rate $= 200.2464$ and new selling rate $= 197.12 \times 1.02 = 201.0624$.

The selling rate is higher than the buying rate by Rs 0.816 (before devaluation the gap was Rs 0.80); both rates have increased by 2%.

(a) In 2 years, interest is compounded every 6 months, so it is calculated 4 times.

(b) Rate per half-year $= \frac{8}{2} = 4\%$.

Amount at the end of the first year (2 half-years): $500000(1.04)^2 = \text{Rs }5,40,800$

After paying Rs 3,00,000 the balance $= 5,40,800 - 300000 = \text{Rs }2,40,800$

Amount at the end of the second year: $240800 \times (1.04)^2 = 240800 \times 1.0816 =$ Rs 2,60,449.28

He must pay Rs 2,60,449.28 to clear the debt.

(c) If no part payment was made, interest for 2 years $= 500000\left[(1.04)^4 - 1\right] = \text{Rs }84,929.28$

Interest actually paid: first year $= \text{Rs }40,800$, second year $= 2,60,449.28 - 240800 = \text{Rs }19,649.28$, total $= \text{Rs }60,449.28$

Difference $= 84,929.28 - 60,449.28 =$ Rs 24,480 more interest would have to be paid if the whole loan were paid at the end of two years.

(a) $P_T$ represents the price (final value) of the land after $T$ years.

(b) From 2080 to 2082 B.S., $T = 2$, $R = 20\%$, $P_T = 1,44,00,000$.

$14400000 = P_0(1.2)^2 = 1.44P_0 \Rightarrow P_0 = \frac{14400000}{1.44} = \textbf{\text{Rs }1,00,00,000}$