Chapter 2: Compound Interest

A person borrowed Rs. $10,00,000$ for a business from a commercial bank. Discuss: a) How much interest should he pay after 2 years at the rate of $8\%$ per annum? b) What total sum should he pay after 2 years, and what is that sum called? c) If he had borrowed for 5 years, how much interest should be paid at the same rate?

The sum of money paid to the bank at the rate of $8\%$ p.a. after 2 years is called simple interest. The total sum of borrowed money and additional interest is called amount. The formula used for the calculation of simple interest is: Simple interest $(SI)=\frac{PTR}{100}$.

Activity 1: A teacher borrowed Rs. $5,00,000$ for 2 years at the interest rate of $12\%$ per annum from a commercial bank. He could not pay the interest at the end of the first year, so he had to pay interest-of-interest of the first year in the second year. The interest to be paid in the first year $I_1=\frac{PTR}{100}=\frac{5,00,000\times 1\times 12}{100}=\text{Rs. }60,000$. Since he could not pay the interest in the first year, the principal for the second year $P_1=\text{Rs. }5,00,000+\text{Rs. }60,000=\text{Rs. }5,60,000$. Thus, the interest for the second year $I_2=\frac{P_{1}TR}{100}=\frac{5,60,000\times 1\times 12}{100}=\text{Rs. }67,200$. Hence, the total interest to be paid by the teacher $=\text{Rs. }60,000+\text{Rs. }67,200=\text{Rs. }1,27,200$. There is a difference between the interests for the first and second year because in the second year, interest is charged on both the original principal and the unpaid first-year interest — i.e. interest is charged on interest.

If the interest of a principal after every year or after a certain period (yearly, half yearly or terminal) is calculated and added to the principal and again the interest is calculated, then the interest so obtained is called compound interest. The sum of the principal and compound interest is called compound amount.

Activity 2: Bishal and Badri each borrowed Rs. $30,000$ from a bank for 3 years at the rate of $10\%$ yearly, with Bishal paying simple interest and Badri paying compound interest. For Bishal (SI): each year's interest is calculated on the same principal $P=\text{Rs. }30,000$: $I_1=I_2=I_3=\frac{30,000\times 10\times 1}{100}=\text{Rs. }3,000$, so total interest $=3(3,000)=\text{Rs. }9,000$. For Badri (CI): the principal changes each year. Year 1: $P_1=\text{Rs. }30,000$, $I_1=\frac{30,000\times 10\times 1}{100}=\text{Rs. }3,000$. Year 2: $P_2=30,000+3,000=\text{Rs. }33,000$, $I_2=\frac{33,000\times 10\times 1}{100}=\text{Rs. }3,300$. Year 3: $P_3=33,000+3,300=\text{Rs. }36,300$, $I_3=\frac{36,300\times 10\times 1}{100}=\text{Rs. }3,630$. Total interest $=3,000+3,300+3,630=\text{Rs. }9,930$. So Badri (compound interest) pays more — Rs. $9,930$ vs Rs. $9,000$ — even at the same rate, because in compound interest the principal of every year equals the interest plus principal of the preceding year, so interest-of-interest is charged.

While calculating simple interest, the principal is the same for each year. But while calculating compound interest, the principals are different every year (the principal for the second year is the amount of the first year, the principal for the third year is the amount of the second year, etc.). The compound interest of the same principal is more at the same rate and time than the simple interest of the same principal.

Activity 3 (general derivation): If a principal $P$ is deposited at the rate of $R\%$ per annum for $T$ years: interest at the end of year 1, $I_1=\frac{P\times T\times R}{100}=\frac{P\times 1\times R}{100}=\frac{PR}{100}$. Amount at the end of year 1, $A_1=P+I_1=P+\frac{PR}{100}=P\left(1+\frac{R}{100}\right)$. Since the amount at the end of year 1 becomes the principal for year 2: $P_2=P\left(1+\frac{R}{100}\right)$. Interest of year 2, $I_2=\frac{P\left(1+\frac{R}{100}\right)\times 1\times R}{100}=P\left(1+\frac{R}{100}\right)\times \frac{R}{100}$. Amount at the end of year 2, $A_2=P\left(1+\frac{R}{100}\right)+P\left(1+\frac{R}{100}\right)\times \frac{R}{100}=P\left(1+\frac{R}{100}\right)\left(1+\frac{R}{100}\right)=P{\left(1+\frac{R}{100}\right)}^2$. Similarly, principal for year 3 $P_3=P{\left(1+\frac{R}{100}\right)}^2$, interest of year 3 $I_3=P{\left(1+\frac{R}{100}\right)}^2\times \frac{R}{100}$, and amount at end of year 3, $A_3=P_3+I_3=P{\left(1+\frac{R}{100}\right)}^2\left(1+\frac{R}{100}\right)=P{\left(1+\frac{R}{100}\right)}^3$. Following this pattern, the compound amount at the end of $T$ years: $CA=P{\left(1+\frac{R}{100}\right)}^T$. Similarly, compound interest $CI=CA-P=P{\left(1+\frac{R}{100}\right)}^T-P=P\left[{\left(1+\frac{R}{100}\right)}^T-1\right]$.

Activity 4 — variations of the compound interest formula: (a) If the rate of interest is different every year ($R_1\%$, $R_2\%$, $R_3\%$ for the first, second and third years respectively): $CA=P\left(1+\frac{R_1}{100}\right)\left(1+\frac{R_2}{100}\right)\left(1+\frac{R_3}{100}\right)$ and $CI=P\left[\left(1+\frac{R_1}{100}\right)\left(1+\frac{R_2}{100}\right)\left(1+\frac{R_3}{100}\right)-1\right]$. (b) If time is given as $T$ years and $M$ months: $CA=P{\left(1+\frac{R}{100}\right)}^T\left(1+\frac{MR}{1200}\right)$ and $CI=P\left[{\left(1+\frac{R}{100}\right)}^T\left(1+\frac{MR}{1200}\right)-1\right]$. (c) If the interest is calculated half yearly: the annual rate $R\%$ is treated as $\frac{R}{2}\%$ per half-year, and time $T$ years becomes $2T$ half-year periods, so $CA=P{\left(1+\frac{R}{200}\right)}^{2T}$ and $CI=P\left[{\left(1+\frac{R}{200}\right)}^{2T}-1\right]$. A financial institution that releases interest twice a year (e.g. the first of Magh and the first of Shrawan) computes each 6-month interest on the principal plus the previous 6 months' interest — this is called half-yearly compound interest. (d) If the interest is calculated terminally (quarterly): the annual rate $R\%$ is treated as $\frac{R}{4}\%$ per quarter, and time $T$ years becomes $4T$ quarterly periods, so $CA=P{\left(1+\frac{R}{400}\right)}^{4T}$ and $CI=P\left[{\left(1+\frac{R}{400}\right)}^{4T}-1\right]$. Terminal (quarterly) compound interest is computed the same way as yearly and half-yearly compound interest, just with a smaller period and more compounding steps.

Activity 5: Someone plans to invest Rs. $1,00,000$ for a year at $15\%$ compound interest per annum, choosing between yearly, half yearly, or terminal (quarterly) compounding. a) Yearly: $CI=1,00,000\left[{\left(1+\frac{15}{100}\right)}^1-1\right]=1,00,000\left[\frac{115}{100}-1\right]=\text{Rs. }15,000$. b) Half yearly: $CI=1,00,000\left[{\left(1+\frac{15}{200}\right)}^{2\times 1}-1\right]=\text{Rs. }15,562.50$. c) Terminal (quarterly): $CI=1,00,000\left[{\left(1+\frac{15}{400}\right)}^{4\times 1}-1\right]=\text{Rs. }15,865.04$. Out of these three options, the terminal (quarterly) option is best because its interest is highest — Rs. $865.04$ more than yearly and Rs. $562.50$ more than half yearly.

While calculating the interest of the same sum at the same rate of interest for the same time, terminal (quarterly) compound interest is more than half yearly compound interest, and half yearly compound interest is greater than yearly compound interest.

What will be the compound interest and compound amount of Rs. 2,000 at the interest rate of 12% p.a. in 2 years? Find the compound interest without using formula.

Solution: Principal $P_1=\text{Rs. }2,000$, rate $R=12\%$ p.a., time $T=2$ years. At the end of the first year, simple interest $I_1=\frac{P_{1}TR}{100}=\frac{2000\times 1\times 12}{100}=\text{Rs. }240$. Principal for the second year $P_2=$ amount at the end of the first year $=P_1+I_1=\text{Rs. }(2000+240)=\text{Rs. }2,240$. Simple interest for the second year $I_2=\frac{2240\times 1\times 12}{100}=\text{Rs. }268.8$. Thus, compound interest at the end of 2 years $CI=I_1+I_2=240+268.8=\text{Rs. }508.8$. Compound amount $CA=P_1+CI=2000+508.8=\text{Rs. }2,508.8$.

Find the compound interest and compound amount of the borrowed amount of Rs. 25,000 which is paid in exactly 3 years at the rate of yearly compound interest rate 12%.

Solution: Principal $P=\text{Rs. }25,000$, rate $R=12\%$ per year, time $T=3$ years. Using the formula, $CI=P\left[{\left(1+\frac{R}{100}\right)}^T-1\right]=25,000\left[{\left(1+\frac{12}{100}\right)}^3-1\right]=25,000\left[{\left(\frac{112}{100}\right)}^3-1\right]=25,000[1.404928-1]=25,000\times 0.404928=\text{Rs. }10,123.20$. Again, compound amount $CA=P+CI=\text{Rs. }25,000+\text{Rs. }10,123.20=\text{Rs. }35,123.20$.

A man borrowed Rs. 32,000 from his friend at the rate of simple interest of 12.5% per annum. He lent the whole sum to a shopkeeper at the same rate of compound interest. How much more money will he get in 3 years? Find.

Solution: Principal $P=\text{Rs. }32,000$, rate $R=12.5\%$, time $T=3$ years. Case I (simple interest, what he pays his friend): $SI=\frac{PTR}{100}=\frac{32000\times 3\times 12.5}{100}=\text{Rs. }12,000$. Case II (compound interest, what he receives from the shopkeeper): $CI=P\left[{\left(1+\frac{R}{100}\right)}^T-1\right]=32,000\left[{\left(1+\frac{12.5}{100}\right)}^3-1\right]=32,000\left[{\left(\frac{112.5}{100}\right)}^3-1\right]=32,000[1.423828125-1]=32,000\times 0.423828125=\text{Rs. }13,562.50$. Therefore, the extra money the man receives $=CI-SI=13,562.50-12,000=\text{Rs. }1,562.50$.

Sameer decided to invest Rs. 5,000 at the rate of 8% per annum for 2 years. For this, he has two safe alternatives. The first alternative is to get half yearly compound interest and the second alternative is to get yearly compound interest. If you were to suggest him, which alternative would you suggest? Write with reason.

Solution: Principal $P=\text{Rs. }5,000$, rate $R=8\%$, time $T=2$ years. a) First alternative (half yearly): $C{I}_1=P\left[{\left(1+\frac{R}{200}\right)}^{2T}-1\right]=5,000\left[{\left(1+\frac{8}{200}\right)}^{2\times 2}-1\right]=5,000\left[{\left(\frac{208}{200}\right)}^4-1\right]=5,000[1.16985856-1]=5,000\times 0.16985856=\text{Rs. }849.29$. b) Second alternative (yearly): $C{I}_2=P\left[{\left(1+\frac{R}{100}\right)}^T-1\right]=5,000\left[{\left(1+\frac{8}{100}\right)}^2-1\right]=5,000\left[(1.08{)}^2-1\right]=5,000(1.1664-1)=5,000\times 0.1664=\text{Rs. }832$. The difference between half yearly and yearly compound interest $=C{I}_1-C{I}_2=849.29-832=\text{Rs. }17.29$. Since half yearly compound interest is Rs. $17.29$ more than yearly compound interest, I would suggest him to invest in the first alternative (half yearly).

A twelve-grade student invests Rs. 10,000 for 2 years at the rate of yearly compound interest. If the compound interest in 1 year is Rs. 1,200 (a) Find the rate of yearly compound interest. (b) Find the yearly compound amount at the end of the second year.

Solution: a) Principal $P=\text{Rs. }10,000$, compound amount at the end of the first year $CA=\text{Rs. }11,200$, time $T_1=1$ year. Using $CA=P{\left(1+\frac{R}{100}\right)}^T$: $11200=10000{\left(1+\frac{R}{100}\right)}^1$, so $\frac{11200}{10000}=1+\frac{R}{100}$, giving $1.12=1+\frac{R}{100}$, so $0.12\times 100=R$, i.e. $R=12\%$. Thus, the rate of yearly compound interest is $12\%$. b) Compound amount at the end of the second year, time $T=2$ years: $CA=P{\left(1+\frac{R}{100}\right)}^T=10,000{\left(1+\frac{12}{100}\right)}^2=10,000{\left(\frac{112}{100}\right)}^2=\text{Rs. }12,544$.

Find the compound interest and compound amount of Rs. 2,00,000 invested for 3 years such that the rate of interest for the first year is 8% p.a., for the second year it is 10% p.a. and for the third year it is 12%.

Solution: Principal $P=\text{Rs. }2,00,000$, time $T=3$ years, $R_1=8\%$, $R_2=10\%$, $R_3=12\%$. Using the formula, compound amount $CA=P\left(1+\frac{R_1}{100}\right)\left(1+\frac{R_2}{100}\right)\left(1+\frac{R_3}{100}\right)=2,00,000\left(1+\frac{8}{100}\right)\left(1+\frac{10}{100}\right)\left(1+\frac{12}{100}\right)=2,00,000\left(\frac{108}{100}\right)\left(\frac{110}{100}\right)\left(\frac{112}{100}\right)=\text{Rs. }2,66,112$. Thus, compound interest $CI=CA-P=\text{Rs. }2,66,112-\text{Rs. }2,00,000=\text{Rs. }66,112$.

A sum amounts to Rs. 14,520 in 2 years and Rs. 15,972 in 3 years at a certain rate of annual compound interest. Then, (a) Find the rate of compound interest. (b) Find what is the principal.

Solution: Let rate of interest $=R\%$ and principal $P=\text{Rs. }x$. Case I: compound amount $C{A}_1=\text{Rs. }14,520$ in $T=2$ years: $x{\left(1+\frac{R}{100}\right)}^2=14,520$ ...(i). Case II: compound amount $C{A}_2=\text{Rs. }15,972$ in $T=3$ years: $x{\left(1+\frac{R}{100}\right)}^3=15,972$ ...(ii). Dividing (ii) by (i): $\frac{15,972}{14,520}=\frac{x{\left(1+\frac{R}{100}\right)}^3}{x{\left(1+\frac{R}{100}\right)}^2}=1+\frac{R}{100}$, so $1.10=1+\frac{R}{100}$, giving $\frac{R}{100}=0.10$, so $R=10\%$ p.a. Putting $R=10\%$ back into (i): $14520=x{\left(1+\frac{10}{100}\right)}^2=x\times 1.21$, so $x=\frac{14,520}{1.21}=\text{Rs. }12,000$. Therefore, the rate of interest is $10\%$ p.a. and the principal is Rs. $12,000$.

A person deposited Rs. 2,00,000 in a development bank for 2 years to get the half yearly compound interest at the rate of 10% per annum after deducting the 5% tax on the interest. But right after a year, the bank has changed the policy and decided to compute the interest terminally (quarterly) at the same rate of interest. (a) Find the interest of the first year by deducting the tax. (b) What would be the interest of the second year after deducting the tax? (c) What is the difference between interests of the first year and second year after deducting the tax? Find. (d) After deducting the tax, by what percentage does the interest of the first year differ from the interest of the second year?

Solution: Principal $P=\text{Rs. }2,00,000$, rate $R=10\%$ p.a. a) For the first year, half yearly compound interest $C{I}_1=P\left[{\left(1+\frac{R}{200}\right)}^{2T}-1\right]=2,00,000\left[{\left(1+\frac{10}{200}\right)}^{2\times 1}-1\right]=2,00,000\left[{\left(\frac{210}{200}\right)}^2-1\right]=2,00,000[1.1025-1]=2,00,000\times 0.1025=\text{Rs. }20,500$. After deducting $5\%$ tax: $C{I}_1=20,500-20,500\times \frac{5}{100}=20,500-1,025=\text{Rs. }19,475$. So the interest of the first year after deducting tax is Rs. $19,475$. b) Compound amount after 1 year $CA=2,00,000+19,475=\text{Rs. }2,19,475$; this becomes the principal for the second year. Using quarterly compound interest, $C{I}_2=P\left[{\left(1+\frac{R}{400}\right)}^{4T}-1\right]=2,19,475\left[{\left(1+\frac{10}{400}\right)}^{4\times 1}-1\right]=2,19,475[0.1038128906]=\text{Rs. }22,784.33$. After deducting $5\%$ tax: $C{I}_2=22,784.33-22,784.33\times \frac{5}{100}=22,784.33-1,139.21=\text{Rs. }21,645.12$. So the interest after deducting tax is Rs. $21,645.12$. c) The difference in interests $=C{I}_2-C{I}_1=21,645.12-19,475=\text{Rs. }2,170.12$. d) The difference of interests in percentage $=\frac{C{I}_2-C{I}_1}{C{I}_1}\times 100\%=\frac{2170.12}{19475}\times 100\%=11.14\%$. Thus, the interest of the second year differs by $11.14\%$ from that of the first year.

A commercial bank releases a loan of Rs. 52,500 to Babulal and Jibanlal at the rate of yearly 10% compound interest. If the compound amount paid by Babulal in 2 years is the same as the compound amount paid by Jibanlal in 3 years, how much loan did each of them borrow from the bank?

Solution: Let the loan amount of Babulal $P_1=\text{Rs. }x$; the loan amount of Jibanlal $P_2=\text{Rs. }(52,500-x)$. The compound amount to be paid by Babulal in 2 years: $C{A}_1=P_1{\left(1+\frac{R}{100}\right)}^T=x{\left(1+\frac{10}{100}\right)}^2=x{\left(\frac{110}{100}\right)}^2=1.21x$. The compound amount to be paid by Jibanlal in 3 years: $C{A}_2=P_2{\left(1+\frac{R}{100}\right)}^T=(52,500-x){\left(1+\frac{10}{100}\right)}^3=(52,500-x)(1.331)=69,877.5-1.331x$. Since $C{A}_1=C{A}_2$: $1.21x=69,877.5-1.331x$, so $1.21x+1.331x=69,877.5$, giving $2.541x=69,877.5$, so $x=27,500$. Thus, Babulal borrowed Rs. $27,500$ and Jibanlal borrowed Rs. $(52,500-27,500)=\text{Rs. }25,000$.