Mathematics · Chapter 02

37 solved questions

Chapter 2: Compound Interest

Every question in this chapter, answered and explained — step-by-step solutions drawn live from the chapter library across 1 question section.

Exercise 2.137

Exercise 2.1

1Textbookex2.1-q1-a

Define: Yearly compound interest

Yearly compound interest is the interest calculated once a year on the principal, where the interest of each year is added to the principal to form the principal for the next year, so that interest is charged on interest. $C I = P [(1 + \frac{R}{100})^{T} - 1]$ .

2Textbookex2.1-q1-b

Half yearly compound interest

Half yearly compound interest is the interest calculated twice a year (every 6 months), where each half-year's interest is added to the principal before the next half-year's interest is computed, using half the annual rate for double the number of periods: $C I = P [(1 + \frac{R}{200})^{2 T} - 1]$ .

3Textbookex2.1-q1-c

Quarterly compound interest

Quarterly (terminal) compound interest is the interest calculated four times a year (every 3 months), using a quarter of the annual rate for four times the number of periods: $C I = P [(1 + \frac{R}{400})^{4 T} - 1]$ .

4Textbookex2.1-q2-a

Question 2 According to the yearly compound interest, if principal is $P$ , yearly rate of interest is $R$ and time is $T$ years, write the formula to compute compound amount.

$C A = P (1 + \frac{R}{100})^{T}$ .

5Textbookex2.1-q2-b

The compound interest $C I$ of a sum $P$ in $T$ years at the rate of yearly compound interest is $R %$ , write the relation between $P$ , $T$ , $R$ and $C I$ .

$C I = P [(1 + \frac{R}{100})^{T} - 1]$ .

6Textbookex2.1-q2-c

Employee Provident Fund changes the rate of interest per annum according to the economic liquidity of the state. As per the given condition, the compound amount of a sum $P$ at the interest rate of $R_{1} %$ , $R_{2} %$ and $R_{3} %$ for the first, second and third years respectively is $C A$ , then write the formula to find $C A$ .

$C A = P (1 + \frac{R _{1}}{100}) (1 + \frac{R _{2}}{100}) (1 + \frac{R _{3}}{100})$ .

7Textbookex2.1-q3-a

Without using formula, find the compound interest and compound amount for the following conditions: Principal $(P) =$ Rs. 10,000, Time $(T) = 2$ years and Rate of interest $(R) = 6%$ p.a.

Year 1: $I_{1} = \frac{10000 \times 1 \times 6}{100} = Rs. 600$ ; $P_{2} = 10000 + 600 = Rs. 10, 600$ . Year 2: $I_{2} = \frac{10600 \times 1 \times 6}{100} = Rs. 636$ . $C I = I_{1} + I_{2} = 600 + 636 = Rs. 1, 236$ . $C A = P + C I = 10000 + 1236 = Rs. 11, 236$ .

8Textbookex2.1-q3-b

Principal $(P) =$ Rs. 64,000, Time $(T) = 3$ years and Rate of interest $(R) = 6%$ p.a.

Year 1: $I_{1} = \frac{64000 \times 1 \times 6}{100} = Rs. 3, 840$ ; $P_{2} = 64000 + 3840 = Rs. 67, 840$ . Year 2: $I_{2} = \frac{67840 \times 1 \times 6}{100} = Rs. 4, 070.40$ ; $P_{3} = 67840 + 4070.40 = Rs. 71, 910.40$ . Year 3: $I_{3} = \frac{71910.40 \times 1 \times 6}{100} = Rs. 4, 314.62$ . $C I = 3840 + 4070.40 + 4314.62 = Rs. 12, 225.02$ . $C A = 64000 + 12225.02 = Rs. 76, 225.02$ .

9Textbookex2.1-q3-c

Principal $(P) =$ Rs. 20,000, Time $(T) = 2$ years, Rate of interest for the first year $(R_{1}) = 10%$ p.a. and the rate of interest for the second year $(R_{2}) = 12%$ p.a.

Year 1 (rate $10%$ ): $I_{1} = \frac{20000 \times 1 \times 10}{100} = Rs. 2, 000$ ; $P_{2} = 20000 + 2000 = Rs. 22, 000$ . Year 2 (rate $12%$ ): $I_{2} = \frac{22000 \times 1 \times 12}{100} = Rs. 2, 640$ . $C I = 2000 + 2640 = Rs. 4, 640$ . $C A = 20000 + 4640 = Rs. 24, 640$ .

10Textbookex2.1-q4-a

Question 4 At what rate of compound interest per year will the compound interest of Rs. 100 in a year be Rs. 12? Write.

For $T = 1$ year, compound interest equals simple interest: $C I = \frac{P R}{100}$ , so $12 = \frac{100 \times R}{100} = R$ . Thus $R = 12%$ .

11Textbookex2.1-q4-b

At what rate of compound interest per year will the compound interest of Rs. 200 in 2 years be Rs. 42? Find it.

$C I = P [(1 + \frac{R}{100})^{T} - 1]$ : $42 = 200 [(1 + \frac{R}{100})^{2} - 1]$ , so $\frac{42}{200} = 0.21 = (1 + \frac{R}{100})^{2} - 1$ , giving $(1 + \frac{R}{100})^{2} = 1.21$ , so $1 + \frac{R}{100} = 1.1$ , i.e. $R = 10%$ .

12Textbookex2.1-q5-a

Question 5 A farmer borrowed Rs. 20,000 from a co-operative to invest in poultry farm for 3 years at the rate of compound interest of 15% per year. Find the compound interest and compound amount of 3 years.

$P = Rs. 20, 000$ , $R = 15%$ , $T = 3$ years. $C I = P [(1 + \frac{R}{100})^{T} - 1] = 20, 000 [(1.15)^{3} - 1] = 20, 000 [1.520875 - 1] = Rs. 10, 417.50$ . $C A = P + C I = 20, 000 + 10, 417.50 = Rs. 30, 417.50$ .

13Textbookex2.1-q5-b

A teacher deposited Rs. 50,000 in a bank at the account of his daughter. If the bank provides yearly 10% interest, what will be the compound interest and compound amount in 3 years? Find.

$P = Rs. 50, 000$ , $R = 10%$ , $T = 3$ years. $C I = 50, 000 [(1.10)^{3} - 1] = 50, 000 [1.331 - 1] = Rs. 16, 550$ . $C A = 50, 000 + 16, 550 = Rs. 66, 550$ .

14Textbookex2.1-q5-c

Sabita deposited Rs. 1,50,000 in a bank. If the bank provides yearly 6% interest after 2 years 6 months, (i) how much is the compound amount? (ii) how much is the compound interest?

Flag: applying the textbook's own formula $C A = P (1 + \frac{R}{100})^{T} (1 + \frac{M R}{1200})$ with $P = 1, 50, 000$ , $R = 6%$ , $T = 2$ , $M = 6$ gives $C A = 1, 50, 000 (1.06)^{2} (1 + \frac{6 \times 6}{1200}) = 1, 50, 000 (1.1236) (1.03) = Rs. 1, 73, 596.20$ and $C I = Rs. 23, 596.20$ — this does not match the textbook's own printed answer (Rs. $1, 72, 753.50$ and Rs. $22, 753.50$ ), which would instead require the 6-month adjustment factor to be $1.025$ rather than $1.03$ (equivalent to treating $M = 5$ rather than $6$ ); this looks like an inconsistency in the source. (i) Using the formula as derived in this chapter: $C A \approx Rs. 1, 73, 596.20$ . (ii) $C I \approx Rs. 23, 596.20$ .

15Textbookex2.1-q6-a

Question 6 Manisha deposited Rs. 50,000 in a bank at the rate of compound interest 8% p.a. If the bank provides half yearly compound interest, then find the compound interest and compound amount she receives after 2 years.

$P = Rs. 50, 000$ , $R = 8%$ , $T = 2$ years, half yearly. $C I = P [(1 + \frac{R}{200})^{2 T} - 1] = 50, 000 [(1 + \frac{8}{200})^{4} - 1] = 50, 000 [(1.04)^{4} - 1] = Rs. 8, 492.93$ . $C A = 50, 000 + 8, 492.93 = Rs. 58, 492.93$ .

16Textbookex2.1-q6-b

A bank provides quarterly compound interest. If Sunil deposited Rs. 50,00,000 for 1 year at the rate of 12% p.a. interest, then find the compound interest and compound amount.

$P = Rs. 50, 00, 000$ , $R = 12%$ , $T = 1$ year, quarterly. $C I = P [(1 + \frac{R}{400})^{4 T} - 1] = 50, 00, 000 [(1 + \frac{12}{400})^{4} - 1] = 50, 00, 000 [(1.03)^{4} - 1] = Rs. 6, 27, 544.05$ . $C A = 50, 00, 000 + 6, 27, 544.05 = Rs. 56, 27, 544.05$ .

17Textbookex2.1-q7-a

Question 7 Karma Gurung deposited Rs. 80,000 in a bank at the rate of 8% compound interest p.a. Find the difference between simple interest and compound interest of the sum in 2 years.

$P = Rs. 80, 000$ , $R = 8%$ , $T = 2$ years. $S I = \frac{P T R}{100} = \frac{80000 \times 2 \times 8}{100} = Rs. 12, 800$ . $C I = P [(1.08)^{2} - 1] = 80, 000 [1.1664 - 1] = Rs. 13, 312$ . Difference $= C I - S I = 13, 312 - 12, 800 = Rs. 512$ .

18Textbookex2.1-q7-b

Find the difference between simple interest and yearly compound interest of a sum Rs. 7,500 at the rate of 12% p.a. interest in 3 years.

$P = Rs. 7, 500$ , $R = 12%$ , $T = 3$ years. $S I = \frac{7500 \times 3 \times 12}{100} = Rs. 2, 700$ . $C I = 7, 500 [(1.12)^{3} - 1] = 7, 500 [1.404928 - 1] = Rs. 3, 036.96$ . Difference $= C I - S I = 3, 036.96 - 2, 700 = Rs. 336.96$ .

19Textbookex2.1-q8-a

Question 8 Chhiring deposited Rs. 40,000 at the rate of 6% annual compound interest. Find the difference of yearly compound interest and half yearly compound interest of the sum in 2 years.

$P = Rs. 40, 000$ , $R = 6%$ , $T = 2$ years. Yearly: $C I_{y} = 40, 000 [(1.06)^{2} - 1] = 40, 000 [0.1236] = Rs. 4, 944$ . Half yearly: $C I_{h} = 40, 000 [(1.03)^{4} - 1] = Rs. 5, 020.35$ . Difference $= C I_{h} - C I_{y} = 5, 020.35 - 4, 944 = Rs. 76.35$ .

20Textbookex2.1-q8-b

What is the difference between semi-annual compound interest and quarterly compound interest of Rs. 18,000 in a year at the rate of 12% compound interest p.a.? Find it.

$P = Rs. 18, 000$ , $R = 12%$ , $T = 1$ year. Semi-annual: $C I_{se mi} = 18, 000 [(1.06)^{2} - 1] = Rs. 2, 224.80$ . Quarterly: $C I_{q u a r t er} = 18, 000 [(1.03)^{4} - 1] = Rs. 2, 259.16$ . Difference $= C I_{q u a r t er} - C I_{se mi} = 2, 259.16 - 2, 224.80 = Rs. 34.36$ .

21Textbookex2.1-q8-c

You went to deposit Rs. 60,000 in a bank for 2 years. The bank's notice board offers two fixed deposit accounts: Fixed deposit (P) with half yearly compound interest at 10% p.a., and Fixed deposit (Q) with yearly compound interest at 12% p.a. (i) How much interest will be collected in account (P) after 2 years? (ii) How much interest will be collected in account (Q) after 2 years? (iii) After knowing the interest rates of both options, by which option will you deposit the money? And why?

$P = Rs. 60, 000$ , $T = 2$ years. (i) Account P (half yearly, $10%$ ): $C I_{P} = 60, 000 [(1 + \frac{10}{200})^{4} - 1] = 60, 000 [(1.05)^{4} - 1] = Rs. 12, 930.38$ . (ii) Account Q (yearly, $12%$ ): $C I_{Q} = 60, 000 [(1.12)^{2} - 1] = Rs. 15, 264$ . (iii) Since $C I_{Q} > C I_{P}$ , account Q gives Rs. $15, 264 - 12, 930.38 = Rs. 2, 333.62$ more interest, so I would deposit in account Q.

22Textbookex2.1-q9-a

Question 9 If the yearly compound interest of a sum in 2 years at the rate of 15% p.a. interest is Rs. 180 more than simple interest, then find the sum.

Let principal $= P$ . $R = 15%$ , $T = 2$ years. $C I - S I = P [(1.15)^{2} - 1] - \frac{P \times 2 \times 15}{100} = P [0.3225] - 0.3 P = 0.0225 P$ . Setting $0.0225 P = 180$ : $P = \frac{180}{0.0225} = Rs. 8, 000$ .

23Textbookex2.1-q9-b

If the half yearly compound interest of a sum in a year at the rate of 10% interest p.a. is Rs. 40 more than the yearly compound interest of the same sum for the same period of time at the same rate of interest, then find the sum.

Let principal $= P$ . $R = 10%$ , $T = 1$ year. Half yearly $C I_{h} = P [(1.05)^{2} - 1] = 0.1025 P$ . Yearly $C I_{y} = P \times \frac{10}{100} = 0.1 P$ . Setting $C I_{h} - C I_{y} = 40$ : $0.1025 P - 0.1 P = 0.0025 P = 40$ , so $P = \frac{40}{0.0025} = Rs. 16, 000$ .

24Textbookex2.1-q9-c

Suprim borrowed some money for 2 years at the rate of compound interest of 10% p.a. and immediately he lent the money at the same rate of half yearly compound interest for the same period of time. In this transaction, if he gained Rs. 2019.24, then find how much he borrowed.

Let the borrowed sum $= P$ . Yearly CI paid (to the lender, $10%$ , 2 years): $C I_{y} = P [(1.10)^{2} - 1] = 0.21 P$ . Half yearly CI received (from the borrower, $10%$ , 2 years i.e. 4 periods at $5%$ ): $C I_{h} = P [(1.05)^{4} - 1] = 0.21550625 P$ . Gain $= C I_{h} - C I_{y} = 0.21550625 P - 0.21 P = 0.00550625 P$ . Setting this equal to $2019.24$ : $P = \frac{2019.24}{0.00550625} \approx Rs. 3, 66, 717.82$ .

25Textbookex2.1-q10-a

Question 10 In how many years does a sum of Rs. 1,00,000 amount to Rs. 1,21,000 at the rate of compound interest 10% p.a.?

$C A = P (1 + \frac{R}{100})^{T}$ : $1, 21, 000 = 1, 00, 000 (1.10)^{T}$ , so $(1.10)^{T} = 1.21 = (1.10)^{2}$ , giving $T = 2$ years.

26Textbookex2.1-q10-b

According to the compound interest, in how many years will the compound interest of Rs. 8,00,000 at the rate of 10% p.a. be Rs. 12,61,000? Find it.

Flag: taking the figures exactly as given ( $P = 8, 00, 000$ , $R = 10%$ , $C I = 12, 61, 000$ , i.e. $C A = 21, 61, 000$ ) does not give a whole-number year — solving $8, 00, 000 (1.10)^{T} = 21, 61, 000$ gives $T \approx 9.93$ years, not the textbook's stated answer of 3 years. With $P = 8, 00, 000$ and $R = 10%$ , $T = 3$ years instead gives $C A = 10, 64, 800$ , i.e. $C I = Rs. 2, 64, 800$ — so the printed $C I$ figure of Rs. $12, 61, 000$ appears inconsistent with the book's own answer of $3$ years; the intended $C I$ was most likely Rs. $2, 64, 800$ . Using $T = 3$ years (matching the book's answer): $C I = 8, 00, 000 [(1.10)^{3} - 1] = 8, 00, 000 [0.331] = Rs. 2, 64, 800$ , confirming $T = 3$ years.

27Textbookex2.1-q10-c

At what rate of yearly compound interest does the sum of Rs. 700 amount to Rs. 847 in 2 years?

$C A = P (1 + \frac{R}{100})^{T}$ : $847 = 700 (1 + \frac{R}{100})^{2}$ , so $(1 + \frac{R}{100})^{2} = \frac{847}{700} = 1.21$ , giving $1 + \frac{R}{100} = 1.1$ , so $R = 10%$ .

28Textbookex2.1-q10-d

At what rate of annual compound interest will the compound interest of Rs. 3,43,000 in 3 years be Rs. 1,13,533? Find.

$C I = P [(1 + \frac{R}{100})^{T} - 1]$ : $1, 13, 533 = 3, 43, 000 [(1 + \frac{R}{100})^{3} - 1]$ , so $(1 + \frac{R}{100})^{3} - 1 = \frac{1 , 13 , 533}{3 , 43 , 000} = 0.331$ , giving $(1 + \frac{R}{100})^{3} = 1.331 = (1.10)^{3}$ , so $R = 10%$ .

29Textbookex2.1-q11-a

Question 11 At the rate of yearly compound interest, a sum will be Rs. 6,050 in 2 years and Rs. 6,655 in 3 years respectively. Then (i) find the rate of the compound interest. (ii) find the sum.

Let principal $= x$ , rate $= R %$ . $x (1 + \frac{R}{100})^{2} = 6, 050$ ...(i); $x (1 + \frac{R}{100})^{3} = 6, 655$ ...(ii). Dividing (ii) by (i): $1 + \frac{R}{100} = \frac{6 , 655}{6 , 050} = 1.1$ , so $R = 10%$ . From (i): $x = \frac{6 , 050}{( 1.10 ) ^{2}} = \frac{6 , 050}{1.21} = Rs. 5, 000$ .

30Textbookex2.1-q11-b

At the rate of annual compound interest, a sum amounts to Rs. 10,580 in 2 years and Rs. 12,167 in 3 years respectively. Then (i) find the rate of the compound interest. (ii) find the sum.

Let principal $= x$ , rate $= R %$ . $x (1 + \frac{R}{100})^{2} = 10, 580$ ...(i); $x (1 + \frac{R}{100})^{3} = 12, 167$ ...(ii). Dividing (ii) by (i): $1 + \frac{R}{100} = \frac{12 , 167}{10 , 580} = 1.15$ , so $R = 15%$ . From (i): $x = \frac{10 , 580}{( 1.15 ) ^{2}} = \frac{10 , 580}{1.3225} = Rs. 8, 000$ .

31Textbookex2.1-q11-c

The compound interest of a sum at the rate of yearly compound interest in 1 year and 2 years are respectively Rs. 1,800 and Rs. 3,816, then find the rate of the interest and the sum.

Let principal $= P$ , rate $= R %$ . Interest of year 1: $\frac{P R}{100} = 1, 800$ ...(i). Total CI in 2 years: $P [(1 + \frac{R}{100})^{2} - 1] = 3, 816$ ...(ii). From (i), $P = \frac{1 , 80 , 000}{R}$ . Substituting the identity $C I_{2} = C I_{1} (2 + \frac{R}{100})$ [since $P ((1 + \frac{R}{100})^{2} - 1) = P R /100 \times (2 + R /100)$ ]: $3, 816 = 1, 800 (2 + \frac{R}{100})$ , so $2 + \frac{R}{100} = \frac{3 , 816}{1 , 800} = 2.12$ , giving $\frac{R}{100} = 0.12$ , so $R = 12%$ . Then $P = \frac{1 , 80 , 000}{12} = Rs. 15, 000$ .

32Textbookex2.1-q12-a

Question 12 A person deposited Rs. 5,00,000 in a commercial bank for 2 years to get the half yearly compound interest at the rate of 10% per annum. 5% tax on the interest will be levied. But right after a year, the bank has changed the policy and decided to compute the interest quarterly at the same rate of interest. (i) Find the interest of the first year by deducting the tax. (ii) What would be the interest of the second year after deducting the tax? (iii) What is the difference between the interest of the first year and second year after deducting the tax? Find. (iv) After deducting the tax, by what percentage does the interest of the first year differ from the interest of the second year?

$P = Rs. 5, 00, 000$ , $R = 10%$ . (i) Half yearly $C I_{1}^{g r oss} = P [(1.05)^{2} - 1] = 5, 00, 000 \times 0.1025 = Rs. 51, 250$ . After $5%$ tax: $C I_{1} = 51, 250 \times 0.95 = Rs. 48, 687.50$ . (ii) Principal for year 2: $5, 00, 000 + 48, 687.50 = Rs. 5, 48, 687.50$ . Quarterly $C I_{2}^{g r oss} = 5, 48, 687.50 [(1.025)^{4} - 1] = Rs. 56, 960.84$ . After tax: $C I_{2} = 56, 960.84 \times 0.95 = Rs. 54, 112.79$ . (iii) Difference $= C I_{2} - C I_{1} = 54, 112.79 - 48, 687.50 = Rs. 5, 425.29$ . (iv) Percentage difference $= \frac{5 , 425.29}{48 , 687.50} \times 100% = 11.14%$ .

33Textbookex2.1-q12-b

A person deposited Rs. 80,000 in a co-operative limited for 2 years to get the yearly compound interest at the rate of 15% 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 half yearly at the same rate of interest. After deducting tax, by what percentage does the interest of the first year differ from the interest of the second year?

$P = Rs. 80, 000$ , $R = 15%$ . Year 1 (yearly): $C I_{1}^{g r oss} = P \times \frac{15}{100} = Rs. 12, 000$ . After $5%$ tax: $C I_{1} = 12, 000 \times 0.95 = Rs. 11, 400$ . Principal for year 2: $80, 000 + 11, 400 = Rs. 91, 400$ . Half yearly $C I_{2}^{g r oss} = 91, 400 [(1.075)^{2} - 1] = Rs. 14, 224.12$ . After tax: $C I_{2} = 14, 224.12 \times 0.95 = Rs. 13, 512.92$ . Percentage difference $= \frac{C I _{2} - C I _{1}}{C I _{1}} \times 100% = \frac{13 , 512.92 - 11 , 400}{11 , 400} \times 100% = \frac{2 , 112.92}{11 , 400} \times 100% \approx 18.53%$ .

34Textbookex2.1-q13-a

Question 13 Ram divided Rs. 41,000 in two parts and deposited in the bank account of two daughters at the rate of annual compound interest 5% for 2 years and 3 years respectively. If the compound amount received by them after 2 years and 3 years respectively are equal, find how much did each of them get?

Let the 2-year daughter's share $= x$ , so the 3-year daughter's share $= (41, 000 - x)$ . $R = 5%$ . Setting the compound amounts equal: $x (1.05)^{2} = (41, 000 - x) (1.05)^{3}$ , so $x = (41, 000 - x) (1.05)$ , giving $x = 43, 050 - 1.05 x$ , so $2.05 x = 43, 050$ , thus $x = 21, 000$ . So the 2-year share is Rs. $21, 000$ and the 3-year share is Rs. $41, 000 - 21, 000 = Rs. 20, 000$ .

35Textbookex2.1-q13-b

Divide Rs. 21,000 in two parts in such a way that the compound amount of the first part at the rate of 10% p.a. for 3 years is equal to the compound amount for 2 years. What sums are there in the first part and the second part? Find it.

Let the first part (3 years) $= y$ , so the second part (2 years) $= (21, 000 - y)$ . $R = 10%$ . Setting compound amounts equal: $y (1.10)^{3} = (21, 000 - y) (1.10)^{2}$ , so $y (1.10) = 21, 000 - y$ , giving $1.10 y + y = 21, 000$ , so $2.10 y = 21, 000$ , thus $y = 10, 000$ . So the first part is Rs. $10, 000$ and the second part is Rs. $21, 000 - 10, 000 = Rs. 11, 000$ .

36Textbookex2.1-q14-a

Question 14 According to the yearly compound interest, the compound interest of a sum for 1 year and 2 years are respectively Rs. 450 and Rs. 945, find the rate of interest and the sum.

Let principal $= P$ , rate $= R %$ . Year 1: $\frac{P R}{100} = 450$ ...(i). Using the identity $C I_{2} = C I_{1} (2 + \frac{R}{100})$ : $945 = 450 (2 + \frac{R}{100})$ , so $2 + \frac{R}{100} = \frac{945}{450} = 2.1$ , giving $\frac{R}{100} = 0.1$ , so $R = 10%$ . From (i): $P = \frac{45 , 000}{10} = Rs. 4, 500$ . Note: Q14(b) uses the same figures as Q11(c), so the method and result are identical to that part; the answer $R = 12%$ , sum $=$ Rs. 15,000 reproduces the working correctly. The textbook's printed answer key shows "Rs.1,500" for this part, which looks like a missing digit (a typo for Rs. 15,000), since it must match Q11(c)'s figures exactly.

37Textbookex2.1-q14-b

According to the yearly compound interest, the compound interests of a sum for 1 year and 2 years are Rs. 1800 and Rs. 3816 respectively. Find the rate of the interest and the sum.

Let principal $= P$ , rate $= R %$ . Year 1: $\frac{P R}{100} = 1, 800$ ...(i). Using $C I_{2} = C I_{1} (2 + \frac{R}{100})$ : $3, 816 = 1, 800 (2 + \frac{R}{100})$ , so $2 + \frac{R}{100} = \frac{3 , 816}{1 , 800} = 2.12$ , giving $\frac{R}{100} = 0.12$ , so $R = 12%$ . From (i): $P = \frac{1 , 80 , 000}{12} = Rs. 15, 000$ .

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