Selina Concise Solutions for ICSE Class 8 Mathematics Chapter 9 Simple and Compound Interest

Exercise 9(A)

Question 1. Find the interest and the amount on:
(i) Rs. 750 in 3 years 4 months at 10% per annum.
(ii) Rs. 5,000 at 8% per year from 23rd December 2011 to 29th July 2012.
(iii) Rs. 2,600 in 2 years 3 months at 1% per month.
(iv) Rs. 4,000 in \(1 \frac{1}{3}\) years at 2 paise per rupee per month.
Answer:
(i) We are given:
Principal amount (\(P\)) = Rs. 750
Time duration (\(T\)) = 3 years 4 months = \(3 \frac{4}{12}\) years = \(3 \frac{1}{3}\) years = \(\frac{10}{3}\) years
Interest rate (\(R\)) = 10% per annum
Using the simple interest formula:
\(\text{Simple Interest } (I) = \frac{P \times R \times T}{100}\)
\(\implies I = \frac{750 \times 10 \times \frac{10}{3}}{100}\)
\(\implies I = \frac{250 \times 10 \times 10}{100 \times 10}\) = Rs. 250
Now, let's find the total amount (\(A\)):
\(A = P + I\)
\(\implies A = 750 + 250\) = Rs. 1,000

(ii) We are given:
Principal amount (\(P\)) = Rs. 5,000
Interest rate (\(R\)) = 8% per year
Time duration (\(T\)) from December 23, 2011 to July 29, 2012:
Let's calculate the total number of days:
- December: 8 days (from December 23 to December 31)
- January: 31 days
- February: 29 days (2012 is a leap year)
- March: 31 days
- April: 30 days
- May: 31 days
- June: 30 days
- July: 29 days
Total days = 8 + 31 + 29 + 31 + 30 + 31 + 30 + 29 = 219 days
Converting time into years:
\(T = \frac{219}{365}\) years
Now, let's calculate the simple interest:
\(\text{Interest } (I) = \frac{P \times R \times T}{100}\)
\(\implies I = \frac{5000 \times 8 \times 219}{100 \times 365}\)
\(\implies I = 10 \times 8 \times 3\) = Rs. 240
The total accumulated amount is:
\(A = P + I\)
\(\implies A = 5000 + 240\) = Rs. 5,240

(iii) We are given:
Principal amount (\(P\)) = Rs. 2,600
Time duration (\(T\)) = 2 years 3 months = (2 \times 12) + 3 = 27 months
Interest rate (\(R\)) = 1% per month
Using the simple interest formula based on monthly values:
\(\text{Interest } (I) = \frac{P \times T \times R}{100}\)
\(\implies I = \frac{2600 \times 27 \times 1}{100}\)
\(\implies I = 26 \times 27\) = Rs. 702
To find the final amount:
\(A = P + I\)
\(\implies A = 2600 + 702\) = Rs. 3,302

(iv) We are given:
Principal amount (\(P\)) = Rs. 4,000
Time duration (\(T\)) = \(1 \frac{1}{3}\) years = 1 year and 4 months = 16 months
Interest rate (\(R\)) = 2 paise per rupee per month
Since 1 rupee contains 100 paise, this rate is equivalent to:
\(R = 2\%\) per month
Now, we calculate the simple interest:
\(\text{Interest } (I) = \frac{P \times T \times R}{100}\)
\(\implies I = \frac{4000 \times 16 \times 2}{100}\)
\(\implies I = 40 \times 32\) = Rs. 1,280
The total amount to be returned is:
\(A = P + I\)
\(\implies A = 4000 + 1280\) = Rs. 5,280
In simple words: Simple interest is calculated by multiplying the starting money, the rate, and the time together, then dividing by 100. Adding this interest to the starting money gives the total amount.

Exam Tip: Pay close attention to the units of time and interest rate. If the rate is given per annum, convert the time to years; if the rate is per month, convert the time to months before calculating.

Question 2. Rohit borrowed Rs. 24,000 at 7.5 percent per year. How much money will he pay at the end of 4th years to clear his debt ?
Answer:
We are given:
Principal sum borrowed (\(P\)) = Rs. 24,000
Interest rate (\(R\)) = 7.5% per annum
Time period (\(T\)) = 4 years
Using the simple interest formula:
\(\text{Simple Interest } (S.I.) = \frac{P \times T \times R}{100}\)
\(\implies S.I. = \frac{24000 \times 4 \times 7.5}{100}\)
\(\implies S.I. = 240 \times 4 \times 7.5 = 240 \times 30\) = Rs. 7,200
To clear his total debt, the amount Rohit needs to pay is:
\(\text{Total Amount } (A) = \text{Principal } (P) + \text{Simple Interest } (S.I.)\)
\(\implies A = 24000 + 7200\) = Rs. 31,200
In simple words: When you borrow money, you have to pay back the original sum plus the interest. To clear the debt after 4 years, Rohit pays back Rs. 24,000 plus the Rs. 7,200 interest.

Exam Tip: Be careful with the final step. The question asks for the total money to clear the debt, which means you must add the principal and the interest together, not just find the interest.

Question 3. The interest on a certain sum of money is Rs. 1,480 in 2 years and at 10 per cent per year. Find the sum of money.
Answer:
Let us assume the principal sum of money (\(P\)) is Rs. \(x\).
Given:
Interest (\(I\)) = Rs. 1,480
Time duration (\(T\)) = 2 years
Interest rate (\(R\)) = 10% per annum
We use the simple interest formula:
\(\text{Interest } (I) = \frac{P \times T \times R}{100}\)
\(\implies 1480 = \frac{x \times 2 \times 10}{100}\)
\(\implies 1480 = \frac{x \times 20}{100}\)
\(\implies 1480 = \frac{x}{5}\)
\(\implies x = 1480 \times 5\)
\(\implies x = 7400\)
Therefore, the required sum of money is Rs. 7,400.
In simple words: We know the interest, rate, and time, and we want to find the starting money. By putting the values into our formula and working backward, we find that the starting sum of money was Rs. 7,400.

Exam Tip: Whenever the principal is unknown, assume it to be a variable like \(x\) (or Rs. 100), write the formula, and solve the equation to find the value of the variable.

Question 4. On what principal will the simple interest be Rs. 7,008 in 6 years 3 months at 5% per year ?
Answer:
Let the principal amount be Rs. \(P\).
We are given:
Simple Interest (\(I\)) = Rs. 7,008
Time period (\(T\)) = 6 years 3 months = \(6 \frac{3}{12}\) years = \(6 \frac{1}{4}\) years = \(\frac{25}{4}\) years
Interest rate (\(R\)) = 5% per annum
Using the simple interest formula:
\(\text{Simple Interest } (I) = \frac{P \times T \times R}{100}\)
\(\implies 7008 = \frac{P \times \frac{25}{4} \times 5}{100}\)
\(\implies 7008 = \frac{P \times 125}{400}\)
\(\implies P = \frac{7008 \times 400}{125}\)
Simplifying this:
\(\implies P = \frac{7008 \times 16}{5}\)
\(\implies P = \frac{112128}{5}\)
\(\implies P\) = Rs. 22,425.60
Thus, the required principal amount is Rs. 22,425.60.
In simple words: We are looking for the original money that grows to have Rs. 7,008 in interest over 6 years and 3 months at 5% interest. By rearranging our formula, we calculate that the starting amount is Rs. 22,425.60.

Exam Tip: When converting months to years, divide the number of months by 12. Always simplify the fraction (like \( \frac{3}{12} = \frac{1}{4} \)) to make the calculation easier.

Question 5. Find the principal which will amount to Rs. 4,000 in 4 years at 6.25% Per annum.
Answer:
Let the principal sum of money be Rs. \(P\).
We are given:
Total accumulated Amount (\(A\)) = Rs. 4,000
Time period (\(T\)) = 4 years
Interest rate (\(R\)) = 6.25% per annum = \(6 \frac{1}{4}\%\) per annum = \(\frac{25}{4}\%\) per annum
First, let's write down the formula for simple interest:
\(\text{Simple Interest } (I) = \frac{P \times T \times R}{100}\)
\(\implies I = \frac{P \times 4 \times \frac{25}{4}}{100}\)
\(\implies I = \frac{P \times 25}{100}\)
\(\implies I = \frac{P}{4}\)
We know that total Amount (\(A\)) is the sum of Principal and Interest:
\(A = P + I\)
\(\implies 4000 = P + \frac{P}{4}\)
\(\implies 4000 = \frac{5P}{4}\)
Let's solve for \(P\):
\(\implies 5P = 4000 \times 4\)
\(\implies 5P = 16000\)
\(\implies P = \frac{16000}{5}\)
\(\implies P = 3200\)
Therefore, the required principal sum is Rs. 3,200.
In simple words: Since we are given the final amount (which includes both the starting money and the interest earned), we can express the interest as a fraction of the principal. This allows us to set up an equation to find that the initial principal was Rs. 3,200.

Exam Tip: Remember that Amount = Principal + Interest. When you are given the total Amount instead of just the interest, express the interest in terms of \(P\), and then solve for \(P\) using this formula.

Question 6.
(i) At what rate per cent per annum will Rs. 630 produce an interest of Rs. 126 in 4 years ?
(ii) At what rate per cent per year will a sum double itself in \(6 \frac{1}{4}\) years ?
Answer:
(i) We are given:
Principal (\(P\)) = Rs. 630
Simple Interest (\(I\)) = Rs. 126
Time duration (\(T\)) = 4 years
Let's find the rate of interest (\(R\)) per annum:
Using the formula:
\(R = \frac{100 \times I}{P \times T}\)
\(\implies R = \frac{100 \times 126}{630 \times 4}\)
Since \(126 \times 5 = 630\):
\(\implies R = \frac{100}{5 \times 4}\)
\(\implies R = \frac{100}{20}\)
\(\implies R = 5\%\)
So, the required rate is 5% per annum.

(ii) Let us assume the principal sum of money (\(P\)) is Rs. 100.
Since the sum doubles itself:
Total Amount (\(A\)) = \(2 \times 100\) = Rs. 200
Therefore, the interest earned is:
\(I = A - P\)
\(\implies I = 200 - 100\) = Rs. 100
Time period (\(T\)) = \(6 \frac{1}{4}\) years = \(\frac{25}{4}\) years
We calculate the rate of interest (\(R\)):
\(R = \frac{100 \times I}{P \times T}\)
\(\implies R = \frac{100 \times 100}{100 \times \frac{25}{4}}\)
\(\implies R = \frac{100 \times 4}{25}\)
\(\implies R = 4 \times 4\)
\(\implies R = 16\%\)
Hence, the rate of interest is 16% per year.
In simple words: (i) We use the rearranged formula to find that a rate of 5% is needed to earn Rs. 126 from Rs. 630 in 4 years. (ii) For a sum to double, the interest earned must equal the original principal. Over six and a quarter years, this requires a rate of 16% each year.

Exam Tip: For "doubling" or "tripling" problems where no principal is given, assuming the starting principal to be Rs. 100 is an extremely simple and effective strategy to avoid complex variables.

Question 7.
(i) In how many years will Rs.950 produce Rs.399 as simple interest at 7% ?
(ii) Find the time in which Rs.1200 will amount to Rs.1536 at 3.5% per year.
Answer:
(i) We are given:
Principal amount (\(P\)) = Rs. 950
Simple Interest (\(I\)) = Rs. 399
Interest rate (\(R\)) = 7% per annum
We want to find the time period (\(T\)):
Using the formula:
\(T = \frac{100 \times I}{P \times R}\)
\(\implies T = \frac{100 \times 399}{950 \times 7}\)
Simplifying:
\(\implies T = \frac{10 \times 57}{95}\)
\(\implies T = \frac{570}{95}\) = 6 years
Thus, it will take 6 years.

(ii) We are given:
Principal amount (\(P\)) = Rs. 1,200
Total accumulated Amount (\(A\)) = Rs. 1,536
Interest rate (\(R\)) = 3.5% per year
First, find the simple interest earned:
\(I = A - P\)
\(\implies I = 1536 - 1200\) = Rs. 336
Now, calculate the time period (\(T\)):
\(T = \frac{100 \times I}{P \times R}\)
\(\implies T = \frac{100 \times 336}{1200 \times 3.5}\)
\(\implies T = \frac{336}{12 \times 3.5}\)
\(\implies T = \frac{336}{42}\)
\(\implies T\) = 8 years
Therefore, the time required is 8 years.
In simple words: (i) By using the formula for time, we find that it takes 6 years for Rs. 950 to earn Rs. 399 in interest at 7%. (ii) We first subtract the principal from the total amount to find that the interest is Rs. 336. Then we calculate that it takes 8 years to earn this interest at 3.5% annually.

Exam Tip: If the question provides the accumulated Amount and you need to find the time, always calculate the interest first by using the formula Interest = Amount - Principal.

Question 8. The simple interest on a certain sum of money is \(\frac{3}{8}\) of the sum in \(6 \frac{1}{4}\) years. Find the rate percent charged.
Answer:
Let the principal sum (\(P\)) be Rs. 8.
Then, the Simple Interest (\(I\)) is:
\(I = \frac{3}{8} \times 8\) = Rs. 3
Time period (\(T\)) = \(6 \frac{1}{4}\) years = \(\frac{25}{4}\) years
Now, let's calculate the rate of interest (\(R\)):
\(R = \frac{100 \times I}{P \times T}\)
\(\implies R = \frac{100 \times 3}{8 \times \frac{25}{4}}\)
\(\implies R = \frac{100 \times 3 \times 4}{8 \times 25}\)
\(\implies R = \frac{1200}{200}\)
\(\implies R\) = 6%
Hence, the rate percent charged is 6% per annum.
In simple words: If we assume the starting sum is Rs. 8, the interest is Rs. 3. Over six and a quarter years, calculating the rate shows that 6% interest is charged each year.

Exam Tip: When the interest is given as a fraction of the principal, like \(\frac{3}{8}\), assuming the principal to be the denominator of the fraction (Rs. 8) makes the math simple and avoids dealing with fractions during calculation.

Question 9. What sum of money borrowed on 24th May will amount to Rs. 10210.20 on 17th October of the same year at 5 percent per annum simple interest.
Answer:
We are given:
Total amount (\(A\)) = Rs. 10,210.20
Interest rate (\(R\)) = 5% per annum
First, we find the number of days between May 24 and October 17 of the same year:
- May: 7 days (from May 24 to May 31)
- June: 30 days
- July: 31 days
- August: 31 days
- September: 30 days
- October: 17 days (up to October 17)
Total days = 7 + 30 + 31 + 31 + 30 + 17 = 146 days
Let's convert this duration into years:
\(T = \frac{146}{365} = \frac{2}{5}\) years
We know that:
\(A = P + \frac{P \times R \times T}{100}\)
\(\implies A = P \left(1 + \frac{R \times T}{100}\right)\)
Substituting the given values:
\(\implies 10210.20 = P \left(1 + \frac{5 \times \frac{2}{5}}{100}\right)\)
\(\implies 10210.20 = P \left(1 + \frac{2}{100}\right)\)
\(\implies 10210.20 = P \left(\frac{102}{100}\right)\)
\(\implies P = \frac{10210.20 \times 100}{102}\)
\(\implies P = \frac{1021020}{102}\)
\(\implies P\) = Rs. 10,010
Thus, the sum of money borrowed is Rs. 10,010.
In simple words: First, we count the exact days between May 24 and October 17, which is 146 days or \(\frac{2}{5}\) of a year. By setting up our formula with the total amount and solving, we find that the borrowed money was Rs. 10,010.

Exam Tip: Remember that 73 is a key prime factor for day-to-year conversions. 73 days is \(\frac{1}{5}\) of a year, 146 days is \(\frac{2}{5}\), and 219 days is \(\frac{3}{5}\). Recognizing this simplifies the math instantly.

Question 10. In what time will the interest on a certain sum of money at 6% be \(\frac{5}{8}\) of itself ?
Answer:
Let us assume the principal sum (\(P\)) is Rs. 8.
Given:
Simple Interest (\(I\)) = \(\frac{5}{8}\) of the sum = \(\frac{5}{8} \times 8\) = Rs. 5
Interest rate (\(R\)) = 6% per annum
We calculate the time period (\(T\)) using the formula:
\(T = \frac{100 \times I}{P \times R}\)
\(\implies T = \frac{100 \times 5}{8 \times 6}\)
\(\implies T = \frac{500}{48} \text{ years}\)
Simplifying the fraction:
\(\implies T = \frac{125}{12} \text{ years}\)
Converting this into years and months:
\(\implies T = 10 \frac{5}{12} \text{ years} = 10 \text{ years } + \left(\frac{5}{12} \times 12\right) \text{ months}\)
\(\implies T = 10 \text{ years } 5 \text{ months}\)
Thus, the time required is 10 years and 5 months.
In simple words: By assuming the principal is Rs. 8, the interest is Rs. 5. Using the time formula gives us \(\frac{125}{12}\) years, which converts exactly to 10 years and 5 months.

Exam Tip: Always convert fractional years into months by multiplying the fractional part by 12. For example, \(\frac{5}{12}\) of a year is \(\frac{5}{12} \times 12 = 5\) months.

Question 11. Ashok lent out Rs.7000 at 6% and Rs.9500 at 5%. Find his total income from the interest in 3 years.
Answer:
To solve this, we calculate the interest earned from both cases separately:
First Case:
Principal (\(P_1\)) = Rs. 7,000
Rate of interest (\(R_1\)) = 6% per annum
Time period (\(T_1\)) = 3 years
Simple Interest (\(I_1\)) = \(\frac{P_1 \times R_1 \times T_1}{100}\)
\(\implies I_1 = \frac{7000 \times 6 \times 3}{100}\)
\(\implies I_1 = 70 \times 18\) = Rs. 1,260

Second Case:
Principal (\(P_2\)) = Rs. 9,500
Rate of interest (\(R_2\)) = 5% per annum
Time period (\(T_2\)) = 3 years
Simple Interest (\(I_2\)) = \(\frac{P_2 \times R_2 \times T_2}{100}\)
\(\implies I_2 = \frac{9500 \times 5 \times 3}{100}\)
\(\implies I_2 = 95 \times 15\) = Rs. 1,425

Now, let's find the total interest income:
\(\text{Total Income} = I_1 + I_2\)
\(\implies \text{Total Income} = 1260 + 1425\) = Rs. 2,685
Thus, his total income from the interest is Rs. 2,685.
In simple words: We find the interest earned from the Rs. 7,000 loan (which is Rs. 1,260) and the Rs. 9,500 loan (which is Rs. 1,425) separately. Adding these two amounts together gives a total income of Rs. 2,685.

Exam Tip: When a question involves multiple investments or loans, handle each scenario step-by-step as independent problems, then add the resulting interests together at the end.

Question 12. Raj borrows Rs.8,000; out of which Rs. 4500 at 5% and remainder at 6%. Find the total interest paid by him in 4 years.
Answer:
We are given:
Total borrowed amount = Rs. 8,000
Let's divide this into two parts:

First Part:
Principal (\(P_1\)) = Rs. 4,500
Interest rate (\(R_1\)) = 5% per annum
Time period (\(T\)) = 4 years
Simple Interest (\(I_1\)) = \(\frac{P_1 \times R_1 \times T}{100}\)
\(\implies I_1 = \frac{4500 \times 5 \times 4}{100}\)
\(\implies I_1 = 45 \times 20\) = Rs. 900

Second Part:
Remaining principal (\(P_2\)) = Rs. 8,000 - Rs. 4,500 = Rs. 3,500
Interest rate (\(R_2\)) = 6% per annum
Time period (\(T\)) = 4 years
Simple Interest (\(I_2\)) = \(\frac{P_2 \times R_2 \times T}{100}\)
\(\implies I_2 = \frac{3500 \times 6 \times 4}{100}\)
\(\implies I_2 = 35 \times 24\) = Rs. 840

Now, we calculate the total interest paid:
\(\text{Total Interest} = I_1 + I_2\)
\(\implies \text{Total Interest} = 900 + 840\) = Rs. 1,740
Thus, the total interest paid by Raj over 4 years is Rs. 1,740.
In simple words: Raj borrows Rs. 8,000 in two parts. The first part is Rs. 4,500 (earning Rs. 900 in interest), and the rest is Rs. 3,500 (earning Rs. 840 in interest). Adding them together, the total interest is Rs. 1,740.

Exam Tip: Always find the "remainder" principal by subtracting the first part from the total sum before calculating the interest for the second part.

Question 13. Mohan lends Rs.4800 to John for \(4 \frac{1}{2}\) years and Rs.2500 to Shy am for 6 years and receives a total sum of Rs.2196 as interest. Find the rate percent per annum, it being the same in both the cases.
Answer:
Let the common rate of interest per annum be \(x\%\).

First Case (John):
Principal (\(P_1\)) = Rs. 4,800
Time period (\(T_1\)) = \(4 \frac{1}{2}\) years = \(\frac{9}{2}\) years
Simple Interest (\(I_1\)) = \(\frac{P_1 \times T_1 \times R}{100}\)
\(\implies I_1 = \frac{4800 \times \frac{9}{2} \times x}{100}\)
\(\implies I_1 = \frac{4800 \times 9 \times x}{200}\)
\(\implies I_1 = 24 \times 9 \times x = 216x\)

Second Case (Shyam):
Principal (\(P_2\)) = Rs. 2,500
Time period (\(T_2\)) = 6 years
Simple Interest (\(I_2\)) = \(\frac{P_2 \times T_2 \times R}{100}\)
\(\implies I_2 = \frac{2500 \times 6 \times x}{100}\)
\(\implies I_2 = 25 \times 6 \times x = 150x\)

We are given that the total interest received is Rs. 2,196:
\(\implies I_1 + I_2 = 2196\)
\(\implies 216x + 150x = 2196\)
\(\implies 366x = 2196\)
\(\implies x = \frac{2196}{366}\)
\(\implies x = 6\)
Thus, the interest rate per annum is 6%.
In simple words: Since the interest rate is the same for both, we can represent the interest from both loans in terms of \(x\). Adding them together gives \(366x = 2196\), which solves to find that the interest rate is 6% per year.

Exam Tip: When the rate of interest is unknown but identical in multiple cases, assume it to be a common variable \(x\). Express each interest value in terms of \(x\), set up their sum, and solve the linear equation.

Question 14. John lent Rs. 2550 to Mohan at 7.5 per cent per annum. If Mohan discharges the debt after 8 months by giving an old black and white television and Rs. 1422.50; find the price of the television.
Answer:
We are given:
Principal amount lent (\(P\)) = Rs. 2,550
Rate of interest (\(R\)) = 7.5% per annum
Time period (\(T\)) = 8 months = \(\frac{8}{12}\) years = \(\frac{2}{3}\) years
First, let's calculate the simple interest accrued:
\(\text{Simple Interest } (S.I.) = \frac{P \times R \times T}{100}\)
\(\implies S.I. = \frac{2550 \times 7.5 \times \frac{2}{3}}{100}\)
\(\implies S.I. = \frac{2550 \times 5}{100}\)
\(\implies S.I. = \frac{12750}{100}\)
\(\implies S.I.\) = Rs. 127.50

Next, we find the total amount due at the end of 8 months:
\(\text{Total Amount } (A) = P + S.I.\)
\(\implies A = 2550 + 127.50\) = Rs. 2,677.50

Mohan cleared this debt by paying cash and giving a television:
\(\text{Total Amount } (A) = \text{Cash Paid} + \text{Price of Television}\)
\(\implies 2677.50 = 1422.50 + \text{Price of Television}\)
\(\implies \text{Price of Television} = 2677.50 - 1422.50\)
\(\implies \text{Price of Television}\) = Rs. 1,255
Thus, the price of the television is Rs. 1,255.
In simple words: First, we calculate that the total amount Mohan owes after 8 months is Rs. 2,677.50. Since he pays Rs. 1,422.50 in cash and gives a television to clear the rest, the value of the television must be Rs. 1,255.

Exam Tip: When a debt is cleared with a combination of cash and an item, calculate the total amount due first. The value of the item is simply the total amount due minus the cash paid.

Exercise 9(B)

Question 1. The interest on a certain sum of money is 0.24 times of itself in 3 years. Find the rate of interest.
Answer:
Let us assume the principal sum of money borrowed (\(P\)) is Rs. 100.
Given:
Time duration (\(T\)) = 3 years
The simple interest (\(I\)) is 0.24 times the principal sum itself:
\(I = 0.24 \times 100\) = Rs. 24
We need to find the rate of interest (\(R\)) per annum:
Using the simple interest formula:
\(I = \frac{P \times T \times R}{100}\)
Substituting the values:
\(\implies 24 = \frac{100 \times 3 \times R}{100}\)
\(\implies 24 = 3 \times R\)
\(\implies R = \frac{24}{3}\)
\(\implies R = 8\%\)
Hence, the required rate of interest is 8% per annum.
In simple words: If we assume the starting sum is Rs. 100, the interest earned is Rs. 24 over 3 years. By working backward with our simple interest formula, we find the rate is 8% per year.

Exam Tip: "0.24 times of itself" means the interest is \(0.24 \times P\). Assuming \(P = 100\) makes it extremely straightforward because \(0.24 \times 100 = 24\), and the 100s in the formula cancel out nicely.

Question 2. If Rs. 3,750 amount to Rs. 4,620 in 3 years at simple interest. Find:
(i) the rate of interest
(ii) the amount of Rs. 7,500 in \(5 \frac{1}{2}\) years at the same rate of interest
Answer:
(i) We are given:
Principal (\(P\)) = Rs. 3,750
Total Amount (\(A\)) = Rs. 4,620
Time period (\(T\)) = 3 years
First, let's find the interest earned (\(I\)):
\(I = A - P\)
\(\implies I = 4620 - 3750\) = Rs. 870
Now, let's find the rate of interest (\(R\)):
\(R = \frac{100 \times I}{P \times T}\)
\(\implies R = \frac{100 \times 870}{3750 \times 3}\)
\(\implies R = \frac{87000}{11250}\)
\(\implies R = \frac{116}{15}\% = 7 \frac{11}{15}\%\)
So, the interest rate is \(7 \frac{11}{15}\%\) per annum.

(ii) For the second case:
New principal (\(P_2\)) = Rs. 7,500
Time period (\(T_2\)) = \(5 \frac{1}{2}\) years = \(\frac{11}{2}\) years
Rate of interest (\(R\)) = \(\frac{116}{15}\%\) per annum
Let's find the simple interest:
\(I = \frac{P_2 \times T_2 \times R}{100}\)
\(\implies I = \frac{7500 \times \frac{11}{2} \times \frac{116}{15}}{100}\)
\(\implies I = \frac{7500 \times 11 \times 116}{100 \times 2 \times 15}\)
\(\implies I = \frac{75 \times 11 \times 116}{30}\)
\(\implies I = \frac{5 \times 11 \times 116}{2}\)
\(\implies I = 5 \times 11 \times 58\)
\(\implies I = 55 \times 58\) = Rs. 3,190
Now, let's find the final total amount (\(A_2\)):
\(A_2 = P_2 + I\)
\(\implies A_2 = 7500 + 3190\) = Rs. 10,690
Thus, the total amount is Rs. 10,690.
In simple words: (i) First, we find that the interest earned on Rs. 3,750 is Rs. 870 over 3 years, giving an annual rate of \(7 \frac{11}{15}\%\). (ii) Using this same rate, Rs. 7,500 earns Rs. 3,190 in interest over five and a half years, leading to a total amount of Rs. 10,690.

Exam Tip: Keep fractional rates like \(\frac{116}{15}\%\) in their improper fraction form rather than converting to decimal. This allows for clean cancellations in subsequent calculations.

Question 3. A sum of money, lent out at simple interst, doubles itself in 8 years. Find :
(i) the rate of interest
(ii) in how many years will the sum become triple (three times) of itself at the same rate per cent ?
Answer:
(i) Suppose the starting principal (P) is Rs. 100. Since it doubles itself, the final amount (A) becomes Rs. 200.
Therefore, the simple interest earned is:
\( I = \text{Rs. } 200 - \text{Rs. } 100 = \text{Rs. } 100 \)
The time period is given as \( T = 8 \) years.
The formula to find the annual rate of interest (R) is:
\( R = \frac{100 \times I}{P \times T} \)
Substituting these values:
\( R = \frac{100 \times 100}{100 \times 8} \% \)

\( \implies R = \frac{100}{8} \% = \frac{25}{2} \% = 12.5\% \)

(ii) For the second part, the money needs to become three times its original amount. If the principal is Rs. 100, the final amount will be Rs. 300.
Thus, the simple interest required is:
\( I = \text{Rs. } 300 - \text{Rs. } 100 = \text{Rs. } 200 \)
Using the interest rate of \( R = \frac{25}{2}\% \), the time (T) needed is calculated as:
\( T = \frac{100 \times I}{P \times R} \)
\( T = \frac{100 \times 200}{100 \times \frac{25}{2}} \)

\( \implies T = \frac{100 \times 200 \times 2}{100 \times 25} \)

\( \implies T = 16 \) years
Consequently, the sum of money will triple in 16 years.
In simple words: When money doubles, the interest you get is equal to the money you started with. To make the money triple, you need twice that amount of interest, which takes twice as long, or 16 years.

Exam Tip: Choosing Rs. 100 as the initial principal makes calculations very straightforward. Remember to multiply by 2 when dividing by a fraction like 25/2 in the denominator.

Question 4. Rupees 4000 amount to Rs.5000 in 8 years ; in what time will Rs.2100 amount to Rs.2800 at the same rate ?
Answer:
In the first scenario:
The initial principal (P) is Rs. 4000, and the final amount (A) is Rs. 5000.
The simple interest (I) earned during this period is:
\( I = A - P \)

\( \implies I = \text{Rs. } 5000 - \text{Rs. } 4000 = \text{Rs. } 1000 \)
The time duration is \( T = 8 \) years.
The rate of interest (R) is calculated as:
\( R = \frac{100 \times I}{P \times T} \)
Substituting these values:
\( R = \frac{100 \times 1000}{4000 \times 8} \)

\( \implies R = \frac{25}{8} \% \)

In the second scenario:
The starting principal (P) is Rs. 2100, and the target amount (A) is Rs. 2800.
The required interest is:
\( I = \text{Rs. } 2800 - \text{Rs. } 2100 = \text{Rs. } 700 \)
Using the same interest rate of \( R = \frac{25}{8} \% \), the time (T) required is:
\( T = \frac{100 \times I}{P \times R} \)
Substituting the values:
\( T = \frac{100 \times 700}{2100 \times \frac{25}{8}} \)

\( \implies T = \frac{100 \times 700 \times 8}{2100 \times 25} \)

\( \implies T = \frac{32}{3} \text{ years} = 10\frac{2}{3} \text{ years} \)
Converting the fractional year into months:
\( T = 10\text{ years} + \left(\frac{2}{3} \times 12\right) \text{ months} \)

\( \implies T = 10\text{ years} + \frac{24}{3}\text{ months} = 10\text{ years and } 8\text{ months} \)
Consequently, it will take 10 years and 8 months for Rs. 2100 to become Rs. 2800.
In simple words: First, find the interest rate from the first group of numbers. Then, use that rate to calculate how long it takes for the second sum of money to grow.

Exam Tip: Be careful when converting fractional years into months. Multiply the fractional part by 12 to convert it to months accurately.

Question 5. What sum of money lent at 6.5% per annum will produce the same interest in 4 years as Rs.7500 produce in 6 years at 5% per annum ?
Answer:
In the first scenario:
The principal amount (P) is Rs. 7500.
The annual rate of interest (R) is 5%.
The time period (T) is 6 years.
The interest generated is calculated as:
\( \text{Simple Interest} = \frac{P \times R \times T}{100} \)
Substituting these values:
\( \text{Simple Interest} = \text{Rs. } \frac{7500 \times 5 \times 6}{100} \)
\( \text{Simple Interest} = \text{Rs. } 75 \times 5 \times 6 \)

\( \implies \text{Simple Interest} = \text{Rs. } 2250 \)

In the second scenario:
According to the problem, the interest produced must also be Rs. 2250.
The rate of interest (R) is 6.5% per annum.
The time period (T) is 4 years.
The principal (P) is calculated as:
\( P = \frac{100 \times I}{R \times T} \)
Substituting the values:
\( P = \frac{100 \times 2250}{6.5 \times 4} \)

\( \implies P = \text{Rs. } \frac{225000}{26} = \text{Rs. } \frac{112500}{13} \)

\( \implies P = \text{Rs. } 8653.85 \)
Thus, the required principal is Rs. 8653.85.
In simple words: First, find out how much interest Rs. 7500 earns at 5% for 6 years. Then, work backward to find the starting money that makes that same interest at 6.5% in 4 years.

Exam Tip: Double-check calculations with decimal percentages like 6.5%. Converting 6.5 to 13/2 or multiplying the numerator and denominator by 10 makes the arithmetic easier and less prone to errors.

Question 6. A certain sum amounts to Rs.3825 in 4 years and to Rs.4050 in 6 years. Find the rate percent and the sum.
Answer:
The final amount after 6 years is Rs. 4050.
The final amount after 4 years is Rs. 3825.
By subtracting these values, we get the interest accumulated over the 2-year difference:
\( \text{Interest for 2 years} = \text{Rs. } 4050 - \text{Rs. } 3825 = \text{Rs. } 225 \)
This means the interest for 1 year is:
\( \text{Interest for 1 year} = \text{Rs. } \frac{225}{2} \)
Therefore, the interest earned over 4 years is:
\( \text{Interest for 4 years} = \text{Rs. } \frac{225}{2} \times 4 = \text{Rs. } 450 \)
Now, we find the starting principal (P) by subtracting the 4-year interest from the 4-year final amount:
\( P = A - I \)

\( \implies P = \text{Rs. } 3825 - \text{Rs. } 450 = \text{Rs. } 3375 \)
To find the annual rate of interest (R):
We have \( P = \text{Rs. } 3375 \), \( I = \text{Rs. } 450 \), and \( T = 4 \) years.
\( R = \frac{100 \times I}{P \times T} \)
Substituting these values:
\( R = \frac{100 \times 450}{3375 \times 4} \% \)

\( \implies R = \frac{45000}{13500} \% = \frac{450}{135} \% \)

\( \implies R = \frac{10}{3} \% = 3\frac{1}{3}\% \)
Thus, the starting sum is Rs. 3375 and the rate of interest is \( 3\frac{1}{3}\% \).
In simple words: Simple interest is the same every year. Find the difference in the final amounts over two years to get the interest for two years. Use this to find the starting money and the rate.

Exam Tip: Since simple interest remains constant each year, subtracting two different final amounts directly gives you the interest earned during that time gap. This is a very common exam question type.

Question 7. At what rafepercent of simple interest will the interest on Rs.3750 be one-fifth of itself in 4 years ? To what will it amount in 15 years ?
Answer:
The principal amount (P) is Rs. 3750.
The interest (I) after 4 years is one-fifth of the principal:
\( I = \text{Rs. } 3750 \times \frac{1}{5} = \text{Rs. } 750 \)
The time period is \( T = 4 \) years.
The rate of interest (R) is calculated as:
\( R = \frac{100 \times I}{P \times T} \)
Substituting these values:
\( R = \frac{100 \times 750}{3750 \times 4} \% \)

\( \implies R = 5\% \)
Now, to find the final amount in 15 years:
The interest for 4 years is Rs. 750.
So, the interest for 1 year is:
\( \text{Interest for 1 year} = \text{Rs. } \frac{750}{4} \)
Therefore, the interest accumulated over 15 years is:
\( \text{Interest for 15 years} = \text{Rs. } \frac{750}{4} \times 15 \)

\( \implies \text{Interest for 15 years} = \text{Rs. } \frac{11250}{4} = \text{Rs. } 2812.50 \)
The final amount after 15 years is:
\( A = P + I \)

\( \implies A = \text{Rs. } 3750 + \text{Rs. } 2812.50 = \text{Rs. } 6562.50 \)
Hence, the rate of interest is 5% and the final amount after 15 years is Rs. 6562.50.
In simple words: First, find the interest, which is 1/5th of the starting money. Use that to find the rate, then calculate the total interest for 15 years and add it to the starting money.

Exam Tip: When the interest is a fraction "of itself", it means "of the principal". Alternatively, you can just assume the principal is Rs. 5 and the interest is Rs. 1 to find the rate quickly.

Question 8. On what date will Rs. 1950 lent on 5th January, 2011 amount to Rs. 2125.50 at 5 percent per annum simple interest?
Answer:
The starting principal (P) is Rs. 1950.
The final amount (A) is Rs. 2125.50.
The annual rate of interest (R) is 5%.
First, find the simple interest earned (I):
\( I = A - P \)

\( \implies I = \text{Rs. } 2125.50 - \text{Rs. } 1950 = \text{Rs. } 175.50 \)
Now, calculate the time period (T) in years:
\( T = \frac{100 \times I}{P \times R} \)
Substituting these values:
\( T = \frac{100 \times 175.50}{1950 \times 5} \)

\( \implies T = \frac{17550}{9750} = \frac{1755}{975} = \frac{117}{65} \)

\( \implies T = \frac{9}{5} \text{ years} = 1\frac{4}{5} \text{ years} \)
Converting the fractional year to days:
\( \frac{4}{5} \text{ years} = \frac{4}{5} \times 365 \text{ days} = 292 \text{ days} \)
So, the total duration is 1 year and 292 days.
Starting from 5th January, 2011:
Adding 1 year brings us to 5th January, 2012.
Now, add 292 days within the leap year 2012 (February has 29 days):
- Remaining days in January = \( 31 - 5 = 26 \) days
- February = 29 days
- March = 31 days
- April = 30 days
- May = 31 days
- June = 30 days
- July = 31 days
- August = 31 days
- September = 30 days
- October = 23 days (since \( 26 + 29 + 31 + 30 + 31 + 30 + 31 + 31 + 30 = 269 \) days, and \( 292 - 269 = 23 \) days)
Thus, the required date is 23rd October, 2012.
In simple words: First, find how much interest was made. Use that to calculate the time, which is 1 year and 292 days. Add this to the starting date, remembering that 2012 is a leap year.

Exam Tip: Remember to check if the year in question is a leap year. Since 2012 is divisible by 4, it is a leap year with 29 days in February, which is critical for finding the correct final date.

Question 9. If the interest on Rs.2400 be more than the interest on Rs.2000 by Rs.60 in 3 years at the same rate percent ; find the rate.
Answer:
Let the rate of interest be \( x\% \).
First scenario:
Principal (\( P_1 \)) = Rs. 2400
Time (T) = 3 years
\( \text{Interest}_1 = \frac{2400 \times x \times 3}{100} = 72x \)
Second scenario:
Principal (\( P_2 \)) = Rs. 2000
Time (T) = 3 years
\( \text{Interest}_2 = \frac{2000 \times x \times 3}{100} = 60x \)
Based on the given condition, the difference in interest is Rs. 60:
\( 72x = 60x + 60 \)

\( \implies 72x - 60x = 60 \)

\( \implies 12x = 60 \)

\( \implies x = \frac{60}{12} \)

\( \implies x = 5 \)
Thus, the rate of interest is 5%.
In simple words: Write equations for the interest from both amounts of money. Since one interest is Rs. 60 more than the other, subtract them to solve for the interest rate.

Exam Tip: Alternatively, you can solve this by looking at the difference in principal: Rs. 2400 - Rs. 2000 = Rs. 400. The extra interest of Rs. 60 is generated by this Rs. 400 in 3 years. Solving for the rate of Rs. 400 producing Rs. 60 in 3 years gives 5% instantly!

Question 10. Divide Rs. 15,600 into two parts such that the interest on one at 5 percent for 5 years may be equal to that on the other at \( 4\frac{1}{2} \) per cent for 6 years.
Answer:
Let the first part be Rs. \( x \).
Thus, the second part will be Rs. \( (15,600 - x) \).
According to the given condition, the simple interest of both parts is equal:
\( \frac{x \times 5 \times 5}{100} = \frac{(15,600 - x) \times \frac{9}{2} \times 6}{100} \)
Multiplying both sides by 100:
\( 25x = (15,600 - x) \times 27 \)

\( \implies 25x = 27 \times 15,600 - 27x \)

\( \implies 25x + 27x = 27 \times 15,600 \)

\( \implies 52x = 27 \times 15,600 \)

\( \implies x = \frac{27 \times 15,600}{52} \)

\( \implies x = 27 \times 300 \)

\( \implies x = 8100 \)
So, the first part is Rs. 8,100.
The second part is:
\( \text{Rs. } 15,600 - \text{Rs. } 8,100 = \text{Rs. } 7,500 \)
Therefore, the two parts are Rs. 8,100 and Rs. 7,500.
In simple words: Split the total money into two parts, \( x \) and \( 15,600 - x \). Set their interest formulas equal to each other and solve for \( x \) to find both parts.

Exam Tip: Instead of multiplying large numbers like 27 by 15,600, keep them in factored form. This allows for easy cancellation later when dividing by 52, saving time and preventing arithmetic errors.

Exercise 9(C)

Question 1. A sum of Rs. 8,000 is invested for 2 years at 10% per annum compound interest. Calculate:
(i) interest for the first year.
(ii) principal for the second year.
(iii) interest for the second year.
(iv) final amount at the end of second year
(v) compound interest earned in 2 years.
Answer:
(i) For the first year:
Principal (P) = Rs. 8,000
Rate (R) = 10% per annum
Interest for the first year is:
\( I_1 = \frac{8,000 \times 10 \times 1}{100} = \text{Rs. } 800 \)

(ii) Principal for the second year:
The ending amount of the first year becomes the principal for the second year:
\( \text{Principal for Year 2} = P + I_1 \)

\( \implies \text{Principal for Year 2} = \text{Rs. } 8,000 + \text{Rs. } 800 = \text{Rs. } 8,800 \)

(iii) Interest for the second year is:
\( I_2 = \frac{8,800 \times 10 \times 1}{100} = \text{Rs. } 880 \)

(iv) Final amount at the end of the second year is:
\( \text{Amount} = \text{Principal for Year 2} + I_2 \)

\( \implies \text{Amount} = \text{Rs. } 8,800 + \text{Rs. } 880 = \text{Rs. } 9,680 \)

(v) Compound interest earned in 2 years:
\( \text{Compound Interest} = \text{Final Amount} - \text{Initial Principal} \)

\( \implies \text{Compound Interest} = \text{Rs. } 9,680 - \text{Rs. } 8,000 = \text{Rs. } 1,680 \)
In simple words: Compound interest is calculated year-by-year. The interest from the first year is added to the starting money to find the new starting money for the second year.

Exam Tip: For step-by-step compound interest questions without using the formula, remember that the final amount of one year is always used as the principal for the next year.

Question 2. A man borrowed Rs. 20,000 for 2 years at 8% per year compound interest. Calculate :
(i) the interest of the first year.
(ii) the interest of the second year.
(iii) the final amount at the end of second year.
(iv) the compound interest of two years.
Answer:
(i) For the first year:
Principal (P) = Rs. 20,000
Rate (R) = 8% per annum
Interest for the first year is:
\( I_1 = \frac{20,000 \times 8 \times 1}{100} = \text{Rs. } 1,600 \)

(ii) The principal for the second year is:
\( \text{Principal for Year 2} = \text{Rs. } 20,000 + \text{Rs. } 1,600 = \text{Rs. } 21,600 \)
Interest for the second year is:
\( I_2 = \frac{21,600 \times 8 \times 1}{100} = 216 \times 8 = \text{Rs. } 1,728 \)

(iii) Final amount at the end of the second year:
\( A = \text{Principal for Year 2} + I_2 \)

\( \implies A = \text{Rs. } 21,600 + \text{Rs. } 1,728 = \text{Rs. } 23,328 \)

(iv) Total compound interest for the two years:
\( \text{Compound Interest} = \text{Final Amount} - \text{Initial Principal} \)

\( \implies \text{Compound Interest} = \text{Rs. } 23,328 - \text{Rs. } 20,000 = \text{Rs. } 3,328 \)
In simple words: Find the first year's interest and add it to the starting amount. Then find 8% of this new total to get the second year's interest, and add them up.

Exam Tip: Be neat in your calculations. Checking that the second year's interest is higher than the first year's is a quick way to ensure your compound interest steps are correct.

Question 3. Calculate the amount and the compound interest on Rs. 12,000 in 2 years and at 10% per year.
Answer:
For the first year:
Principal (P) = Rs. 12,000
Rate (R) = 10% per annum
Interest for the first year:
\( I_1 = \frac{12,000 \times 10 \times 1}{100} = \text{Rs. } 1,200 \)
Amount at the end of the first year:
\( A_1 = \text{Rs. } 12,000 + \text{Rs. } 1,200 = \text{Rs. } 13,200 \)

For the second year:
Principal (P) = Rs. 13,200
Rate (R) = 10% per annum
Interest for the second year:
\( I_2 = \frac{13,200 \times 10 \times 1}{100} = \text{Rs. } 1,320 \)
Amount at the end of the second year:
\( A_2 = \text{Rs. } 13,200 + \text{Rs. } 1,320 = \text{Rs. } 14,520 \)
Thus, the final amount is Rs. 14,520.
The total compound interest is:
\( \text{Compound Interest} = \text{Final Amount} - \text{Initial Principal} \)

\( \implies \text{Compound Interest} = \text{Rs. } 14,520 - \text{Rs. } 12,000 = \text{Rs. } 2,520 \)
In simple words: Find the interest for the first year, add it to the starting money, then find the interest for the second year using that new amount.

Exam Tip: You can verify your compound interest by adding the individual years' interests together: Rs. 1200 + Rs. 1320 = Rs. 2520, which matches the result from subtracting the initial principal from the final amount.

Question 4. Calculate the amount and the compound interest on Rs. 10,000 in 3 years at 8% per annum.
Answer:
For the first year:
Principal (P) = Rs. 10,000
Rate (R) = 8% per annum
Interest for the first year:
\( I_1 = \frac{10,000 \times 8 \times 1}{100} = \text{Rs. } 800 \)
Amount at the end of the first year:
\( A_1 = \text{Rs. } 10,000 + \text{Rs. } 800 = \text{Rs. } 10,800 \)

For the second year:
Principal (P) = Rs. 10,800
Rate (R) = 8% per annum
Interest for the second year:
\( I_2 = \frac{10,800 \times 8 \times 1}{100} = \text{Rs. } 864 \)
Amount at the end of the second year:
\( A_2 = \text{Rs. } 10,800 + \text{Rs. } 864 = \text{Rs. } 11,664 \)

For the third year:
Principal (P) = Rs. 11,664
Rate (R) = 8% per annum
Interest for the third year:
\( I_3 = \frac{11,664 \times 8 \times 1}{100} = \text{Rs. } 933.12 \)
Amount at the end of the third year:
\( A_3 = \text{Rs. } 11,664 + \text{Rs. } 933.12 = \text{Rs. } 12,597.12 \)
Thus, the final amount is Rs. 12,597.12.
The total compound interest earned is:
\( \text{Compound Interest} = \text{Final Amount} - \text{Initial Principal} \)

\( \implies \text{Compound Interest} = \text{Rs. } 12,597.12 - \text{Rs. } 10,000 = \text{Rs. } 2,597.12 \)
In simple words: Do the interest calculation three times. Each year, start with the previous year's ending amount. Finally, subtract the starting money from the ending money.

Exam Tip: Be extremely careful with decimals in the third year. Do not round off intermediate calculations, as even small rounding errors can affect the final decimal digits of your answer.

Question 5. Calculate the compound interest on Rs. 5,000 in 2 years ; if the rates of interest for successive years be 10% and 12% respectively.
Answer:
For the first year:
Principal (P) = Rs. 5,000
Rate for the first year (\( R_1 \)) = 10%
Interest for the first year:
\( I_1 = \frac{5,000 \times 10 \times 1}{100} = \text{Rs. } 500 \)
Amount at the end of the first year:
\( A_1 = \text{Rs. } 5,000 + \text{Rs. } 500 = \text{Rs. } 5,500 \)

For the second year:
Principal (P) = Rs. 5,500
Rate for the second year (\( R_2 \)) = 12%
Interest for the second year:
\( I_2 = \frac{5,500 \times 12 \times 1}{100} = 55 \times 12 = \text{Rs. } 660 \)
Amount at the end of the second year:
\( A_2 = \text{Rs. } 5,500 + \text{Rs. } 660 = \text{Rs. } 6,160 \)
The total compound interest earned is:
\( \text{Compound Interest} = \text{Final Amount} - \text{Initial Principal} \)

\( \implies \text{Compound Interest} = \text{Rs. } 6,160 - \text{Rs. } 5,000 = \text{Rs. } 1,160 \)
In simple words: This problem has a different rate for each year. Find 10% interest for the first year and add it to Rs. 5,000, then find 12% interest on that new sum.

Exam Tip: Pay attention to the phrase "successive years". This means you must use the first rate for the first year, and then switch to the second rate for the second year's calculation.

Question 6. Calculate the compound interest on Rs. 15,000 in 3 years ; if the rates of interest for successive years be 6%, 8% and 10% respectively.
Answer:
For the first year:
Principal (P) = Rs. 15,000
Rate for the first year (\( R_1 \)) = 6%
Interest for the first year:
\( I_1 = \frac{15,000 \times 6 \times 1}{100} = \text{Rs. } 900 \)
Amount at the end of the first year:
\( A_1 = \text{Rs. } 15,000 + \text{Rs. } 900 = \text{Rs. } 15,900 \)

For the second year:
Principal (P) = Rs. 15,900
Rate for the second year (\( R_2 \)) = 8%
Interest for the second year:
\( I_2 = \frac{15,900 \times 8 \times 1}{100} = 159 \times 8 = \text{Rs. } 1,272 \)
Amount at the end of the second year:
\( A_2 = \text{Rs. } 15,900 + \text{Rs. } 1,272 = \text{Rs. } 17,172 \)

For the third year:
Principal (P) = Rs. 17,172
Rate for the third year (\( R_3 \)) = 10%
Interest for the third year:
\( I_3 = \frac{17,172 \times 10 \times 1}{100} = \text{Rs. } 1,717.20 \)
Amount at the end of the third year:
\( A_3 = \text{Rs. } 17,172 + \text{Rs. } 1,717.20 = \text{Rs. } 18,889.20 \)
The total compound interest earned is:
\( \text{Compound Interest} = \text{Final Amount} - \text{Initial Principal} \)

\( \implies \text{Compound Interest} = \text{Rs. } 18,889.20 - \text{Rs. } 15,000 = \text{Rs. } 3,889.20 \)
In simple words: Work out the interest year-by-year using a different rate each time: 6% first, then 8%, and finally 10%. Subtract the very first amount from the final amount to find the total interest.

Exam Tip: For successive rates, ensure you apply the rates in the exact order given (6%, then 8%, then 10%). Misordering the rates will result in an incorrect year-by-year working, even if the final mathematical product would be identical.

Question 7. Mohan borrowed Rs. 16,000 for 3 years at 5% per annum compound interest. Calculate the amount that Mohan will pay at the end of 3 years.
Answer:
For the first year:
Principal (P) = Rs. 16,000
Rate (R) = 5% per annum
Interest for the first year:
\( I_1 = \frac{16,000 \times 5 \times 1}{100} = \text{Rs. } 800 \)
Amount at the end of the first year:
\( A_1 = \text{Rs. } 16,000 + \text{Rs. } 800 = \text{Rs. } 16,800 \)

For the second year:
Principal (P) = Rs. 16,800
Rate (R) = 5% per annum
Interest for the second year:
\( I_2 = \frac{16,800 \times 5 \times 1}{100} = 168 \times 5 = \text{Rs. } 840 \)
Amount at the end of the second year:
\( A_2 = \text{Rs. } 16,800 + \text{Rs. } 840 = \text{Rs. } 17,640 \)

For the third year:
Principal (P) = Rs. 17,640
Rate (R) = 5% per annum
Interest for the third year:
\( I_3 = \frac{17,640 \times 5 \times 1}{100} = \text{Rs. } 882 \)
Amount at the end of the third year:
\( A_3 = \text{Rs. } 17,640 + \text{Rs. } 882 = \text{Rs. } 18,522 \)
Hence, the final amount Mohan needs to pay after 3 years is Rs. 18,522.
In simple words: Find 5% interest each year, add it to the principal, and use that new total for the next year. Repeat this three times to get the final amount.

Exam Tip: Since the question only asks for the final amount, you do not need to subtract the initial principal at the end. Always read carefully to see whether the question asks for the "Amount" or the "Compound Interest".

Question 8. Rekha borrowed Rs. 40,000 for 3 years at 10% per annum compound interest. Calculate the interest paid by her for the second year.
Answer:
For the first year:
Principal (P) = Rs. 40,000
Rate (R) = 10% per annum
Interest for the first year:
\( I_1 = \frac{40,000 \times 10 \times 1}{100} = \text{Rs. } 4,000 \)
Amount at the end of the first year:
\( A_1 = \text{Rs. } 40,000 + \text{Rs. } 4,000 = \text{Rs. } 44,000 \)

For the second year:
The starting principal for the second year is Rs. 44,000.
The rate is 10% per annum.
So, the interest earned specifically during the second year is:
\( I_2 = \frac{44,000 \times 10 \times 1}{100} = \text{Rs. } 4,400 \)
Thus, the interest paid by Rekha in the second year is Rs. 4,400.
In simple words: Find the first year's interest, add it to Rs. 40,000, and then calculate 10% of that new amount. That gives you the second year's interest.

Exam Tip: This question asks specifically for "the interest paid for the second year" (which is Rs. 4,400), not the total compound interest over 3 years. Be careful not to waste time calculating the third year's interest.

Question 9. Calculate the compound interest for the second year on Rs. 15000 invested for 5 years at 6% per annum.
Answer:
For the first year:
Principal (P) = Rs. 15,000
Rate (R) = 6% per annum
Interest for the first year:
\( I_1 = \frac{15,000 \times 6 \times 1}{100} = \text{Rs. } 900 \)
Amount at the end of the first year:
\( A_1 = \text{Rs. } 15,000 + \text{Rs. } 900 = \text{Rs. } 15,900 \)

For the second year:
The starting principal is now Rs. 15,900.
The interest earned in the second year is:
\( I_2 = \frac{15,900 \times 6 \times 1}{100} = 159 \times 6 = \text{Rs. } 954 \)
Therefore, the compound interest specifically for the second year is Rs. 954.
In simple words: Calculate 6% of Rs. 15,000 to find the first year's interest. Add it to the starting money, then find 6% of that new sum to get the second year's interest.

Exam Tip: Even though the investment period is 5 years, the question only asks for the interest of the second year. You only need to perform two year-by-year steps to arrive at the answer.

Question 10. A man invests Rs. 9600 at 10% per annum compound interest for 3 years. Calculate :
(i) the interest for the first year.
(ii) the amount at the end of the first year.
(iii) the interest for the second year.
(iv) the interest for the third year.
Answer:
For the first year, the principal amount \( (P) \) is Rs. 9,600, the rate of interest \( (R) \) is 10% per annum, and the time period \( (T) \) is 1 year.
(i) \( \text{Interest for the first year} = \frac{P \times R \times T}{100} = \frac{9600 \times 10 \times 1}{100} = \text{Rs. } 960 \)
(ii) \( \text{Amount after the first year} = \text{Principal} + \text{Interest} = \text{Rs. } 9600 + \text{Rs. } 960 = \text{Rs. } 10,560 \)
(iii) For the second year, the starting principal becomes the closing amount of the first year, which is Rs. 10,560.
\( \text{Interest for the second year} = \frac{10560 \times 10 \times 1}{100} = \text{Rs. } 1,056 \)
(iv) To find the interest for the third year, we first determine the accumulated amount at the end of the second year:
\( \text{Amount after the second year} = \text{Rs. } 10560 + \text{Rs. } 1056 = \text{Rs. } 11,616 \)
This value of Rs. 11,616 becomes our principal for the third year.
\( \text{Interest for the third year} = \frac{11616 \times 10 \times 1}{100} = \text{Rs. } 1,161.60 \)
In simple words: To calculate compound interest year-by-year, find the interest for the first year and add it to the starting money. This new total becomes the starting money for the next year, and you repeat the steps.

Exam Tip: Always remember that in compound interest calculated yearly, the closing amount of any year serves as the opening principal for the following year.

Question 11. A person invests Rs. 5,000 for two years at a certain rate of interest compounded annually. At the end of one year, this sum amounts to Rs. 5,600. Calculate :
(i) the rate of interest per year.
(ii) the amount at the end of the second year.
Answer:
We are given a starting principal \( (P) \) of Rs. 5,000. At the end of the first year, the total amount \( (A) \) becomes Rs. 5,600. Therefore, the interest earned in the first year is:
\( \text{Interest} = A - P = \text{Rs. } 5600 - \text{Rs. } 5000 = \text{Rs. } 600 \)
(i) To find the annual interest rate \( (R) \):
\( R = \frac{\text{Interest} \times 100}{P \times T} = \frac{600 \times 100}{5000 \times 1} = 12\% \text{ per annum} \)
(ii) The accumulated amount of Rs. 5,600 at the end of the first year serves as the principal for the second year. The interest for this second year is:
\( \text{Interest} = \frac{5600 \times 12 \times 1}{100} = \text{Rs. } 672 \)
So, the final amount after two years is:
\( \text{Amount} = \text{Rs. } 5600 + \text{Rs. } 672 = \text{Rs. } 6,272 \)
In simple words: First, find how much interest was made in the first year to work out the percentage rate. Then, use that same rate on the new total to find the interest and amount for the second year.

Exam Tip: To calculate the interest rate, make sure to use the initial principal of Rs. 5,000, not the first year's ending amount.

Question 12. Calculate the difference between the compound interest and the simple interest on Rs. 7,500 in two years and at 8% per annum.
Answer:
We have a principal \( (P) \) of Rs. 7,500, an annual rate \( (R) \) of 8%, and a time period \( (T) \) of 2 years.
First, let us calculate the Simple Interest (S.I.) for these 2 years:
\( \text{Simple Interest} = \frac{P \times R \times T}{100} = \frac{7500 \times 8 \times 2}{100} = \text{Rs. } 1,200 \)
Next, we calculate the Compound Interest (C.I.) step-by-step:
For the first year:
\( \text{Interest for Year 1} = \frac{7500 \times 8 \times 1}{100} = \text{Rs. } 600 \)
\( \text{Amount at the end of Year 1} = \text{Rs. } 7500 + \text{Rs. } 600 = \text{Rs. } 8,100 \)
For the second year, the starting principal is Rs. 8,100:
\( \text{Interest for Year 2} = \frac{8100 \times 8 \times 1}{100} = \text{Rs. } 648 \)
\( \text{Total Compound Interest for 2 years} = \text{Rs. } 600 + \text{Rs. } 648 = \text{Rs. } 1,248 \)
Finally, find the difference between C.I. and S.I.:
\( \text{Difference} = \text{C.I.} - \text{S.I.} = \text{Rs. } 1248 - \text{Rs. } 1200 = \text{Rs. } 48 \)
In simple words: Work out the simple interest for two years first. Then, find the compound interest by calculating interest year-by-year. Subtract the simple interest from the compound interest to get the difference.

Exam Tip: You can double-check your answer using the formula for the difference between C.I. and S.I. for 2 years: \( \text{Difference} = P \left(\frac{R}{100}\right)^2 \).

Question 13. Calculate the difference between the compound interest and the simple interest on Rs. 8,000 in three years and at 10% per annum.
Answer:
Let the principal \( (P) \) be Rs. 8,000, the rate of interest \( (R) \) be 10% per annum, and the duration \( (T) \) be 3 years.
First, find the total Simple Interest (S.I.) over the 3-year term:
\( \text{S.I. for 3 years} = \frac{P \times R \times T}{100} = \frac{8000 \times 10 \times 3}{100} = \text{Rs. } 2,400 \)
Now, let us calculate the Compound Interest (C.I.) annually:
For the first year:
\( \text{Interest} = \frac{8000 \times 10 \times 1}{100} = \text{Rs. } 800 \)
\( \text{Amount at the end of year 1} = \text{Rs. } 8000 + \text{Rs. } 800 = \text{Rs. } 8,800 \)
For the second year:
\( \text{Principal} = \text{Rs. } 8,800 \)
\( \text{Interest} = \frac{8800 \times 10 \times 1}{100} = \text{Rs. } 880 \)
\( \text{Amount at the end of year 2} = \text{Rs. } 8800 + \text{Rs. } 880 = \text{Rs. } 9,680 \)
For the third year:
\( \text{Principal} = \text{Rs. } 9,680 \)
\( \text{Interest} = \frac{9680 \times 10 \times 1}{100} = \text{Rs. } 968 \)
Calculate the sum of interest over the three years to get the total C.I.:
\( \text{Total C.I.} = \text{Rs. } 800 + \text{Rs. } 880 + \text{Rs. } 968 = \text{Rs. } 2,648 \)
Subtract Simple Interest from Compound Interest to find the difference:
\( \text{Difference} = \text{C.I.} - \text{S.I.} = \text{Rs. } 2648 - \text{Rs. } 2400 = \text{Rs. } 248 \)
In simple words: Calculate the simple interest for three years, then calculate the compound interest year-by-year for three years. The final answer is the difference between these two interest totals.

Exam Tip: Ensure you add up the three individual yearly interests correctly to find the total compound interest before calculating the difference.

Question 14. Rohit borrowed Rs. 40,000 for 2 years at 10% per annum C.I. and Manish borrowed the same sum for the same time at 10.5% per annum simple interest. Which of these two gets less interest and by how much?
Answer:
Let us evaluate the two borrowing scenarios separately.
Case 1: Rohit's Compound Interest (C.I.)
The borrowed principal \( (P) \) is Rs. 40,000, at an annual rate \( (R) \) of 10% for 2 years.
Interest for the first year:
\( \text{Interest for Year 1} = \frac{40000 \times 10 \times 1}{100} = \text{Rs. } 4,000 \)
Amount after the first year:
\( \text{Amount} = \text{Rs. } 40000 + \text{Rs. } 4000 = \text{Rs. } 44,000 \)
This amount becomes the starting principal for the second year. Interest for the second year:
\( \text{Interest for Year 2} = \frac{44000 \times 10 \times 1}{100} = \text{Rs. } 4,400 \)
Total compound interest paid by Rohit:
\( \text{Total C.I.} = \text{Rs. } 4000 + \text{Rs. } 4400 = \text{Rs. } 8,400 \)

Case 2: Manish's Simple Interest (S.I.)
The borrowed principal \( (P) \) is Rs. 40,000, at an annual rate \( (R) \) of 10.5% for 2 years.
Simple interest paid by Manish:
\( \text{S.I.} = \frac{P \times R \times T}{100} = \frac{40000 \times 10.5 \times 2}{100} = \text{Rs. } 8,400 \)

Comparing both calculations, both Rohit and Manish pay the exact same amount of interest.
In simple words: Rohit pays compound interest which totals Rs. 8,400, and Manish pays simple interest which also totals Rs. 8,400. This means both of them pay the same amount of interest.

Exam Tip: Pay close attention to the rates: Rohit has a 10% compound rate and Manish has a 10.5% simple rate. Clearly show the step-by-step working for both scenarios to earn full marks.

Question 15. Mr. Sharma borrowed Rs. 24,000 at 13% p.a. simple interest and an equal sum at 12% p.a. compound interest. Find the total interest earned by Mr. Sharma in 2 years.
Answer:
We will calculate the interest for each scenario and then add them together.
First Investment: Simple Interest (S.I.)
The principal is Rs. 24,000 at a rate of 13% per annum for 2 years.
\( \text{Simple Interest} = \frac{24000 \times 13 \times 2}{100} = \text{Rs. } 6,240 \)

Second Investment: Compound Interest (C.I.)
The principal is Rs. 24,000 at a rate of 12% per annum for 2 years.
For the first year:
\( \text{Interest for Year 1} = \frac{24000 \times 12 \times 1}{100} = \text{Rs. } 2,880 \)
\( \text{Amount after Year 1} = \text{Rs. } 24000 + \text{Rs. } 2,880 = \text{Rs. } 26,880 \)
For the second year, using Rs. 26,880 as the principal:
\( \text{Interest for Year 2} = \frac{26880 \times 12 \times 1}{100} = \frac{322560}{100} = \text{Rs. } 3,225.60 \)
\( \text{Total C.I. over 2 years} = \text{Rs. } 2880 + \text{Rs. } 3225.60 = \text{Rs. } 6,105.60 \)

Total Interest:
\( \text{Total Interest} = \text{Simple Interest} + \text{Compound Interest} = \text{Rs. } 6240 + \text{Rs. } 6105.60 = \text{Rs. } 12,345.60 \)
In simple words: Calculate the simple interest on the first sum and the compound interest on the second sum. Adding these two interests together gives the total interest of Rs. 12,345.60.

Exam Tip: Be careful with calculations involving decimals, particularly in the second-year compound interest where you must divide 322560 by 100.

Question 16. Peter borrows Rs. 12,000 for 2 years at 10% p.a. compound interest. He repays Rs. 8,000 at the end of first year. Find:
(i) the amount at the end of first year, before making the repayment.
(ii) the amount at the end of first year, after making the repayment.
(iii) the principal for the second year.
(iv) the amount to be paid at the end of second year, to clear the account.
Answer:
Let the starting principal \( (P) \) be Rs. 12,000 and the annual interest rate \( (R) \) be 10%. Let us calculate the interest for the first year:
\( \text{Interest for Year 1} = \frac{12000 \times 10 \times 1}{100} = \text{Rs. } 1,200 \)
(i) The total accumulated amount at the end of the first year, prior to repayment, is:
\( \text{Amount} = \text{Rs. } 12000 + \text{Rs. } 1200 = \text{Rs. } 13,200 \)
(ii) Since Peter pays back Rs. 8,000 at the end of this first year, the remaining balance is:
\( \text{Remaining Balance} = \text{Rs. } 13200 - \text{Rs. } 8000 = \text{Rs. } 5,200 \)
(iii) The principal for the second year is this remaining balance:
\( \text{Principal for Year 2} = \text{Rs. } 5,200 \)
(iv) Now we compute the interest on this new principal for the second year:
\( \text{Interest for Year 2} = \frac{5200 \times 10 \times 1}{100} = \text{Rs. } 520 \)
To completely settle the debt at the end of the second year, the total amount payable is:
\( \text{Final Amount to clear the debt} = \text{Rs. } 5200 + \text{Rs. } 520 = \text{Rs. } 5,720 \)
In simple words: Add the first year's interest to the loan to find the total before payment. Subtract the repayment of Rs. 8,000 to find the new starting balance. Calculate interest on this balance for the second year and add it to get the final payoff amount.

Exam Tip: For repayment problems, always subtract the payment from the total outstanding amount at the end of the year before using that balance as the new principal.

Question 17. Gautam takes a loan of Rs. 16,000 for 2 years at 15% p.a. compound interest. He repays Rs. 9,000 at the end of first year. How mucH must he pay at the end of second year to clear the debt?
Answer:
The initial loan amount \( (P) \) is Rs. 16,000 at a rate \( (R) \) of 15% per annum. Let us calculate the interest accrued during the first year:
\( \text{Interest for Year 1} = \frac{16000 \times 15 \times 1}{100} = \text{Rs. } 2,400 \)
The total outstanding debt at the end of the first year is:
\( \text{Amount} = \text{Rs. } 16000 + \text{Rs. } 2400 = \text{Rs. } 18,400 \)
After Gautam pays back Rs. 9,000, the remaining balance is:
\( \text{Remaining Balance} = \text{Rs. } 18400 - \text{Rs. } 9000 = \text{Rs. } 9,400 \)
This balance of Rs. 9,400 acts as the principal for the second year. Interest for the second year:
\( \text{Interest for Year 2} = \frac{9400 \times 15 \times 1}{100} = \text{Rs. } 1,410 \)
To fully clear the loan at the end of the second year, Gautam must pay:
\( \text{Final Amount} = \text{Rs. } 9400 + \text{Rs. } 1410 = \text{Rs. } 10,810 \)
In simple words: Gautam's debt grows to Rs. 18,400 after the first year. He pays back Rs. 9,000, leaving Rs. 9,400. This remaining amount earns interest for another year, making the final payment Rs. 10,810.

Exam Tip: Always clearly label each step: first-year interest, outstanding balance, remaining principal after payment, and final payment amount to ensure maximum clarity.

Question 18. A certain sum of money, invested for 5 years at 8% p.a. simple interest, earns an interest of Rs. 12,000. Find:
(i) the sum of money.
(ii) the compound interest earned by this money in two years and at 10% p.a. compound interest.
Answer:
We are given the simple interest \( (I) \) of Rs. 12,000, an investment period \( (T) \) of 5 years, and a rate \( (R) \) of 8% per annum.
(i) To determine the original principal sum \( (P) \):
\( \text{Sum} = \frac{\text{Interest} \times 100}{\text{Rate} \times \text{Time}} = \frac{12000 \times 100}{8 \times 5} = \text{Rs. } 30,000 \)
(ii) Now, we calculate the compound interest on this sum of Rs. 30,000 for 2 years at a rate of 10% per annum.
For the first year:
\( \text{Interest for Year 1} = \frac{30000 \times 10 \times 1}{100} = \text{Rs. } 3,000 \)
\( \text{Amount after Year 1} = \text{Rs. } 30000 + \text{Rs. } 3000 = \text{Rs. } 33,000 \)
For the second year:
\( \text{Principal for Year 2} = \text{Rs. } 33,000 \)
\( \text{Interest for Year 2} = \frac{33000 \times 10 \times 1}{100} = \text{Rs. } 3,300 \)
The total compound interest earned over the two years is:
\( \text{Compound Interest} = \text{Rs. } 3000 + \text{Rs. } 3300 = \text{Rs. } 6,300 \)
In simple words: First, use the simple interest formula backwards to find the starting sum of Rs. 30,000. Next, calculate the compound interest on this Rs. 30,000 for two years at a rate of 10% per annum.

Exam Tip: Verify your calculated principal value carefully, as any mistake in part (i) will carry over and make your calculations in part (ii) incorrect.

Question 19. Find the amount and the C.I. on Rs. 12,000 at 10% per annum compounded half-yearly.
Answer:
We are given a principal \( (P) \) of Rs. 12,000 and an annual interest rate \( (r) \) of 10%. The compounding is done half-yearly over a period of 1 year. The formula for half-yearly compounding is:
\( \text{Amount} = P \times \left(1 + \frac{r}{2 \times 100}\right)^{n \times 2} \)
Here, \( n = 1 \) year, so the exponent is \( 1 \times 2 = 2 \) half-years. Substituting the given values:
\( \text{Amount} = 12000 \times \left(1 + \frac{10}{200}\right)^{2} \)
\( \implies \text{Amount} = 12000 \times \left(\frac{210}{200}\right)^{2} \)
\( \implies \text{Amount} = 12000 \times \frac{21}{20} \times \frac{21}{20} = \text{Rs. } 13,230 \)
Now, let us find the Compound Interest (C.I.):
\( \text{C.I.} = \text{Amount} - \text{Principal} = \text{Rs. } 13230 - \text{Rs. } 12000 = \text{Rs. } 1,230 \)
In simple words: When interest is compounded half-yearly, we halve the yearly interest rate to 5% and double the number of time periods. For 1 year, we calculate the compound interest for 2 half-years.

Exam Tip: When compounding half-yearly, always adjust the rate by dividing by 2 and the time periods by multiplying by 2 before plugging them into the formula.

Question 20. Find the amount and the C.I. on Rs. 8,000 in 1 1/2 years at 20% per year compounded half yearly.
Answer:
We have a principal \( (P) \) of Rs. 8,000, an annual rate of interest \( (r) \) of 20%, and a time period of \( 1\frac{1}{2} \) years \( = \frac{3}{2} \) years.
Using the formula for compounding half-yearly:
\( \text{Amount} = \text{Principal} \times \left(1 + \frac{r}{200}\right)^{n \times 2} \)
Let us substitute the values:
\( \text{Amount} = 8000 \times \left(1 + \frac{20}{200}\right)^{\frac{3}{2} \times 2} \)
\( \implies \text{Amount} = 8000 \times \left(\frac{220}{200}\right)^{3} \)
\( \implies \text{Amount} = 8000 \times \frac{11}{10} \times \frac{11}{10} \times \frac{11}{10} = \text{Rs. } 10,648 \)
The Compound Interest is calculated as:
\( \text{C.I.} = \text{Amount} - \text{Principal} = \text{Rs. } 10648 - \text{Rs. } 8000 = \text{Rs. } 2,648 \)
In simple words: For a period of 1 and a half years compounded half-yearly, there are exactly 3 half-years. We apply a half-yearly interest rate of 10% (which is half of 20%) three times.

Exam Tip: Ensure you simplify the fraction \( \frac{220}{200} \) to \( \frac{11}{10} \) first to make calculating the cube much quicker and less prone to errors.

Question 21. Find the amount and the compound interest on Rs. 24,000 for 2 years at 10% per annum compounded yearly.
Answer:
We are given a principal \( (P) \) of Rs. 24,000, an annual rate \( (r) \) of 10%, and a term of 2 years with yearly compounding.
The formula for yearly compounding is:
\( \text{Amount} = P \times \left(1 + \frac{r}{100}\right)^{n} \)
Substituting the values:
\( \text{Amount} = 24000 \times \left(1 + \frac{10}{100}\right)^{2} \)
\( \implies \text{Amount} = 24000 \times \left(\frac{11}{10}\right)^{2} \)
\( \implies \text{Amount} = 24000 \times \frac{111}{100} = 24000 \times 1.21 = \text{Rs. } 29,040 \)
To find the Compound Interest:
\( \text{C.I.} = \text{Amount} - \text{Principal} = \text{Rs. } 29040 - \text{Rs. } 24000 = \text{Rs. } 5,040 \)
In simple words: For yearly compounding over two years, we apply a 10% interest rate to the initial sum, and then apply that same 10% rate to the new total at the end of the first year.

Exam Tip: Be careful not to confuse yearly compounding with half-yearly compounding. Make sure you divide the rate by 100 (not 200) and use the exact number of years as the exponent.

Question 22. Find the amount and the compound interest on Rs. 16,000 for 3 years at 5% per annum compounded annually.
Answer:
The given principal \( (P) \) is Rs. 16,000, the interest rate \( (r) \) is 5% per annum, and the time \( (n) \) is 3 years. Since compounding is annual:
\( \text{Amount} = P \times \left(1 + \frac{r}{100}\right)^{n} \)
Substituting the values into our formula:
\( \text{Amount} = 16000 \times \left(1 + \frac{5}{100}\right)^{3} \)
\( \implies \text{Amount} = 16000 \times \left(\frac{21}{20}\right)^{3} \)
\( \implies \text{Amount} = 16000 \times \frac{9261}{8000} = 2 \times 9261 = \text{Rs. } 18,522 \)
Now, let us calculate the Compound Interest:
\( \text{C.I.} = \text{Amount} - \text{Principal} = \text{Rs. } 18522 - \text{Rs. } 16000 = \text{Rs. } 2,522 \)
In simple words: We calculate the compounding factor for three years at a rate of 5%. We then multiply this factor by our starting money of Rs. 16,000 to get our final total.

Exam Tip: Factoring the denominator of the cubed fraction (since \( 20^3 = 8000 \)) allows you to cleanly divide the principal Rs. 16,000 by 8,000, leaving a simple multiplication step of \( 2 \times 9261 \).

Question 23. Find the amount and the compound interest on Rs. 20,000 for 1 1/2 years at 10% per annum compounded half-yearly.
Answer:
The given principal \( (P) \) is Rs. 20,000, the annual interest rate \( (r) \) is 10%, and the time period is \( 1\frac{1}{2} \) years \( = \frac{3}{2} \) years. Using the formula for half-yearly compounding:
\( \text{Amount} = P \times \left(1 + \frac{r}{200}\right)^{n \times 2} \)
Substituting the given values:
\( \text{Amount} = 20000 \times \left(1 + \frac{10}{200}\right)^{\frac{3}{2} \times 2} \)
\( \implies \text{Amount} = 20000 \times \left(\frac{210}{200}\right)^{3} \)
\( \implies \text{Amount} = 20000 \times \frac{21}{20} \times \frac{21}{20} \times \frac{21}{20} = \text{Rs. } 23,152.50 \)
To find the Compound Interest:
\( \text{C.I.} = \text{Amount} - \text{Principal} = \text{Rs. } 23152.50 - \text{Rs. } 20000 = \text{Rs. } 3,152.50 \)
In simple words: Compounding half-yearly for 1 and a half years means we have 3 compounding periods. We apply half of the yearly rate, which is 5%, across these 3 periods.

Exam Tip: Remember that when compounding half-yearly, the exponent is the total number of half-years (which is 3 for 1.5 years), and the rate is divided by 2.

Question 24. Find the amount and the compound interest on Rs. 32,000 for 1 year at 20% per annum compounded half-yearly.
Answer:
We are given a principal \( (P) \) of Rs. 32,000, an annual interest rate \( (r) \) of 20%, and a time period of 1 year. The compounding is done half-yearly. Applying the half-yearly compounding formula:
\( \text{Amount} = P \times \left(1 + \frac{r}{200}\right)^{n \times 2} \)
Substituting the given values:
\( \text{Amount} = 32000 \times \left(1 + \frac{20}{200}\right)^{1 \times 2} \)
\( \implies \text{Amount} = 32000 \times \left(\frac{11}{10}\right)^{2} \)
\( \implies \text{Amount} = 32000 \times \frac{11}{10} \times \frac{11}{10} = \text{Rs. } 38,720 \)
Now, let us calculate the Compound Interest:
\( \text{C.I.} = \text{Amount} - \text{Principal} = \text{Rs. } 38720 - \text{Rs. } 32000 = \text{Rs. } 6,720 \)
In simple words: For a 1-year loan compounded half-yearly, there are 2 half-years. We halve the yearly rate of 20% to get 10% and apply it for those 2 periods.

Exam Tip: Keep calculations simple by reducing the fraction inside the bracket: \( 1 + \frac{20}{200} \) becomes \( 1 + \frac{1}{10} = \frac{11}{10} \).

Question 25. Find the amount and the compound interest on Rs. 4,000 in 2 years, if the rate of interest for first year is 10% and for the second year is 15%.
Answer:
We have a principal \( (P) \) of Rs. 4,000 and a 2-year term with different rates of interest: \( R_1 = 10\% \) for the first year and \( R_2 = 15\% \) for the second year. The formula for varying interest rates is:
\( \text{Amount} = P \times \left(1 + \frac{R_1}{100}\right) \times \left(1 + \frac{R_2}{100}\right) \)
Substituting the given values:
\( \text{Amount} = 4000 \times \left(1 + \frac{10}{100}\right) \times \left(1 + \frac{15}{100}\right) \)
\( \implies \text{Amount} = 4000 \times \frac{11}{10} \times \frac{23}{20} = \text{Rs. } 5,060 \)
Now, let us calculate the Compound Interest:
\( \text{C.I.} = \text{Amount} - \text{Principal} = \text{Rs. } 5060 - \text{Rs. } 4000 = \text{Rs. } 1,060 \)
In simple words: When interest rates change each year, calculate the growth factor for each rate and multiply them by the starting money. This gives the total amount at the end.

Exam Tip: For varying rates of interest, avoid using the standard power-based formula. Instead, use the product of individual year factors: \( P(1 + \frac{R_1}{100})(1 + \frac{R_2}{100}) \).

Question 26. Find the amount and the compound interest on Rs. 10,000 in 3 years, if the rates of interest for the successive years are 10%, 15% and 20% respectively.
Answer:
We are given a principal \( (P) \) of Rs. 10,000 for a duration of 3 years with successive annual rates of \( R_1 = 10\% \), \( R_2 = 15\% \), and \( R_3 = 20\% \). The compound interest formula for successive rates is:
\( \text{Amount} = P \times \left(1 + \frac{R_1}{100}\right) \times \left(1 + \frac{R_2}{100}\right) \times \left(1 + \frac{R_3}{100}\right) \)
Substituting the given values:
\( \text{Amount} = 10000 \times \left(1 + \frac{10}{100}\right) \times \left(1 + \frac{15}{100}\right) \times \left(1 + \frac{20}{100}\right) \)
\( \implies \text{Amount} = 10000 \times \frac{11}{10} \times \frac{23}{20} \times \frac{6}{5} = \text{Rs. } 15,180 \)
To find the Compound Interest:
\( \text{C.I.} = \text{Amount} - \text{Principal} = \text{Rs. } 15180 - \text{Rs. } 10000 = \text{Rs. } 5,180 \)
In simple words: Multiply the initial sum by each year's specific interest growth factor. This gives the final amount, from which you can subtract the starting money to find the compound interest.

Exam Tip: Simplify fractions like \( \frac{120}{100} \) to \( \frac{6}{5} \) and \( \frac{115}{100} \) to \( \frac{23}{20} \) before multiplying, which saves time during exams.