Compound Interest

Q1: Calculate the amount and the compound interest on ₹25000 for 2 years at 8% per annum, compounded annually.

i. Calculate Interest and Amount for the First Year

Step 1: Identify the given values for the first year.
Principal for the first year \( (P_1) = \text{₹}25000 \)
Rate of Interest \( (R) = 8\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
Step 2: Calculate the Simple Interest for the first year.
Formula: \( \text{Interest} = \frac{P \times R \times T}{100} \)
\( \text{Interest for Year 1 } (I_1) = \frac{25000 \times 8 \times 1}{100} \)
\( I_1 = 250 \times 8 \)
\( I_1 = \text{₹}2000 \)
Step 3: Find the Amount at the end of the first year.
Formula: \( \text{Amount} = \text{Principal} + \text{Interest} \)
\( \text{Amount after Year 1 } (A_1) = 25000 + 2000 \)
\( A_1 = \text{₹}27000 \)

ii. Calculate Interest and Amount for the Second Year

Step 1: Use the amount of the first year as the principal for the second year.
Principal for the second year \( (P_2) = \text{₹}27000 \)
Rate of Interest \( (R) = 8\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
Step 2: Calculate the Simple Interest for the second year.
\( \text{Interest for Year 2 } (I_2) = \frac{27000 \times 8 \times 1}{100} \)
\( I_2 = 270 \times 8 \)
\( I_2 = \text{₹}2160 \)
Step 3: Find the final Amount at the end of the second year.
\( \text{Final Amount } (A_2) = P_2 + I_2 \)
\( A_2 = 27000 + 2160 \)
\( A_2 = \text{₹}29160 \)

iii. Calculate Total Compound Interest

Step 1: Sum up the interest of both years to get the final compound interest.
Formula: \( \text{Total Compound Interest (CI)} = I_1 + I_2 \)
\( \text{CI} = 2000 + 2160 \)
\( \text{CI} = \text{₹}4160 \)
Answer:Amount = ₹29160, Compound Interest = ₹4160

Q2: Rohit borrows ₹62500 from Arun for 2 years at 10% per annum, simple interest. He immediately lends out this sum to Kunal at 10% per annum for the same period, compounded annually. Calculate Rohit’s profit in the transaction at the end of two years.

i. Calculate Simple Interest Paid by Rohit to Arun

Step 1: Identify the given values for the loan from Arun.
Principal \( (P) = \text{₹}62500 \)
Rate of Interest \( (R) = 10\% \text{ per annum} \)
Time \( (T) = 2 \text{ years} \)
Step 2: Compute the total Simple Interest (SI) paid after 2 years.
Formula: \( \text{SI} = \frac{P \times R \times T}{100} \)
\( \text{SI} = \frac{62500 \times 10 \times 2}{100} \)
\( \text{SI} = 625 \times 20 \)
\( \text{SI} = \text{₹}12500 \)

ii. Calculate Compound Interest Received from Kunal Year by Year

Step 1: Calculate the interest and amount for the first year.
Principal for Year 1 \( (P_1) = \text{₹}62500 \)
\( \text{Interest for Year 1 } (I_1) = \frac{62500 \times 10 \times 1}{100} \)
\( I_1 = \text{₹}6250 \)
Amount at the end of Year 1 \( (A_1) = 62500 + 6250 = \text{₹}68750 \)
Step 2: Calculate the interest for the second year using the new principal.
Principal for Year 2 \( (P_2) = A_1 = \text{₹}68750 \)
\( \text{Interest for Year 2 } (I_2) = \frac{68750 \times 10 \times 1}{100} \)
\( I_2 = \text{₹}6875 \)
Step 3: Find the total Compound Interest (CI) earned over 2 years.
Formula: \( \text{Total CI} = I_1 + I_2 \)
\( \text{Total CI} = 6250 + 6875 \)
\( \text{Total CI} = \text{₹}13125 \)

iii. Calculate Rohit’s Profit

Step 1: Subtract the Simple Interest paid from the Compound Interest earned.
Formula: \( \text{Profit} = \text{Total CI} – \text{Total SI} \)
\( \text{Profit} = 13125 – 12500 \)
\( \text{Profit} = \text{₹}625 \)
Answer:Rohit’s profit in the transaction is ₹625

Q3: A man invests ₹10000 for 3 years at a certain rate Of interest, compounded annually. At the end of one year, it amounts to ₹11200. Calculate:

i. the rate of interest per annum

Step 1: Identify the principal and the interest earned during the first year.
Principal for the first year \( (P_1) = \text{₹}10000 \)
Amount at the end of the first year \( (A_1) = \text{₹}11200 \)
Interest for the first year \( (I_1) = A_1 – P_1 \)
\( I_1 = 11200 – 10000 = \text{₹}1200 \)
Step 2: Compute the rate of interest using the simple interest relation for 1 year.
Formula: \( \text{Interest} = \frac{P \times R \times T}{100} \)
\( 1200 = \frac{10000 \times R \times 1}{100} \)
\( 1200 = 100 \times R \)
\( R = \frac{1200}{100} = 12\% \)
Answer:The rate of interest per annum is 12%

ii. the interest accrued in the second year

Step 1: Determine the principal for the second year.
Principal for the second year \( (P_2) = A_1 = \text{₹}11200 \)
Rate of Interest \( (R) = 12\% \)
Step 2: Calculate the interest earned in the second year.
\( \text{Interest for Year 2 } (I_2) = \frac{11200 \times 12 \times 1}{100} \)
\( I_2 = 112 \times 12 \)
\( I_2 = \text{₹}1344 \)
Answer:The interest accrued in the second year is ₹1344

iii. the amount at the end of the third year

Step 1: Find the accumulated amount at the end of the second year.
Amount after Year 2 \( (A_2) = P_2 + I_2 \)
\( A_2 = 11200 + 1344 = \text{₹}12544 \)
Step 2: Determine the interest earned in the third year using the second year’s amount as principal.
Principal for the third year \( (P_3) = A_2 = \text{₹}12544 \)
\( \text{Interest for Year 3 } (I_3) = \frac{12544 \times 12 \times 1}{100} \)
\( I_3 = \frac{150528}{100} = \text{₹}1505.28 \)
Step 3: Calculate the final total amount at the end of the third year.
Amount after Year 3 \( (A_3) = P_3 + I_3 \)
\( A_3 = 12544 + 1505.28 = \text{₹}14049.28 \)
Answer:The amount at the end of the third year is ₹14049.28

Q4: Sudhakar borrows ₹22500 at 10% per annum, compounded annually. If he repays ₹11250 at the end of first year and ₹12550 at the end of the second year, find the amount of loan outstanding against him at the end of the third year.

i. Calculate Loan Balance at the End of the First Year

Step 1: Identify given values and compute interest for the first year.
Principal for Year 1 \( (P_1) = \text{₹}22500 \)
Rate of Interest \( (R) = 10\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
\( \text{Interest for Year 1 } (I_1) = \frac{22500 \times 10 \times 1}{100} \)
\( I_1 = \text{₹}2250 \)
Step 2: Find the total amount due before repayment at the end of Year 1.
\( \text{Amount before repayment } (A_1) = P_1 + I_1 \)
\( A_1 = 22500 + 2250 \)
\( A_1 = \text{₹}24750 \)
Step 3: Deduct the first year’s repayment to find the outstanding principal.
Repayment at the end of Year 1 = \( \text{₹}11250 \)
\( \text{Remaining Principal for Year 2 } (P_2) = 24750 – 11250 \)
\( P_2 = \text{₹}13500 \)

ii. Calculate Loan Balance at the End of the Second Year

Step 1: Compute interest on the remaining principal for the second year.
\( \text{Interest for Year 2 } (I_2) = \frac{13500 \times 10 \times 1}{100} \)
\( I_2 = \text{₹}1350 \)
Step 2: Find the total amount due before repayment at the end of Year 2.
\( \text{Amount before repayment } (A_2) = P_2 + I_2 \)
\( A_2 = 13500 + 1350 \)
\( A_2 = \text{₹}14850 \)
Step 3: Deduct the second year’s repayment to find the final principal for the third year.
Repayment at the end of Year 2 = \( \text{₹}12550 \)
\( \text{Remaining Principal for Year 3 } (P_3) = 14850 – 12550 \)
\( P_3 = \text{₹}2300 \)

iii. Calculate Outstanding Loan Amount at the End of the Third Year

Step 1: Compute interest for the third year on the leftover balance.
\( \text{Interest for Year 3 } (I_3) = \frac{2300 \times 10 \times 1}{100} \)
\( I_3 = \text{₹}230 \)
Step 2: Combine the third-year principal and interest to find the final outstanding amount.
\( \text{Outstanding Loan Amount } (A_3) = P_3 + I_3 \)
\( A_3 = 2300 + 230 \)
\( A_3 = \text{₹}2530 \)
Answer:The amount of loan outstanding at the end of the third year is ₹2530

Q5: A man borrows ₹15000 at 12% per annum, compounded annually. If he repays ₹4400 at the end of each year, find the amount outstanding against him at the beginning of the third year.

i. Calculate Loan Balance at the End of the First Year

Step 1: Identify the given values for the first year and calculate the interest.
Principal for Year 1 \( (P_1) = \text{₹}15000 \)
Rate of Interest \( (R) = 12\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
\( \text{Interest for Year 1 } (I_1) = \frac{15000 \times 12 \times 1}{100} \)
\( I_1 = 150 \times 12 \)
\( I_1 = \text{₹}1800 \)
Step 2: Compute the total amount due at the end of the first year before repayment.
\( \text{Amount before repayment } (A_1) = P_1 + I_1 \)
\( A_1 = 15000 + 1800 \)
\( A_1 = \text{₹}16800 \)
Step 3: Deduct the first year’s repayment to find the remaining balance.
Repayment at the end of Year 1 = \( \text{₹}4400 \)
\( \text{Outstanding Principal at the beginning of Year 2 } (P_2) = 16800 – 4400 \)
\( P_2 = \text{₹}12400 \)

ii. Calculate Loan Balance at the End of the Second Year

Step 1: Calculate the interest on the remaining principal for the second year.
Principal for Year 2 \( (P_2) = \text{₹}12400 \)
\( \text{Interest for Year 2 } (I_2) = \frac{12400 \times 12 \times 1}{100} \)
\( I_2 = 124 \times 12 \)
\( I_2 = \text{₹}1488 \)
Step 2: Compute the total amount due at the end of the second year before repayment.
\( \text{Amount before repayment } (A_2) = P_2 + I_2 \)
\( A_2 = 12400 + 1488 \)
\( A_2 = \text{₹}13888 \)
Step 3: Deduct the second year’s repayment to find the outstanding amount at the beginning of the third year.
Repayment at the end of Year 2 = \( \text{₹}4400 \)
\( \text{Amount outstanding at the beginning of Year 3} = 13888 – 4400 \)
\( \text{Amount outstanding at the beginning of Year 3} = \text{₹}9488 \)
Answer:The amount outstanding against him at the beginning of the third year is ₹9488

Q6: Mr. Ravi borrows ₹16,000 for 2 years. The rate of interest for the two successive years are 10% and 12% respectively. If he repays ₹5,600 at the end of first year, find the amount outstanding at the end of the second year.

i. Calculate Loan Balance at the End of the First Year

Step 1: Identify given values and calculate the interest for the first year.
Principal for Year 1 \( (P_1) = \text{₹}16000 \)
Rate of Interest for Year 1 \( (R_1) = 10\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
\( \text{Interest for Year 1 } (I_1) = \frac{16000 \times 10 \times 1}{100} \)
\( I_1 = \text{₹}1600 \)
Step 2: Find the total amount due at the end of the first year before repayment.
\( \text{Amount before repayment } (A_1) = P_1 + I_1 \)
\( A_1 = 16000 + 1600 \)
\( A_1 = \text{₹}17600 \)
Step 3: Deduct the first year’s repayment to find the remaining principal.
Repayment at the end of Year 1 = \( \text{₹}5600 \)
\( \text{Remaining Principal for Year 2 } (P_2) = 17600 – 5600 \)
\( P_2 = \text{₹}12000 \)

ii. Calculate Loan Balance at the End of the Second Year

Step 1: Calculate the interest on the remaining principal for the second year using the new rate.
Principal for Year 2 \( (P_2) = \text{₹}12000 \)
Rate of Interest for Year 2 \( (R_2) = 12\% \text{ per annum} \)
\( \text{Interest for Year 2 } (I_2) = \frac{12000 \times 12 \times 1}{100} \)
\( I_2 = 120 \times 12 \)
\( I_2 = \text{₹}1440 \)
Step 2: Combine the second-year principal and interest to find the final outstanding amount.
\( \text{Outstanding Amount at the end of Year 2 } (A_2) = P_2 + I_2 \)
\( A_2 = 12000 + 1440 \)
\( A_2 = \text{₹}13440 \)
Answer:The amount outstanding at the end of the second year is ₹13440

Q7: Calculate the amount of ₹30000 at the end of 2 years 4 months, compounded annually at 10% per annum.

i. Calculate Interest and Amount for the First Year

Step 1: Identify the given values for the first year.
Principal for the first year \( (P_1) = \text{₹}30000 \)
Rate of Interest \( (R) = 10\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
Step 2: Calculate the Simple Interest for the first year.
Formula: \( \text{Interest} = \frac{P \times R \times T}{100} \)
\( \text{Interest for Year 1 } (I_1) = \frac{30000 \times 10 \times 1}{100} \)
\( I_1 = 300 \times 10 \)
\( I_1 = \text{₹}3000 \)
Step 3: Find the Amount at the end of the first year.
Formula: \( \text{Amount} = \text{Principal} + \text{Interest} \)
\( \text{Amount after Year 1 } (A_1) = 30000 + 3000 \)
\( A_1 = \text{₹}33000 \)

ii. Calculate Interest and Amount for the Second Year

Step 1: Use the amount of the first year as the principal for the second year.
Principal for the second year \( (P_2) = \text{₹}33000 \)
Time \( (T) = 1 \text{ year} \)
Step 2: Calculate the Simple Interest for the second year.
\( \text{Interest for Year 2 } (I_2) = \frac{33000 \times 10 \times 1}{100} \)
\( I_2 = 330 \times 10 \)
\( I_2 = \text{₹}3300 \)
Step 3: Find the Amount at the end of the second year.
\( \text{Amount after Year 2 } (A_2) = P_2 + I_2 \)
\( A_2 = 33000 + 3300 \)
\( A_2 = \text{₹}36300 \)

iii. Calculate Interest for the Remaining 4 Months

Step 1: Convert the remaining time period from months to years.
\( \text{Time } (T_3) = 4 \text{ months} = \frac{4}{12} \text{ year} = \frac{1}{3} \text{ year} \)
Step 2: Use the amount of the second year as the principal for the remaining period.
Principal for the final period \( (P_3) = \text{₹}36300 \)
\( \text{Interest for 4 months } (I_3) = \frac{36300 \times 10 \times \frac{1}{3}}{100} \)
\( I_3 = \frac{36300 \times 10}{3 \times 100} \)
\( I_3 = \frac{363000}{300} \)
\( I_3 = \text{₹}1210 \)
Step 3: Calculate the final total amount at the end of 2 years 4 months.
\( \text{Total Final Amount } (A_3) = P_3 + I_3 \)
\( A_3 = 36300 + 1210 \)
\( A_3 = \text{₹}37510 \)
Answer:The total amount is ₹37510

Q8: Calculate the amount of ₹31250 at the end of \(2\frac{1}{2}\) years, compounded annually at 8% per annum.

i. Calculate Interest and Amount for the First Year

Step 1: Identify the given values for the first year.
Principal for the first year \( (P_1) = \text{₹}31250 \)
Rate of Interest \( (R) = 8\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
Step 2: Calculate the Simple Interest for the first year.
Formula: \( \text{Interest} = \frac{P \times R \times T}{100} \)
\( \text{Interest for Year 1 } (I_1) = \frac{31250 \times 8 \times 1}{100} \)
\( I_1 = \frac{250000}{100} \)
\( I_1 = \text{₹}2500 \)
Step 3: Find the Amount at the end of the first year.
Formula: \( \text{Amount} = \text{Principal} + \text{Interest} \)
\( \text{Amount after Year 1 } (A_1) = 31250 + 2500 \)
\( A_1 = \text{₹}33750 \)

ii. Calculate Interest and Amount for the Second Year

Step 1: Use the amount of the first year as the principal for the second year.
Principal for the second year \( (P_2) = A_1 = \text{₹}33750 \)
Time \( (T) = 1 \text{ year} \)
Step 2: Calculate the Simple Interest for the second year.
\( \text{Interest for Year 2 } (I_2) = \frac{33750 \times 8 \times 1}{100} \)
\( I_2 = \frac{270000}{100} \)
\( I_2 = \text{₹}2700 \)
Step 3: Find the Amount at the end of the second year.
\( \text{Amount after Year 2 } (A_2) = P_2 + I_2 \)
\( A_2 = 33750 + 2700 \)
\( A_2 = \text{₹}36450 \)

iii. Calculate Interest for the Remaining Half Year

Step 1: Identify the time period for the final fraction of the year.
\( \text{Time } (T_3) = \frac{1}{2} \text{ year} \)
Step 2: Use the amount of the second year as the principal for the half-year period.
Principal for the final period \( (P_3) = A_2 = \text{₹}36450 \)
\( \text{Interest for } \frac{1}{2} \text{ year } (I_3) = \frac{36450 \times 8 \times \frac{1}{2}}{100} \)
\( I_3 = \frac{36450 \times 4}{100} \)
\( I_3 = \frac{145800}{100} \)
\( I_3 = \text{₹}1458 \)
Step 3: Calculate the final total amount at the end of \(2\frac{1}{2}\) years.
\( \text{Total Final Amount } (A_3) = P_3 + I_3 \)
\( A_3 = 36450 + 1458 \)
\( A_3 = \text{₹}37908 \)
Answer:The total amount is ₹37908

Q9: Calculate the amount and the compound interest on ₹15000 for 2 years compounded annually, the rates of interest for the successive years being 8% and 9% per annum respectively.

i. Calculate Interest and Amount for the First Year

Step 1: Identify the given values for the first year.
Principal for the first year \( (P_1) = \text{₹}15000 \)
Rate of Interest for Year 1 \( (R_1) = 8\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
Step 2: Calculate the Simple Interest for the first year.
Formula: \( \text{Interest} = \frac{P \times R \times T}{100} \)
\( \text{Interest for Year 1 } (I_1) = \frac{15000 \times 8 \times 1}{100} \)
\( I_1 = 150 \times 8 \)
\( I_1 = \text{₹}1200 \)
Step 3: Find the Amount at the end of the first year.
Formula: \( \text{Amount} = \text{Principal} + \text{Interest} \)
\( \text{Amount after Year 1 } (A_1) = 15000 + 1200 \)
\( A_1 = \text{₹}16200 \)

ii. Calculate Interest and Amount for the Second Year

Step 1: Use the amount of the first year as the principal for the second year.
Principal for the second year \( (P_2) = A_1 = \text{₹}16200 \)
Rate of Interest for Year 2 \( (R_2) = 9\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
Step 2: Calculate the Simple Interest for the second year.
\( \text{Interest for Year 2 } (I_2) = \frac{16200 \times 9 \times 1}{100} \)
\( I_2 = 162 \times 9 \)
\( I_2 = \text{₹}1458 \)
Step 3: Find the Final Amount at the end of the second year.
\( \text{Final Amount } (A_2) = P_2 + I_2 \)
\( A_2 = 16200 + 1458 \)
\( A_2 = \text{₹}17658 \)

iii. Calculate Total Compound Interest

Step 1: Find the total compound interest earned over the 2 years.
Formula: \( \text{Total Compound Interest (CI)} = I_1 + I_2 \)
\( \text{CI} = 1200 + 1458 \)
\( \text{CI} = \text{₹}2658 \)
Answer:Amount = ₹17658, Compound Interest = ₹2658

Q10: Calculate the amount and the compound interest on ₹25000 for 3 years, compounded annually, the rates of interest for the successive years being 8%, 9% and 10% respectively.

i. Calculate Interest and Amount for the First Year

Step 1: Identify the given values for the first year.
Principal for the first year \( (P_1) = \text{₹}25000 \)
Rate of Interest for Year 1 \( (R_1) = 8\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
Step 2: Calculate the Simple Interest for the first year.
Formula: \( \text{Interest} = \frac{P \times R \times T}{100} \)
\( \text{Interest for Year 1 } (I_1) = \frac{25000 \times 8 \times 1}{100} \)
\( I_1 = 250 \times 8 \)
\( I_1 = \text{₹}2000 \)
Step 3: Find the Amount at the end of the first year.
Formula: \( \text{Amount} = \text{Principal} + \text{Interest} \)
\( \text{Amount after Year 1 } (A_1) = 25000 + 2000 \)
\( A_1 = \text{₹}27000 \)

ii. Calculate Interest and Amount for the Second Year

Step 1: Use the amount of the first year as the principal for the second year.
Principal for the second year \( (P_2) = A_1 = \text{₹}27000 \)
Rate of Interest for Year 2 \( (R_2) = 9\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
Step 2: Calculate the Simple Interest for the second year.
\( \text{Interest for Year 2 } (I_2) = \frac{27000 \times 9 \times 1}{100} \)
\( I_2 = 270 \times 9 \)
\( I_2 = \text{₹}2430 \)
Step 3: Find the Amount at the end of the second year.
\( \text{Amount after Year 2 } (A_2) = P_2 + I_2 \)
\( A_2 = 27000 + 2430 \)
\( A_2 = \text{₹}29430 \)

iii. Calculate Interest and Amount for the Third Year

Step 1: Use the amount of the second year as the principal for the third year.
Principal for the third year \( (P_3) = A_2 = \text{₹}29430 \)
Rate of Interest for Year 3 \( (R_3) = 10\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
Step 2: Calculate the Simple Interest for the third year.
\( \text{Interest for Year 3 } (I_3) = \frac{29430 \times 10 \times 1}{100} \)
\( I_3 = 294.3 \times 10 \)
\( I_3 = \text{₹}2943 \)
Step 3: Find the Final Amount at the end of the third year.
\( \text{Final Amount } (A_3) = P_3 + I_3 \)
\( A_3 = 29430 + 2943 \)
\( A_3 = \text{₹}32373 \)

iv. Calculate Total Compound Interest

Step 1: Find the total compound interest earned over the 3 years.
Formula: \( \text{Total Compound Interest (CI)} = I_1 + I_2 + I_3 \)
\( \text{CI} = 2000 + 2430 + 2943 \)
\( \text{CI} = \text{₹}7373 \)
Answer:Amount = ₹32373, Compound Interest = ₹7373

Q11: Peter invested ₹240000 for 2 years at 10% per annum compounded annually. If 20% of the accrued interest at the end of each year is deducted as income tax, find the amount he received at the end of 2 years.

i. Calculate Net Interest and Amount for the First Year

Step 1: Identify given values for the first year and compute the gross interest.
Principal for the first year \( (P_1) = \text{₹}240000 \)
Rate of Interest \( (R) = 10\% \text{ per annum} \)
Time \( (T) = 1 \text{ year} \)
\( \text{Gross Interest for Year 1 } (I_1) = \frac{240000 \times 10 \times 1}{100} \)
\( I_1 = \text{₹}24000 \)
Step 2: Calculate and deduct the 20% income tax from the accrued interest.
\( \text{Income Tax for Year 1} = 20\% \text{ of } 24000 = \frac{20}{100} \times 24000 \)
\( \text{Income Tax for Year 1} = \text{₹}4800 \)
\( \text{Net Interest for Year 1 } (N_1) = 24000 – 4800 \)
\( N_1 = \text{₹}19200 \)
Step 3: Find the net amount at the end of the first year.
\( \text{Amount after Year 1 } (A_1) = P_1 + N_1 \)
\( A_1 = 240000 + 19200 \)
\( A_1 = \text{₹}259200 \)

ii. Calculate Net Interest and Amount for the Second Year

Step 1: Use the net amount of the first year as the principal for the second year.
Principal for the second year \( (P_2) = A_1 = \text{₹}259200 \)
\( \text{Gross Interest for Year 2 } (I_2) = \frac{259200 \times 10 \times 1}{100} \)
\( I_2 = \text{₹}25920 \)
Step 2: Calculate and deduct the 20% income tax from the second year’s interest.
\( \text{Income Tax for Year 2} = 20\% \text{ of } 25920 = \frac{20}{100} \times 25920 \)
\( \text{Income Tax for Year 2} = \text{₹}5184 \)
\( \text{Net Interest for Year 2 } (N_2) = 25920 – 5184 \)
\( N_2 = \text{₹}20736 \)
Step 3: Find the final net amount at the end of the second year.
\( \text{Final Amount Received } (A_2) = P_2 + N_2 \)
\( A_2 = 259200 + 20736 \)
\( A_2 = \text{₹}279936 \)
Answer:The amount he received at the end of 2 years is ₹279936

Q12: Find the amount and the compound interest on ₹10000 for 1 year at 12% per annum, compounded half-yearly.

i. Adjust Rate and Time for Half-Yearly Compounding

Step 1: Convert the annual rate of interest to a half-yearly rate.
Annual Rate \( (R) = 12\% \text{ per annum} \)
\( \text{Half-yearly Rate } (R_{\text{half}}) = \frac{12\%}{2} = 6\% \text{ per half-year} \)
Step 2: Determine the total number of half-yearly periods in 1 year.
\( \text{Total Time} = 1 \text{ year} = 2 \text{ half-years} \)
We will now calculate the interest step-by-step for 2 successive half-yearly periods.

ii. Calculate Interest and Amount for the First Half-Year

Step 1: Identify the given values for the first half-year.
Principal for the first half-year \( (P_1) = \text{₹}10000 \)
Rate of Interest = \( 6\% \text{ per half-year} \)
Time \( (T) = 1 \text{ half-year} \)
Step 2: Calculate the Simple Interest for the first half-year.
Formula: \( \text{Interest} = \frac{P \times R \times T}{100} \)
\( \text{Interest for Period 1 } (I_1) = \frac{10000 \times 6 \times 1}{100} \)
\( I_1 = 100 \times 6 \)
\( I_1 = \text{₹}600 \)
Step 3: Find the Amount at the end of the first half-year.
Formula: \( \text{Amount} = \text{Principal} + \text{Interest} \)
\( \text{Amount after Period 1 } (A_1) = 10000 + 600 \)
\( A_1 = \text{₹}10600 \)

iii. Calculate Interest and Amount for the Second Half-Year

Step 1: Use the amount of the first half-year as the principal for the second half-year.
Principal for the second half-year \( (P_2) = A_1 = \text{₹}10600 \)
Step 2: Calculate the Simple Interest for the second half-year.
\( \text{Interest for Period 2 } (I_2) = \frac{10600 \times 6 \times 1}{100} \)
\( I_2 = 106 \times 6 \)
\( I_2 = \text{₹}636 \)
Step 3: Find the final total Amount at the end of 1 year.
\( \text{Final Amount } (A_2) = P_2 + I_2 \)
\( A_2 = 10600 + 636 \)
\( A_2 = \text{₹}11236 \)

iv. Calculate Total Compound Interest

Step 1: Sum up the interest of both half-yearly periods to find total compound interest.
Formula: \( \text{Total Compound Interest (CI)} = I_1 + I_2 \)
\( \text{CI} = 600 + 636 \)
\( \text{CI} = \text{₹}1236 \)
Answer:Amount = ₹11236, Compound Interest = ₹1236

Q13: Find the amount and the compound interest on ₹64000 for \(1\frac{1}{2}\) years at 15% per annum, compounded half-yearly.

i. Adjust Rate and Time for Half-Yearly Compounding

Step 1: Convert the annual rate of interest to a half-yearly rate.
Annual Rate \( (R) = 15\% \text{ per annum} \)
\( \text{Half-yearly Rate } (R_{\text{half}}) = \frac{15\%}{2} = 7.5\% \text{ per half-year} \)
Step 2: Determine the total number of half-yearly periods in \(1\frac{1}{2}\) years.
\( \text{Total Time} = 1\frac{1}{2} \text{ years} = \frac{3}{2} \text{ years} = 3 \text{ half-years} \)
We will calculate the interest step-by-step for 3 successive half-yearly periods.

ii. Calculate Interest and Amount for the First Half-Year

Step 1: Identify the given values for the first half-year.
Principal for the first half-year \( (P_1) = \text{₹}64000 \)
Rate of Interest = \( 7.5\% \text{ per half-year} \)
Time \( (T) = 1 \text{ half-year} \)
Step 2: Calculate the Simple Interest for the first half-year.
Formula: \( \text{Interest} = \frac{P \times R \times T}{100} \)
\( \text{Interest for Period 1 } (I_1) = \frac{64000 \times 7.5 \times 1}{100} \)
\( I_1 = 640 \times 7.5 \)
\( I_1 = \text{₹}4800 \)
Step 3: Find the Amount at the end of the first half-year.
Formula: \( \text{Amount} = \text{Principal} + \text{Interest} \)
\( \text{Amount after Period 1 } (A_1) = 64000 + 4800 \)
\( A_1 = \text{₹}68800 \)

iii. Calculate Interest and Amount for the Second Half-Year

Step 1: Use the amount of the first half-year as the principal for the second half-year.
Principal for the second half-year \( (P_2) = A_1 = \text{₹}68800 \)
Step 2: Calculate the Simple Interest for the second half-year.
\( \text{Interest for Period 2 } (I_2) = \frac{68800 \times 7.5 \times 1}{100} \)
\( I_2 = 688 \times 7.5 \)
\( I_2 = \text{₹}5160 \)
Step 3: Find the Amount at the end of the second half-year.
\( \text{Amount after Period 2 } (A_2) = P_2 + I_2 \)
\( A_2 = 68800 + 5160 \)
\( A_2 = \text{₹}73960 \)

iv. Calculate Interest and Amount for the Third Half-Year

Step 1: Use the amount of the second half-year as the principal for the third half-year.
Principal for the third half-year \( (P_3) = A_2 = \text{₹}73960 \)
Step 2: Calculate the Simple Interest for the third half-year.
\( \text{Interest for Period 3 } (I_3) = \frac{73960 \times 7.5 \times 1}{100} \)
\( I_3 = 739.6 \times 7.5 \)
\( I_3 = \text{₹}5547 \)
Step 3: Find the final total Amount at the end of \(1\frac{1}{2}\) years.
\( \text{Final Amount } (A_3) = P_3 + I_3 \)
\( A_3 = 73960 + 5547 \)
\( A_3 = \text{₹}79507 \)

v. Calculate Total Compound Interest

Step 1: Sum up the interest of all three half-yearly periods to find total compound interest.
Formula: \( \text{Total Compound Interest (CI)} = I_1 + I_2 + I_3 \)
\( \text{CI} = 4800 + 5160 + 5547 \)
\( \text{CI} = \text{₹}15507 \)
Answer:Amount = ₹79507, Compound Interest = ₹15507

Q14: The simple interest on a sum of money for 2 years at 10% p.a. is ₹1700. Find:

i. the sum of money

Step 1: Identify the given values for the Simple Interest transaction.
Simple Interest \( (\text{SI}) = \text{₹}1700 \)
Rate of Interest \( (R) = 10\% \text{ per annum} \)
Time \( (T) = 2 \text{ years} \)
Step 2: Use the Simple Interest formula to find the Principal (sum of money).
Formula: \( \text{SI} = \frac{P \times R \times T}{100} \)
\( 1700 = \frac{P \times 10 \times 2}{100} \)
\( 1700 = \frac{20P}{100} \)
\( 1700 = \frac{P}{5} \)
\( P = 1700 \times 5 \)
\( P = \text{₹}8500 \)
Answer:The sum of money is ₹8500

ii. the compound interest on this sum for 1 year, payable half-yearly at the same rate.

Step 1: Adjust the rate and time for half-yearly compounding.
Principal \( (P_1) = \text{₹}8500 \)
Annual Rate = \( 10\% \text{ per annum} \)
\( \text{Half-yearly Rate } (R_{\text{half}}) = \frac{10\%}{2} = 5\% \text{ per half-year} \)
\( \text{Total Time} = 1 \text{ year} = 2 \text{ half-years} \)
Step 2: Calculate the interest for the first half-year period.
\( \text{Interest for Period 1 } (I_1) = \frac{8500 \times 5 \times 1}{100} \)
\( I_1 = 85 \times 5 \)
\( I_1 = \text{₹}425 \)
\( \text{Amount after Period 1 } (A_1) = 8500 + 425 = \text{₹}8925 \)
Step 3: Calculate the interest for the second half-year period using the updated principal.
Principal for Period 2 \( (P_2) = A_1 = \text{₹}8925 \)
\( \text{Interest for Period 2 } (I_2) = \frac{8925 \times 5 \times 1}{100} \)
\( I_2 = \frac{44625}{100} \)
\( I_2 = \text{₹}446.25 \)
Step 4: Find the total compound interest collected over both periods.
Formula: \( \text{Total Compound Interest (CI)} = I_1 + I_2 \)
\( \text{CI} = 425 + 446.25 \)
\( \text{CI} = \text{₹}871.25 \)
Answer:The compound interest is ₹871.25